[{"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/Basic.lean", "full_name": "Prod.swap_bijective", "start": [209, 1], "end": [210, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasurableSet.exists_lt_isClosed_of_ne_top", "start": [592, 1], "end": [595, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_simpleFunc_larger_space", "start": [1849, 1], "end": [1855, 10], "traced_tactics": [{"tactic": "simp_rw [\u2190 f.coe_toLargerSpace_eq hm]", "annotated_tactic": ["simp_rw [\u2190 f.coe_toLargerSpace_eq hm]", []], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int : Integrable \u2191f\n\u22a2 \u222b (x : \u03b2), \u2191f x \u2202\u03bc = \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x})) \u2022 x", "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int : Integrable \u2191f\n\u22a2 \u222b (x : \u03b2), \u2191(SimpleFunc.toLargerSpace hm f) x \u2202\u03bc =\n    \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x"}, {"tactic": "have hf_int : Integrable (f.toLargerSpace hm) \u03bc := by rwa [SimpleFunc.coe_toLargerSpace_eq]", "annotated_tactic": ["have hf_int : <a>Integrable</a> (f.toLargerSpace hm) \u03bc := by rwa [<a>SimpleFunc.coe_toLargerSpace_eq</a>]", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.SimpleFunc.coe_toLargerSpace_eq", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 40]}]], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int : Integrable \u2191f\n\u22a2 \u222b (x : \u03b2), \u2191(SimpleFunc.toLargerSpace hm f) x \u2202\u03bc =\n    \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x", "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int\u271d : Integrable \u2191f\nhf_int : Integrable \u2191(SimpleFunc.toLargerSpace hm f)\n\u22a2 \u222b (x : \u03b2), \u2191(SimpleFunc.toLargerSpace hm f) x \u2202\u03bc =\n    \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x"}, {"tactic": "rw [SimpleFunc.integral_eq_sum _ hf_int]", "annotated_tactic": ["rw [<a>SimpleFunc.integral_eq_sum</a> _ hf_int]", [{"full_name": "MeasureTheory.SimpleFunc.integral_eq_sum", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1403, 9], "def_end_pos": [1403, 35]}]], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int\u271d : Integrable \u2191f\nhf_int : Integrable \u2191(SimpleFunc.toLargerSpace hm f)\n\u22a2 \u222b (x : \u03b2), \u2191(SimpleFunc.toLargerSpace hm f) x \u2202\u03bc =\n    \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x", "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int\u271d : Integrable \u2191f\nhf_int : Integrable \u2191(SimpleFunc.toLargerSpace hm f)\n\u22a2 \u2211 x in SimpleFunc.range (SimpleFunc.toLargerSpace hm f),\n      ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x =\n    \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x"}, {"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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int\u271d : Integrable \u2191f\nhf_int : Integrable \u2191(SimpleFunc.toLargerSpace hm f)\n\u22a2 \u2211 x in SimpleFunc.range (SimpleFunc.toLargerSpace hm f),\n      ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x =\n    \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x", "state_after": "no goals"}, {"tactic": "rwa [SimpleFunc.coe_toLargerSpace_eq]", "annotated_tactic": ["rwa [<a>SimpleFunc.coe_toLargerSpace_eq</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_toLargerSpace_eq", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 40]}]], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192\u209b F\nhf_int : Integrable \u2191f\n\u22a2 Integrable \u2191(SimpleFunc.toLargerSpace hm f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "full_name": "MeasureTheory.Martingale.stoppedValue_min_ae_eq_condexp", "start": [196, 1], "end": [221, 86], "traced_tactics": [{"tactic": "refine'\n  (h.stoppedValue_ae_eq_condexp_of_le h\u03c4 (h\u03c3.min h\u03c4) (fun x => min_le_right _ _) h\u03c4_le).trans _", "annotated_tactic": ["refine'\n    (h.stoppedValue_ae_eq_condexp_of_le h\u03c4 (h\u03c3.min h\u03c4) (fun x => <a>min_le_right</a> _ _) h\u03c4_le).<a>trans</a> _", [{"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"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\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 (stoppedValue f fun x => min (\u03c3 x) (\u03c4 x)) =\u1d50[\u03bc] \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[\u03bc]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]"}, {"tactic": "refine' ae_of_ae_restrict_of_ae_restrict_compl {x | \u03c3 x \u2264 \u03c4 x} _ _", "annotated_tactic": ["refine' <a>ae_of_ae_restrict_of_ae_restrict_compl</a> {x | \u03c3 x \u2264 \u03c4 x} _ _", [{"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": "\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[\u03bc]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c3 x \u2264 \u03c4 x},\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c3 x \u2264 \u03c4 x}\u1d9c,\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x"}, {"tactic": "exact condexp_min_stopping_time_ae_eq_restrict_le h\u03c3 h\u03c4", "annotated_tactic": ["exact <a>condexp_min_stopping_time_ae_eq_restrict_le</a> h\u03c3 h\u03c4", [{"full_name": "MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1210, 9], "def_end_pos": [1210, 52]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c3 x \u2264 \u03c4 x},\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x", "state_after": "no goals"}, {"tactic": "apply Filter.EventuallyEq.trans _ ((condexp_min_stopping_time_ae_eq_restrict_le h\u03c4 h\u03c3).trans _)", "annotated_tactic": ["apply <a>Filter.EventuallyEq.trans</a> _ ((<a>condexp_min_stopping_time_ae_eq_restrict_le</a> h\u03c4 h\u03c3).<a>trans</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_min_stopping_time_ae_eq_restrict_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1210, 9], "def_end_pos": [1210, 52]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[stoppedValue f\n        \u03c4|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03a9 \u2192 E\n\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[stoppedValue f\n        \u03c4|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[?m.74018|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))]\n\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[?m.74018|IsStoppingTime.measurableSpace h\u03c4] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]"}, {"tactic": "rw [ae_restrict_iff' (h\u03c3.measurableSpace_le _ (h\u03c3.measurableSet_le_stopping_time h\u03c4).compl)]", "annotated_tactic": ["rw [<a>ae_restrict_iff'</a> (h\u03c3.measurableSpace_le _ (h\u03c3.measurableSet_le_stopping_time h\u03c4).<a>compl</a>)]", [{"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.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u03bc[stoppedValue f\n        \u03c4|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c3 x \u2264 \u03c4 x}\u1d9c,\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u03bc[stoppedValue f\n        \u03c4|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x"}, {"tactic": "rw [Filter.EventuallyEq, ae_restrict_iff'] at this", "annotated_tactic": ["rw [<a>Filter.EventuallyEq</a>, <a>ae_restrict_iff'</a>] at this", [{"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": [2572, 9], "def_end_pos": [2572, 25]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u03bc[stoppedValue f\n        \u03c4|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x},\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 MeasurableSet {x | \u03c4 x \u2264 \u03c3 x}"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x},\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 MeasurableSet {x | \u03c4 x \u2264 \u03c3 x}", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x},\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 MeasurableSet {x | \u03c4 x \u2264 \u03c3 x}\n\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x"}, {"tactic": "filter_upwards [this] with x hx hx_mem", "annotated_tactic": ["filter_upwards [this] with x hx hx_mem", []], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x", "state_after": "case h\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nx : \u03a9\nhx :\n  \u03c4 x \u2264 \u03c3 x \u2192\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nhx_mem : x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c\n\u22a2 (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x"}, {"tactic": "simp only [Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx_mem", "annotated_tactic": ["simp only [<a>Set.mem_compl_iff</a>, <a>Set.mem_setOf_eq</a>, <a>not_le</a>] at hx_mem", [{"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 h\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nx : \u03a9\nhx :\n  \u03c4 x \u2264 \u03c3 x \u2192\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nhx_mem : x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c\n\u22a2 (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x", "state_after": "case h\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nx : \u03a9\nhx :\n  \u03c4 x \u2264 \u03c3 x \u2192\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nhx_mem : \u03c4 x < \u03c3 x\n\u22a2 (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x"}, {"tactic": "exact hx hx_mem.le", "annotated_tactic": ["exact hx hx_mem.le", []], "state_before": "case h\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202\u03bc,\n    x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n        (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nx : \u03a9\nhx :\n  \u03c4 x \u2264 \u03c3 x \u2192\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\nhx_mem : \u03c4 x < \u03c3 x\n\u22a2 (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x", "state_after": "no goals"}, {"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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis :\n  \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x},\n    (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =\n      (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x\n\u22a2 MeasurableSet {x | \u03c4 x \u2264 \u03c3 x}", "state_after": "no goals"}, {"tactic": "exact stoppedValue f \u03c4", "annotated_tactic": ["exact <a>stoppedValue</a> f \u03c4", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03a9 \u2192 E", "state_after": "no goals"}, {"tactic": "rw [IsStoppingTime.measurableSpace_min h\u03c3, IsStoppingTime.measurableSpace_min h\u03c4, inf_comm]", "annotated_tactic": ["rw [<a>IsStoppingTime.measurableSpace_min</a> h\u03c3, <a>IsStoppingTime.measurableSpace_min</a> h\u03c4, <a>inf_comm</a>]", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSpace_min", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [591, 9], "def_end_pos": [591, 28]}, {"full_name": "MeasureTheory.IsStoppingTime.measurableSpace_min", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [591, 9], "def_end_pos": [591, 28]}, {"full_name": "inf_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [500, 9], "def_end_pos": [500, 17]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[stoppedValue f\n        \u03c4|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))]", "state_after": "no goals"}, {"tactic": "rw [h1]", "annotated_tactic": ["rw [h1]", []], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nh1 : \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \u03c4\n\u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]", "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nh1 : \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \u03c4\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}] \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]"}, {"tactic": "exact (condexp_stoppedValue_stopping_time_ae_eq_restrict_le h h\u03c4 h\u03c3 h\u03c4_le).symm", "annotated_tactic": ["exact (<a>condexp_stoppedValue_stopping_time_ae_eq_restrict_le</a> h h\u03c4 h\u03c3 h\u03c4_le).<a>symm</a>", [{"full_name": "MeasureTheory.Martingale.condexp_stoppedValue_stopping_time_ae_eq_restrict_le", "def_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "def_pos": [163, 9], "def_end_pos": [163, 61]}, {"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\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nh1 : \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \u03c4\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}] \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]", "state_after": "no goals"}, {"tactic": "refine' condexp_of_stronglyMeasurable h\u03c4.measurableSpace_le _ _", "annotated_tactic": ["refine' <a>condexp_of_stronglyMeasurable</a> h\u03c4.measurableSpace_le _ _", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}]], "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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 StronglyMeasurable (stoppedValue f \u03c4)\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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 Integrable (stoppedValue f \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'_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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 StronglyMeasurable (stoppedValue f \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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 Measurable (stoppedValue f \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'_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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 Measurable (stoppedValue f \u03c4)", "state_after": "no goals"}, {"tactic": "exact integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le", "annotated_tactic": ["exact <a>integrable_stoppedValue</a> \u03b9 h\u03c4 h.integrable h\u03c4_le", [{"full_name": "MeasureTheory.integrable_stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [975, 9], "def_end_pos": [975, 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\u271d : \u03b9\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nn : \u03b9\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_sf_min :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 Integrable (stoppedValue f \u03c4)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.lintegral_mono", "start": [1105, 1], "end": [1112, 95], "traced_tactics": [{"tactic": "rw [sup_of_le_right hfg]", "annotated_tactic": ["rw [<a>sup_of_le_right</a> hfg]", [{"full_name": "sup_of_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [200, 22], "def_end_pos": [200, 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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfg : f \u2264 g\nh\u03bc\u03bd : \u03bc \u2264 \u03bd\n\u22a2 lintegral (f \u2294 g) \u03bc = lintegral g \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "full_name": "MeasureTheory.Measure.restrict_singleton", "start": [67, 1], "end": [73, 13], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 restrict \u03bc {a} = \u2191\u2191\u03bc {a} \u2022 dirac a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "by_cases ha : a \u2208 s", "annotated_tactic": ["by_cases ha : a \u2208 s", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "have : s \u2229 {a} = {a} := by simpa", "annotated_tactic": ["have : s \u2229 {a} = {a} := by simpa", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\nthis : s \u2229 {a} = {a}\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\nthis : s \u2229 {a} = {a}\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 s \u2229 {a} = {a}", "state_after": "no goals"}, {"tactic": "have : s \u2229 {a} = \u2205 := inter_singleton_eq_empty.2 ha", "annotated_tactic": ["have : s \u2229 {a} = \u2205 := <a>inter_singleton_eq_empty</a>.2 ha", [{"full_name": "Set.inter_singleton_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 33]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\nthis : s \u2229 {a} = \u2205\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\nthis : s \u2229 {a} = \u2205\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.Integrable.tendsto_set_integral_nhds_zero", "start": [1000, 1], "end": [1003, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Classical.not_exists_not", "start": [691, 1], "end": [691, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_one", "start": [611, 1], "end": [613, 34], "traced_tactics": [{"tactic": "rw [val_one_eq_one_mod]", "annotated_tactic": ["rw [<a>val_one_eq_one_mod</a>]", [{"full_name": "ZMod.val_one_eq_one_mod", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [607, 9], "def_end_pos": [607, 27]}]], "state_before": "n : \u2115\ninst\u271d : Fact (1 < n)\n\u22a2 val 1 = 1", "state_after": "n : \u2115\ninst\u271d : Fact (1 < n)\n\u22a2 1 % n = 1"}, {"tactic": "exact Nat.mod_eq_of_lt Fact.out", "annotated_tactic": ["exact <a>Nat.mod_eq_of_lt</a> <a>Fact.out</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": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [118, 3], "def_end_pos": [118, 6]}]], "state_before": "n : \u2115\ninst\u271d : Fact (1 < n)\n\u22a2 1 % n = 1", "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_swap", "start": [499, 1], "end": [501, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.snorm'_lim_eq_lintegral_liminf", "start": [1298, 1], "end": [1310, 54], "traced_tactics": [{"tactic": "suffices h_no_pow :\n    (\u222b\u207b a, (\u2016f_lim a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, atTop.liminf fun m => (\u2016f m a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc by\n  rw [snorm', h_no_pow]", "annotated_tactic": ["suffices h_no_pow :\n      (\u222b\u207b a, (\u2016f_lim a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, atTop.liminf fun m => (\u2016f m a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc by\n    rw [<a>snorm'</a>, h_no_pow]", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}]], "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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\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 = (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (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 : \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 : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f_lim a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc"}, {"tactic": "refine' lintegral_congr_ae (h_lim.mono fun a ha => _)", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> (h_lim.mono fun a ha => _)", [{"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\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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f_lim a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \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\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 : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 (fun a => \u2191\u2016f_lim a\u2016\u208a ^ p) a = (fun a => liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop) a"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 (fun a => \u2191\u2016f_lim a\u2016\u208a ^ p) a = (fun a => liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop) a", "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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 \u2191\u2016f_lim a\u2016\u208a ^ p = liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop"}, {"tactic": "rw [Tendsto.liminf_eq]", "annotated_tactic": ["rw [<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": "\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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 \u2191\u2016f_lim a\u2016\u208a ^ p = liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop", "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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 Tendsto (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop (\ud835\udcdd (\u2191\u2016f_lim a\u2016\u208a ^ p))"}, {"tactic": "simp_rw [ENNReal.coe_rpow_of_nonneg _ hp_nonneg, ENNReal.tendsto_coe]", "annotated_tactic": ["simp_rw [<a>ENNReal.coe_rpow_of_nonneg</a> _ hp_nonneg, <a>ENNReal.tendsto_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.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\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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 Tendsto (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop (\ud835\udcdd (\u2191\u2016f_lim a\u2016\u208a ^ 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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 Tendsto (fun a_1 => \u2016f a_1 a\u2016\u208a ^ p) atTop (\ud835\udcdd (\u2016f_lim a\u2016\u208a ^ p))"}, {"tactic": "refine' ((NNReal.continuous_rpow_const hp_nonneg).tendsto \u2016f_lim a\u2016\u208a).comp _", "annotated_tactic": ["refine' ((<a>NNReal.continuous_rpow_const</a> hp_nonneg).<a>tendsto</a> \u2016f_lim a\u2016\u208a).<a>comp</a> _", [{"full_name": "NNReal.continuous_rpow_const", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "def_pos": [478, 9], "def_end_pos": [478, 30]}, {"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": "\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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 Tendsto (fun a_1 => \u2016f a_1 a\u2016\u208a ^ p) atTop (\ud835\udcdd (\u2016f_lim a\u2016\u208a ^ 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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 Tendsto (fun a_1 => \u2016f a_1 a\u2016\u208a) atTop (\ud835\udcdd \u2016f_lim a\u2016\u208a)"}, {"tactic": "exact (continuous_nnnorm.tendsto (f_lim a)).comp ha", "annotated_tactic": ["exact (continuous_nnnorm.tendsto (f_lim a)).<a>comp</a> ha", [{"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\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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\na : \u03b1\nha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a))\n\u22a2 Tendsto (fun a_1 => \u2016f a_1 a\u2016\u208a) atTop (\ud835\udcdd \u2016f_lim a\u2016\u208a)", "state_after": "no goals"}, {"tactic": "rw [snorm', h_no_pow]", "annotated_tactic": ["rw [<a>snorm'</a>, h_no_pow]", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}]], "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\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 G\np : \u211d\nhp_nonneg : 0 \u2264 p\nf_lim : \u03b1 \u2192 G\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nh_no_pow : \u222b\u207b (a : \u03b1), \u2191\u2016f_lim a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc\n\u22a2 snorm' f_lim p \u03bc = (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Trunc.induction_on", "start": [502, 11], "end": [503, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.mem_insert_of_mem", "start": [567, 1], "end": [578, 37], "traced_tactics": [{"tactic": "match e : zoom (cmp v) t with\n| (nil, p) =>\n  let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e\n  simp [\u2190 mem_toList, h\u2081] at h\n  simp [\u2190 mem_toList, h\u2082]; cases h <;> simp [*]\n| (node .., p) =>\n  let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e\n  simp [\u2190 mem_toList, h\u2081] at h\n  simp [\u2190 mem_toList, h\u2082]; rcases h with h|h|h <;> simp [*]\n  exact .inr (Path.zoom_zoomed\u2081 e)", "annotated_tactic": ["match e : <a>zoom</a> (cmp v) t with\n  | (<a>nil</a>, p) =>\n    let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_nil</a> ht e\n    simp [\u2190 <a>mem_toList</a>, h\u2081] at h\n    simp [\u2190 <a>mem_toList</a>, h\u2082]; cases h <;> simp [*]\n  | (<a>node</a> .., p) =>\n    let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_node</a> ht e\n    simp [\u2190 <a>mem_toList</a>, h\u2081] at h\n    simp [\u2190 <a>mem_toList</a>, h\u2082]; rcases h with h|h|h <;> simp [*]\n    exact .inr (<a>Path.zoom_zoomed\u2081</a> e)", [{"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.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}, {"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": "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]}, {"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": "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": "Std.RBNode.Path.zoom_zoomed\u2081", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [218, 9], "def_end_pos": [218, 21]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "no goals"}, {"tactic": "let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e", "annotated_tactic": ["let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_nil</a> ht e", [{"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]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2081] at h", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2081] at h", [{"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]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2082]", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2082]", [{"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]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq"}, {"tactic": "cases h <;> simp [*]", "annotated_tactic": ["cases h <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq", "state_after": "no goals"}, {"tactic": "let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e", "annotated_tactic": ["let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_node</a> ht e", [{"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\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2081] at h", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2081] at h", [{"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]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2082]", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2082]", [{"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]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq"}, {"tactic": "rcases h with h|h|h <;> simp [*]", "annotated_tactic": ["rcases h with h|h|h <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq", "state_after": "case inr.inl\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' = v\u271d\n\u22a2 (v\u271d \u2208 w\u271d\u00b9 \u2228 v\u271d = v \u2228 v\u271d \u2208 w\u271d) \u2228 cmp v v\u271d = Ordering.eq"}, {"tactic": "exact .inr (Path.zoom_zoomed\u2081 e)", "annotated_tactic": ["exact .inr (<a>Path.zoom_zoomed\u2081</a> e)", [{"full_name": "Std.RBNode.Path.zoom_zoomed\u2081", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [218, 9], "def_end_pos": [218, 21]}]], "state_before": "case inr.inl\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' = v\u271d\n\u22a2 (v\u271d \u2208 w\u271d\u00b9 \u2228 v\u271d = v \u2228 v\u271d \u2208 w\u271d) \u2228 cmp v v\u271d = Ordering.eq", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "OrderIso.essInf_apply", "start": [187, 1], "end": [189, 52], "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_addChar_eq", "start": [125, 1], "end": [125, 97], "traced_tactics": [{"tactic": "rw [\u2190 zero_addChar_byteIdx]", "annotated_tactic": ["rw [\u2190 <a>zero_addChar_byteIdx</a>]", [{"full_name": "String.Pos.zero_addChar_byteIdx", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [122, 9], "def_end_pos": [122, 29]}]], "state_before": "c : Char\n\u22a2 0 + c = { byteIdx := csize c }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec.fix", "start": [851, 1], "end": [861, 42], "traced_tactics": [{"tactic": "let F : \u03b1 \u2192 \u2115 \u2192. Sum \u03c3 \u03b1 := fun a n =>\n  n.rec (some (Sum.inr a)) fun _ IH => IH.bind fun s => Sum.casesOn s (fun _ => Part.some s) f", "annotated_tactic": ["let F : \u03b1 \u2192 \u2115 \u2192. <a>Sum</a> \u03c3 \u03b1 := fun a n =>\n    n.rec (<a>some</a> (<a>Sum.inr</a> a)) fun _ IH => IH.bind fun s => <a>Sum.casesOn</a> s (fun _ => <a>Part.some</a> s) f", [{"full_name": "Sum", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"full_name": "Part.some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}, {"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": "Sum.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"full_name": "Part.some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}]], "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 \u2295 \u03b1\nhf : Partrec f\n\u22a2 Partrec (PFun.fix 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\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\n\u22a2 Partrec (PFun.fix f)"}, {"tactic": "have hF : Partrec\u2082 F :=\n  Partrec.nat_rec snd (sum_inr.comp fst).partrec\n    (sum_casesOn_right (snd.comp snd) (snd.comp <| snd.comp fst).to\u2082 (hf.comp snd).to\u2082).to\u2082", "annotated_tactic": ["have hF : <a>Partrec\u2082</a> F :=\n    <a>Partrec.nat_rec</a> <a>snd</a> (sum_inr.comp <a>fst</a>).<a>partrec</a>\n      (<a>sum_casesOn_right</a> (snd.comp <a>snd</a>) (snd.comp <| snd.comp <a>fst</a>).<a>to\u2082</a> (hf.comp <a>snd</a>).<a>to\u2082</a>).<a>to\u2082</a>", [{"full_name": "Partrec\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [241, 5], "def_end_pos": [241, 13]}, {"full_name": "Partrec.nat_rec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.partrec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [264, 19], "def_end_pos": [264, 37]}, {"full_name": "Partrec.sum_casesOn_right", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [787, 9], "def_end_pos": [787, 26]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [294, 9], "def_end_pos": [294, 12]}, {"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": "\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 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\n\u22a2 Partrec (PFun.fix 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\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\n\u22a2 Partrec (PFun.fix f)"}, {"tactic": "let p a n := @Part.map _ Bool (fun s => Sum.casesOn s (fun _ => true) fun _ => false) (F a n)", "annotated_tactic": ["let p a n := @<a>Part.map</a> _ <a>Bool</a> (fun s => <a>Sum.casesOn</a> s (fun _ => <a>true</a>) fun _ => <a>false</a>) (F a n)", [{"full_name": "Part.map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [434, 5], "def_end_pos": [434, 8]}, {"full_name": "Bool", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}, {"full_name": "Sum.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"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": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 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\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\n\u22a2 Partrec (PFun.fix 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\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\n\u22a2 Partrec (PFun.fix f)"}, {"tactic": "have hp : Partrec\u2082 p :=\n  hF.map ((sum_casesOn Computable.id (const true).to\u2082 (const false).to\u2082).comp snd).to\u2082", "annotated_tactic": ["have hp : <a>Partrec\u2082</a> p :=\n    hF.map ((<a>sum_casesOn</a> <a>Computable.id</a> (<a>const</a> <a>true</a>).<a>to\u2082</a> (<a>const</a> <a>false</a>).<a>to\u2082</a>).<a>comp</a> <a>snd</a>).<a>to\u2082</a>", [{"full_name": "Partrec\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [241, 5], "def_end_pos": [241, 13]}, {"full_name": "Computable.sum_casesOn", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [713, 9], "def_end_pos": [713, 20]}, {"full_name": "Computable.id", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [298, 19], "def_end_pos": [298, 21]}, {"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [284, 9], "def_end_pos": [284, 14]}, {"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": "Computable.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [294, 9], "def_end_pos": [294, 12]}, {"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [284, 9], "def_end_pos": [284, 14]}, {"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": "Computable.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [294, 9], "def_end_pos": [294, 12]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [294, 9], "def_end_pos": [294, 12]}]], "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 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\n\u22a2 Partrec (PFun.fix 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\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\n\u22a2 Partrec (PFun.fix f)"}, {"tactic": "exact (hp.rfind.bind (hF.bind (sum_casesOn_right snd snd.to\u2082 none.to\u2082).to\u2082).to\u2082).of_eq fun a =>\n  ext fun b => by simp; apply fix_aux f", "annotated_tactic": ["exact (hp.rfind.bind (hF.bind (<a>sum_casesOn_right</a> <a>snd</a> snd.to\u2082 none.to\u2082).<a>to\u2082</a>).<a>to\u2082</a>).<a>of_eq</a> fun a =>\n    <a>ext</a> fun b => by simp; apply <a>fix_aux</a> f", [{"full_name": "Partrec.sum_casesOn_right", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [787, 9], "def_end_pos": [787, 26]}, {"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]}, {"full_name": "Partrec.of_eq", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [430, 9], "def_end_pos": [430, 14]}, {"full_name": "Part.ext", "def_path": "Mathlib/Data/Part.lean", "def_pos": [116, 9], "def_end_pos": [116, 12]}, {"full_name": "Partrec.fix_aux", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "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 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\n\u22a2 Partrec (PFun.fix f)", "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\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 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\na : \u03b1\nb : \u03c3\n\u22a2 (b \u2208\n      Part.bind (Nat.rfind (p a)) fun b =>\n        Part.bind (F (a, b).1 (a, b).2) fun b_1 =>\n          Sum.casesOn ((a, b), b_1).2 (fun b_2 => Part.some (((a, b), b_1), b_2).2) fun b => Part.none) \u2194\n    b \u2208 PFun.fix f a", "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\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\na : \u03b1\nb : \u03c3\n\u22a2 (\u2203 a_1,\n      ((\u2203 a_2,\n            Sum.inl a_2 \u2208\n              Nat.rec (Part.some (Sum.inr a))\n                (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2227\n          \u2200 {m : \u2115},\n            m < a_1 \u2192\n              \u2203 b,\n                Sum.inr b \u2208\n                  Nat.rec (Part.some (Sum.inr a))\n                    (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m) \u2227\n        Sum.inl b \u2208\n          Nat.rec (Part.some (Sum.inr a))\n            (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2194\n    b \u2208 PFun.fix f a"}, {"tactic": "apply fix_aux f", "annotated_tactic": ["apply <a>fix_aux</a> f", [{"full_name": "Partrec.fix_aux", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "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 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n  fun a n =>\n    Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\na : \u03b1\nb : \u03c3\n\u22a2 (\u2203 a_1,\n      ((\u2203 a_2,\n            Sum.inl a_2 \u2208\n              Nat.rec (Part.some (Sum.inr a))\n                (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2227\n          \u2200 {m : \u2115},\n            m < a_1 \u2192\n              \u2203 b,\n                Sum.inr b \u2208\n                  Nat.rec (Part.some (Sum.inr a))\n                    (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m) \u2227\n        Sum.inl b \u2208\n          Nat.rec (Part.some (Sum.inr a))\n            (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2194\n    b \u2208 PFun.fix f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Ioc.coe_ne_one", "start": [270, 1], "end": [271, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.insert_inj", "start": [1232, 1], "end": [1234, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.IsUniform.integral_eq", "start": [391, 1], "end": [407, 58], "traced_tactics": [{"tactic": "haveI := hasPDF hns hnt huX", "annotated_tactic": ["haveI := <a>hasPDF</a> hns hnt huX", [{"full_name": "MeasureTheory.pdf.IsUniform.hasPDF", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [320, 9], "def_end_pos": [320, 15]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\n\u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis : HasPDF X \u2119\n\u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x"}, {"tactic": "haveI := huX.isProbabilityMeasure hns hnt hms", "annotated_tactic": ["haveI := huX.isProbabilityMeasure hns hnt hms", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis : HasPDF X \u2119\n\u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\n\u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x"}, {"tactic": "rw [\u2190 integral_mul_eq_integral]", "annotated_tactic": ["rw [\u2190 <a>integral_mul_eq_integral</a>]", [{"full_name": "MeasureTheory.pdf.integral_mul_eq_integral", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [288, 9], "def_end_pos": [288, 33]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\n\u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\n\u22a2 \u222b (x : \u211d), x * ENNReal.toReal (pdf (fun x => X x) \u2119 x) = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x"}, {"tactic": "rw [integral_congr_ae (Filter.EventuallyEq.mul (ae_eq_refl _) (pdf_toReal_ae_eq huX))]", "annotated_tactic": ["rw [<a>integral_congr_ae</a> (<a>Filter.EventuallyEq.mul</a> (<a>ae_eq_refl</a> _) (<a>pdf_toReal_ae_eq</a> huX))]", [{"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.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": "MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [336, 9], "def_end_pos": [336, 25]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\n\u22a2 \u222b (x : \u211d), x * ENNReal.toReal (pdf (fun x => X x) \u2119 x) = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\n\u22a2 \u222b (a : \u211d), a * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) a) =\n    ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x"}, {"tactic": "simp_rw [this, \u2190 s.indicator_mul_right fun x => x, integral_indicator hms]", "annotated_tactic": ["simp_rw [this, \u2190 s.indicator_mul_right fun x => x, <a>integral_indicator</a> hms]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d\u00b9 : HasPDF X \u2119\nthis\u271d : IsProbabilityMeasure \u2119\nthis :\n  \u2200 (x : \u211d),\n    x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =\n      x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x\n\u22a2 \u222b (a : \u211d), a * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) a) =\n    ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d\u00b9 : HasPDF X \u2119\nthis\u271d : IsProbabilityMeasure \u2119\nthis :\n  \u2200 (x : \u211d),\n    x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =\n      x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x\n\u22a2 \u222b (a : \u211d) in s, a * (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) a = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x"}, {"tactic": "change \u222b x in s, x * (volume s)\u207b\u00b9.toReal \u2022 (1 : \u211d) = _", "annotated_tactic": ["change \u222b x in s, x * (<a>volume</a> s)\u207b\u00b9.<a>toReal</a> \u2022 (1 : \u211d) = _", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d\u00b9 : HasPDF X \u2119\nthis\u271d : IsProbabilityMeasure \u2119\nthis :\n  \u2200 (x : \u211d),\n    x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =\n      x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x\n\u22a2 \u222b (a : \u211d) in s, a * (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) a = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d\u00b9 : HasPDF X \u2119\nthis\u271d : IsProbabilityMeasure \u2119\nthis :\n  \u2200 (x : \u211d),\n    x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =\n      x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x\n\u22a2 \u222b (x : \u211d) in s, x * ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x"}, {"tactic": "rw [integral_mul_right, mul_comm, smul_eq_mul, mul_one]", "annotated_tactic": ["rw [<a>integral_mul_right</a>, <a>mul_comm</a>, <a>smul_eq_mul</a>, <a>mul_one</a>]", [{"full_name": "MeasureTheory.integral_mul_right", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [928, 9], "def_end_pos": [928, 27]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 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_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d\u00b9 : HasPDF X \u2119\nthis\u271d : IsProbabilityMeasure \u2119\nthis :\n  \u2200 (x : \u211d),\n    x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =\n      x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x\n\u22a2 \u222b (x : \u211d) in s, x * ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x", "state_after": "no goals"}, {"tactic": "by_cases hx : x \u2208 s", "annotated_tactic": ["by_cases hx : x \u2208 s", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\nx : \u211d\n\u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\nx : \u211d\nhx : x \u2208 s\n\u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\nx : \u211d\nhx : \u00acx \u2208 s\n\u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x"}, {"tactic": "simp [Set.indicator_of_mem hx]", "annotated_tactic": ["simp [<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]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\nx : \u211d\nhx : x \u2208 s\n\u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x", "state_after": "no goals"}, {"tactic": "simp [Set.indicator_of_not_mem hx]", "annotated_tactic": ["simp [<a>Set.indicator_of_not_mem</a> hx]", [{"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\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : 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\nhnt : \u2191\u2191volume s \u2260 \u22a4\nhuX : IsUniform X s \u2119\nthis\u271d : HasPDF X \u2119\nthis : IsProbabilityMeasure \u2119\nx : \u211d\nhx : \u00acx \u2208 s\n\u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bodd_neg", "start": [61, 1], "end": [69, 18], "traced_tactics": [{"tactic": "cases n with\n| ofNat =>\n  rw [\u2190negOfNat_eq, bodd_negOfNat]\n  simp\n| negSucc n =>\n  rw [neg_negSucc, bodd_coe, Nat.bodd_succ]\n  change (!Nat.bodd n) = !(bodd n)\n  rw [bodd_coe]", "annotated_tactic": ["cases n with\n  | <a>ofNat</a> =>\n    rw [\u2190<a>negOfNat_eq</a>, <a>bodd_negOfNat</a>]\n    simp\n  | <a>negSucc</a> n =>\n    rw [<a>neg_negSucc</a>, <a>bodd_coe</a>, <a>Nat.bodd_succ</a>]\n    change (!<a>Nat.bodd</a> n) = !(<a>bodd</a> n)\n    rw [<a>bodd_coe</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]}, {"full_name": "Int.negOfNat_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}, {"full_name": "Int.bodd_negOfNat", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [55, 9], "def_end_pos": [55, 22]}, {"full_name": "Int.negSucc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 12]}, {"full_name": "Int.neg_negSucc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [41, 23], "def_end_pos": [41, 34]}, {"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_succ", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [68, 9], "def_end_pos": [68, 18]}, {"full_name": "Nat.bodd", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Int.bodd", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [29, 5], "def_end_pos": [29, 9]}, {"full_name": "Int.bodd_coe", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [42, 9], "def_end_pos": [42, 17]}]], "state_before": "n : \u2124\n\u22a2 bodd (-n) = bodd n", "state_after": "no goals"}, {"tactic": "rw [\u2190negOfNat_eq, bodd_negOfNat]", "annotated_tactic": ["rw [\u2190<a>negOfNat_eq</a>, <a>bodd_negOfNat</a>]", [{"full_name": "Int.negOfNat_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}, {"full_name": "Int.bodd_negOfNat", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [55, 9], "def_end_pos": [55, 22]}]], "state_before": "case ofNat\na\u271d : \u2115\n\u22a2 bodd (-ofNat a\u271d) = bodd (ofNat a\u271d)", "state_after": "case ofNat\na\u271d : \u2115\n\u22a2 Nat.bodd a\u271d = bodd (ofNat a\u271d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case ofNat\na\u271d : \u2115\n\u22a2 Nat.bodd a\u271d = bodd (ofNat a\u271d)", "state_after": "no goals"}, {"tactic": "rw [neg_negSucc, bodd_coe, Nat.bodd_succ]", "annotated_tactic": ["rw [<a>neg_negSucc</a>, <a>bodd_coe</a>, <a>Nat.bodd_succ</a>]", [{"full_name": "Int.neg_negSucc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [41, 23], "def_end_pos": [41, 34]}, {"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_succ", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [68, 9], "def_end_pos": [68, 18]}]], "state_before": "case negSucc\nn : \u2115\n\u22a2 bodd (- -[n+1]) = bodd -[n+1]", "state_after": "case negSucc\nn : \u2115\n\u22a2 (!Nat.bodd n) = bodd -[n+1]"}, {"tactic": "change (!Nat.bodd n) = !(bodd n)", "annotated_tactic": ["change (!<a>Nat.bodd</a> n) = !(<a>bodd</a> n)", [{"full_name": "Nat.bodd", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Int.bodd", "def_path": "Mathlib/Init/Data/Int/Bitwise.lean", "def_pos": [29, 5], "def_end_pos": [29, 9]}]], "state_before": "case negSucc\nn : \u2115\n\u22a2 (!Nat.bodd n) = bodd -[n+1]", "state_after": "case negSucc\nn : \u2115\n\u22a2 (!Nat.bodd n) = !bodd \u2191n"}, {"tactic": "rw [bodd_coe]", "annotated_tactic": ["rw [<a>bodd_coe</a>]", [{"full_name": "Int.bodd_coe", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [42, 9], "def_end_pos": [42, 17]}]], "state_before": "case negSucc\nn : \u2115\n\u22a2 (!Nat.bodd n) = !bodd \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.measurable_measure_condCdf", "start": [981, 1], "end": [1003, 45], "traced_tactics": [{"tactic": "rw [Measure.measurable_measure]", "annotated_tactic": ["rw [<a>Measure.measurable_measure</a>]", [{"full_name": "MeasureTheory.Measure.measurable_measure", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [71, 9], "def_end_pos": [71, 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)\n\u22a2 Measurable fun a => StieltjesFunction.measure (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)\n\u22a2 \u2200 (s : Set \u211d), MeasurableSet s \u2192 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s"}, {"tactic": "refine' fun s hs => ?_", "annotated_tactic": ["refine' fun s 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)\n\u22a2 \u2200 (s : Set \u211d), MeasurableSet s \u2192 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) 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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s"}, {"tactic": "refine' MeasurableSpace.induction_on_inter\n  (C := fun s => Measurable fun b \u21a6 StieltjesFunction.measure (condCdf \u03c1 b) s)\n  (borel_eq_generateFrom_Iic \u211d) isPiSystem_Iic _ _ _ _ hs", "annotated_tactic": ["refine' <a>MeasurableSpace.induction_on_inter</a>\n    (C := fun s => <a>Measurable</a> fun b \u21a6 <a>StieltjesFunction.measure</a> (<a>condCdf</a> \u03c1 b) s)\n    (<a>borel_eq_generateFrom_Iic</a> \u211d) <a>isPiSystem_Iic</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": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "StieltjesFunction.measure", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [344, 27], "def_end_pos": [344, 34]}, {"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "borel_eq_generateFrom_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 34]}, {"full_name": "isPiSystem_Iic", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [156, 9], "def_end_pos": [156, 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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) 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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) \u2205\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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t\n\ncase refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u211d),\n    MeasurableSet t \u2192\n      (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t \u2192\n        (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t\u1d9c\n\ncase refine'_4\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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)) \u2192\n          (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (\u22c3 i, f i)"}, {"tactic": "simp only [measure_empty, measurable_const]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>measurable_const</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) \u2205", "state_after": "no goals"}, {"tactic": "rintro S \u27e8u, rfl\u27e9", "annotated_tactic": ["rintro S \u27e8u, rfl\u27e9", []], "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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t", "state_after": "case refine'_2.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)\ns : Set \u211d\nhs : MeasurableSet s\nu : \u211d\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (Iic u)"}, {"tactic": "simp_rw [measure_condCdf_Iic \u03c1 _ u]", "annotated_tactic": ["simp_rw [<a>measure_condCdf_Iic</a> \u03c1 _ u]", [{"full_name": "ProbabilityTheory.measure_condCdf_Iic", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [965, 9], "def_end_pos": [965, 28]}]], "state_before": "case refine'_2.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)\ns : Set \u211d\nhs : MeasurableSet s\nu : \u211d\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (Iic u)", "state_after": "case refine'_2.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)\ns : Set \u211d\nhs : MeasurableSet s\nu : \u211d\n\u22a2 Measurable fun b => ENNReal.ofReal (\u2191(condCdf \u03c1 b) u)"}, {"tactic": "exact (measurable_condCdf \u03c1 u).ennreal_ofReal", "annotated_tactic": ["exact (<a>measurable_condCdf</a> \u03c1 u).<a>ennreal_ofReal</a>", [{"full_name": "ProbabilityTheory.measurable_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [854, 9], "def_end_pos": [854, 27]}, {"full_name": "Measurable.ennreal_ofReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2005, 9], "def_end_pos": [2005, 34]}]], "state_before": "case refine'_2.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)\ns : Set \u211d\nhs : MeasurableSet s\nu : \u211d\n\u22a2 Measurable fun b => ENNReal.ofReal (\u2191(condCdf \u03c1 b) u)", "state_after": "no goals"}, {"tactic": "intro t ht ht_cd_meas", "annotated_tactic": ["intro t ht ht_cd_meas", []], "state_before": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u211d),\n    MeasurableSet t \u2192\n      (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t \u2192\n        (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t\u1d9c", "state_after": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c"}, {"tactic": "have :\n  (fun a => (condCdf \u03c1 a).measure t\u1d9c) =\n    (fun a => (condCdf \u03c1 a).measure univ) - fun a => (condCdf \u03c1 a).measure t := by\n  ext1 a\n  rw [measure_compl ht (measure_ne_top (condCdf \u03c1 a).measure _), Pi.sub_apply]", "annotated_tactic": ["have :\n      (fun a => (<a>condCdf</a> \u03c1 a).<a>measure</a> t\u1d9c) =\n        (fun a => (<a>condCdf</a> \u03c1 a).<a>measure</a> <a>univ</a>) - fun a => (<a>condCdf</a> \u03c1 a).<a>measure</a> t := by\n      ext1 a\n      rw [<a>measure_compl</a> ht (<a>measure_ne_top</a> (<a>condCdf</a> \u03c1 a).<a>measure</a> _), <a>Pi.sub_apply</a>]", [{"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "StieltjesFunction.measure", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [344, 27], "def_end_pos": [344, 34]}, {"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "StieltjesFunction.measure", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [344, 27], "def_end_pos": [344, 34]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "StieltjesFunction.measure", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [344, 27], "def_end_pos": [344, 34]}, {"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": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "StieltjesFunction.measure", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [344, 27], "def_end_pos": [344, 34]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c", "state_after": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\nthis :\n  (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n    (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c"}, {"tactic": "simp_rw [this, measure_condCdf_univ \u03c1]", "annotated_tactic": ["simp_rw [this, <a>measure_condCdf_univ</a> \u03c1]", [{"full_name": "ProbabilityTheory.measure_condCdf_univ", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [971, 9], "def_end_pos": [971, 29]}]], "state_before": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\nthis :\n  (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n    (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c", "state_after": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\nthis :\n  (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n    (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n\u22a2 Measurable ((fun a => 1) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)"}, {"tactic": "exact Measurable.sub measurable_const ht_cd_meas", "annotated_tactic": ["exact <a>Measurable.sub</a> <a>measurable_const</a> ht_cd_meas", [{"full_name": "Measurable.sub", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [333, 3], "def_end_pos": [333, 14]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case refine'_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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\nthis :\n  (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n    (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n\u22a2 Measurable ((fun a => 1) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)", "state_after": "no goals"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\n\u22a2 (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n    (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t", "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)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c =\n    ((fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a =>\n        \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)\n      a"}, {"tactic": "rw [measure_compl ht (measure_ne_top (condCdf \u03c1 a).measure _), Pi.sub_apply]", "annotated_tactic": ["rw [<a>measure_compl</a> ht (<a>measure_ne_top</a> (<a>condCdf</a> \u03c1 a).<a>measure</a> _), <a>Pi.sub_apply</a>]", [{"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": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "StieltjesFunction.measure", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [344, 27], "def_end_pos": [344, 34]}, {"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\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c =\n    ((fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a =>\n        \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)\n      a", "state_after": "no goals"}, {"tactic": "intro f hf_disj hf_meas hf_cd_meas", "annotated_tactic": ["intro f hf_disj hf_meas hf_cd_meas", []], "state_before": "case refine'_4\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)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)) \u2192\n          (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (\u22c3 i, f i)", "state_after": "case refine'_4\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)\ns : Set \u211d\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (\u22c3 i, f i)"}, {"tactic": "simp_rw [measure_iUnion hf_disj hf_meas]", "annotated_tactic": ["simp_rw [<a>measure_iUnion</a> hf_disj hf_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 refine'_4\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)\ns : Set \u211d\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (\u22c3 i, f i)", "state_after": "case refine'_4\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)\ns : Set \u211d\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)\n\u22a2 Measurable fun b => \u2211' (i : \u2115), \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (f i)"}, {"tactic": "exact Measurable.ennreal_tsum hf_cd_meas", "annotated_tactic": ["exact <a>Measurable.ennreal_tsum</a> hf_cd_meas", [{"full_name": "Measurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2136, 9], "def_end_pos": [2136, 32]}]], "state_before": "case refine'_4\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)\ns : Set \u211d\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)\n\u22a2 Measurable fun b => \u2211' (i : \u2115), \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (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\u2082_toFinmap", "start": [134, 1], "end": [136, 33], "traced_tactics": [{"tactic": "cases s\u2081", "annotated_tactic": ["cases s\u2081", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\ns\u2081 s\u2082 : AList \u03b2\nf : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3\nH : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082\n\u22a2 liftOn\u2082 \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 f H = f s\u2081 s\u2082", "state_after": "case mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\ns\u2082 : AList \u03b2\nf : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3\nH : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082\nentries\u271d : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys entries\u271d\n\u22a2 liftOn\u2082 \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 \u27e6s\u2082\u27e7 f H =\n    f { entries := entries\u271d, nodupKeys := nodupKeys\u271d } s\u2082"}, {"tactic": "cases s\u2082", "annotated_tactic": ["cases s\u2082", []], "state_before": "case mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\ns\u2082 : AList \u03b2\nf : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3\nH : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082\nentries\u271d : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys entries\u271d\n\u22a2 liftOn\u2082 \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 \u27e6s\u2082\u27e7 f H =\n    f { entries := entries\u271d, nodupKeys := nodupKeys\u271d } s\u2082", "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\nf : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3\nH : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082\nentries\u271d\u00b9 : List (Sigma \u03b2)\nnodupKeys\u271d\u00b9 : NodupKeys entries\u271d\u00b9\nentries\u271d : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys entries\u271d\n\u22a2 liftOn\u2082 \u27e6{ entries := entries\u271d\u00b9, nodupKeys := nodupKeys\u271d\u00b9 }\u27e7 \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 f H =\n    f { entries := entries\u271d\u00b9, nodupKeys := nodupKeys\u271d\u00b9 } { entries := entries\u271d, nodupKeys := nodupKeys\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\n\u03b3 : Type u_1\nf : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3\nH : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082\nentries\u271d\u00b9 : List (Sigma \u03b2)\nnodupKeys\u271d\u00b9 : NodupKeys entries\u271d\u00b9\nentries\u271d : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys entries\u271d\n\u22a2 liftOn\u2082 \u27e6{ entries := entries\u271d\u00b9, nodupKeys := nodupKeys\u271d\u00b9 }\u27e7 \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 f H =\n    f { entries := entries\u271d\u00b9, nodupKeys := nodupKeys\u271d\u00b9 } { entries := entries\u271d, nodupKeys := nodupKeys\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.rename_bind\u2081", "start": [250, 1], "end": [252, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.FTCFilter.finiteAt_inner", "start": [218, 1], "end": [220, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_const", "start": [638, 1], "end": [645, 37], "traced_tactics": [{"tactic": "cases h\u03b1 : isEmpty_or_nonempty \u03b1", "annotated_tactic": ["cases h\u03b1 : <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\nE : Type u_2\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\nx : F\nm : MeasurableSpace \u03b1\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) 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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : IsEmpty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) 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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : Nonempty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x"}, {"tactic": "have h_univ_empty : (univ : Set \u03b1) = \u2205 := Subsingleton.elim _ _", "annotated_tactic": ["have h_univ_empty : (<a>univ</a> : <a>Set</a> \u03b1) = \u2205 := <a>Subsingleton.elim</a> _ _", [{"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]}, {"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\nE : Type u_2\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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : IsEmpty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) 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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : IsEmpty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\nh_univ_empty : Set.univ = \u2205\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x"}, {"tactic": "rw [h_univ_empty, hT_empty]", "annotated_tactic": ["rw [h_univ_empty, hT_empty]", []], "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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : IsEmpty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\nh_univ_empty : Set.univ = \u2205\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) 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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : IsEmpty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\nh_univ_empty : Set.univ = \u2205\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u21910 x"}, {"tactic": "simp only [setToSimpleFunc, ContinuousLinearMap.zero_apply, sum_empty,\n  range_eq_empty_of_isEmpty]", "annotated_tactic": ["simp only [<a>setToSimpleFunc</a>, <a>ContinuousLinearMap.zero_apply</a>, <a>sum_empty</a>,\n      <a>range_eq_empty_of_isEmpty</a>]", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [281, 5], "def_end_pos": [281, 20]}, {"full_name": "ContinuousLinearMap.zero_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [644, 9], "def_end_pos": [644, 19]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [535, 9], "def_end_pos": [535, 34]}]], "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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : IsEmpty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\nh_univ_empty : Set.univ = \u2205\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u21910 x", "state_after": "no goals"}, {"tactic": "exact setToSimpleFunc_const' T x", "annotated_tactic": ["exact <a>setToSimpleFunc_const'</a> T x", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_const'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [632, 9], "def_end_pos": [632, 31]}]], "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\nx : F\nm : MeasurableSpace \u03b1\nh\u271d : Nonempty \u03b1\nh\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1)\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral", "start": [1705, 1], "end": [1757, 65], "traced_tactics": [{"tactic": "induction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing L", "annotated_tactic": ["induction' f using <a>MeasureTheory.SimpleFunc.induction</a> with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing L", [{"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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "simp only [hs, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter,\n  piecewise_eq_indicator, lintegral_indicator, lintegral_const, Measure.restrict_apply',\n  ENNReal.coe_indicator, Function.const_apply] at hL", "annotated_tactic": ["simp only [hs, <a>const_zero</a>, <a>coe_piecewise</a>, <a>coe_const</a>, <a>SimpleFunc.coe_zero</a>, <a>univ_inter</a>,\n      <a>piecewise_eq_indicator</a>, <a>lintegral_indicator</a>, <a>lintegral_const</a>, <a>Measure.restrict_apply'</a>,\n      <a>ENNReal.coe_indicator</a>, <a>Function.const_apply</a>] at hL", [{"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.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Set.piecewise_eq_indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [52, 3], "def_end_pos": [52, 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.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": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}]], "state_before": "case h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "have c_ne_zero : c \u2260 0 := by\n  intro hc\n  simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL", "annotated_tactic": ["have c_ne_zero : c \u2260 0 := by\n      intro hc\n      simp only [hc, <a>ENNReal.coe_zero</a>, <a>zero_mul</a>, <a>not_lt_zero</a>] at hL", [{"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}]], "state_before": "case h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "obtain \u27e8t, ht, ts, mlt, t_top\u27e9 :\n  \u2203 t : Set \u03b1, MeasurableSet t \u2227 t \u2286 s \u2227 L / \u2191c < \u03bc t \u2227 \u03bc t < \u221e :=\n  Measure.exists_subset_measure_lt_top hs this", "annotated_tactic": ["obtain \u27e8t, ht, ts, mlt, t_top\u27e9 :\n      \u2203 t : <a>Set</a> \u03b1, <a>MeasurableSet</a> t \u2227 t \u2286 s \u2227 L / \u2191c < \u03bc t \u2227 \u03bc t < \u221e :=\n      <a>Measure.exists_subset_measure_lt_top</a> hs this", [{"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": "case h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case h_ind.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "refine' \u27e8piecewise t ht (const \u03b1 c) (const \u03b1 0), fun x => _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>piecewise</a> t ht (<a>const</a> \u03b1 c) (<a>const</a> \u03b1 0), fun x => _, _, _\u27e9", [{"full_name": "MeasureTheory.SimpleFunc.piecewise", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [222, 5], "def_end_pos": [222, 14]}, {"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.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}]], "state_before": "case h_ind.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n      \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case h_ind.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\nx : \u03b1\n\u22a2 \u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n\ncase h_ind.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc < \u22a4\n\ncase h_ind.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc"}, {"tactic": "intro hc", "annotated_tactic": ["intro hc", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\n\u22a2 c \u2260 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nhc : c = 0\n\u22a2 False"}, {"tactic": "simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL", "annotated_tactic": ["simp only [hc, <a>ENNReal.coe_zero</a>, <a>zero_mul</a>, <a>not_lt_zero</a>] at hL", [{"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nhc : c = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rwa [ENNReal.div_lt_iff, mul_comm]", "annotated_tactic": ["rwa [<a>ENNReal.div_lt_iff</a>, <a>mul_comm</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": "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\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 L / \u2191c < \u2191\u2191\u03bc s", "state_after": "case h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2191c \u2260 0 \u2228 L \u2260 0\n\ncase ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4"}, {"tactic": "simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]", "annotated_tactic": ["simp only [c_ne_zero, <a>Ne.def</a>, <a>coe_eq_zero</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.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"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 h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2191c \u2260 0 \u2228 L \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]", "annotated_tactic": ["simp only [<a>Ne.def</a>, <a>coe_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.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]}, {"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\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "refine indicator_le_indicator_of_subset ts (fun x => ?_) x", "annotated_tactic": ["refine <a>indicator_le_indicator_of_subset</a> ts (fun x => ?_) x", [{"full_name": "Set.indicator_le_indicator_of_subset", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [849, 3], "def_end_pos": [849, 14]}]], "state_before": "case h_ind.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\nx : \u03b1\n\u22a2 \u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x", "state_after": "case h_ind.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\nx\u271d x : \u03b1\n\u22a2 0 \u2264 \u2191(const \u03b1 c) x"}, {"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 h_ind.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\nx\u271d x : \u03b1\n\u22a2 0 \u2264 \u2191(const \u03b1 c) x", "state_after": "no goals"}, {"tactic": "simp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter,\n  piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,\n  lintegral_const, Measure.restrict_apply', ENNReal.mul_lt_top ENNReal.coe_ne_top t_top.ne]", "annotated_tactic": ["simp only [ht, <a>const_zero</a>, <a>coe_piecewise</a>, <a>coe_const</a>, <a>SimpleFunc.coe_zero</a>, <a>univ_inter</a>,\n        <a>piecewise_eq_indicator</a>, <a>ENNReal.coe_indicator</a>, <a>Function.const_apply</a>, <a>lintegral_indicator</a>,\n        <a>lintegral_const</a>, <a>Measure.restrict_apply'</a>, <a>ENNReal.mul_lt_top</a> <a>ENNReal.coe_ne_top</a> t_top.ne]", [{"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.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"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_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"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": "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": [1567, 9], "def_end_pos": [1567, 24]}, {"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]}]], "state_before": "case h_ind.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero,\n  piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,\n  lintegral_const, Measure.restrict_apply', univ_inter]", "annotated_tactic": ["simp only [ht, <a>const_zero</a>, <a>coe_piecewise</a>, <a>coe_const</a>, <a>SimpleFunc.coe_zero</a>,\n        <a>piecewise_eq_indicator</a>, <a>ENNReal.coe_indicator</a>, <a>Function.const_apply</a>, <a>lintegral_indicator</a>,\n        <a>lintegral_const</a>, <a>Measure.restrict_apply'</a>, <a>univ_inter</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]}, {"full_name": "ENNReal.coe_indicator", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}, {"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": "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": [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]}]], "state_before": "case h_ind.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc", "state_after": "case h_ind.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 L < \u2191c * \u2191\u2191\u03bc t"}, {"tactic": "rwa [mul_comm, \u2190 ENNReal.div_lt_iff]", "annotated_tactic": ["rwa [<a>mul_comm</a>, \u2190 <a>ENNReal.div_lt_iff</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.div_lt_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1658, 19], "def_end_pos": [1658, 29]}]], "state_before": "case h_ind.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 L < \u2191c * \u2191\u2191\u03bc t", "state_after": "case h_ind.intro.intro.intro.intro.refine'_3.h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2191c \u2260 0 \u2228 L \u2260 0\n\ncase h_ind.intro.intro.intro.intro.refine'_3.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4"}, {"tactic": "simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]", "annotated_tactic": ["simp only [c_ne_zero, <a>Ne.def</a>, <a>coe_eq_zero</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.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"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 h_ind.intro.intro.intro.intro.refine'_3.h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2191c \u2260 0 \u2228 L \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]", "annotated_tactic": ["simp only [<a>Ne.def</a>, <a>coe_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.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]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "case h_ind.intro.intro.intro.intro.refine'_3.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "replace hL : L < \u222b\u207b x, f\u2081 x \u2202\u03bc + \u222b\u207b x, f\u2082 x \u2202\u03bc", "annotated_tactic": ["replace hL : L < \u222b\u207b x, f\u2081 x \u2202\u03bc + \u222b\u207b x, f\u2082 x \u2202\u03bc", []], "state_before": "case h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case hL\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "by_cases hf\u2081 : \u222b\u207b x, f\u2081 x \u2202\u03bc = 0", "annotated_tactic": ["by_cases hf\u2081 : \u222b\u207b x, f\u2081 x \u2202\u03bc = 0", []], "state_before": "case h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "by_cases hf\u2082 : \u222b\u207b x, f\u2082 x \u2202\u03bc = 0", "annotated_tactic": ["by_cases hf\u2082 : \u222b\u207b x, f\u2082 x \u2202\u03bc = 0", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "obtain \u27e8L\u2081, L\u2082, hL\u2081, hL\u2082, hL\u27e9 :\n  \u2203 L\u2081 L\u2082 : \u211d\u22650\u221e, (L\u2081 < \u222b\u207b x, f\u2081 x \u2202\u03bc) \u2227 (L\u2082 < \u222b\u207b x, f\u2082 x \u2202\u03bc) \u2227 L < L\u2081 + L\u2082 :=\n  ENNReal.exists_lt_add_of_lt_add hL hf\u2081 hf\u2082", "annotated_tactic": ["obtain \u27e8L\u2081, L\u2082, hL\u2081, hL\u2082, hL\u27e9 :\n      \u2203 L\u2081 L\u2082 : \u211d\u22650\u221e, (L\u2081 < \u222b\u207b x, f\u2081 x \u2202\u03bc) \u2227 (L\u2082 < \u222b\u207b x, f\u2082 x \u2202\u03bc) \u2227 L < L\u2081 + L\u2082 :=\n      <a>ENNReal.exists_lt_add_of_lt_add</a> hL hf\u2081 hf\u2082", [{"full_name": "ENNReal.exists_lt_add_of_lt_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [690, 9], "def_end_pos": [690, 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\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case neg.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "rcases h\u2081 hL\u2081 with \u27e8g\u2081, g\u2081_le, g\u2081_top, hg\u2081\u27e9", "annotated_tactic": ["rcases h\u2081 hL\u2081 with \u27e8g\u2081, g\u2081_le, g\u2081_top, hg\u2081\u27e9", []], "state_before": "case neg.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case neg.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "rcases h\u2082 hL\u2082 with \u27e8g\u2082, g\u2082_le, g\u2082_top, hg\u2082\u27e9", "annotated_tactic": ["rcases h\u2082 hL\u2082 with \u27e8g\u2082, g\u2082_le, g\u2082_top, hg\u2082\u27e9", []], "state_before": "case neg.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case neg.intro.intro.intro.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "refine' \u27e8g\u2081 + g\u2082, fun x => add_le_add (g\u2081_le x) (g\u2082_le x), _, _\u27e9", "annotated_tactic": ["refine' \u27e8g\u2081 + g\u2082, fun x => <a>add_le_add</a> (g\u2081_le x) (g\u2082_le x), _, _\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 neg.intro.intro.intro.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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc < \u22a4\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc"}, {"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": "case hL\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [hf\u2081, zero_add] at hL", "annotated_tactic": ["simp only [hf\u2081, <a>zero_add</a>] at hL", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "rcases h\u2082 hL with \u27e8g, g_le, g_top, gL\u27e9", "annotated_tactic": ["rcases h\u2082 hL with \u27e8g, g_le, g_top, gL\u27e9", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "refine' \u27e8g, fun x => (g_le x).trans _, g_top, gL\u27e9", "annotated_tactic": ["refine' \u27e8g, fun x => (g_le x).<a>trans</a> _, g_top, gL\u27e9", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nx : \u03b1\n\u22a2 \u2191f\u2082 x \u2264 \u2191(f\u2081 + f\u2082) x"}, {"tactic": "simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_left, zero_le']", "annotated_tactic": ["simp only [<a>SimpleFunc.coe_add</a>, <a>Pi.add_apply</a>, <a>le_add_iff_nonneg_left</a>, <a>zero_le'</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]}, {"full_name": "le_add_iff_nonneg_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [465, 30], "def_end_pos": [465, 52]}, {"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nx : \u03b1\n\u22a2 \u2191f\u2082 x \u2264 \u2191(f\u2081 + f\u2082) x", "state_after": "no goals"}, {"tactic": "simp only [hf\u2082, add_zero] at hL", "annotated_tactic": ["simp only [hf\u2082, <a>add_zero</a>] at hL", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "rcases h\u2081 hL with \u27e8g, g_le, g_top, gL\u27e9", "annotated_tactic": ["rcases h\u2081 hL with \u27e8g, g_le, g_top, gL\u27e9", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "refine' \u27e8g, fun x => (g_le x).trans _, g_top, gL\u27e9", "annotated_tactic": ["refine' \u27e8g, fun x => (g_le x).<a>trans</a> _, g_top, gL\u27e9", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nx : \u03b1\n\u22a2 \u2191f\u2081 x \u2264 \u2191(f\u2081 + f\u2082) x"}, {"tactic": "simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_right, zero_le']", "annotated_tactic": ["simp only [<a>SimpleFunc.coe_add</a>, <a>Pi.add_apply</a>, <a>le_add_iff_nonneg_right</a>, <a>zero_le'</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]}, {"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'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nx : \u03b1\n\u22a2 \u2191f\u2081 x \u2264 \u2191(f\u2081 + f\u2082) x", "state_after": "no goals"}, {"tactic": "apply lt_of_le_of_lt _ (add_lt_top.2 \u27e8g\u2081_top, g\u2082_top\u27e9)", "annotated_tactic": ["apply <a>lt_of_le_of_lt</a> _ (<a>add_lt_top</a>.2 \u27e8g\u2081_top, g\u2082_top\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": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc"}, {"tactic": "rw [\u2190 lintegral_add_left g\u2081.measurable.coe_nnreal_ennreal]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_left</a> g\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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc"}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply hL.trans ((ENNReal.add_lt_add hg\u2081 hg\u2082).trans_le _)", "annotated_tactic": ["apply hL.trans ((<a>ENNReal.add_lt_add</a> hg\u2081 hg\u2082).<a>trans_le</a> _)", [{"full_name": "ENNReal.add_lt_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [931, 19], "def_end_pos": [931, 29]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) 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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc"}, {"tactic": "rw [\u2190 lintegral_add_left g\u2081.measurable.coe_nnreal_ennreal]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_left</a> g\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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) 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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc"}, {"tactic": "simp only [coe_add, Pi.add_apply, ENNReal.coe_add, le_rfl]", "annotated_tactic": ["simp only [<a>coe_add</a>, <a>Pi.add_apply</a>, <a>ENNReal.coe_add</a>, <a>le_rfl</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]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n  \u2200 {L : \u211d\u22650\u221e},\n    L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) 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.Ico_subset_Iic_self", "start": [457, 1], "end": [458, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.rnDeriv_smul", "start": [1152, 1], "end": [1161, 59], "traced_tactics": [{"tactic": "refine'\n  Integrable.ae_eq_of_withDensity\u1d65_eq (integrable_rnDeriv _ _)\n    ((integrable_rnDeriv _ _).smul r) _", "annotated_tactic": ["refine'\n    <a>Integrable.ae_eq_of_withDensity\u1d65_eq</a> (<a>integrable_rnDeriv</a> _ _)\n      ((<a>integrable_rnDeriv</a> _ _).<a>smul</a> r) _", [{"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": 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[132, 22]}]], "state_before": "a b : \u2124\n\u22a2 Fintype.card \u2191(Set.uIcc a b) = natAbs (b - a) + 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.addHaar_sphere_of_ne_zero", "start": [515, 1], "end": [520, 73], "traced_tactics": [{"tactic": "rcases hr.lt_or_lt with (h | h)", "annotated_tactic": ["rcases hr.lt_or_lt with (h | 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 : Set E\nx : E\nr : \u211d\nhr : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0", "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 : Set E\nx : E\nr : \u211d\nhr : r \u2260 0\nh : r < 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0\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\nx : E\nr : \u211d\nhr : r \u2260 0\nh : 0 < r\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0"}, {"tactic": "simp only [empty_diff, measure_empty, \u2190 closedBall_diff_ball, closedBall_eq_empty.2 h]", "annotated_tactic": ["simp only [<a>empty_diff</a>, <a>measure_empty</a>, \u2190 <a>closedBall_diff_ball</a>, <a>closedBall_eq_empty</a>.2 h]", [{"full_name": "Set.empty_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1921, 9], "def_end_pos": [1921, 19]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "Metric.closedBall_diff_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [580, 9], "def_end_pos": [580, 29]}, {"full_name": "Metric.closedBall_eq_empty", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [525, 9], "def_end_pos": [525, 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\ns : Set E\nx : E\nr : \u211d\nhr : r \u2260 0\nh : r < 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 closedBall_diff_ball,\n  measure_diff ball_subset_closedBall measurableSet_ball measure_ball_lt_top.ne,\n  addHaar_ball_of_pos \u03bc _ h, addHaar_closedBall \u03bc _ h.le, tsub_self]", "annotated_tactic": ["rw [\u2190 <a>closedBall_diff_ball</a>,\n      <a>measure_diff</a> <a>ball_subset_closedBall</a> <a>measurableSet_ball</a> measure_ball_lt_top.ne,\n      <a>addHaar_ball_of_pos</a> \u03bc _ h, <a>addHaar_closedBall</a> \u03bc _ h.le, <a>tsub_self</a>]", [{"full_name": "Metric.closedBall_diff_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [580, 9], "def_end_pos": [580, 29]}, {"full_name": "MeasureTheory.measure_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [252, 9], "def_end_pos": [252, 21]}, {"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}, {"full_name": "measurableSet_ball", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1676, 9], "def_end_pos": [1676, 27]}, {"full_name": "MeasureTheory.Measure.addHaar_ball_of_pos", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [446, 9], "def_end_pos": [446, 28]}, {"full_name": "MeasureTheory.Measure.addHaar_closedBall", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [502, 9], "def_end_pos": [502, 27]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}]], "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\nx : E\nr : \u211d\nhr : r \u2260 0\nh : 0 < r\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 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.findExistsOneDivLT_spec", "start": [110, 9], "end": [113, 70], "traced_tactics": [{"tactic": "rw [findExistsOneDivLT, dif_pos hi]", "annotated_tactic": ["rw [<a>findExistsOneDivLT</a>, <a>dif_pos</a> hi]", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [107, 13], "def_end_pos": [107, 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.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT 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 MeasureTheory.SignedMeasure.ExistsOneDivLT s i (Nat.find (_ : \u2203 n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n))"}, {"tactic": "convert Nat.find_spec (existsNatOneDivLTMeasure_of_not_negative hi)", "annotated_tactic": ["convert <a>Nat.find_spec</a> (<a>existsNatOneDivLTMeasure_of_not_negative</a> hi)", [{"full_name": "Nat.find_spec", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [717, 19], "def_end_pos": [717, 28]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.existsNatOneDivLTMeasure_of_not_negative", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [99, 17], "def_end_pos": [99, 57]}]], "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.ExistsOneDivLT s i (Nat.find (_ : \u2203 n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSet.measurableSet_limsup", "start": [2132, 1], "end": [2134, 74], "traced_tactics": [{"tactic": "simpa only [\u2190 blimsup_true] using measurableSet_blimsup fun n _ => hs n", "annotated_tactic": ["simpa only [\u2190 <a>blimsup_true</a>] using <a>measurableSet_blimsup</a> fun n _ => hs n", [{"full_name": "Filter.blimsup_true", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [465, 9], "def_end_pos": [465, 21]}, {"full_name": "MeasurableSet.measurableSet_blimsup", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [2118, 9], "def_end_pos": [2118, 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 (limsup s atTop)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "volume_image_subtype_coe", "start": [4190, 1], "end": [4192, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "IsUnit.stronglyMeasurable_const_smul_iff", "start": [524, 8], "end": [528, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "IsometryEquiv.measurePreserving_hausdorffMeasure", "start": [920, 1], "end": [921, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.exists_forall_norm_le", "start": [287, 1], "end": [288, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.induction", "start": [924, 1], "end": [934, 39], "traced_tactics": [{"tactic": "refine' fun f => (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed _", "annotated_tactic": ["refine' fun f => (<a>Lp.simpleFunc.denseRange</a> hp_ne_top).<a>induction_on</a> f h_closed _", [{"full_name": "MeasureTheory.Lp.simpleFunc.denseRange", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [786, 19], "def_end_pos": [786, 29]}, {"full_name": "DenseRange.induction_on", "def_path": "Mathlib/Topology/DenseEmbedding.lean", "def_pos": [334, 9], "def_end_pos": [334, 32]}]], "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\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\n\u22a2 \u2200 (f : { x // x \u2208 Lp E p }), 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\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc E p \u03bc) }), P \u2191a"}, {"tactic": "refine' Lp.simpleFunc.induction (\u03b1 := \u03b1) (E := E) (lt_of_lt_of_le zero_lt_one _i.elim).ne'\n  hp_ne_top _ _", "annotated_tactic": ["refine' <a>Lp.simpleFunc.induction</a> (\u03b1 := \u03b1) (E := E) (<a>lt_of_lt_of_le</a> <a>zero_lt_one</a> _i.elim).<a>ne'</a>\n    hp_ne_top _ _", [{"full_name": "MeasureTheory.Lp.simpleFunc.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [715, 19], "def_end_pos": [715, 28]}, {"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": "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\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\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc E p \u03bc) }), P \u2191a", "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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      P \u2191(simpleFunc.toLp f hf) \u2192 P \u2191(simpleFunc.toLp g hg) \u2192 P \u2191(simpleFunc.toLp f hf + simpleFunc.toLp g hg)"}, {"tactic": "exact fun c s => h_ind c", "annotated_tactic": ["exact fun c s => h_ind c", []], "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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)", "state_after": "no goals"}, {"tactic": "exact fun f g hf hg => h_add hf hg", "annotated_tactic": ["exact fun f g hf hg => h_add hf hg", []], "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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 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 f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      P \u2191(simpleFunc.toLp f hf) \u2192 P \u2191(simpleFunc.toLp g hg) \u2192 P \u2191(simpleFunc.toLp f hf + simpleFunc.toLp g hg)", "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", "start": [292, 1], "end": [294, 17], "traced_tactics": [{"tactic": "rw [comp_eq_mk]", "annotated_tactic": ["rw [<a>comp_eq_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.comp_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [287, 9], "def_end_pos": [287, 19]}]], "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\nhg : Continuous g\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(comp 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\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\nhg : Continuous g\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(mk (g \u2218 \u2191f) (_ : AEStronglyMeasurable (fun x => g (\u2191f x)) \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\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\nhg : Continuous g\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(mk (g \u2218 \u2191f) (_ : AEStronglyMeasurable (fun x => g (\u2191f x)) \u03bc)) =\u1d50[\u03bc] g \u2218 \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.sum_prob_mem_Ioc_le", "start": [232, 1], "end": [311, 64], "traced_tactics": [{"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 N : \u2115\nhKN : K \u2264 N\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": "\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\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": "haveI : IsProbabilityMeasure \u03c1 := isProbabilityMeasure_map hint.aemeasurable", "annotated_tactic": ["haveI : <a>IsProbabilityMeasure</a> \u03c1 := <a>isProbabilityMeasure_map</a> hint.aemeasurable", [{"full_name": "MeasureTheory.IsProbabilityMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3026, 7], "def_end_pos": [3026, 27]}, {"full_name": "MeasureTheory.isProbabilityMeasure_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3065, 9], "def_end_pos": [3065, 33]}]], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\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": "\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\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": "have B : \u2200 a b, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b _ in Set.Ioc a b, (1 : \u211d) \u2202\u03c1) := by\n  intro a b\n  rw [ofReal_set_integral_one \u03c1 _,\n    Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc]\n  rfl", "annotated_tactic": ["have B : \u2200 a b, \u2119 {\u03c9 | X \u03c9 \u2208 <a>Set.Ioc</a> a b} = <a>ENNReal.ofReal</a> (\u222b _ in <a>Set.Ioc</a> a b, (1 : \u211d) \u2202\u03c1) := by\n    intro a b\n    rw [<a>ofReal_set_integral_one</a> \u03c1 _,\n      <a>Measure.map_apply_of_aemeasurable</a> hint.aemeasurable <a>measurableSet_Ioc</a>]\n    rfl", [{"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.ofReal_set_integral_one", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [199, 9], "def_end_pos": [199, 32]}, {"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_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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\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": "\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\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": "calc\n  \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc (j : \u211d) N} =\n      \u2211 j in range K, ENNReal.ofReal (\u222b _ in Set.Ioc (j : \u211d) N, (1 : \u211d) \u2202\u03c1) := by simp_rw [B]\n  _ = ENNReal.ofReal (\u2211 j in range K, \u222b _ in Set.Ioc (j : \u211d) N, (1 : \u211d) \u2202\u03c1) := by\n    rw [ENNReal.ofReal_sum_of_nonneg]\n    simp only [integral_const, Algebra.id.smul_eq_mul, mul_one, ENNReal.toReal_nonneg,\n      imp_true_iff]\n  _ = ENNReal.ofReal (\u2211 j in range K, \u222b _ in (j : \u211d)..N, (1 : \u211d) \u2202\u03c1) := by\n    congr 1\n    refine' sum_congr rfl fun j hj => _\n    rw [intervalIntegral.integral_of_le (Nat.cast_le.2 ((mem_range.1 hj).le.trans hKN))]\n  _ \u2264 ENNReal.ofReal (\ud835\udd3c[X] + 1) := ENNReal.ofReal_le_ofReal A", "annotated_tactic": ["calc\n    \u2211 j in <a>range</a> K, \u2119 {\u03c9 | X \u03c9 \u2208 <a>Set.Ioc</a> (j : \u211d) N} =\n        \u2211 j in <a>range</a> K, <a>ENNReal.ofReal</a> (\u222b _ in <a>Set.Ioc</a> (j : \u211d) N, (1 : \u211d) \u2202\u03c1) := by simp_rw [B]\n    _ = <a>ENNReal.ofReal</a> (\u2211 j in <a>range</a> K, \u222b _ in <a>Set.Ioc</a> (j : \u211d) N, (1 : \u211d) \u2202\u03c1) := by\n      rw [<a>ENNReal.ofReal_sum_of_nonneg</a>]\n      simp only [<a>integral_const</a>, <a>Algebra.id.smul_eq_mul</a>, <a>mul_one</a>, <a>ENNReal.toReal_nonneg</a>,\n        <a>imp_true_iff</a>]\n    _ = <a>ENNReal.ofReal</a> (\u2211 j in <a>range</a> K, \u222b _ in (j : \u211d)..N, (1 : \u211d) \u2202\u03c1) := by\n      congr 1\n      refine' <a>sum_congr</a> <a>rfl</a> fun j hj => _\n      rw [<a>intervalIntegral.integral_of_le</a> (<a>Nat.cast_le</a>.2 ((<a>mem_range</a>.1 hj).le.trans hKN))]\n    _ \u2264 <a>ENNReal.ofReal</a> (\ud835\udd3c[X] + 1) := <a>ENNReal.ofReal_le_ofReal</a> A", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"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": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"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_sum_of_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}, {"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": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"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]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"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 : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\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": "apply sum_congr rfl fun j hj => ?_", "annotated_tactic": ["apply <a>sum_congr</a> <a>rfl</a> fun j hj => ?_", [{"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\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1"}, {"tactic": "rw [intervalIntegral.sum_integral_adjacent_intervals_Ico ((mem_range.1 hj).le.trans hKN)]", "annotated_tactic": ["rw [<a>intervalIntegral.sum_integral_adjacent_intervals_Ico</a> ((<a>mem_range</a>.1 hj).le.trans hKN)]", [{"full_name": "intervalIntegral.sum_integral_adjacent_intervals_Ico", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [906, 9], "def_end_pos": [906, 44]}, {"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\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u2200 (k : \u2115), k \u2208 Set.Ico j N \u2192 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1)"}, {"tactic": "intro k _", "annotated_tactic": ["intro k _", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u2200 (k : \u2115), k \u2208 Set.Ico j N \u2192 IntervalIntegrable (fun x => 1) \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\nk : \u2115\na\u271d : k \u2208 Set.Ico j N\n\u22a2 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1)"}, {"tactic": "exact continuous_const.intervalIntegrable _ _", "annotated_tactic": ["exact continuous_const.intervalIntegrable _ _", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\nk : \u2115\na\u271d : k \u2208 Set.Ico j N\n\u22a2 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "simp_rw [sum_sigma']", "annotated_tactic": ["simp_rw [<a>sum_sigma'</a>]", [{"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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 j in range K, \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 =\n    \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 x in Finset.sigma (range K) fun a => Ico a N, \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), 1 \u2202Measure.map X \u2119 =\n    \u2211 x in Finset.sigma (range N) fun a => range (min (a + 1) K), \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 x in Finset.sigma (range K) fun a => Ico a N, \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), 1 \u2202Measure.map X \u2119 =\n    \u2211 x in Finset.sigma (range N) fun a => range (min (a + 1) K), \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a N),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ico a N\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a 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.97272) =\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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.97271) =\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a N),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) K)"}, {"tactic": "simp only [mem_sigma, mem_range, mem_Ico] at hij", "annotated_tactic": ["simp only [<a>mem_sigma</a>, <a>mem_range</a>, <a>mem_Ico</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_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [331, 9], "def_end_pos": [331, 16]}]], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N\nhij : i < K \u2227 i \u2264 j \u2227 j < N\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 (min (a + 1) K)"}, {"tactic": "simp only [hij, Nat.lt_succ_iff.2 hij.2.1, mem_sigma, mem_range, lt_min_iff, and_self_iff]", "annotated_tactic": ["simp only [hij, <a>Nat.lt_succ_iff</a>.2 hij.2.1, <a>mem_sigma</a>, <a>mem_range</a>, <a>lt_min_iff</a>, <a>and_self_iff</a>]", [{"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}, {"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": "lt_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [53, 9], "def_end_pos": [53, 19]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N\nhij : i < K \u2227 i \u2264 j \u2227 j < N\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 (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)),\n    (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ico a N", "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ico a N"}, {"tactic": "simp only [mem_sigma, mem_range, lt_min_iff] at hij", "annotated_tactic": ["simp only [<a>mem_sigma</a>, <a>mem_range</a>, <a>lt_min_iff</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": "lt_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [53, 9], "def_end_pos": [53, 19]}]], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ico a N", "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\nhij : i < N \u2227 j < i + 1 \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 => Ico a N"}, {"tactic": "simp only [hij, Nat.lt_succ_iff.1 hij.2.1, mem_sigma, mem_range, mem_Ico, and_self_iff]", "annotated_tactic": ["simp only [hij, <a>Nat.lt_succ_iff</a>.1 hij.2.1, <a>mem_sigma</a>, <a>mem_range</a>, <a>mem_Ico</a>, <a>and_self_iff</a>]", [{"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 20]}, {"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_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [331, 9], "def_end_pos": [331, 16]}, {"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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\nhij : i < N \u2227 j < i + 1 \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 => Ico a N", "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 => Ico a N) =\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 => Ico a N) =\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 => Ico a N) =\n    { fst := i, snd := j }", "state_after": "no goals"}, {"tactic": "apply sum_le_sum fun i _ => ?_", "annotated_tactic": ["apply <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 : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264\n    \u2211 i in range N, (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1"}, {"tactic": "simp only [Nat.cast_add, Nat.cast_one, sum_const, card_range, nsmul_eq_mul, Nat.cast_min]", "annotated_tactic": ["simp only [<a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>sum_const</a>, <a>card_range</a>, <a>nsmul_eq_mul</a>, <a>Nat.cast_min</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": "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": "Nat.cast_min", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [49, 9], "def_end_pos": [49, 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 min (\u2191i + 1) \u2191K * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119"}, {"tactic": "refine' mul_le_mul_of_nonneg_right (min_le_left _ _) _", "annotated_tactic": ["refine' <a>mul_le_mul_of_nonneg_right</a> (<a>min_le_left</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": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 min (\u2191i + 1) \u2191K * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119", "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 \u2264 \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119"}, {"tactic": "apply intervalIntegral.integral_nonneg", "annotated_tactic": ["apply <a>intervalIntegral.integral_nonneg</a>", [{"full_name": "intervalIntegral.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1368, 9], "def_end_pos": [1368, 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 \u2264 \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119", "state_after": "case hab\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191i \u2264 \u2191i + 1\n\ncase 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2200 (u : \u211d), u \u2208 Set.Icc (\u2191i) (\u2191i + 1) \u2192 0 \u2264 1"}, {"tactic": "simp only [le_add_iff_nonneg_right, zero_le_one]", "annotated_tactic": ["simp only [<a>le_add_iff_nonneg_right</a>, <a>zero_le_one</a>]", [{"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": "case hab\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191i \u2264 \u2191i + 1", "state_after": "no goals"}, {"tactic": "simp only [zero_le_one, imp_true_iff]", "annotated_tactic": ["simp only [<a>zero_le_one</a>, <a>imp_true_iff</a>]", [{"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"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 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2200 (u : \u211d), u \u2208 Set.Icc (\u2191i) (\u2191i + 1) \u2192 0 \u2264 1", "state_after": "no goals"}, {"tactic": "apply sum_le_sum fun i _ => ?_", "annotated_tactic": ["apply <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 : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 i in range N, (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u2211 i in range N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1"}, {"tactic": "have I : (i : \u211d) \u2264 (i + 1 : \u2115) := by\n  simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]", "annotated_tactic": ["have I : (i : \u211d) \u2264 (i + 1 : \u2115) := by\n          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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1"}, {"tactic": "simp_rw [intervalIntegral.integral_of_le I, \u2190 integral_mul_left]", "annotated_tactic": ["simp_rw [<a>intervalIntegral.integral_of_le</a> I, \u2190 <a>integral_mul_left</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": "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\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 \u222b (a : \u211d) in Set.Ioc \u2191i \u2191(i + 1), (\u2191i + 1) * 1 \u2202Measure.map X \u2119 \u2264\n    \u222b (x : \u211d) in Set.Ioc \u2191i \u2191(i + 1), x + 1 \u2202Measure.map X \u2119"}, {"tactic": "apply set_integral_mono_on", "annotated_tactic": ["apply <a>set_integral_mono_on</a>", [{"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": "\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 \u222b (a : \u211d) in Set.Ioc \u2191i \u2191(i + 1), (\u2191i + 1) * 1 \u2202Measure.map X \u2119 \u2264\n    \u222b (x : \u211d) in Set.Ioc \u2191i \u2191(i + 1), x + 1 \u2202Measure.map X \u2119", "state_after": "case 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 IntegrableOn (fun a => (\u2191i + 1) * 1) (Set.Ioc \u2191i \u2191(i + 1))\n\ncase hg\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 IntegrableOn (fun a => a + 1) (Set.Ioc \u2191i \u2191(i + 1))\n\ncase hs\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 MeasurableSet (Set.Ioc \u2191i \u2191(i + 1))\n\ncase 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 \u2200 (x : \u211d), x \u2208 Set.Ioc \u2191i \u2191(i + 1) \u2192 (\u2191i + 1) * 1 \u2264 x + 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191i \u2264 \u2191(i + 1)", "state_after": "no goals"}, {"tactic": "exact continuous_const.integrableOn_Ioc", "annotated_tactic": ["exact continuous_const.integrableOn_Ioc", []], "state_before": "case 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 IntegrableOn (fun a => (\u2191i + 1) * 1) (Set.Ioc \u2191i \u2191(i + 1))", "state_after": "no goals"}, {"tactic": "exact (continuous_id.add continuous_const).integrableOn_Ioc", "annotated_tactic": ["exact (continuous_id.add <a>continuous_const</a>).<a>integrableOn_Ioc</a>", [{"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}, {"full_name": "Continuous.integrableOn_Ioc", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [411, 9], "def_end_pos": [411, 36]}]], "state_before": "case hg\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 IntegrableOn (fun a => a + 1) (Set.Ioc \u2191i \u2191(i + 1))", "state_after": "no goals"}, {"tactic": "exact measurableSet_Ioc", "annotated_tactic": ["exact <a>measurableSet_Ioc</a>", [{"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "case hs\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 MeasurableSet (Set.Ioc \u2191i \u2191(i + 1))", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 \u2200 (x : \u211d), x \u2208 Set.Ioc \u2191i \u2191(i + 1) \u2192 (\u2191i + 1) * 1 \u2264 x + 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191i \u2191(i + 1)\n\u22a2 (\u2191i + 1) * 1 \u2264 x + 1"}, {"tactic": "simp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx", "annotated_tactic": ["simp only [<a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>Set.mem_Ioc</a>] at hx", [{"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": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}]], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191i \u2191(i + 1)\n\u22a2 (\u2191i + 1) * 1 \u2264 x + 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\nx : \u211d\nhx : \u2191i < x \u2227 x \u2264 \u2191i + 1\n\u22a2 (\u2191i + 1) * 1 \u2264 x + 1"}, {"tactic": "simp [hx.1.le]", "annotated_tactic": ["simp [hx.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": "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\nx : \u211d\nhx : \u2191i < x \u2227 x \u2264 \u2191i + 1\n\u22a2 (\u2191i + 1) * 1 \u2264 x + 1", "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 i in range N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nk : \u2115\nx\u271d : k < N\n\u22a2 IntervalIntegrable (fun x => x + 1) \u03c1 \u2191k \u2191(k + 1)"}, {"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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \u2202\u03c1", "state_after": "no goals"}, {"tactic": "exact (continuous_id.add continuous_const).intervalIntegrable _ _", "annotated_tactic": ["exact (continuous_id.add <a>continuous_const</a>).<a>intervalIntegrable</a> _ _", [{"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}, {"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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nk : \u2115\nx\u271d : k < N\n\u22a2 IntervalIntegrable (fun x => x + 1) \u03c1 \u2191k \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "rw [intervalIntegral.integral_add]", "annotated_tactic": ["rw [<a>intervalIntegral.integral_add</a>]", [{"full_name": "intervalIntegral.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [578, 16], "def_end_pos": [578, 28]}]], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x \u2202\u03c1 + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1", "state_after": "case 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 IntervalIntegrable (fun x => x) \u03c1 0 \u2191N\n\ncase hg\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 IntervalIntegrable (fun x => 1) \u03c1 0 \u2191N"}, {"tactic": "exact continuous_id.intervalIntegrable _ _", "annotated_tactic": ["exact continuous_id.intervalIntegrable _ _", []], "state_before": "case 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 IntervalIntegrable (fun x => x) \u03c1 0 \u2191N", "state_after": "no goals"}, {"tactic": "exact continuous_const.intervalIntegrable _ _", "annotated_tactic": ["exact continuous_const.intervalIntegrable _ _", []], "state_before": "case hg\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 IntervalIntegrable (fun x => 1) \u03c1 0 \u2191N", "state_after": "no goals"}, {"tactic": "rw [integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]", "annotated_tactic": ["rw [<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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, x \u2202\u03c1 + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 = (\u222b (a : \u03a9), truncation X (\u2191N) a) + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1", "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": "\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 (\u222b (a : \u03a9), X a) + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 1"}, {"tactic": "rw [intervalIntegral.integral_of_le (Nat.cast_nonneg _)]", "annotated_tactic": ["rw [<a>intervalIntegral.integral_of_le</a> (<a>Nat.cast_nonneg</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": "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in Set.Ioc 0 \u2191N, 1 \u2202\u03c1 \u2264 1"}, {"tactic": "simp only [integral_const, Measure.restrict_apply', measurableSet_Ioc, Set.univ_inter,\n  Algebra.id.smul_eq_mul, mul_one]", "annotated_tactic": ["simp only [<a>integral_const</a>, <a>Measure.restrict_apply'</a>, <a>measurableSet_Ioc</a>, <a>Set.univ_inter</a>,\n          <a>Algebra.id.smul_eq_mul</a>, <a>mul_one</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": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"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_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in Set.Ioc 0 \u2191N, 1 \u2202\u03c1 \u2264 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 1"}, {"tactic": "rw [\u2190 ENNReal.one_toReal]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.one_toReal</a>]", [{"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 ENNReal.toReal 1"}, {"tactic": "exact ENNReal.toReal_mono ENNReal.one_ne_top prob_le_one", "annotated_tactic": ["exact <a>ENNReal.toReal_mono</a> <a>ENNReal.one_ne_top</a> <a>prob_le_one</a>", [{"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}, {"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.prob_le_one", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3055, 9], "def_end_pos": [3055, 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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 ENNReal.toReal 1", "state_after": "no goals"}, {"tactic": "intro a b", "annotated_tactic": ["intro a b", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\n\u22a2 \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\na b : \u211d\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)"}, {"tactic": "rw [ofReal_set_integral_one \u03c1 _,\n  Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc]", "annotated_tactic": ["rw [<a>ofReal_set_integral_one</a> \u03c1 _,\n      <a>Measure.map_apply_of_aemeasurable</a> hint.aemeasurable <a>measurableSet_Ioc</a>]", [{"full_name": "MeasureTheory.ofReal_set_integral_one", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [199, 9], "def_end_pos": [199, 32]}, {"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_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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\na b : \u211d\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\na b : \u211d\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = \u2191\u2191\u2119 (X \u207b\u00b9' Set.Ioc a b)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\na b : \u211d\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = \u2191\u2191\u2119 (X \u207b\u00b9' Set.Ioc a b)", "state_after": "no goals"}, {"tactic": "simp_rw [B]", "annotated_tactic": ["simp_rw [B]", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} = \u2211 j in range K, ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1)", "state_after": "no goals"}, {"tactic": "rw [ENNReal.ofReal_sum_of_nonneg]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_sum_of_nonneg</a>]", [{"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 : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) =\n    ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2200 (i : \u2115), i \u2208 range K \u2192 0 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191i \u2191N, 1 \u2202\u03c1"}, {"tactic": "simp only [integral_const, Algebra.id.smul_eq_mul, mul_one, ENNReal.toReal_nonneg,\n  imp_true_iff]", "annotated_tactic": ["simp only [<a>integral_const</a>, <a>Algebra.id.smul_eq_mul</a>, <a>mul_one</a>, <a>ENNReal.toReal_nonneg</a>,\n        <a>imp_true_iff</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": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2200 (i : \u2115), i \u2208 range K \u2192 0 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191i \u2191N, 1 \u2202\u03c1", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) =\n    ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1)", "state_after": "case e_r\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1"}, {"tactic": "refine' sum_congr rfl fun j hj => _", "annotated_tactic": ["refine' <a>sum_congr</a> <a>rfl</a> fun j hj => _", [{"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": "case e_r\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1", "state_after": "case e_r\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1"}, {"tactic": "rw [intervalIntegral.integral_of_le (Nat.cast_le.2 ((mem_range.1 hj).le.trans hKN))]", "annotated_tactic": ["rw [<a>intervalIntegral.integral_of_le</a> (<a>Nat.cast_le</a>.2 ((<a>mem_range</a>.1 hj).le.trans hKN))]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 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_r\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 N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.strictMono_iff", "start": [162, 1], "end": [166, 8], "traced_tactics": [{"tactic": "classical\nsimp only [strictMono_iff_forall_covby, covby_iff, forall_exists_index, and_imp]\naesop", "annotated_tactic": ["classical\n  simp only [<a>strictMono_iff_forall_covby</a>, <a>covby_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]\n  aesop", [{"full_name": "strictMono_iff_forall_covby", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1286, 7], "def_end_pos": [1286, 34]}, {"full_name": "Finset.covby_iff", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [144, 7], "def_end_pos": [144, 16]}, {"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": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 StrictMono f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi)", "state_after": "no goals"}, {"tactic": "simp only [strictMono_iff_forall_covby, covby_iff, forall_exists_index, and_imp]", "annotated_tactic": ["simp only [<a>strictMono_iff_forall_covby</a>, <a>covby_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]", [{"full_name": "strictMono_iff_forall_covby", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1286, 7], "def_end_pos": [1286, 34]}, {"full_name": "Finset.covby_iff", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [144, 7], "def_end_pos": [144, 16]}, {"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": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 StrictMono f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi)", "state_after": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (a b : Finset \u03b1) (x : \u03b1) (x_1 : \u00acx \u2208 a), b = cons x a x_1 \u2192 f a < f b) \u2194\n    \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi)"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (a b : Finset \u03b1) (x : \u03b1) (x_1 : \u00acx \u2208 a), b = cons x a x_1 \u2192 f a < f b) \u2194\n    \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi)", "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.empty_eq", "start": [36, 9], "end": [36, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.coeff_map", "start": [1300, 1], "end": [1317, 20], "traced_tactics": [{"tactic": "apply MvPolynomial.induction_on p <;> clear p", "annotated_tactic": ["apply <a>MvPolynomial.induction_on</a> p <;> clear 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\u271d q : MvPolynomial \u03c3 R\nf : R \u2192+* S\u2081\np : MvPolynomial \u03c3 R\n\u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)", "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\n\u22a2 \u2200 (a : R) (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (\u2191C a)) = \u2191f (coeff m (\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\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)) \u2192\n      (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)) \u2192\n        \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (p + q)) = \u2191f (coeff m (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\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)) \u2192\n      \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (p * X n)) = \u2191f (coeff m (p * X n))"}, {"tactic": "intro r m", "annotated_tactic": ["intro r m", []], "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\n\u22a2 \u2200 (a : R) (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (\u2191C a)) = \u2191f (coeff m (\u2191C a))", "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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) (\u2191C r)) = \u2191f (coeff m (\u2191C r))"}, {"tactic": "rw [map_C]", "annotated_tactic": ["rw [<a>map_C</a>]", [{"full_name": "MvPolynomial.map_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1238, 9], "def_end_pos": [1238, 14]}]], "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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) (\u2191C r)) = \u2191f (coeff m (\u2191C r))", "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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191C (\u2191f r)) = \u2191f (coeff m (\u2191C r))"}, {"tactic": "simp only [coeff_C]", "annotated_tactic": ["simp only [<a>coeff_C</a>]", [{"full_name": "MvPolynomial.coeff_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [659, 9], "def_end_pos": [659, 16]}]], "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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191C (\u2191f r)) = \u2191f (coeff m (\u2191C r))", "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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if 0 = m then \u2191f r else 0) = \u2191f (if 0 = m then r else 0)"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if 0 = m then \u2191f r else 0) = \u2191f (if 0 = m then r else 0)", "state_after": "case pos\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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : 0 = m\n\u22a2 \u2191f r = \u2191f r\n\ncase neg\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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : \u00ac0 = m\n\u22a2 0 = \u2191f 0"}, {"tactic": "rw [f.map_zero]", "annotated_tactic": ["rw [f.map_zero]", []], "state_before": "case neg\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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : \u00ac0 = m\n\u22a2 0 = \u2191f 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\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\u271d : \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\nr : R\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : 0 = m\n\u22a2 \u2191f r = \u2191f r", "state_after": "no goals"}, {"tactic": "intro p q hp hq m", "annotated_tactic": ["intro p q hp hq m", []], "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\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)) \u2192\n      (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)) \u2192\n        \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (p + q)) = \u2191f (coeff m (p + q))", "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\u271d : \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\np q : MvPolynomial \u03c3 R\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nhq : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) (p + q)) = \u2191f (coeff m (p + q))"}, {"tactic": "simp only [hp, hq, (map f).map_add, coeff_add]", "annotated_tactic": ["simp only [hp, hq, (<a>map</a> f).<a>map_add</a>, <a>coeff_add</a>]", [{"full_name": "MvPolynomial.map", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1228, 5], "def_end_pos": [1228, 8]}, {"full_name": "RingHom.map_add", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [564, 19], "def_end_pos": [564, 26]}, {"full_name": "MvPolynomial.coeff_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [614, 9], "def_end_pos": [614, 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\u271d : \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\np q : MvPolynomial \u03c3 R\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nhq : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) (p + q)) = \u2191f (coeff m (p + q))", "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\u271d : \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\np q : MvPolynomial \u03c3 R\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nhq : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191f (coeff m p) + \u2191f (coeff m q) = \u2191f (coeff m p + coeff m q)"}, {"tactic": "rw [f.map_add]", "annotated_tactic": ["rw [f.map_add]", []], "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\u271d : \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\np q : MvPolynomial \u03c3 R\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nhq : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191f (coeff m p) + \u2191f (coeff m q) = \u2191f (coeff m p + coeff m q)", "state_after": "no goals"}, {"tactic": "intro p i hp m", "annotated_tactic": ["intro p i hp m", []], "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\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)) \u2192\n      \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (p * X n)) = \u2191f (coeff m (p * X n))", "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 a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) (p * X i)) = \u2191f (coeff m (p * X i))"}, {"tactic": "simp only [hp, (map f).map_mul, map_X]", "annotated_tactic": ["simp only [hp, (<a>map</a> f).<a>map_mul</a>, <a>map_X</a>]", [{"full_name": "MvPolynomial.map", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1228, 5], "def_end_pos": [1228, 8]}, {"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.map_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1243, 9], "def_end_pos": [1243, 14]}]], "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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) (p * X i)) = \u2191f (coeff m (p * X i))", "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 a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) p * X i) = \u2191f (coeff m (p * X i))"}, {"tactic": "simp only [hp, mem_support_iff, coeff_mul_X']", "annotated_tactic": ["simp only [hp, <a>mem_support_iff</a>, <a>coeff_mul_X'</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_mul_X'", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [802, 9], "def_end_pos": [802, 21]}]], "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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m (\u2191(map f) p * X i) = \u2191f (coeff m (p * X i))", "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 a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if i \u2208 m.support then \u2191f (coeff (m - fun\u2080 | i => 1) p) else 0) =\n    \u2191f (if i \u2208 m.support then coeff (m - fun\u2080 | i => 1) p else 0)"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if i \u2208 m.support then \u2191f (coeff (m - fun\u2080 | i => 1) p) else 0) =\n    \u2191f (if i \u2208 m.support then coeff (m - fun\u2080 | i => 1) p else 0)", "state_after": "case pos\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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : i \u2208 m.support\n\u22a2 \u2191f (coeff (m - fun\u2080 | i => 1) p) = \u2191f (coeff (m - fun\u2080 | i => 1) p)\n\ncase neg\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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : \u00aci \u2208 m.support\n\u22a2 0 = \u2191f 0"}, {"tactic": "rw [f.map_zero]", "annotated_tactic": ["rw [f.map_zero]", []], "state_before": "case neg\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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : \u00aci \u2208 m.support\n\u22a2 0 = \u2191f 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\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\u271d : \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\np : MvPolynomial \u03c3 R\ni : \u03c3\nhp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)\nm : \u03c3 \u2192\u2080 \u2115\nh\u271d : i \u2208 m.support\n\u22a2 \u2191f (coeff (m - fun\u2080 | i => 1) p) = \u2191f (coeff (m - fun\u2080 | i => 1) p)", "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_lipschitz_estimate", "start": [390, 1], "end": [405, 40], "traced_tactics": [{"tactic": "simp only [\u2190 \u03bc.testAgainstNN_const (nndist f g), \u2190 testAgainstNN_add, \u2190 ENNReal.coe_le_coe,\n  BoundedContinuousFunction.coe_add, const_apply, ENNReal.coe_add, Pi.add_apply,\n  coe_nnreal_ennreal_nndist, testAgainstNN_coe_eq]", "annotated_tactic": ["simp only [\u2190 \u03bc.testAgainstNN_const (<a>nndist</a> f g), \u2190 <a>testAgainstNN_add</a>, \u2190 <a>ENNReal.coe_le_coe</a>,\n    <a>BoundedContinuousFunction.coe_add</a>, <a>const_apply</a>, <a>ENNReal.coe_add</a>, <a>Pi.add_apply</a>,\n    <a>coe_nnreal_ennreal_nndist</a>, <a>testAgainstNN_coe_eq</a>]", [{"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 9]}, {"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_add", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [372, 9], "def_end_pos": [372, 26]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "BoundedContinuousFunction.coe_add", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [683, 9], "def_end_pos": [683, 16]}, {"full_name": "BoundedContinuousFunction.const_apply", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [294, 3], "def_end_pos": [294, 47]}, {"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": "coe_nnreal_ennreal_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 34]}, {"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 g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc f \u2264 testAgainstNN \u03bc g + nndist f g * mass \u03bc", "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191g \u03c9) + edist f g \u2202\u2191\u03bc"}, {"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": "\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191g \u03c9) + edist f g \u2202\u2191\u03bc", "state_after": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g"}, {"tactic": "have le_dist : \u2200 \u03c9, dist (f \u03c9) (g \u03c9) \u2264 nndist f g := BoundedContinuousFunction.dist_coe_le_dist", "annotated_tactic": ["have le_dist : \u2200 \u03c9, <a>dist</a> (f \u03c9) (g \u03c9) \u2264 <a>nndist</a> f g := <a>BoundedContinuousFunction.dist_coe_le_dist</a>", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 9]}, {"full_name": "BoundedContinuousFunction.dist_coe_le_dist", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [169, 9], "def_end_pos": [169, 25]}]], "state_before": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g", "state_after": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g"}, {"tactic": "intro \u03c9", "annotated_tactic": ["intro \u03c9", []], "state_before": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g", "state_after": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9"}, {"tactic": "have le' : f \u03c9 \u2264 g \u03c9 + nndist f g := by\n  apply (NNReal.le_add_nndist (f \u03c9) (g \u03c9)).trans\n  rw [add_le_add_iff_left]\n  exact dist_le_coe.mp (le_dist \u03c9)", "annotated_tactic": ["have le' : f \u03c9 \u2264 g \u03c9 + <a>nndist</a> f g := by\n    apply (<a>NNReal.le_add_nndist</a> (f \u03c9) (g \u03c9)).<a>trans</a>\n    rw [<a>add_le_add_iff_left</a>]\n    exact dist_le_coe.mp (le_dist \u03c9)", [{"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 9]}, {"full_name": "NNReal.le_add_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1716, 9], "def_end_pos": [1716, 29]}, {"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_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9", "state_after": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9"}, {"tactic": "have le : (f \u03c9 : \u211d\u22650\u221e) \u2264 (g \u03c9 : \u211d\u22650\u221e) + nndist f g := by\n  rw [\u2190 ENNReal.coe_add];\n  exact ENNReal.coe_mono le'", "annotated_tactic": ["have le : (f \u03c9 : \u211d\u22650\u221e) \u2264 (g \u03c9 : \u211d\u22650\u221e) + <a>nndist</a> f g := by\n    rw [\u2190 <a>ENNReal.coe_add</a>];\n    exact <a>ENNReal.coe_mono</a> le'", [{"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 9]}, {"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_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [363, 9], "def_end_pos": [363, 17]}]], "state_before": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9", "state_after": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\nle : \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g)\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9"}, {"tactic": "rwa [coe_nnreal_ennreal_nndist] at le", "annotated_tactic": ["rwa [<a>coe_nnreal_ennreal_nndist</a>] at le", [{"full_name": "coe_nnreal_ennreal_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 34]}]], "state_before": "case hfg\n\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\nle : \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g)\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9", "state_after": "no goals"}, {"tactic": "apply (NNReal.le_add_nndist (f \u03c9) (g \u03c9)).trans", "annotated_tactic": ["apply (<a>NNReal.le_add_nndist</a> (f \u03c9) (g \u03c9)).<a>trans</a>", [{"full_name": "NNReal.le_add_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1716, 9], "def_end_pos": [1716, 29]}, {"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\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g", "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 \u2191g \u03c9 + nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191g \u03c9 + nndist f g"}, {"tactic": "rw [add_le_add_iff_left]", "annotated_tactic": ["rw [<a>add_le_add_iff_left</a>]", [{"full_name": "add_le_add_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 \u2191g \u03c9 + nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191g \u03c9 + nndist f g", "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 nndist f g"}, {"tactic": "exact dist_le_coe.mp (le_dist \u03c9)", "annotated_tactic": ["exact dist_le_coe.mp (le_dist \u03c9)", []], "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 nndist f g", "state_after": "no goals"}, {"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": "\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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g)", "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9 + nndist f g)"}, {"tactic": "exact ENNReal.coe_mono le'", "annotated_tactic": ["exact <a>ENNReal.coe_mono</a> le'", [{"full_name": "ENNReal.coe_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [363, 9], "def_end_pos": [363, 17]}]], "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 g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9 + nndist f g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "IsUnit.aemeasurable_const_smul_iff", "start": [746, 8], "end": [749, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.hausdorffMeasure_smul\u2080", "start": [803, 1], "end": [816, 71], "traced_tactics": [{"tactic": "intro r _ s", "annotated_tactic": ["intro r _ s", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\n\u22a2 \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr\u271d : \ud835\udd5c\nhr : r\u271d \u2260 0\ns\u271d : Set E\nr : \ud835\udd5c\na\u271d : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s"}, {"tactic": "simp only [NNReal.rpow_eq_pow, ENNReal.smul_def, \u2190 ENNReal.coe_rpow_of_nonneg _ hd, smul_eq_mul]", "annotated_tactic": ["simp only [<a>NNReal.rpow_eq_pow</a>, <a>ENNReal.smul_def</a>, \u2190 <a>ENNReal.coe_rpow_of_nonneg</a> _ hd, <a>smul_eq_mul</a>]", [{"full_name": "NNReal.rpow_eq_pow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [42, 9], "def_end_pos": [42, 20]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"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": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr\u271d : \ud835\udd5c\nhr : r\u271d \u2260 0\ns\u271d : Set E\nr : \ud835\udd5c\na\u271d : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr\u271d : \ud835\udd5c\nhr : r\u271d \u2260 0\ns\u271d : Set E\nr : \ud835\udd5c\na\u271d : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 \u2191\u2016r\u2016\u208a ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "exact (lipschitzWith_smul (\u03b2 := E) r).hausdorffMeasure_image_le hd s", "annotated_tactic": ["exact (<a>lipschitzWith_smul</a> (\u03b2 := E) r).<a>hausdorffMeasure_image_le</a> hd s", [{"full_name": "lipschitzWith_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [46, 9], "def_end_pos": [46, 27]}, {"full_name": "LipschitzWith.hausdorffMeasure_image_le", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [794, 9], "def_end_pos": [794, 34]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr\u271d : \ud835\udd5c\nhr : r\u271d \u2260 0\ns\u271d : Set E\nr : \ud835\udd5c\na\u271d : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 \u2191\u2016r\u2016\u208a ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "no goals"}, {"tactic": "refine' le_antisymm (this hr s) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (this hr s) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) = NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u2264 \u2191\u2191\u03bcH[d] (r \u2022 s)"}, {"tactic": "rw [\u2190 ENNReal.le_inv_smul_iff]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.le_inv_smul_iff</a>]", [{"full_name": "ENNReal.le_inv_smul_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1714, 19], "def_end_pos": [1714, 34]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u2264 \u2191\u2191\u03bcH[d] (r \u2022 s)", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 (NNReal.rpow \u2016r\u2016\u208a d)\u207b\u00b9 \u2022 \u2191\u2191\u03bcH[d] (r \u2022 s)\n\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 (NNReal.rpow \u2016r\u2016\u208a d)\u207b\u00b9 \u2022 \u2191\u2191\u03bcH[d] (r \u2022 s)\n\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type 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\u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0"}, {"tactic": "refine' Eq.trans_le _ (this (inv_ne_zero hr) (r \u2022 s))", "annotated_tactic": ["refine' <a>Eq.trans_le</a> _ (this (<a>inv_ne_zero</a> hr) (r \u2022 s))", [{"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 18]}, {"full_name": "inv_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [49, 9], "def_end_pos": [49, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 \u2191(\u2016r\u207b\u00b9\u2016\u208a ^ d) * \u2191\u2191\u03bcH[d] (r \u2022 s)", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s = \u2191\u2191\u03bcH[d] (r\u207b\u00b9 \u2022 r \u2022 s)"}, {"tactic": "rw [inv_smul_smul\u2080 hr]", "annotated_tactic": ["rw [<a>inv_smul_smul\u2080</a> hr]", [{"full_name": "inv_smul_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [192, 9], "def_end_pos": [192, 23]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c 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"full_name": "Partrec.rfind", "start": [565, 1], "end": [573, 20], "traced_tactics": [{"tactic": "cases' e : decode (\u03b1 := \u03b1) n with a <;> simp [e, Nat.rfind_zero_none, map_id']", "annotated_tactic": ["cases' e : <a>decode</a> (\u03b1 := \u03b1) n with a <;> simp [e, <a>Nat.rfind_zero_none</a>, <a>map_id'</a>]", [{"full_name": "Encodable.decode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [51, 3], "def_end_pos": [51, 9]}, {"full_name": "Nat.rfind_zero_none", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [118, 9], "def_end_pos": [118, 24]}, {"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\n\u22a2 (Nat.rfind fun n_1 =>\n      (fun m => decide (m = 0)) <$>\n        Part.bind \u2191(decode (Nat.pair n n_1)) fun a =>\n          Part.map encode ((fun a => Part.map (fun b => bif (a, b).2 then 0 else 1) (p a.1 a.2)) a)) =\n    Part.bind \u2191(decode n) fun a => Part.map encode ((fun a => Nat.rfind (p a)) a)", "state_after": "case some\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Nat.rfind fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) =\n    Nat.rfind (p a)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case some\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Nat.rfind fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) =\n    Nat.rfind (p a)", "state_after": "case some.e_p\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) = p a"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case some.e_p\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) = p a", "state_after": "case some.e_p.h\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n)) = p a n"}, {"tactic": "simp only [map_map, Function.comp]", "annotated_tactic": ["simp only [<a>map_map</a>, <a>Function.comp</a>]", [{"full_name": "Part.map_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [559, 9], "def_end_pos": [559, 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 some.e_p.h\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n)) = p a n", "state_after": "case some.e_p.h\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 Part.map (fun x => decide ((bif x then 0 else 1) = 0)) (p a n) = p a n"}, {"tactic": "refine map_id' (fun b => ?_) _", "annotated_tactic": ["refine <a>map_id'</a> (fun b => ?_) _", [{"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "state_before": "case some.e_p.h\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\np : \u03b1 \u2192 \u2115 \u2192. 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Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\nb : Bool\n\u22a2 decide ((bif b then 0 else 1) = 0) = b"}, {"tactic": "cases b <;> rfl", "annotated_tactic": ["cases b <;> rfl", []], "state_before": "case some.e_p.h\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\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\nb : Bool\n\u22a2 decide ((bif b then 0 else 1) = 0) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.cardinal_generateMeasurable_le_continuum", "start": [176, 1], "end": [181, 79], "traced_tactics": [{"tactic": "rw [\u2190 continuum_power_aleph0]", "annotated_tactic": ["rw [\u2190 <a>continuum_power_aleph0</a>]", [{"full_name": "Cardinal.continuum_power_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [213, 9], "def_end_pos": [213, 31]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\nhs : #\u2191s \u2264 \ud835\udd20\n\u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\nhs : #\u2191s \u2264 \ud835\udd20\n\u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20 ^ \u2135\u2080"}, {"tactic": "exact_mod_cast power_le_power_right (max_le hs (nat_lt_continuum 2).le)", "annotated_tactic": ["exact_mod_cast <a>power_le_power_right</a> (<a>max_le</a> hs (<a>nat_lt_continuum</a> 2).<a>le</a>)", [{"full_name": "Cardinal.power_le_power_right", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 29]}, {"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "Cardinal.nat_lt_continuum", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [87, 9], "def_end_pos": [87, 25]}, {"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\ns : Set (Set \u03b1)\nhs : #\u2191s \u2264 \ud835\udd20\n\u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20 ^ \u2135\u2080", "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_integral_tendsto", "start": [680, 1], "end": [699, 40], "traced_tactics": [{"tactic": "refine' \u27e8_, tendsto_of_forall_integral_tendsto\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>tendsto_of_forall_integral_tendsto</a>\u27e9", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_of_forall_integral_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [650, 9], "def_end_pos": [650, 43]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))"}, {"tactic": "rw [tendsto_iff_forall_lintegral_tendsto]", "annotated_tactic": ["rw [<a>tendsto_iff_forall_lintegral_tendsto</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [532, 9], "def_end_pos": [532, 45]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))) \u2192\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))"}, {"tactic": "intro h f", "annotated_tactic": ["intro h f", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))) \u2192\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))"}, {"tactic": "simp_rw [BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub]", "annotated_tactic": ["simp_rw [<a>BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub</a>]", [{"full_name": "BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [49, 9], "def_end_pos": [49, 44]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))"}, {"tactic": "set f_pos := f.nnrealPart with _def_f_pos", "annotated_tactic": ["set f_pos := f.nnrealPart with _def_f_pos", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))"}, {"tactic": "set f_neg := (-f).nnrealPart with _def_f_neg", "annotated_tactic": ["set f_neg := (-f).<a>nnrealPart</a> with _def_f_neg", [{"full_name": "BoundedContinuousFunction.nnrealPart", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [1594, 5], "def_end_pos": [1594, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))"}, {"tactic": "have tends_pos := (ENNReal.tendsto_toReal (f_pos.lintegral_lt_top_of_nnreal \u03bc).ne).comp (h f_pos)", "annotated_tactic": ["have tends_pos := (<a>ENNReal.tendsto_toReal</a> (f_pos.lintegral_lt_top_of_nnreal \u03bc).<a>ne</a>).<a>comp</a> (h f_pos)", [{"full_name": "ENNReal.tendsto_toReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc)))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))"}, {"tactic": "have tends_neg := (ENNReal.tendsto_toReal (f_neg.lintegral_lt_top_of_nnreal \u03bc).ne).comp (h f_neg)", "annotated_tactic": ["have tends_neg := (<a>ENNReal.tendsto_toReal</a> (f_neg.lintegral_lt_top_of_nnreal \u03bc).<a>ne</a>).<a>comp</a> (h f_neg)", [{"full_name": "ENNReal.tendsto_toReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc)))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc)))\ntends_neg :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))"}, {"tactic": "have aux :\n  \u2200 g : \u03a9 \u2192\u1d47 \u211d\u22650,\n    (ENNReal.toReal \u2218 fun i : \u03b3 => \u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : Measure \u03a9)) = fun i : \u03b3 =>\n      (\u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : Measure \u03a9)).toReal :=\n  fun _ => rfl", "annotated_tactic": ["have aux :\n    \u2200 g : \u03a9 \u2192\u1d47 \u211d\u22650,\n      (<a>ENNReal.toReal</a> \u2218 fun i : \u03b3 => \u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : <a>Measure</a> \u03a9)) = fun i : \u03b3 =>\n        (\u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : <a>Measure</a> \u03a9)).<a>toReal</a> :=\n    fun _ => <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": "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.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": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc)))\ntends_neg :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc)))\ntends_neg :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)))\naux :\n  \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),\n    (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))"}, {"tactic": "simp_rw [aux, BoundedContinuousFunction.toReal_lintegral_coe_eq_integral] at tends_pos tends_neg", "annotated_tactic": ["simp_rw [aux, <a>BoundedContinuousFunction.toReal_lintegral_coe_eq_integral</a>] at tends_pos tends_neg", [{"full_name": "BoundedContinuousFunction.toReal_lintegral_coe_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [60, 9], "def_end_pos": [60, 41]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc)))\ntends_neg :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)))\naux :\n  \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),\n    (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\naux :\n  \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),\n    (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i))\ntends_pos : Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc))\ntends_neg :\n  Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))"}, {"tactic": "exact Tendsto.sub tends_pos tends_neg", "annotated_tactic": ["exact <a>Tendsto.sub</a> tends_pos tends_neg", [{"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \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\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\naux :\n  \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),\n    (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i))\ntends_pos : Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc))\ntends_neg :\n  Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\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.min_eq_right", "start": [706, 11], "end": [707, 49], "traced_tactics": [{"tactic": "rw [Int.min_comm a b]", "annotated_tactic": ["rw [<a>Int.min_comm</a> a b]", [{"full_name": "Int.min_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [690, 19], "def_end_pos": [690, 27]}]], "state_before": "a b : Int\nh : b \u2264 a\n\u22a2 min a b = b", "state_after": "a b : Int\nh : b \u2264 a\n\u22a2 min b a = b"}, {"tactic": "exact Int.min_eq_left h", "annotated_tactic": ["exact <a>Int.min_eq_left</a> h", [{"full_name": "Int.min_eq_left", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [704, 19], "def_end_pos": [704, 30]}]], "state_before": "a b : Int\nh : b \u2264 a\n\u22a2 min b a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_iUnion", "start": [185, 1], "end": [189, 24], "traced_tactics": [{"tactic": "haveI := fun a => (ht a).to_subtype", "annotated_tactic": ["haveI := fun a => (ht a).<a>to_subtype</a>", [{"full_name": "Set.Countable.to_subtype", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [46, 33], "def_end_pos": [46, 53]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nt : \u03b9 \u2192 Set \u03b1\ninst\u271d : Countable \u03b9\nht : \u2200 (i : \u03b9), Set.Countable (t i)\n\u22a2 Set.Countable (\u22c3 i, t i)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nt : \u03b9 \u2192 Set \u03b1\ninst\u271d : Countable \u03b9\nht : \u2200 (i : \u03b9), Set.Countable (t i)\nthis : \u2200 (a : \u03b9), Countable \u2191(t a)\n\u22a2 Set.Countable (\u22c3 i, t i)"}, {"tactic": "rw [iUnion_eq_range_psigma]", "annotated_tactic": ["rw [<a>iUnion_eq_range_psigma</a>]", [{"full_name": "Set.iUnion_eq_range_psigma", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1412, 9], "def_end_pos": [1412, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nt : \u03b9 \u2192 Set \u03b1\ninst\u271d : Countable \u03b9\nht : \u2200 (i : \u03b9), Set.Countable (t i)\nthis : \u2200 (a : \u03b9), Countable \u2191(t a)\n\u22a2 Set.Countable (\u22c3 i, t i)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nt : \u03b9 \u2192 Set \u03b1\ninst\u271d : Countable \u03b9\nht : \u2200 (i : \u03b9), Set.Countable (t i)\nthis : \u2200 (a : \u03b9), Countable \u2191(t a)\n\u22a2 Set.Countable (range fun a => \u2191a.snd)"}, {"tactic": "apply countable_range", "annotated_tactic": ["apply <a>countable_range</a>", [{"full_name": "Set.countable_range", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nt : \u03b9 \u2192 Set \u03b1\ninst\u271d : Countable \u03b9\nht : \u2200 (i : \u03b9), Set.Countable (t i)\nthis : \u2200 (a : \u03b9), Countable \u2191(t a)\n\u22a2 Set.Countable (range fun a => \u2191a.snd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "BddBelow.finite_of_bddAbove", "start": [315, 1], "end": [319, 49], "traced_tactics": [{"tactic": "let \u27e8a, ha\u27e9 := h\u2080", "annotated_tactic": ["let \u27e8a, ha\u27e9 := h\u2080", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\n\u22a2 Set.Finite s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\n\u22a2 Set.Finite s"}, {"tactic": "let \u27e8b, hb\u27e9 := h\u2081", "annotated_tactic": ["let \u27e8b, hb\u27e9 := h\u2081", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\n\u22a2 Set.Finite s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 Set.Finite s"}, {"tactic": "classical exact \u27e8Set.fintypeOfMemBounds ha hb\u27e9", "annotated_tactic": ["classical exact \u27e8<a>Set.fintypeOfMemBounds</a> ha hb\u27e9", [{"full_name": "Set.fintypeOfMemBounds", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [310, 5], "def_end_pos": [310, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 Set.Finite s", "state_after": "no goals"}, {"tactic": "exact \u27e8Set.fintypeOfMemBounds ha hb\u27e9", "annotated_tactic": ["exact \u27e8<a>Set.fintypeOfMemBounds</a> ha hb\u27e9", [{"full_name": "Set.fintypeOfMemBounds", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [310, 5], "def_end_pos": [310, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 Set.Finite s", "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_eq_ite", "start": [170, 1], "end": [171, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "full_name": "BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub", "start": [49, 1], "end": [53, 6], "traced_tactics": [{"tactic": "simp only [f.self_eq_nnrealPart_sub_nnrealPart_neg, Pi.sub_apply, integral_sub,\n           integrable_of_nnreal]", "annotated_tactic": ["simp only [f.self_eq_nnrealPart_sub_nnrealPart_neg, <a>Pi.sub_apply</a>, <a>integral_sub</a>,\n             <a>integrable_of_nnreal</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.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}, {"full_name": "BoundedContinuousFunction.integrable_of_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [42, 9], "def_end_pos": [42, 29]}]], "state_before": "X : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\n\u22a2 \u222b (x : X), \u2191f x \u2202\u03bc = \u222b (x : X), \u2191(\u2191(nnrealPart f) x) \u2202\u03bc - \u222b (x : X), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u03bc", "state_after": "X : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\n\u22a2 \u222b (a : X), (NNReal.toReal \u2218 \u2191(nnrealPart f)) a \u2202\u03bc - \u222b (a : X), (NNReal.toReal \u2218 \u2191(nnrealPart (-f))) a \u2202\u03bc =\n    \u222b (x : X), \u2191(\u2191(nnrealPart f) x) \u2202\u03bc - \u222b (x : X), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u03bc"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "X : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\n\u22a2 \u222b (a : X), (NNReal.toReal \u2218 \u2191(nnrealPart f)) a \u2202\u03bc - \u222b (a : X), (NNReal.toReal \u2218 \u2191(nnrealPart (-f))) a \u2202\u03bc =\n    \u222b (x : X), \u2191(\u2191(nnrealPart f) x) \u2202\u03bc - \u222b (x : X), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "full_name": "MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensity\u1d65_rnDeriv_eq", "start": [109, 1], "end": [111, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.mod_zero", "start": [1647, 11], "end": [1651, 21], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "n : Num\n\u22a2 mod n 0 = n", "state_after": "case zero\n\n\u22a2 mod zero 0 = zero\n\ncase pos\na\u271d : PosNum\n\u22a2 mod (pos a\u271d) 0 = pos a\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\n\u22a2 mod zero 0 = zero", "state_after": "no goals"}, {"tactic": "simp [Num.mod]", "annotated_tactic": ["simp [<a>Num.mod</a>]", [{"full_name": "Num.mod", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [613, 5], "def_end_pos": [613, 8]}]], "state_before": "case pos\na\u271d : PosNum\n\u22a2 mod (pos a\u271d) 0 = pos a\u271d", "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.exists_set_sigmaFinite", "start": [1026, 1], "end": [1046, 35], "traced_tactics": [{"tactic": "rcases hf with \u27e8fs, hT_lt_top, h_approx\u27e9", "annotated_tactic": ["rcases hf with \u27e8fs, hT_lt_top, h_approx\u27e9", []], "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 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nhf : FinStronglyMeasurable f \u03bc\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)", "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)"}, {"tactic": "let T n := support (fs n)", "annotated_tactic": ["let T n := <a>support</a> (fs n)", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}]], "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)", "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)"}, {"tactic": "have hT_meas : \u2200 n, MeasurableSet (T n) := fun n => SimpleFunc.measurableSet_support (fs n)", "annotated_tactic": ["have hT_meas : \u2200 n, <a>MeasurableSet</a> (T n) := fun n => <a>SimpleFunc.measurableSet_support</a> (fs n)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.SimpleFunc.measurableSet_support", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1159, 9], "def_end_pos": [1159, 30]}]], "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)", "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)"}, {"tactic": "let t := \u22c3 n, T n", "annotated_tactic": ["let t := \u22c3 n, T n", []], "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)", "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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)"}, {"tactic": "refine' \u27e8t, MeasurableSet.iUnion hT_meas, _, _\u27e9", "annotated_tactic": ["refine' \u27e8t, <a>MeasurableSet.iUnion</a> hT_meas, _, _\u27e9", [{"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\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t)", "state_after": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\n\ncase intro.intro.refine'_2\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 SigmaFinite (Measure.restrict \u03bc t)"}, {"tactic": "have h_fs_zero : \u2200 n, \u2200 x \u2208 t\u1d9c, fs n x = 0 := by\n  intro n x hxt\n  rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt\n  simpa using hxt n", "annotated_tactic": ["have h_fs_zero : \u2200 n, \u2200 x \u2208 t\u1d9c, fs n x = 0 := by\n      intro n x hxt\n      rw [<a>Set.mem_compl_iff</a>, <a>Set.mem_iUnion</a>, <a>not_exists</a>] at hxt\n      simpa using hxt n", [{"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_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}]], "state_before": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0", "state_after": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nh_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0"}, {"tactic": "refine' fun x hxt => tendsto_nhds_unique (h_approx x) _", "annotated_tactic": ["refine' fun x hxt => <a>tendsto_nhds_unique</a> (h_approx x) _", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "state_before": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nh_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0", "state_after": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nh_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0\nx : \u03b1\nhxt : x \u2208 t\u1d9c\n\u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [funext fun n => h_fs_zero n x hxt]", "annotated_tactic": ["rw [<a>funext</a> fun n => h_fs_zero n x hxt]", [{"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 intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nh_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0\nx : \u03b1\nhxt : x \u2208 t\u1d9c\n\u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd 0)", "state_after": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nh_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0\nx : \u03b1\nhxt : x \u2208 t\u1d9c\n\u22a2 Tendsto (fun n => 0) atTop (\ud835\udcdd 0)"}, {"tactic": "exact tendsto_const_nhds", "annotated_tactic": ["exact <a>tendsto_const_nhds</a>", [{"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "case intro.intro.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nh_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0\nx : \u03b1\nhxt : x \u2208 t\u1d9c\n\u22a2 Tendsto (fun n => 0) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "intro n x hxt", "annotated_tactic": ["intro n x hxt", []], "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 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0", "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 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\nx : \u03b1\nhxt : x \u2208 t\u1d9c\n\u22a2 \u2191(fs n) x = 0"}, {"tactic": "rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt", "annotated_tactic": ["rw [<a>Set.mem_compl_iff</a>, <a>Set.mem_iUnion</a>, <a>not_exists</a>] at hxt", [{"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_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}]], "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 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\nx : \u03b1\nhxt : x \u2208 t\u1d9c\n\u22a2 \u2191(fs n) x = 0", "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 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\nx : \u03b1\nhxt : \u2200 (x_1 : \u2115), \u00acx \u2208 T x_1\n\u22a2 \u2191(fs n) x = 0"}, {"tactic": "simpa using hxt n", "annotated_tactic": ["simpa using hxt 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 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\nx : \u03b1\nhxt : \u2200 (x_1 : \u2115), \u00acx \u2208 T x_1\n\u22a2 \u2191(fs n) x = 0", "state_after": "no goals"}, {"tactic": "refine' \u27e8\u27e8\u27e8fun n => t\u1d9c \u222a T n, fun _ => trivial, fun n => _, _\u27e9\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8\u27e8fun n => t\u1d9c \u222a T n, fun _ => <a>trivial</a>, fun n => _, _\u27e9\u27e9\u27e9", [{"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.refine'_2\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 SigmaFinite (Measure.restrict \u03bc t)", "state_after": "case intro.intro.refine'_2.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\n\u22a2 \u2191\u2191(Measure.restrict \u03bc t) ((fun n => t\u1d9c \u222a T n) n) < \u22a4\n\ncase intro.intro.refine'_2.refine'_2\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u22c3 i, (fun n => t\u1d9c \u222a T n) i = univ"}, {"tactic": "rw [Measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,\n  Set.compl_inter_self t, Set.empty_union]", "annotated_tactic": ["rw [<a>Measure.restrict_apply'</a> (<a>MeasurableSet.iUnion</a> hT_meas), <a>Set.union_inter_distrib_right</a>,\n        <a>Set.compl_inter_self</a> t, <a>Set.empty_union</a>]", [{"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"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.compl_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1672, 9], "def_end_pos": [1672, 25]}, {"full_name": "Set.empty_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}]], "state_before": "case intro.intro.refine'_2.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\n\u22a2 \u2191\u2191(Measure.restrict \u03bc t) ((fun n => t\u1d9c \u222a T n) n) < \u22a4", "state_after": "case intro.intro.refine'_2.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (T n \u2229 \u22c3 b, T b) < \u22a4"}, {"tactic": "exact (measure_mono (Set.inter_subset_left _ _)).trans_lt (hT_lt_top n)", "annotated_tactic": ["exact (<a>measure_mono</a> (<a>Set.inter_subset_left</a> _ _)).<a>trans_lt</a> (hT_lt_top 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.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": "case intro.intro.refine'_2.refine'_1\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (T n \u2229 \u22c3 b, T b) < \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.union_iUnion t\u1d9c T]", "annotated_tactic": ["rw [\u2190 <a>Set.union_iUnion</a> t\u1d9c T]", [{"full_name": "Set.union_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [653, 9], "def_end_pos": [653, 21]}]], "state_before": "case intro.intro.refine'_2.refine'_2\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 \u22c3 i, (fun n => t\u1d9c \u222a T n) i = univ", "state_after": "case intro.intro.refine'_2.refine'_2\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 t\u1d9c \u222a \u22c3 i, T i = univ"}, {"tactic": "exact Set.compl_union_self _", "annotated_tactic": ["exact <a>Set.compl_union_self</a> _", [{"full_name": "Set.compl_union_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1744, 9], "def_end_pos": [1744, 25]}]], "state_before": "case intro.intro.refine'_2.refine'_2\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b2\nfs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2\nhT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4\nh_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nT : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n)\nhT_meas : \u2200 (n : \u2115), MeasurableSet (T n)\nt : Set \u03b1 := \u22c3 n, T n\n\u22a2 t\u1d9c \u222a \u22c3 i, T i = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.map_preimage_singleton", "start": [324, 1], "end": [326, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le.intro_sub", "start": [558, 1], "end": [559, 32], "traced_tactics": [{"tactic": "simp [le_def, h]", "annotated_tactic": ["simp [<a>le_def</a>, h]", [{"full_name": "Int.le_def", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [554, 9], "def_end_pos": [554, 15]}]], "state_before": "a b : Int\nn : Nat\nh : b - a = \u2191n\n\u22a2 a \u2264 b", "state_after": "a b : Int\nn : Nat\nh : b - a = \u2191n\n\u22a2 NonNeg \u2191n"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "a b : Int\nn : Nat\nh : b - a = \u2191n\n\u22a2 NonNeg \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.piecewise_empty", "start": [252, 1], "end": [253, 60], "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\ninst\u271d : MeasurableSpace \u03b1\nf g : \u03b1 \u2192\u209b \u03b2\n\u22a2 \u2191(piecewise \u2205 (_ : MeasurableSet \u2205) f g) = \u2191g", "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 g : \u03b1 \u2192\u209b \u03b2\n\u22a2 Set.piecewise \u2205 \u2191f \u2191g = \u2191g"}, {"tactic": "convert Set.piecewise_empty f g", "annotated_tactic": ["convert <a>Set.piecewise_empty</a> f g", [{"full_name": "Set.piecewise_empty", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}]], "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 g : \u03b1 \u2192\u209b \u03b2\n\u22a2 Set.piecewise \u2205 \u2191f \u2191g = \u2191g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snormEssSup_smul_measure", "start": [635, 1], "end": [638, 31], "traced_tactics": [{"tactic": "simp_rw [snormEssSup]", "annotated_tactic": ["simp_rw [<a>snormEssSup</a>]", [{"full_name": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 snormEssSup f (c \u2022 \u03bc) = snormEssSup f \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\nhc : c \u2260 0\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) (c \u2022 \u03bc) = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc"}, {"tactic": "exact essSup_smul_measure hc", "annotated_tactic": ["exact <a>essSup_smul_measure</a> hc", [{"full_name": "essSup_smul_measure", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [208, 9], "def_end_pos": [208, 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 : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) (c \u2022 \u03bc) = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_liminf'", "start": [1456, 1], "end": [1505, 37], "traced_tactics": [{"tactic": "have : Countable (Subtype p) := Encodable.nonempty_encodable.1 hv.countable", "annotated_tactic": ["have : <a>Countable</a> (<a>Subtype</a> p) := <a>Encodable.nonempty_encodable</a>.1 hv.countable", [{"full_name": "Countable", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}, {"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Encodable.nonempty_encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 27]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v"}, {"tactic": "rcases isEmpty_or_nonempty (Subtype p) with hp|hp", "annotated_tactic": ["rcases <a>isEmpty_or_nonempty</a> (<a>Subtype</a> p) with hp|hp", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}, {"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v", "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : IsEmpty (Subtype p)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v\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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v"}, {"tactic": "by_cases H : \u2203 (j : Subtype p), s j = \u2205", "annotated_tactic": ["by_cases H : \u2203 (j : <a>Subtype</a> p), s j = \u2205", [{"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}]], "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u2203 j, s \u2191j = \u2205\n\u22a2 Measurable fun x => liminf (fun i => f i x) v\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\n\u22a2 Measurable fun x => liminf (fun i => f i x) v"}, {"tactic": "simp_rw [hv.liminf_eq_ite, if_neg H]", "annotated_tactic": ["simp_rw [hv.liminf_eq_ite, <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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\n\u22a2 Measurable fun x => liminf (fun i => f i x) v", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x"}, {"tactic": "have : \u2200 i, Countable (s i) := fun i \u21a6 countable_coe_iff.2 (hs i)", "annotated_tactic": ["have : \u2200 i, <a>Countable</a> (s i) := fun i \u21a6 <a>countable_coe_iff</a>.2 (hs i)", [{"full_name": "Countable", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}, {"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x"}, {"tactic": "let m : Subtype p \u2192 Set \u03b4 := fun j \u21a6 {x | BddBelow (range (fun (i : s j) \u21a6 f i x))}", "annotated_tactic": ["let m : <a>Subtype</a> p \u2192 <a>Set</a> \u03b4 := fun j \u21a6 {x | <a>BddBelow</a> (<a>range</a> (fun (i : s j) \u21a6 f i x))}", [{"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "BddBelow", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x"}, {"tactic": "have m_meas : \u2200 j, MeasurableSet (m j) :=\n  fun j \u21a6 measurableSet_bddBelow_range (fun (i : s j) \u21a6 hf i)", "annotated_tactic": ["have m_meas : \u2200 j, <a>MeasurableSet</a> (m j) :=\n    fun j \u21a6 <a>measurableSet_bddBelow_range</a> (fun (i : s j) \u21a6 hf i)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_bddBelow_range", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1335, 7], "def_end_pos": [1335, 35]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x"}, {"tactic": "have mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), x \u2209 m j} := by\n  rw [setOf_forall]\n  exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl)", "annotated_tactic": ["have mc_meas : <a>MeasurableSet</a> {x | \u2200 (j : <a>Subtype</a> p), x \u2209 m j} := by\n    rw [<a>setOf_forall</a>]\n    exact <a>MeasurableSet.iInter</a> (fun j \u21a6 (m_meas j).<a>compl</a>)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x"}, {"tactic": "apply Measurable.piecewise mc_meas measurable_const", "annotated_tactic": ["apply <a>Measurable.piecewise</a> mc_meas <a>measurable_const</a>", [{"full_name": "Measurable.piecewise", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [336, 19], "def_end_pos": [336, 39]}, {"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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\n\u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\n\u22a2 Measurable fun x => \u2a06 j, \u2a05 i, f (\u2191i) x"}, {"tactic": "apply measurable_iSup (fun j \u21a6 ?_)", "annotated_tactic": ["apply <a>measurable_iSup</a> (fun j \u21a6 ?_)", [{"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\n\u22a2 Measurable fun x => \u2a06 j, \u2a05 i, f (\u2191i) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b"}, {"tactic": "let reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x \u21a6 liminf_reparam (fun i \u21a6 f i x) s p", "annotated_tactic": ["let reparam : \u03b4 \u2192 <a>Subtype</a> p \u2192 <a>Subtype</a> p := fun x \u21a6 <a>liminf_reparam</a> (fun i \u21a6 f i x) s p", [{"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", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Filter.liminf_reparam", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1293, 19], "def_end_pos": [1293, 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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b"}, {"tactic": "let F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x \u21a6 \u2a05 (i : s j), f i x", "annotated_tactic": ["let F0 : <a>Subtype</a> p \u2192 \u03b4 \u2192 \u03b1 := fun j x \u21a6 \u2a05 (i : s j), f i x", [{"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b"}, {"tactic": "have F0_meas : \u2200 j, Measurable (F0 j) := fun j \u21a6 measurable_iInf (fun (i : s j) \u21a6 hf i)", "annotated_tactic": ["have F0_meas : \u2200 j, <a>Measurable</a> (F0 j) := fun j \u21a6 <a>measurable_iInf</a> (fun (i : s j) \u21a6 hf i)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_iInf", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1385, 9], "def_end_pos": [1385, 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b"}, {"tactic": "set F1 : \u03b4 \u2192 \u03b1 := fun x \u21a6 F0 (reparam x j) x with hF1", "annotated_tactic": ["set F1 : \u03b4 \u2192 \u03b1 := fun x \u21a6 F0 (reparam x j) x with hF1", []], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\n\u22a2 Measurable fun b => \u2a05 i, f (\u2191i) 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\n\u22a2 Measurable F1"}, {"tactic": "let g : \u2115 \u2192 Subtype p := choose (exists_surjective_nat (Subtype p))", "annotated_tactic": ["let g : \u2115 \u2192 <a>Subtype</a> p := <a>choose</a> (<a>exists_surjective_nat</a> (<a>Subtype</a> p))", [{"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}, {"full_name": "Classical.choose", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [19, 19], "def_end_pos": [19, 25]}, {"full_name": "exists_surjective_nat", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [59, 9], "def_end_pos": [59, 30]}, {"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\n\u22a2 Measurable F1", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\n\u22a2 Measurable F1"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\n\u22a2 Measurable F1", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\n\u22a2 Measurable fun x =>\n    if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x"}, {"tactic": "apply Measurable.piecewise (m_meas j) (F0_meas j)", "annotated_tactic": ["apply <a>Measurable.piecewise</a> (m_meas j) (F0_meas j)", [{"full_name": "Measurable.piecewise", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [336, 19], "def_end_pos": [336, 39]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\n\u22a2 Measurable fun x =>\n    if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\n\u22a2 Measurable fun x => F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x"}, {"tactic": "apply Measurable.find (fun n \u21a6 F0_meas (g n)) (fun n \u21a6 ?_)", "annotated_tactic": ["apply <a>Measurable.find</a> (fun n \u21a6 F0_meas (g n)) (fun n \u21a6 ?_)", [{"full_name": "Measurable.find", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [814, 9], "def_end_pos": [814, 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\n\u22a2 Measurable fun x => F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\nn : \u2115\n\u22a2 MeasurableSet {x | x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k}"}, {"tactic": "exact (m_meas (g n)).union mc_meas", "annotated_tactic": ["exact (m_meas (g n)).<a>union</a> mc_meas", [{"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nthis :\n  F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x\nn : \u2115\n\u22a2 MeasurableSet {x | x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k}", "state_after": "no goals"}, {"tactic": "simp [hv.liminf_eq_sSup_iUnion_iInter]", "annotated_tactic": ["simp [hv.liminf_eq_sSup_iUnion_iInter]", []], "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : IsEmpty (Subtype p)\n\u22a2 Measurable fun x => liminf (fun i => f i x) v", "state_after": "no goals"}, {"tactic": "simp_rw [hv.liminf_eq_ite, if_pos H, measurable_const]", "annotated_tactic": ["simp_rw [hv.liminf_eq_ite, <a>if_pos</a> H, <a>measurable_const</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": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u2203 j, s \u2191j = \u2205\n\u22a2 Measurable fun x => liminf (fun i => f i x) v", "state_after": "no goals"}, {"tactic": "rw [setOf_forall]", "annotated_tactic": ["rw [<a>setOf_forall</a>]", [{"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [386, 9], "def_end_pos": [386, 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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\n\u22a2 MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\n\u22a2 MeasurableSet (\u22c2 i, {x | \u00acx \u2208 m i})"}, {"tactic": "exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl)", "annotated_tactic": ["exact <a>MeasurableSet.iInter</a> (fun j \u21a6 (m_meas j).<a>compl</a>)", [{"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\n\u22a2 MeasurableSet (\u22c2 i, {x | \u00acx \u2208 m i})", "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\n\u22a2 \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k"}, {"tactic": "by_cases H : \u2203 k, x \u2208 m k", "annotated_tactic": ["by_cases H : \u2203 k, x \u2208 m k", []], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH\u271d : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nH : \u2203 k, x \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH\u271d : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nH : \u00ac\u2203 k, x \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k"}, {"tactic": "rcases H with \u27e8k, hk\u27e9", "annotated_tactic": ["rcases H with \u27e8k, hk\u27e9", []], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH\u271d : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nH : \u2203 k, x \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "state_after": "case pos.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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nk : Subtype p\nhk : x \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k"}, {"tactic": "rcases choose_spec (exists_surjective_nat (Subtype p)) k with \u27e8n, rfl\u27e9", "annotated_tactic": ["rcases <a>choose_spec</a> (<a>exists_surjective_nat</a> (<a>Subtype</a> p)) k with \u27e8n, rfl\u27e9", [{"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": "exists_surjective_nat", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [59, 9], "def_end_pos": [59, 30]}, {"full_name": "Subtype", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 11], "def_end_pos": [560, 18]}]], "state_before": "case pos.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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nk : Subtype p\nhk : x \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "state_after": "case pos.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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nn : \u2115\nhk : x \u2208 m (choose (_ : \u2203 f, Function.Surjective f) n)\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k"}, {"tactic": "exact \u27e8n, Or.inl hk\u27e9", "annotated_tactic": ["exact \u27e8n, <a>Or.inl</a> hk\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 pos.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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nn : \u2115\nhk : x \u2208 m (choose (_ : \u2203 f, Function.Surjective f) n)\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH\u271d : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nH : \u00ac\u2203 k, x \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH\u271d : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nH : \u2200 (k : Subtype p), \u00acx \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k"}, {"tactic": "exact \u27e80, Or.inr H\u27e9", "annotated_tactic": ["exact \u27e80, <a>Or.inr</a> H\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 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH\u271d : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nx : \u03b4\nH : \u2200 (k : Subtype p), \u00acx \u2208 m k\n\u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\n\u22a2 F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\n\u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x"}, {"tactic": "have A : reparam x j = if x \u2208 m j then j else g (Nat.find (Z x)) := rfl", "annotated_tactic": ["have A : reparam x j = if x \u2208 m j then j else g (<a>Nat.find</a> (Z x)) := <a>rfl</a>", [{"full_name": "Nat.find", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [713, 15], "def_end_pos": [713, 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 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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\n\u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\n\u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x"}, {"tactic": "split_ifs with hjx", "annotated_tactic": ["split_ifs with hjx", []], "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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\n\u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : x \u2208 m j\n\u22a2 F1 x = F0 j x\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : \u00acx \u2208 m j\n\u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x"}, {"tactic": "have : reparam x j = j := by rw [A, if_pos hjx]", "annotated_tactic": ["have : reparam x j = j := by rw [A, <a>if_pos</a> hjx]", [{"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : x \u2208 m j\n\u22a2 F1 x = F0 j x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : x \u2208 m j\nthis : reparam x j = j\n\u22a2 F1 x = F0 j x"}, {"tactic": "simp only [hF1, this]", "annotated_tactic": ["simp only [hF1, 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : x \u2208 m j\nthis : reparam x j = j\n\u22a2 F1 x = F0 j x", "state_after": "no goals"}, {"tactic": "rw [A, if_pos hjx]", "annotated_tactic": ["rw [A, <a>if_pos</a> hjx]", [{"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : x \u2208 m j\n\u22a2 reparam x j = j", "state_after": "no goals"}, {"tactic": "have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx]", "annotated_tactic": ["have : reparam x j = g (<a>Nat.find</a> (Z x)) := by rw [A, <a>if_neg</a> hjx]", [{"full_name": "Nat.find", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [713, 15], "def_end_pos": [713, 19]}, {"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : \u00acx \u2208 m j\n\u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : \u00acx \u2208 m j\nthis : reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\n\u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x"}, {"tactic": "simp only [hF1, this]", "annotated_tactic": ["simp only [hF1, 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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d\u00b9 : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : \u00acx \u2208 m j\nthis : reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\n\u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x", "state_after": "no goals"}, {"tactic": "rw [A, if_neg hjx]", "annotated_tactic": ["rw [A, <a>if_neg</a> hjx]", [{"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\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\n\u03b9' : Type u_7\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nv : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhv : HasCountableBasis v p s\nhs : \u2200 (j : \u03b9'), Set.Countable (s j)\nthis\u271d : Countable (Subtype p)\nhp : Nonempty (Subtype p)\nH : \u00ac\u2203 j, s \u2191j = \u2205\nthis : \u2200 (i : \u03b9'), Countable \u2191(s i)\nm : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)}\nm_meas : \u2200 (j : Subtype p), MeasurableSet (m j)\nmc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j}\nj : Subtype p\nreparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p\nF0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x\nF0_meas : \u2200 (j : Subtype p), Measurable (F0 j)\nF1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x\nhF1 : F1 = fun x => F0 (reparam x j) x\ng : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f)\nZ : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k\nx : \u03b4\nA : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))\nhjx : \u00acx \u2208 m j\n\u22a2 reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Finite.cast_ncard_eq", "start": [477, 1], "end": [478, 87], "traced_tactics": [{"tactic": "rwa [ncard, ENat.coe_toNat_eq_self, ne_eq, encard_eq_top_iff, Set.Infinite, not_not]", "annotated_tactic": ["rwa [<a>ncard</a>, <a>ENat.coe_toNat_eq_self</a>, <a>ne_eq</a>, <a>encard_eq_top_iff</a>, <a>Set.Infinite</a>, <a>not_not</a>]", [{"full_name": "Set.ncard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [471, 19], "def_end_pos": [471, 24]}, {"full_name": "ENat.coe_toNat_eq_self", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [157, 9], "def_end_pos": [157, 26]}, {"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": "Set.encard_eq_top_iff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [136, 17], "def_end_pos": [136, 34]}, {"full_name": "Set.Infinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [142, 15], "def_end_pos": [142, 23]}, {"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\ns t : Set \u03b1\nhs : Set.Finite s\n\u22a2 \u2191(ncard s) = encard s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.card_dvd_card_image\u2082_right", "start": [521, 1], "end": [537, 35], "traced_tactics": [{"tactic": "induction' s using Finset.induction with a s _ ih", "annotated_tactic": ["induction' s using <a>Finset.induction</a> with a s _ ih", [{"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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : 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\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id\n\u22a2 card t \u2223 card (image\u2082 f s t)", "state_after": "case empty\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : 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\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id\nhf : \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 Injective (f a)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191\u2205) id\n\u22a2 card t \u2223 card (image\u2082 f \u2205 t)\n\ncase insert\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)) \u2192\n    PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id \u2192 card t \u2223 card (image\u2082 f s t)\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\n\u22a2 card t \u2223 card (image\u2082 f (insert a s) t)"}, {"tactic": "specialize ih (forall_of_forall_insert hf)\n  (hs.subset <| Set.image_subset _ <| coe_subset.2 <| subset_insert _ _)", "annotated_tactic": ["specialize ih (<a>forall_of_forall_insert</a> hf)\n    (hs.subset <| <a>Set.image_subset</a> _ <| <a>coe_subset</a>.2 <| <a>subset_insert</a> _ _)", [{"full_name": "Finset.forall_of_forall_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3143, 9], "def_end_pos": [3143, 32]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"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_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 22]}]], "state_before": "case insert\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)) \u2192\n    PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id \u2192 card t \u2223 card (image\u2082 f s t)\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\n\u22a2 card t \u2223 card (image\u2082 f (insert a s) t)", "state_after": "case insert\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\n\u22a2 card t \u2223 card (image\u2082 f (insert a s) t)"}, {"tactic": "rw [image\u2082_insert_left]", "annotated_tactic": ["rw [<a>image\u2082_insert_left</a>]", [{"full_name": "Finset.image\u2082_insert_left", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [180, 9], "def_end_pos": [180, 27]}]], "state_before": "case insert\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\n\u22a2 card t \u2223 card (image\u2082 f (insert a s) t)", "state_after": "case insert\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)"}, {"tactic": "by_cases h : Disjoint (image (f a) t) (image\u2082 f s t)", "annotated_tactic": ["by_cases h : <a>Disjoint</a> (<a>image</a> (f a) t) (<a>image\u2082</a> f s t)", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [313, 5], "def_end_pos": [313, 10]}, {"full_name": "Finset.image\u2082", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [39, 5], "def_end_pos": [39, 11]}]], "state_before": "case insert\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)", "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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : Disjoint (image (f a) t) (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s 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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : \u00acDisjoint (image (f a) t) (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)"}, {"tactic": "simp_rw [\u2190 biUnion_image_left, disjoint_biUnion_right, not_forall] at h", "annotated_tactic": ["simp_rw [\u2190 <a>biUnion_image_left</a>, <a>disjoint_biUnion_right</a>, <a>not_forall</a>] at h", [{"full_name": "Finset.biUnion_image_left", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [288, 9], "def_end_pos": [288, 27]}, {"full_name": "Finset.disjoint_biUnion_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3754, 9], "def_end_pos": [3754, 31]}, {"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 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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : \u00acDisjoint (image (f a) t) (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)", "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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : \u2203 x x_1, \u00acDisjoint (image (f a) t) (image (f x) t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)"}, {"tactic": "obtain \u27e8b, hb, h\u27e9 := h", "annotated_tactic": ["obtain \u27e8b, hb, h\u27e9 := h", []], "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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : \u2203 x x_1, \u00acDisjoint (image (f a) t) (image (f x) t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)", "state_after": "case neg.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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb\u271d b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nb : \u03b1\nhb : b \u2208 s\nh : \u00acDisjoint (image (f a) t) (image (f b) t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)"}, {"tactic": "rwa [union_eq_right.2]", "annotated_tactic": ["rwa [<a>union_eq_right</a>.2]", [{"full_name": "Finset.union_eq_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1509, 15], "def_end_pos": [1509, 29]}]], "state_before": "case neg.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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb\u271d b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nb : \u03b1\nhb : b \u2208 s\nh : \u00acDisjoint (image (f a) t) (image (f b) t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)", "state_after": "case neg.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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb\u271d b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nb : \u03b1\nhb : b \u2208 s\nh : \u00acDisjoint (image (f a) t) (image (f b) t)\n\u22a2 image (fun b => f a b) t \u2286 image\u2082 f s t"}, {"tactic": "exact (hs.eq (Set.mem_image_of_mem _ <| mem_insert_self _ _)\n    (Set.mem_image_of_mem _ <| mem_insert_of_mem hb) h).trans_subset\n  (image_subset_image\u2082_right hb)", "annotated_tactic": ["exact (hs.eq (<a>Set.mem_image_of_mem</a> _ <| <a>mem_insert_self</a> _ _)\n      (<a>Set.mem_image_of_mem</a> _ <| <a>mem_insert_of_mem</a> hb) h).<a>trans_subset</a>\n    (<a>image_subset_image\u2082_right</a> hb)", [{"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": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}, {"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": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 26]}, {"full_name": "Eq.trans_subset", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [652, 7], "def_end_pos": [652, 22]}, {"full_name": "Finset.image_subset_image\u2082_right", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [95, 9], "def_end_pos": [95, 34]}]], "state_before": "case neg.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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb\u271d b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nb : \u03b1\nhb : b \u2208 s\nh : \u00acDisjoint (image (f a) t) (image (f b) t)\n\u22a2 image (fun b => f a b) t \u2286 image\u2082 f s t", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : 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\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id\nhf : \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 Injective (f a)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191\u2205) id\n\u22a2 card t \u2223 card (image\u2082 f \u2205 t)", "state_after": "no goals"}, {"tactic": "rw [card_union_eq h]", "annotated_tactic": ["rw [<a>card_union_eq</a> h]", [{"full_name": "Finset.card_union_eq", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [431, 9], "def_end_pos": [431, 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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : Disjoint (image (f a) t) (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t)", "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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : Disjoint (image (f a) t) (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (f a) t) + card (image\u2082 f s t)"}, {"tactic": "exact (card_image_of_injective _ <| hf _ <| mem_insert_self _ _).symm.dvd.add ih", "annotated_tactic": ["exact (<a>card_image_of_injective</a> _ <| hf _ <| <a>mem_insert_self</a> _ _).symm.dvd.add ih", [{"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.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d\u00b9 a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nhf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a)\nhs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id\na : \u03b1\ns : Finset \u03b1\na\u271d : \u00aca \u2208 s\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1)\nhs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id\nih : card t \u2223 card (image\u2082 f s t)\nh : Disjoint (image (f a) t) (image\u2082 f s t)\n\u22a2 card t \u2223 card (image (f a) t) + card (image\u2082 f s t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_countable", "start": [1488, 1], "end": [1494, 74], "traced_tactics": [{"tactic": "rw [biUnion_of_singleton]", "annotated_tactic": ["rw [<a>biUnion_of_singleton</a>]", [{"full_name": "Set.biUnion_of_singleton", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 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\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in \u22c3 x \u2208 s, {x}, f a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [lintegral_singleton]", "annotated_tactic": ["simp only [<a>lintegral_singleton</a>]", [{"full_name": "MeasureTheory.lintegral_singleton", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1483, 9], "def_end_pos": [1483, 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 : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 \u2211' (a : \u2191s), \u222b\u207b (x : \u03b1) in {\u2191a}, f x \u2202\u03bc = \u2211' (a : \u2191s), f \u2191a * \u2191\u2191\u03bc {\u2191a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "convex_parallelepiped", "start": [127, 1], "end": [129, 54], "traced_tactics": [{"tactic": "rw [parallelepiped_eq_sum_segment]", "annotated_tactic": ["rw [<a>parallelepiped_eq_sum_segment</a>]", [{"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 Convex \u211d (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\n\u22a2 Convex \u211d (\u2211 i : \u03b9, segment \u211d 0 (v i))"}, {"tactic": "exact convex_sum _ fun _i _hi => convex_segment _ _", "annotated_tactic": ["exact <a>convex_sum</a> _ fun _i _hi => <a>convex_segment</a> _ _", [{"full_name": "convex_sum", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [247, 9], "def_end_pos": [247, 19]}, {"full_name": "convex_segment", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 23]}]], "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 Convex \u211d (\u2211 i : \u03b9, segment \u211d 0 (v i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/AddCircle.lean", "full_name": "AddCircle.volume_of_add_preimage_eq", "start": [95, 1], "end": [104, 63], "traced_tactics": [{"tactic": "let G := AddSubgroup.zmultiples u", "annotated_tactic": ["let G := <a>AddSubgroup.zmultiples</a> u", [{"full_name": "AddSubgroup.zmultiples", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)", "state_after": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)"}, {"tactic": "haveI : Fintype G := @Fintype.ofFinite _ hu.finite_zmultiples", "annotated_tactic": ["haveI : <a>Fintype</a> G := @<a>Fintype.ofFinite</a> _ hu.finite_zmultiples", [{"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [54, 7], "def_end_pos": [54, 14]}, {"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [448, 19], "def_end_pos": [448, 35]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)", "state_after": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)"}, {"tactic": "have hsG : \u2200 g : G, (g +\u1d65 s : Set <| AddCircle T) =\u1d50[volume] s := by\n  rintro \u27e8y, hy\u27e9; exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _)", "annotated_tactic": ["have hsG : \u2200 g : G, (g +\u1d65 s : <a>Set</a> <| <a>AddCircle</a> T) =\u1d50[<a>volume</a>] s := by\n    rintro \u27e8y, hy\u27e9; exact (<a>vadd_ae_eq_self_of_mem_zmultiples</a> hs hy : _)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [124, 8], "def_end_pos": [124, 17]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [308, 9], "def_end_pos": [308, 42]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)", "state_after": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\nhsG : \u2200 (g : { x // x \u2208 G }), g +\u1d65 s =\u1da0[ae volume] s\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)"}, {"tactic": "rw [(isAddFundamentalDomain_of_ae_ball I u x hu hI).measure_eq_card_smul_of_vadd_ae_eq_self s hsG,\n  add_order_eq_card_zmultiples' u, Nat.card_eq_fintype_card]", "annotated_tactic": ["rw [(<a>isAddFundamentalDomain_of_ae_ball</a> I u x hu hI).<a>measure_eq_card_smul_of_vadd_ae_eq_self</a> s hsG,\n    <a>add_order_eq_card_zmultiples'</a> u, <a>Nat.card_eq_fintype_card</a>]", [{"full_name": "AddCircle.isAddFundamentalDomain_of_ae_ball", "def_path": "Mathlib/MeasureTheory/Group/AddCircle.lean", "def_pos": [54, 9], "def_end_pos": [54, 42]}, {"full_name": "MeasureTheory.IsAddFundamentalDomain.measure_eq_card_smul_of_vadd_ae_eq_self", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [315, 15], "def_end_pos": [315, 54]}, {"full_name": "add_order_eq_card_zmultiples'", "def_path": "Mathlib/Data/ZMod/Quotient.lean", "def_pos": [207, 15], "def_end_pos": [207, 44]}, {"full_name": "Nat.card_eq_fintype_card", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [42, 9], "def_end_pos": [42, 29]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\nhsG : \u2200 (g : { x // x \u2208 G }), g +\u1d65 s =\u1da0[ae volume] s\n\u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I)", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, hy\u27e9", "annotated_tactic": ["rintro \u27e8y, hy\u27e9", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\n\u22a2 \u2200 (g : { x // x \u2208 G }), g +\u1d65 s =\u1da0[ae volume] s", "state_after": "case mk\nT : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\ny : AddCircle T\nhy : y \u2208 G\n\u22a2 { val := y, property := hy } +\u1d65 s =\u1da0[ae volume] s"}, {"tactic": "exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _)", "annotated_tactic": ["exact (<a>vadd_ae_eq_self_of_mem_zmultiples</a> hs hy : _)", [{"full_name": "MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [308, 9], "def_end_pos": [308, 42]}]], "state_before": "case mk\nT : \u211d\nhT : Fact (0 < T)\ns I : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhs : u +\u1d65 s =\u1da0[ae volume] s\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nthis : Fintype { x // x \u2208 G }\ny : AddCircle T\nhy : y \u2208 G\n\u22a2 { val := y, property := hy } +\u1d65 s =\u1da0[ae volume] 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_gcd_self_left_left", "start": [171, 9], "end": [172, 45], "traced_tactics": [{"tactic": "rw [gcd_comm m n, gcd_gcd_self_left_right]", "annotated_tactic": ["rw [<a>gcd_comm</a> m n, <a>gcd_gcd_self_left_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_left_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [168, 17], "def_end_pos": [168, 40]}]], "state_before": "m n : Nat\n\u22a2 gcd (gcd m n) m = gcd m n", "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", "start": [279, 11], "end": [281, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_mul_le", "start": [317, 1], "end": [318, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.generateFrom_insert_univ", "start": [456, 1], "end": [458, 91], "traced_tactics": [{"tactic": "rw [insert_eq, \u2190 generateFrom_sup_generateFrom, generateFrom_singleton_univ, bot_sup_eq]", "annotated_tactic": ["rw [<a>insert_eq</a>, \u2190 <a>generateFrom_sup_generateFrom</a>, <a>generateFrom_singleton_univ</a>, <a>bot_sup_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": "MeasurableSpace.generateFrom_sup_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [442, 9], "def_end_pos": [442, 38]}, {"full_name": "MeasurableSpace.generateFrom_singleton_univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [451, 9], "def_end_pos": [451, 36]}, {"full_name": "bot_sup_eq", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [454, 9], "def_end_pos": [454, 19]}]], "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\nS : Set (Set \u03b1)\n\u22a2 generateFrom (insert univ S) = generateFrom S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "full_name": "MeasureTheory.Measure.sum_smul_dirac", "start": [87, 1], "end": [88, 94], "traced_tactics": [{"tactic": "simpa using (map_eq_sum \u03bc id measurable_id).symm", "annotated_tactic": ["simpa using (<a>map_eq_sum</a> \u03bc <a>id</a> <a>measurable_id</a>).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.map_eq_sum", "def_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "def_pos": [77, 9], "def_end_pos": [77, 19]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "measurable_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [550, 9], "def_end_pos": [550, 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\n\u03b2 : Type ?u.17474\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\n\u03bc : Measure \u03b1\n\u22a2 (sum fun a => \u2191\u2191\u03bc {a} \u2022 dirac a) = \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.classes_mkClasses", "start": [243, 1], "end": [248, 32], "traced_tactics": [{"tactic": "rwa [show s = b from hs.symm \u25b8 Set.ext fun x =>\n  \u27e8fun hx => symm' (mkClasses c hc.2) hx b hm hb, fun hx b' hc' hx' =>\n      eq_of_mem_eqv_class hc.2 hm hx hc' hx' \u25b8 hb\u27e9]", "annotated_tactic": ["rwa [show s = b from hs.symm \u25b8 <a>Set.ext</a> fun x =>\n      \u27e8fun hx => <a>symm'</a> (<a>mkClasses</a> c hc.2) hx b hm hb, fun hx b' hc' hx' =>\n          <a>eq_of_mem_eqv_class</a> hc.2 hm hx hc' hx' \u25b8 hb\u27e9]", [{"full_name": "Set.ext", "def_path": "Mathlib/Init/Set.lean", "def_pos": [54, 9], "def_end_pos": [54, 12]}, {"full_name": "Setoid.symm'", "def_path": "Mathlib/Data/Setoid/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 14]}, {"full_name": "Setoid.mkClasses", "def_path": "Mathlib/Data/Setoid/Partition.lean", "def_pos": [55, 5], "def_end_pos": [55, 14]}, {"full_name": "Setoid.eq_of_mem_eqv_class", "def_path": "Mathlib/Data/Setoid/Partition.lean", "def_pos": [47, 9], "def_end_pos": [47, 28]}]], "state_before": "\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhc : IsPartition c\ns : Set \u03b1\nx\u271d : s \u2208 classes (mkClasses c (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b))\ny : \u03b1\nhs : s = {x | Rel (mkClasses c (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) x y}\nb : Set \u03b1\nhm : b \u2208 c\nhb : y \u2208 b\n_hy : \u2200 (y_1 : Set \u03b1), y_1 \u2208 c \u2192 y \u2208 y_1 \u2192 y_1 = b\n\u22a2 s \u2208 c", "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_Ioo_of_neg", "start": [662, 1], "end": [664, 66], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Ioo_of_neg a b h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Ioo_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_Ioo_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [576, 9], "def_end_pos": [576, 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' Ioo a b = Ioo (b / c) (a / c)", "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_ne", "start": [919, 1], "end": [920, 56], "traced_tactics": [{"tactic": "simp only [set_eq_modifyNth, get?_modifyNth_ne _ _ h]", "annotated_tactic": ["simp only [<a>set_eq_modifyNth</a>, <a>get?_modifyNth_ne</a> _ _ h]", [{"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_ne", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [877, 17], "def_end_pos": [877, 34]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nm n : Nat\nl : List \u03b1\nh : m \u2260 n\n\u22a2 get? (set l m a) n = get? l n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_inj_right", "start": [167, 1], "end": [168, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq", "start": [533, 1], "end": [542, 78], "traced_tactics": [{"tactic": "rw [\u2190 sub_ae_eq_zero]", "annotated_tactic": ["rw [\u2190 <a>sub_ae_eq_zero</a>]", [{"full_name": "MeasureTheory.sub_ae_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2654, 9], "def_end_pos": [2654, 23]}]], "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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (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 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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f - g =\u1d50[\u03bc] 0"}, {"tactic": "have hfg' : \u2200 s : Set \u03b1, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 := by\n  intro s hs h\u03bcs\n  rw [integral_sub' hf.integrableOn hg.integrableOn]\n  exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", "annotated_tactic": ["have hfg' : \u2200 s : <a>Set</a> \u03b1, <a>MeasurableSet</a> s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 := by\n    intro s hs h\u03bcs\n    rw [<a>integral_sub'</a> hf.integrableOn hg.integrableOn]\n    exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", [{"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.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f - g =\u1d50[\u03bc] 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n\u22a2 f - g =\u1d50[\u03bc] 0"}, {"tactic": "exact Integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'", "annotated_tactic": ["exact <a>Integrable.ae_eq_zero_of_forall_set_integral_eq_zero</a> (hf.sub hg) hfg'", [{"full_name": "MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [520, 9], "def_end_pos": [520, 61]}]], "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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n\u22a2 f - g =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0"}, {"tactic": "rw [integral_sub' hf.integrableOn hg.integrableOn]", "annotated_tactic": ["rw [<a>integral_sub'</a> hf.integrableOn hg.integrableOn]", [{"full_name": "MeasureTheory.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, (f - 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc - \u222b (a : \u03b1) in s, g a \u2202\u03bc = 0"}, {"tactic": "exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", "annotated_tactic": ["exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", []], "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 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc - \u222b (a : \u03b1) in s, g a \u2202\u03bc = 0", "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_add_left_iff_pos", "start": [333, 11], "end": [334, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "manyOneEquiv_up", "start": [289, 1], "end": [290, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.one_mul", "start": [726, 11], "end": [727, 33], "traced_tactics": [{"tactic": "rw [Fin.mul_comm, Fin.mul_one]", "annotated_tactic": ["rw [<a>Fin.mul_comm</a>, <a>Fin.mul_one</a>]", [{"full_name": "Fin.mul_comm", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [723, 19], "def_end_pos": [723, 27]}, {"full_name": "Fin.mul_one", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [718, 19], "def_end_pos": [718, 26]}]], "state_before": "n : Nat\nk : Fin (n + 1)\n\u22a2 1 * k = k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.surjOn_iff_exists_map_subtype", "start": [775, 1], "end": [781, 47], "traced_tactics": [{"tactic": "rw [hfg, hx, Subtype.coe_mk]", "annotated_tactic": ["rw [hfg, hx, <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\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\u271d g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u2203 t' g, t \u2286 t' \u2227 Surjective g \u2227 \u2200 (x : \u2191s), f \u2191x = \u2191(g x)\ny : \u03b2\nhy : y \u2208 t\nt' : Set \u03b2\ng : \u2191s \u2192 \u2191t'\nhtt' : t \u2286 t'\nhg : Surjective g\nhfg : \u2200 (x : \u2191s), f \u2191x = \u2191(g x)\nx : \u2191s\nhx : g x = { val := y, property := (_ : y \u2208 t') }\n\u22a2 f \u2191x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.tendstoInMeasure_iff_tendsto_Lp", "start": [623, 1], "end": [630, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.coe_finset_sum", "start": [99, 1], "end": [100, 30], "traced_tactics": []}, {"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_aux1", "start": [934, 1], "end": [1036, 41], "traced_tactics": [{"tactic": "choose \u03b4 h\u03b4 using this", "annotated_tactic": ["choose \u03b4 h\u03b4 using 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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nthis :\n  \u2200 (A : E \u2192L[\u211d] E),\n    \u2203 \u03b4,\n      0 < \u03b4 \u2227\n        (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n          \u2200 (t : Set E) (g : E \u2192 E),\n            ApproximatesLinearOn g A t \u03b4 \u2192\n              ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\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": "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\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": "obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, -\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, -\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": "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\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": "case 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\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": "have s_eq : s = \u22c3 n, s \u2229 t n := by\n  rw [\u2190 inter_iUnion]\n  exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)", "annotated_tactic": ["have s_eq : s = \u22c3 n, s \u2229 t n := by\n    rw [\u2190 <a>inter_iUnion</a>]\n    exact <a>Subset.antisymm</a> (<a>subset_inter</a> <a>Subset.rfl</a> t_cover) (<a>inter_subset_left</a> _ _)", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"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]}, {"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.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\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": "case 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\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": "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200 (A : E \u2192L[\u211d] E),\n    \u2203 \u03b4,\n      0 < \u03b4 \u2227\n        (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n          \u2200 (t : Set E) (g : E \u2192 E),\n            ApproximatesLinearOn g A t \u03b4 \u2192\n              ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "obtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, dist B A < \u03b4' \u2192 dist B.det A.det < \u2191\u03b5 :=\n  continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos", "annotated_tactic": ["obtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, <a>dist</a> B A < \u03b4' \u2192 <a>dist</a> B.det A.det < \u2191\u03b5 :=\n      <a>continuousAt_iff</a>.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos", [{"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": "Metric.continuousAt_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1061, 9], "def_end_pos": [1061, 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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "let \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (half_pos \u03b4'pos).le\u27e9", "annotated_tactic": ["let \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (<a>half_pos</a> \u03b4'pos).<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": "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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "have I'' : \u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |B.det - A.det| \u2264 \u2191\u03b5 := by\n  intro B hB\n  rw [\u2190 Real.dist_eq]\n  apply (h\u03b4' B _).le\n  rw [dist_eq_norm]\n  exact hB.trans_lt (half_lt_self \u03b4'pos)", "annotated_tactic": ["have I'' : \u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |B.det - A.det| \u2264 \u2191\u03b5 := by\n      intro B hB\n      rw [\u2190 <a>Real.dist_eq</a>]\n      apply (h\u03b4' B _).<a>le</a>\n      rw [<a>dist_eq_norm</a>]\n      exact hB.trans_lt (<a>half_lt_self</a> \u03b4'pos)", [{"full_name": "Real.dist_eq", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "half_lt_self", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [525, 11], "def_end_pos": [525, 23]}]], "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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "rcases eq_or_ne A.det 0 with (hA | hA)", "annotated_tactic": ["rcases <a>eq_or_ne</a> A.det 0 with (hA | hA)", [{"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\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.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\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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A = 0\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n\ncase intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "rcases ((mul_le_addHaar_image_of_lt_det \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, \u03b4pos\u27e9", "annotated_tactic": ["rcases ((<a>mul_le_addHaar_image_of_lt_det</a> \u03bc A I).<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8\u03b4, h, \u03b4pos\u27e9", [{"full_name": "MeasureTheory.mul_le_addHaar_image_of_lt_det", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [392, 9], "def_end_pos": [392, 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 intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.inr.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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "refine' \u27e8min \u03b4 \u03b4'', lt_min \u03b4pos (half_pos \u03b4'pos), _, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>min</a> \u03b4 \u03b4'', <a>lt_min</a> \u03b4pos (<a>half_pos</a> \u03b4'pos), _, _\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": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}]], "state_before": "case intro.intro.inr.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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.inr.intro.intro.refine'_1\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n\ncase intro.intro.inr.intro.intro.refine'_2\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n    ApproximatesLinearOn g A t (min \u03b4 \u03b4'') \u2192\n      ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "intro B hB", "annotated_tactic": ["intro B hB", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5"}, {"tactic": "rw [\u2190 Real.dist_eq]", "annotated_tactic": ["rw [\u2190 <a>Real.dist_eq</a>]", [{"full_name": "Real.dist_eq", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) \u2264 \u2191\u03b5"}, {"tactic": "apply (h\u03b4' B _).le", "annotated_tactic": ["apply (h\u03b4' B _).<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": "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) \u2264 \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 dist B A < \u03b4'"}, {"tactic": "rw [dist_eq_norm]", "annotated_tactic": ["rw [<a>dist_eq_norm</a>]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 dist B A < \u03b4'", "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 \u2016B - A\u2016 < \u03b4'"}, {"tactic": "exact hB.trans_lt (half_lt_self \u03b4'pos)", "annotated_tactic": ["exact hB.trans_lt (<a>half_lt_self</a> \u03b4'pos)", [{"full_name": "half_lt_self", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [525, 11], "def_end_pos": [525, 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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 \u2016B - A\u2016 < \u03b4'", "state_after": "no goals"}, {"tactic": "refine' \u27e8\u03b4'', half_pos \u03b4'pos, I'', _\u27e9", "annotated_tactic": ["refine' \u27e8\u03b4'', <a>half_pos</a> \u03b4'pos, I'', _\u27e9", [{"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.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\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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A = 0\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.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\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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A = 0\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n    ApproximatesLinearOn g A t \u03b4'' \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "simp only [hA, forall_const, zero_mul, ENNReal.ofReal_zero, imp_true_iff,\n  zero_le, abs_zero]", "annotated_tactic": ["simp only [hA, <a>forall_const</a>, <a>zero_mul</a>, <a>ENNReal.ofReal_zero</a>, <a>imp_true_iff</a>,\n        <a>zero_le</a>, <a>abs_zero</a>]", [{"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"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": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}]], "state_before": "case intro.intro.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\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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A = 0\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n    ApproximatesLinearOn g A t \u03b4'' \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.ofReal, ENNReal.coe_sub]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal</a>, <a>ENNReal.coe_sub</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.coe_sub", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1137, 17], "def_end_pos": [1137, 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191m < ENNReal.ofReal |ContinuousLinearMap.det 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 : 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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|)"}, {"tactic": "apply ENNReal.sub_lt_self ENNReal.coe_ne_top", "annotated_tactic": ["apply <a>ENNReal.sub_lt_self</a> <a>ENNReal.coe_ne_top</a>", [{"full_name": "ENNReal.sub_lt_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1213, 19], "def_end_pos": [1213, 30]}, {"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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|)", "state_after": "case ha\u2080\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2260 0\n\ncase hb\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191\u03b5 \u2260 0"}, {"tactic": "simpa only [abs_nonpos_iff, Real.toNNReal_eq_zero, ENNReal.coe_eq_zero, Ne.def] using hA", "annotated_tactic": ["simpa only [<a>abs_nonpos_iff</a>, <a>Real.toNNReal_eq_zero</a>, <a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>] using hA", [{"full_name": "abs_nonpos_iff", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [185, 9], "def_end_pos": [185, 23]}, {"full_name": "Real.toNNReal_eq_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [633, 9], "def_end_pos": [633, 25]}, {"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]}]], "state_before": "case ha\u2080\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [\u03b5pos.ne', ENNReal.coe_eq_zero, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [\u03b5pos.ne', <a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>, <a>not_false_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": "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 hb\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191\u03b5 \u2260 0", "state_after": "no goals"}, {"tactic": "intro B hB", "annotated_tactic": ["intro B hB", []], "state_before": "case intro.intro.inr.intro.intro.refine'_1\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5", "state_after": "case intro.intro.inr.intro.intro.refine'_1\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5"}, {"tactic": "apply I'' _ (hB.trans _)", "annotated_tactic": ["apply I'' _ (hB.trans _)", []], "state_before": "case intro.intro.inr.intro.intro.refine'_1\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 \u2191(min \u03b4 \u03b4'') \u2264 \u2191\u03b4''"}, {"tactic": "simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]", "annotated_tactic": ["simp only [<a>le_refl</a>, <a>NNReal.coe_min</a>, <a>min_le_iff</a>, <a>or_true_iff</a>]", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "NNReal.coe_min", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [597, 9], "def_end_pos": [597, 16]}, {"full_name": "min_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [43, 9], "def_end_pos": [43, 19]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}]], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 \u2191(min \u03b4 \u03b4'') \u2264 \u2191\u03b4''", "state_after": "no goals"}, {"tactic": "intro t g htg", "annotated_tactic": ["intro t g htg", []], "state_before": "case intro.intro.inr.intro.intro.refine'_2\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n    ApproximatesLinearOn g A t (min \u03b4 \u03b4'') \u2192\n      ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.inr.intro.intro.refine'_2\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "rcases eq_or_ne (\u03bc t) \u221e with (ht | ht)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (\u03bc t) \u221e with (ht | ht)", [{"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.inr.intro.intro.refine'_2\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.inr.intro.intro.refine'_2.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\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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t = \u22a4\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n\ncase intro.intro.inr.intro.intro.refine'_2.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "have := h t g (htg.mono_num (min_le_left _ _))", "annotated_tactic": ["have := h t g (htg.mono_num (<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": "case intro.intro.inr.intro.intro.refine'_2.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.inr.intro.intro.refine'_2.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : \u2191m * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t"}, {"tactic": "rwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this", "annotated_tactic": ["rwa [<a>ENNReal.coe_sub</a>, <a>ENNReal.sub_mul</a>, <a>tsub_le_iff_right</a>] at this", [{"full_name": "ENNReal.coe_sub", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1137, 17], "def_end_pos": [1137, 24]}, {"full_name": "ENNReal.sub_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1235, 9], "def_end_pos": [1235, 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 intro.intro.inr.intro.intro.refine'_2.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : \u2191m * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "state_after": "case intro.intro.inr.intro.intro.refine'_2.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : (\u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5) * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t)\n\u22a2 0 < \u2191\u03b5 \u2192 \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2192 \u2191\u2191\u03bc t \u2260 \u22a4"}, {"tactic": "simp only [ht, imp_true_iff, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [ht, <a>imp_true_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 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 intro.intro.inr.intro.intro.refine'_2.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : (\u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5) * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t)\n\u22a2 0 < \u2191\u03b5 \u2192 \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2192 \u2191\u2191\u03bc t \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [ht, \u03b5pos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne.def,\n  not_false_iff, _root_.add_top]", "annotated_tactic": ["simp only [ht, \u03b5pos.ne', <a>ENNReal.mul_top</a>, <a>ENNReal.coe_eq_zero</a>, <a>le_top</a>, <a>Ne.def</a>,\n          <a>not_false_iff</a>, <a>_root_.add_top</a>]", [{"full_name": "ENNReal.mul_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [581, 17], "def_end_pos": [581, 24]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"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": "add_top", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 16]}]], "state_before": "case intro.intro.inr.intro.intro.refine'_2.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\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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t = \u22a4\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t", "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\n\u22a2 s = \u22c3 n, 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\n\u22a2 s = s \u2229 \u22c3 i, t i"}, {"tactic": "exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)", "annotated_tactic": ["exact <a>Subset.antisymm</a> (<a>subset_inter</a> <a>Subset.rfl</a> t_cover) (<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.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]}, {"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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\n\u22a2 s = s \u2229 \u22c3 i, t i", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [s_eq]", "annotated_tactic": ["conv_lhs => rw [s_eq]", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in \u22c3 n, s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in \u22c3 n, s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 t n)"}, {"tactic": "exact fun n => hs.inter (t_meas n)", "annotated_tactic": ["exact fun n => hs.inter (t_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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)", "state_after": "no goals"}, {"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 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 t 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264\n    \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \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": "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264\n    \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 \u2200\u1d50 (a : E) \u2202Measure.restrict \u03bc (s \u2229 t n),\n    ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5"}, {"tactic": "filter_upwards [(ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n    (hf' x hx.1).mono (inter_subset_left _ _)]", "annotated_tactic": ["filter_upwards [(ht n).<a>norm_fderiv_sub_le</a> \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n          (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _)]", [{"full_name": "ApproximatesLinearOn.norm_fderiv_sub_le", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [465, 9], "def_end_pos": [465, 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": "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 \u2200\u1d50 (a : E) \u2202Measure.restrict \u03bc (s \u2229 t n),\n    ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 \u2200 (a : E),\n    \u2016f' a - A n\u2016\u208a \u2264 \u03b4 (A n) \u2192\n      ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 \u2200 (a : E),\n    \u2016f' a - A n\u2016\u208a \u2264 \u03b4 (A n) \u2192\n      ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5"}, {"tactic": "have I : |(f' x).det| \u2264 |(A n).det| + \u03b5 :=\n  calc\n    |(f' x).det| = |(A n).det + ((f' x).det - (A n).det)| := by congr 1; abel\n    _ \u2264 |(A n).det| + |(f' x).det - (A n).det| := (abs_add _ _)\n    _ \u2264 |(A n).det| + \u03b5 := add_le_add le_rfl ((h\u03b4 (A n)).2.1 _ hx)", "annotated_tactic": ["have I : |(f' x).<a>det</a>| \u2264 |(A n).<a>det</a>| + \u03b5 :=\n        calc\n          |(f' x).<a>det</a>| = |(A n).<a>det</a> + ((f' x).<a>det</a> - (A n).<a>det</a>)| := by congr 1; abel\n          _ \u2264 |(A n).<a>det</a>| + |(f' x).<a>det</a> - (A n).<a>det</a>| := (<a>abs_add</a> _ _)\n          _ \u2264 |(A n).<a>det</a>| + \u03b5 := <a>add_le_add</a> <a>le_rfl</a> ((h\u03b4 (A n)).2.1 _ hx)", [{"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 22]}, {"full_name": "abs_add", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [266, 9], "def_end_pos": [266, 16]}, {"full_name": "ContinuousLinearMap.det", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [22, 19], "def_end_pos": [22, 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": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5"}, {"tactic": "calc\n  ENNReal.ofReal |(f' x).det| \u2264 ENNReal.ofReal (|(A n).det| + \u03b5) :=\n    ENNReal.ofReal_le_ofReal I\n  _ = ENNReal.ofReal |(A n).det| + \u03b5 := by\n    simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]", "annotated_tactic": ["calc\n        <a>ENNReal.ofReal</a> |(f' x).<a>det</a>| \u2264 <a>ENNReal.ofReal</a> (|(A n).<a>det</a>| + \u03b5) :=\n          <a>ENNReal.ofReal_le_ofReal</a> I\n        _ = <a>ENNReal.ofReal</a> |(A n).<a>det</a>| + \u03b5 := by\n          simp only [<a>ENNReal.ofReal_add</a>, <a>abs_nonneg</a>, <a>NNReal.zero_le_coe</a>, <a>ENNReal.ofReal_coe_nnreal</a>]", [{"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": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"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_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "NNReal.zero_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [602, 9], "def_end_pos": [602, 20]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}]], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 |ContinuousLinearMap.det (f' x)| =\n    |ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))|", "state_after": "case e_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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ContinuousLinearMap.det (f' x) =\n    ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case e_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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ContinuousLinearMap.det (f' x) =\n    ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_add</a>, <a>abs_nonneg</a>, <a>NNReal.zero_le_coe</a>, <a>ENNReal.ofReal_coe_nnreal</a>]", [{"full_name": "ENNReal.ofReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2025, 9], "def_end_pos": [2025, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "NNReal.zero_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [602, 9], "def_end_pos": [602, 20]}, {"full_name": "ENNReal.ofReal_coe_nnreal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [212, 17], "def_end_pos": [212, 34]}]], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal (|ContinuousLinearMap.det (A n)| + \u2191\u03b5) = ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "simp only [set_lintegral_const, lintegral_add_right _ measurable_const]", "annotated_tactic": ["simp only [<a>set_lintegral_const</a>, <a>lintegral_add_right</a> _ <a>measurable_const</a>]", [{"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_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]}]], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc =\n    \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n))", "state_after": "no goals"}, {"tactic": "refine' ENNReal.tsum_le_tsum fun n => add_le_add_right _ _", "annotated_tactic": ["refine' <a>ENNReal.tsum_le_tsum</a> fun n => <a>add_le_add_right</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": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) \u2264\n    \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)"}, {"tactic": "exact (h\u03b4 (A n)).2.2 _ _ (ht n)", "annotated_tactic": ["exact (h\u03b4 (A n)).2.2 _ _ (ht 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\nn : \u2115\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [s_eq]", "annotated_tactic": ["conv_rhs => rw [s_eq]", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 t n) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n)"}, {"tactic": "rw [image_iUnion, measure_iUnion]", "annotated_tactic": ["rw [<a>image_iUnion</a>, <a>measure_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": "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 t n) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n)\n\ncase 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i))"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n)\n\ncase 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i))", "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i))\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n)"}, {"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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)\n\ncase 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)\n\ncase 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)", "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)"}, {"tactic": "rw [\u2190 ENNReal.tsum_mul_left, \u2190 ENNReal.tsum_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.tsum_mul_left</a>, \u2190 <a>ENNReal.tsum_add</a>]", [{"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}, {"full_name": "ENNReal.tsum_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [823, 19], "def_end_pos": [823, 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\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)", "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (a : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n    \u2211' (a : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a))", "state_after": "case e_f\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 (fun n => \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = fun a =>\n    \u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a)"}, {"tactic": "ext1 i", "annotated_tactic": ["ext1 i", []], "state_before": "case e_f\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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 (fun n => \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = fun a =>\n    \u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a)", "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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t i)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) = \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i)"}, {"tactic": "rw [mul_assoc, two_mul, add_assoc]", "annotated_tactic": ["rw [<a>mul_assoc</a>, <a>two_mul</a>, <a>add_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "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\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t i)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) = \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i)", "state_after": "no goals"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))", "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => f '' (s \u2229 t i)) i j"}, {"tactic": "apply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)", "annotated_tactic": ["apply <a>Disjoint.image</a> _ hf (<a>inter_subset_left</a> _ _) (<a>inter_subset_left</a> _ _)", [{"full_name": "Disjoint.image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [757, 9], "def_end_pos": [757, 30]}, {"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 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => f '' (s \u2229 t i)) i j", "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 t i) (s \u2229 t j)"}, {"tactic": "exact Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _) (t_disj hij)", "annotated_tactic": ["exact <a>Disjoint.mono</a> (<a>inter_subset_right</a> _ _) (<a>inter_subset_right</a> _ _) (t_disj hij)", [{"full_name": "Disjoint.mono", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}, {"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]}]], "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\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 t i) (s \u2229 t j)", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni : \u2115\n\u22a2 MeasurableSet (f '' (s \u2229 t i))"}, {"tactic": "exact\n  measurable_image_of_fderivWithin (hs.inter (t_meas i))\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n    (hf.mono (inter_subset_left _ _))", "annotated_tactic": ["exact\n          <a>measurable_image_of_fderivWithin</a> (hs.inter (t_meas i))\n            (fun x hx => (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _))\n            (hf.mono (<a>inter_subset_left</a> _ _))", [{"full_name": "MeasureTheory.measurable_image_of_fderivWithin", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [778, 9], "def_end_pos": [778, 41]}, {"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_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\ni : \u2115\n\u22a2 MeasurableSet (f '' (s \u2229 t i))", "state_after": "no goals"}, {"tactic": "exact pairwise_disjoint_mono t_disj fun i => inter_subset_right _ _", "annotated_tactic": ["exact <a>pairwise_disjoint_mono</a> t_disj fun i => <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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 t n)", "state_after": "no goals"}, {"tactic": "exact fun i => hs.inter (t_meas i)", "annotated_tactic": ["exact fun i => hs.inter (t_meas i)", []], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\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 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n        \u2200 (t : Set E) (g : E \u2192 E),\n          ApproximatesLinearOn g A t (\u03b4 A) \u2192\n            ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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))\ns_eq : s = \u22c3 n, s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Measurable.nndist", "start": [1728, 1], "end": [1730, 45], "traced_tactics": []}, {"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_one_of_density_one", "start": [831, 1], "end": [852, 65], "traced_tactics": [{"tactic": "refine this.congr fun r => ?_", "annotated_tactic": ["refine this.congr fun r => ?_", []], "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\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 1)", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)", "state_after": "case e_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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t))"}, {"tactic": "apply measure_toMeasurable_inter_of_sigmaFinite", "annotated_tactic": ["apply <a>measure_toMeasurable_inter_of_sigmaFinite</a>", [{"full_name": "MeasureTheory.Measure.measure_toMeasurable_inter_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3608, 9], "def_end_pos": [3608, 50]}]], "state_before": "case e_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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t))", "state_after": "case e_a.hs\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 MeasurableSet ({x} + r \u2022 t)"}, {"tactic": "simp only [image_add_left, singleton_add]", "annotated_tactic": ["simp only [<a>image_add_left</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": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}]], "state_before": "case e_a.hs\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 MeasurableSet ({x} + r \u2022 t)", "state_after": "case e_a.hs\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 MeasurableSet ((fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 t))"}, {"tactic": "apply (continuous_add_left (-x)).measurable (ht.const_smul\u2080 r)", "annotated_tactic": ["apply (<a>continuous_add_left</a> (-x)).<a>measurable</a> (ht.const_smul\u2080 r)", [{"full_name": "continuous_add_left", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}, {"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}]], "state_before": "case e_a.hs\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 MeasurableSet ((fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 t))", "state_after": "no goals"}, {"tactic": "apply\n  tendsto_addHaar_inter_smul_one_of_density_one_aux \u03bc _ (measurableSet_toMeasurable _ _) _ _\n    t ht h't h''t", "annotated_tactic": ["apply\n      <a>tendsto_addHaar_inter_smul_one_of_density_one_aux</a> \u03bc _ (<a>measurableSet_toMeasurable</a> _ _) _ _\n        t ht h't h''t", [{"full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_aux", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [775, 9], "def_end_pos": [775, 58]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)"}, {"tactic": "apply tendsto_of_tendsto_of_tendsto_of_le_of_le' h tendsto_const_nhds", "annotated_tactic": ["apply <a>tendsto_of_tendsto_of_tendsto_of_le_of_le'</a> h <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": [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]}]], "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)", "state_after": "case hgf\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b)\n\ncase hfh\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 1"}, {"tactic": "refine' eventually_of_forall fun r => mul_le_mul_right' _ _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun r => <a>mul_le_mul_right'</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": "mul_le_mul_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 26]}]], "state_before": "case hgf\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b)", "state_after": "case hgf\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r)"}, {"tactic": "exact measure_mono (inter_subset_inter_left _ (subset_toMeasurable _ _))", "annotated_tactic": ["exact <a>measure_mono</a> (<a>inter_subset_inter_left</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.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "case hgf\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r)", "state_after": "no goals"}, {"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": "case hfh\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 1", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) \u2264 1"}, {"tactic": "rintro r -", "annotated_tactic": ["rintro 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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) \u2264 1", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) \u2264 1"}, {"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\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) \u2264 1", "state_after": "case h.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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 1 * \u2191\u2191\u03bc (closedBall x r)"}, {"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": "case h.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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 1 * \u2191\u2191\u03bc (closedBall x r)", "state_after": "case h.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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "exact measure_mono (inter_subset_right _ _)", "annotated_tactic": ["exact <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": "case h.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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (closedBall x r)", "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_swap", "start": [403, 1], "end": [405, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_product_right", "start": [1003, 1], "end": [1005, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_of_finite", "start": [308, 1], "end": [309, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coeFn_negPart", "start": [1264, 1], "end": [1265, 78], "traced_tactics": [{"tactic": "rw [h, \u2190 max_neg_neg, neg_zero]", "annotated_tactic": ["rw [h, \u2190 <a>max_neg_neg</a>, <a>neg_zero</a>]", [{"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\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\ng : E \u2192 F\nc : \u211d\u22650\nf : { x // x \u2208 Lp \u211d p }\na : \u03b1\nh : \u2191\u2191(negPart f) a = max (-\u2191\u2191f a) 0\n\u22a2 \u2191\u2191(negPart f) a = -min (\u2191\u2191f a) 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_add'", "start": [194, 1], "end": [198, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.lintegral_swapLeft", "start": [725, 1], "end": [727, 53], "traced_tactics": [{"tactic": "rw [swapLeft, lintegral_comap _ measurable_swap a]", "annotated_tactic": ["rw [<a>swapLeft</a>, <a>lintegral_comap</a> _ <a>measurable_swap</a> a]", [{"full_name": "ProbabilityTheory.kernel.swapLeft", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [712, 5], "def_end_pos": [712, 13]}, {"full_name": "ProbabilityTheory.kernel.lintegral_comap", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [642, 9], "def_end_pos": [642, 24]}, {"full_name": "measurable_swap", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [760, 9], "def_end_pos": [760, 24]}]], "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\u271d : \u03b3 \u2192 \u03b1\n\u03ba : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\na : \u03b2 \u00d7 \u03b1\ng : \u03b3 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (c : \u03b3), g c \u2202\u2191(swapLeft \u03ba) a = \u222b\u207b (c : \u03b3), g c \u2202\u2191\u03ba (Prod.swap a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.measure_Ici", "start": [430, 1], "end": [439, 66], "traced_tactics": [{"tactic": "refine' tendsto_nhds_unique (tendsto_measure_Ico_atTop _ _) _", "annotated_tactic": ["refine' <a>tendsto_nhds_unique</a> (<a>tendsto_measure_Ico_atTop</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_measure_Ico_atTop", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2749, 9], "def_end_pos": [2749, 34]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ici x) = ofReal (l - leftLim (\u2191f) x)", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => \u2191\u2191(StieltjesFunction.measure f) (Ico x x_1)) atTop (\ud835\udcdd (ofReal (l - leftLim (\u2191f) x)))"}, {"tactic": "simp_rw [measure_Ico]", "annotated_tactic": ["simp_rw [<a>measure_Ico</a>]", [{"full_name": "StieltjesFunction.measure_Ico", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [413, 9], "def_end_pos": [413, 20]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => \u2191\u2191(StieltjesFunction.measure f) (Ico x x_1)) atTop (\ud835\udcdd (ofReal (l - leftLim (\u2191f) x)))", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => ofReal (leftLim (\u2191f) x_1 - leftLim (\u2191f) x)) atTop (\ud835\udcdd (ofReal (l - leftLim (\u2191f) x)))"}, {"tactic": "refine' ENNReal.tendsto_ofReal (Tendsto.sub_const _ _)", "annotated_tactic": ["refine' <a>ENNReal.tendsto_ofReal</a> (<a>Tendsto.sub_const</a> _ _)", [{"full_name": "ENNReal.tendsto_ofReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [108, 9], "def_end_pos": [108, 23]}, {"full_name": "Filter.Tendsto.sub_const", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1095, 15], "def_end_pos": [1095, 24]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => ofReal (leftLim (\u2191f) x_1 - leftLim (\u2191f) x)) atTop (\ud835\udcdd (ofReal (l - leftLim (\u2191f) x)))", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_le1 : \u2200 x, f (x - 1) \u2264 leftLim f x := fun x => Monotone.le_leftLim f.mono (sub_one_lt x)", "annotated_tactic": ["have h_le1 : \u2200 x, f (x - 1) \u2264 <a>leftLim</a> f x := fun x => <a>Monotone.le_leftLim</a> f.mono (<a>sub_one_lt</a> x)", [{"full_name": "Function.leftLim", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [48, 19], "def_end_pos": [48, 35]}, {"full_name": "Monotone.le_leftLim", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [125, 9], "def_end_pos": [125, 19]}, {"full_name": "sub_one_lt", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1166, 9], "def_end_pos": [1166, 19]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l)", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\n\u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_le2 : \u2200 x, leftLim f x \u2264 f x := fun x => Monotone.leftLim_le f.mono le_rfl", "annotated_tactic": ["have h_le2 : \u2200 x, <a>leftLim</a> f x \u2264 f x := fun x => <a>Monotone.leftLim_le</a> f.mono <a>le_rfl</a>", [{"full_name": "Function.leftLim", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [48, 19], "def_end_pos": [48, 35]}, {"full_name": "Monotone.leftLim_le", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [110, 9], "def_end_pos": [110, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\n\u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l)", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\n\u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l)"}, {"tactic": "refine' tendsto_of_tendsto_of_tendsto_of_le_of_le (hf.comp _) hf h_le1 h_le2", "annotated_tactic": ["refine' <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> (hf.comp _) hf h_le1 h_le2", [{"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]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\n\u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l)", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\n\u22a2 Tendsto (fun i => i - 1) atTop atTop"}, {"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": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\n\u22a2 Tendsto (fun i => i - 1) atTop atTop", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\n\u22a2 \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 a - 1"}, {"tactic": "exact fun y => \u27e8y + 1, fun z hyz => by rwa [le_sub_iff_add_le]\u27e9", "annotated_tactic": ["exact fun y => \u27e8y + 1, fun z hyz => by rwa [<a>le_sub_iff_add_le</a>]\u27e9", [{"full_name": "le_sub_iff_add_le", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [740, 3], "def_end_pos": [740, 14]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\n\u22a2 \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 a - 1", "state_after": "no goals"}, {"tactic": "rwa [le_sub_iff_add_le]", "annotated_tactic": ["rwa [<a>le_sub_iff_add_le</a>]", [{"full_name": "le_sub_iff_add_le", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [740, 3], "def_end_pos": [740, 14]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atTop (\ud835\udcdd l)\nx : \u211d\nh_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x\nh_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x\ny z : \u211d\nhyz : y + 1 \u2264 z\n\u22a2 y \u2264 z - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.lipschitzWith_pos_part", "start": [1225, 1], "end": [1226, 89], "traced_tactics": [{"tactic": "simp [Real.dist_eq, abs_max_sub_max_le_abs]", "annotated_tactic": ["simp [<a>Real.dist_eq</a>, <a>abs_max_sub_max_le_abs</a>]", [{"full_name": "Real.dist_eq", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 21]}, {"full_name": "abs_max_sub_max_le_abs", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [103, 9], "def_end_pos": [103, 31]}]], "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\ng : E \u2192 F\nc : \u211d\u22650\nx y : \u211d\n\u22a2 dist (max x 0) (max y 0) \u2264 \u21911 * dist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.Nontrivial.ne_singleton", "start": [815, 1], "end": [816, 48], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\nhs : Finset.Nontrivial s\n\u22a2 s \u2260 {a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nhs : Finset.Nontrivial {a}\n\u22a2 False"}, {"tactic": "exact not_nontrivial_singleton hs", "annotated_tactic": ["exact <a>not_nontrivial_singleton</a> hs", [{"full_name": "Finset.not_nontrivial_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [812, 9], "def_end_pos": [812, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nhs : Finset.Nontrivial {a}\n\u22a2 False", "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.haarContent_self", "start": [562, 1], "end": [563, 86], "traced_tactics": [{"tactic": "simp_rw [\u2190 ENNReal.coe_one, haarContent_apply, ENNReal.coe_eq_coe, chaar_self]", "annotated_tactic": ["simp_rw [\u2190 <a>ENNReal.coe_one</a>, <a>haarContent_apply</a>, <a>ENNReal.coe_eq_coe</a>, <a>chaar_self</a>]", [{"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"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": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}, {"full_name": "MeasureTheory.Measure.haar.chaar_self", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 19]}]], "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\n\u22a2 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts = 1", "state_after": "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\n\u22a2 { val := 1, property := (_ : (fun r => 0 \u2264 r) 1) } = 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\n\u22a2 { val := 1, property := (_ : (fun r => 0 \u2264 r) 1) } = 1", "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_eq_stieltjes_id", "start": [56, 1], "end": [72, 46], "traced_tactics": [{"tactic": "haveI : IsAddLeftInvariant StieltjesFunction.id.measure :=\n  \u27e8fun a =>\n    Eq.symm <|\n      Real.measure_ext_Ioo_rat fun p q => by\n        simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,\n          sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim,\n          StieltjesFunction.id_apply, id.def, preimage_const_add_Ioo]\u27e9", "annotated_tactic": ["haveI : <a>IsAddLeftInvariant</a> StieltjesFunction.id.measure :=\n    \u27e8fun a =>\n      <a>Eq.symm</a> <|\n        <a>Real.measure_ext_Ioo_rat</a> fun p q => by\n          simp only [<a>Measure.map_apply</a> (<a>measurable_const_add</a> a) <a>measurableSet_Ioo</a>,\n            <a>sub_sub_sub_cancel_right</a>, <a>StieltjesFunction.measure_Ioo</a>, <a>StieltjesFunction.id_leftLim</a>,\n            <a>StieltjesFunction.id_apply</a>, <a>id.def</a>, <a>preimage_const_add_Ioo</a>]\u27e9", [{"full_name": "MeasureTheory.Measure.IsAddLeftInvariant", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [47, 7], "def_end_pos": [47, 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": "Real.measure_ext_Ioo_rat", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1941, 9], "def_end_pos": [1941, 28]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasurableAdd.measurable_const_add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [62, 3], "def_end_pos": [62, 23]}, {"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"full_name": "sub_sub_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [795, 30], "def_end_pos": [795, 54]}, {"full_name": "StieltjesFunction.measure_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [396, 9], "def_end_pos": [396, 20]}, {"full_name": "StieltjesFunction.id_leftLim", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [101, 9], "def_end_pos": [101, 19]}, {"full_name": "StieltjesFunction.id_apply", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [92, 3], "def_end_pos": [92, 8]}, {"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.preimage_const_add_Ioo", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [75, 9], "def_end_pos": [75, 31]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id"}, {"tactic": "have A : StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped = 1 := by\n  change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis \u211d \u211d)) = 1\n  rcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis \u211d \u211d) with (H | H) <;>\n    simp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n      StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one]", "annotated_tactic": ["have A : StieltjesFunction.id.measure (<a>stdOrthonormalBasis</a> \u211d \u211d).toBasis.parallelepiped = 1 := by\n    change StieltjesFunction.id.measure (<a>parallelepiped</a> (<a>stdOrthonormalBasis</a> \u211d \u211d)) = 1\n    rcases <a>parallelepiped_orthonormalBasis_one_dim</a> (<a>stdOrthonormalBasis</a> \u211d \u211d) with (H | H) <;>\n      simp only [H, <a>StieltjesFunction.measure_Icc</a>, <a>StieltjesFunction.id_apply</a>, <a>id.def</a>, <a>tsub_zero</a>,\n        <a>StieltjesFunction.id_leftLim</a>, <a>sub_neg_eq_add</a>, <a>zero_add</a>, <a>ENNReal.ofReal_one</a>]", [{"full_name": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}, {"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}, {"full_name": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}, {"full_name": "parallelepiped_orthonormalBasis_one_dim", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [84, 9], "def_end_pos": [84, 48]}, {"full_name": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}, {"full_name": "StieltjesFunction.measure_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [385, 9], "def_end_pos": [385, 20]}, {"full_name": "StieltjesFunction.id_apply", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [92, 3], "def_end_pos": [92, 8]}, {"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": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 18]}, {"full_name": "StieltjesFunction.id_leftLim", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [101, 9], "def_end_pos": [101, 19]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"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\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id"}, {"tactic": "conv_rhs =>\n  rw [addHaarMeasure_unique StieltjesFunction.id.measure\n      (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped, A]", "annotated_tactic": ["conv_rhs =>\n    rw [<a>addHaarMeasure_unique</a> StieltjesFunction.id.measure\n        (<a>stdOrthonormalBasis</a> \u211d \u211d).toBasis.parallelepiped, 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": "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 : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n\u22a2 volume = 1 \u2022 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d)))"}, {"tactic": "simp only [volume, Basis.addHaar, one_smul]", "annotated_tactic": ["simp only [<a>volume</a>, <a>Basis.addHaar</a>, <a>one_smul</a>]", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Basis.addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [220, 1], "def_end_pos": [223, 42]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n\u22a2 volume = 1 \u2022 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d)))", "state_after": "no goals"}, {"tactic": "simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,\n  sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim,\n  StieltjesFunction.id_apply, id.def, preimage_const_add_Ioo]", "annotated_tactic": ["simp only [<a>Measure.map_apply</a> (<a>measurable_const_add</a> a) <a>measurableSet_Ioo</a>,\n            <a>sub_sub_sub_cancel_right</a>, <a>StieltjesFunction.measure_Ioo</a>, <a>StieltjesFunction.id_leftLim</a>,\n            <a>StieltjesFunction.id_apply</a>, <a>id.def</a>, <a>preimage_const_add_Ioo</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": "MeasurableAdd.measurable_const_add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [62, 3], "def_end_pos": [62, 23]}, {"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"full_name": "sub_sub_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [795, 30], "def_end_pos": [795, 54]}, {"full_name": "StieltjesFunction.measure_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [396, 9], "def_end_pos": [396, 20]}, {"full_name": "StieltjesFunction.id_leftLim", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [101, 9], "def_end_pos": [101, 19]}, {"full_name": "StieltjesFunction.id_apply", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [92, 3], "def_end_pos": [92, 8]}, {"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.preimage_const_add_Ioo", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [75, 9], "def_end_pos": [75, 31]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\np q : \u211a\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (Ioo \u2191p \u2191q) =\n    \u2191\u2191(Measure.map (fun x => a + x) (StieltjesFunction.measure StieltjesFunction.id)) (Ioo \u2191p \u2191q)", "state_after": "no goals"}, {"tactic": "change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis \u211d \u211d)) = 1", "annotated_tactic": ["change StieltjesFunction.id.measure (<a>parallelepiped</a> (<a>stdOrthonormalBasis</a> \u211d \u211d)) = 1", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}, {"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 : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d)) = 1"}, {"tactic": "rcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis \u211d \u211d) with (H | H) <;>\n  simp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n    StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one]", "annotated_tactic": ["rcases <a>parallelepiped_orthonormalBasis_one_dim</a> (<a>stdOrthonormalBasis</a> \u211d \u211d) with (H | H) <;>\n      simp only [H, <a>StieltjesFunction.measure_Icc</a>, <a>StieltjesFunction.id_apply</a>, <a>id.def</a>, <a>tsub_zero</a>,\n        <a>StieltjesFunction.id_leftLim</a>, <a>sub_neg_eq_add</a>, <a>zero_add</a>, <a>ENNReal.ofReal_one</a>]", [{"full_name": "parallelepiped_orthonormalBasis_one_dim", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [84, 9], "def_end_pos": [84, 48]}, {"full_name": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}, {"full_name": "StieltjesFunction.measure_Icc", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [385, 9], "def_end_pos": [385, 20]}, {"full_name": "StieltjesFunction.id_apply", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [92, 3], "def_end_pos": [92, 8]}, {"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": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 18]}, {"full_name": "StieltjesFunction.id_leftLim", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [101, 9], "def_end_pos": [101, 19]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"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\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d)) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.measurePreserving_sumPiEquivProdPi_symm", "start": [767, 1], "end": [776, 29], "traced_tactics": [{"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\u2074 : Fintype \u03b9\nm\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b3 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b2 : Fintype \u03b9'\n\u03c0 : \u03b9 \u2295 \u03b9' \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9 \u2295 \u03b9') \u2192 MeasurableSpace (\u03c0 i)\n\u03bc : (i : \u03b9 \u2295 \u03b9') \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), SigmaFinite (\u03bc i)\n\u22a2 Measure.map (\u2191(MeasurableEquiv.symm (MeasurableEquiv.sumPiEquivProdPi \u03c0)))\n      (Measure.prod (Measure.pi fun i => \u03bc (Sum.inl i)) (Measure.pi fun i => \u03bc (Sum.inr i))) =\n    Measure.pi \u03bc", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b3 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b2 : Fintype \u03b9'\n\u03c0 : \u03b9 \u2295 \u03b9' \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9 \u2295 \u03b9') \u2192 MeasurableSpace (\u03c0 i)\n\u03bc : (i : \u03b9 \u2295 \u03b9') \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), SigmaFinite (\u03bc i)\ns : (i : \u03b9 \u2295 \u03b9') \u2192 Set (\u03c0 i)\nx\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm (MeasurableEquiv.sumPiEquivProdPi \u03c0)))\n            (Measure.prod (Measure.pi fun i => \u03bc (Sum.inl i)) (Measure.pi fun i => \u03bc (Sum.inr i))))\n      (Set.pi univ s) =\n    \u220f i : \u03b9 \u2295 \u03b9', \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "simp_rw [MeasurableEquiv.map_apply, MeasurableEquiv.coe_sumPiEquivProdPi_symm,\n  Equiv.sumPiEquivProdPi_symm_preimage_univ_pi, Measure.prod_prod, Measure.pi_pi,\n  Fintype.prod_sum_type]", "annotated_tactic": ["simp_rw [<a>MeasurableEquiv.map_apply</a>, <a>MeasurableEquiv.coe_sumPiEquivProdPi_symm</a>,\n      <a>Equiv.sumPiEquivProdPi_symm_preimage_univ_pi</a>, <a>Measure.prod_prod</a>, <a>Measure.pi_pi</a>,\n      <a>Fintype.prod_sum_type</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.coe_sumPiEquivProdPi_symm", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 34]}, {"full_name": "Equiv.sumPiEquivProdPi_symm_preimage_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [958, 9], "def_end_pos": [958, 47]}, {"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": "MeasureTheory.Measure.pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [394, 9], "def_end_pos": [394, 14]}, {"full_name": "Fintype.prod_sum_type", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [292, 9], "def_end_pos": [292, 30]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u2074 : Fintype \u03b9\nm\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b3 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\ninst\u271d\u00b2 : Fintype \u03b9'\n\u03c0 : \u03b9 \u2295 \u03b9' \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9 \u2295 \u03b9') \u2192 MeasurableSpace (\u03c0 i)\n\u03bc : (i : \u03b9 \u2295 \u03b9') \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), SigmaFinite (\u03bc i)\ns : (i : \u03b9 \u2295 \u03b9') \u2192 Set (\u03c0 i)\nx\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm (MeasurableEquiv.sumPiEquivProdPi \u03c0)))\n            (Measure.prod (Measure.pi fun i => \u03bc (Sum.inl i)) (Measure.pi fun i => \u03bc (Sum.inr i))))\n      (Set.pi univ s) =\n    \u220f i : \u03b9 \u2295 \u03b9', \u2191\u2191(\u03bc i) (s i)", "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": [1987, 1], "end": [1991, 48], "traced_tactics": [{"tactic": "refine' snorm_one_le_of_le hfint hfint' _", "annotated_tactic": ["refine' <a>snorm_one_le_of_le</a> hfint hfint' _", [{"full_name": "MeasureTheory.snorm_one_le_of_le", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 27]}]], "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\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 r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * ENNReal.ofReal r", "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\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 r\n\u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191(Real.toNNReal r)"}, {"tactic": "simp only [Real.coe_toNNReal', le_max_iff]", "annotated_tactic": ["simp only [<a>Real.coe_toNNReal'</a>, <a>le_max_iff</a>]", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "le_max_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [38, 9], "def_end_pos": [38, 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 : 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\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 r\n\u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191(Real.toNNReal r)", "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\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 r\n\u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 r \u2228 f \u03c9 \u2264 0"}, {"tactic": "filter_upwards [hf] with \u03c9 h\u03c9 using Or.inl h\u03c9", "annotated_tactic": ["filter_upwards [hf] with \u03c9 h\u03c9 using <a>Or.inl</a> h\u03c9", [{"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\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\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 r\n\u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 r \u2228 f \u03c9 \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.union_val", "start": [1368, 1], "end": [1369, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter_of_tendsto_ae", "start": [473, 1], "end": [477, 43], "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_iff", "start": [51, 1], "end": [63, 84], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "d\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\n\u22a2 normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082 \u2194 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh : normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081\n\ncase mpr\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh : n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081\n\u22a2 normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082"}, {"tactic": "simp only [normalize_eq, mk'.injEq] at h", "annotated_tactic": ["simp only [<a>normalize_eq</a>, mk'.injEq] at h", [{"full_name": "Rat.normalize_eq", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [25, 9], "def_end_pos": [25, 21]}]], "state_before": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh : normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081"}, {"tactic": "have' hn\u2081 := Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left n\u2081.natAbs d\u2081", "annotated_tactic": ["have' hn\u2081 := <a>Int.ofNat_dvd_left</a>.2 <| <a>Nat.gcd_dvd_left</a> n\u2081.natAbs d\u2081", [{"full_name": "Int.ofNat_dvd_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [662, 9], "def_end_pos": [662, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}]], "state_before": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081"}, {"tactic": "have' hn\u2082 := Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left n\u2082.natAbs d\u2082", "annotated_tactic": ["have' hn\u2082 := <a>Int.ofNat_dvd_left</a>.2 <| <a>Nat.gcd_dvd_left</a> n\u2082.natAbs d\u2082", [{"full_name": "Int.ofNat_dvd_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [662, 9], "def_end_pos": [662, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}]], "state_before": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\nhn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081"}, {"tactic": "have' hd\u2081 := Int.ofNat_dvd.2 <| Nat.gcd_dvd_right n\u2081.natAbs d\u2081", "annotated_tactic": ["have' hd\u2081 := <a>Int.ofNat_dvd</a>.2 <| <a>Nat.gcd_dvd_right</a> n\u2081.natAbs d\u2081", [{"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"full_name": "Nat.gcd_dvd_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}]], "state_before": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\nhn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\nhn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082\nhd\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 \u2191d\u2081\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081"}, {"tactic": "have' hd\u2082 := Int.ofNat_dvd.2 <| Nat.gcd_dvd_right n\u2082.natAbs d\u2082", "annotated_tactic": ["have' hd\u2082 := <a>Int.ofNat_dvd</a>.2 <| <a>Nat.gcd_dvd_right</a> n\u2082.natAbs d\u2082", [{"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"full_name": "Nat.gcd_dvd_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}]], "state_before": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\nhn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082\nhd\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 \u2191d\u2081\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\nhn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082\nhd\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 \u2191d\u2081\nhd\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 \u2191d\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081"}, {"tactic": "rw [\u2190 Int.ediv_mul_cancel (Int.dvd_trans hd\u2082 (Int.dvd_mul_left ..)),\n  Int.mul_ediv_assoc _ hd\u2082, \u2190 Int.ofNat_ediv, \u2190 h.2, Int.ofNat_ediv,\n  \u2190 Int.mul_ediv_assoc _ hd\u2081, Int.mul_ediv_assoc' _ hn\u2081,\n  Int.mul_right_comm, h.1, Int.ediv_mul_cancel hn\u2082]", "annotated_tactic": ["rw [\u2190 <a>Int.ediv_mul_cancel</a> (<a>Int.dvd_trans</a> hd\u2082 (<a>Int.dvd_mul_left</a> ..)),\n      <a>Int.mul_ediv_assoc</a> _ hd\u2082, \u2190 <a>Int.ofNat_ediv</a>, \u2190 h.2, <a>Int.ofNat_ediv</a>,\n      \u2190 <a>Int.mul_ediv_assoc</a> _ hd\u2081, <a>Int.mul_ediv_assoc'</a> _ hn\u2081,\n      <a>Int.mul_right_comm</a>, h.1, <a>Int.ediv_mul_cancel</a> hn\u2082]", [{"full_name": "Int.ediv_mul_cancel", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [704, 19], "def_end_pos": [704, 34]}, {"full_name": "Int.dvd_trans", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [598, 19], "def_end_pos": [598, 28]}, {"full_name": "Int.dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [614, 19], "def_end_pos": [614, 31]}, {"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": "Int.ofNat_ediv", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [25, 28], "def_end_pos": [25, 38]}, {"full_name": "Int.ofNat_ediv", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [25, 28], "def_end_pos": [25, 38]}, {"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": "Int.mul_ediv_assoc'", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [727, 19], "def_end_pos": [727, 34]}, {"full_name": "Int.mul_right_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [404, 19], "def_end_pos": [404, 33]}, {"full_name": "Int.ediv_mul_cancel", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [704, 19], "def_end_pos": [704, 34]}]], "state_before": "case mp\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh :\n  n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227\n    d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082\nhn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081\nhn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082\nhd\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 \u2191d\u2081\nhd\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 \u2191d\u2082\n\u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081", "state_after": "no goals"}, {"tactic": "rw [\u2190 normalize_mul_right _ z\u2082, \u2190 normalize_mul_left z\u2082 z\u2081, Int.mul_comm d\u2081, h]", "annotated_tactic": ["rw [\u2190 <a>normalize_mul_right</a> _ z\u2082, \u2190 <a>normalize_mul_left</a> z\u2082 z\u2081, <a>Int.mul_comm</a> d\u2081, h]", [{"full_name": "Rat.normalize_mul_right", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [47, 9], "def_end_pos": [47, 28]}, {"full_name": "Rat.normalize_mul_left", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [41, 9], "def_end_pos": [41, 27]}, {"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": "case mpr\nd\u2081 d\u2082 : Nat\nn\u2081 n\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nh : n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081\n\u22a2 normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict", "start": [839, 1], "end": [850, 83], "traced_tactics": [{"tactic": "refine'\n  integrableOn_iUnion_of_summable_integral_norm (fun i => (s i).isCompact.isClosed.measurableSet)\n    (fun i => (map_continuous f).continuousOn.integrableOn_compact (s i).isCompact)\n    (summable_of_nonneg_of_le (fun \u03b9 => integral_nonneg fun x => norm_nonneg _) (fun i => _) hf)", "annotated_tactic": ["refine'\n    <a>integrableOn_iUnion_of_summable_integral_norm</a> (fun i => (s i).isCompact.isClosed.measurableSet)\n      (fun i => (<a>map_continuous</a> f).continuousOn.integrableOn_compact (s i).<a>isCompact</a>)\n      (<a>summable_of_nonneg_of_le</a> (fun \u03b9 => <a>integral_nonneg</a> fun x => <a>norm_nonneg</a> _) (fun i => _) hf)", [{"full_name": "MeasureTheory.integrableOn_iUnion_of_summable_integral_norm", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [819, 9], "def_end_pos": [819, 54]}, {"full_name": "ContinuousMapClass.map_continuous", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 17]}, {"full_name": "TopologicalSpace.Compacts.isCompact", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [52, 19], "def_end_pos": [52, 28]}, {"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": "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\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : Countable \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nf : C(\u03b1, E)\ns : \u03b2 \u2192 Compacts \u03b1\nhf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))\n\u22a2 IntegrableOn (\u2191f) (\u22c3 i, \u2191(s i))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : Countable \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nf : C(\u03b1, E)\ns : \u03b2 \u2192 Compacts \u03b1\nhf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))\ni : \u03b2\n\u22a2 \u222b (a : \u03b1) in \u2191(s i), \u2016\u2191f a\u2016 \u2202\u03bc \u2264 \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))"}, {"tactic": "rw [\u2190 (Real.norm_of_nonneg (integral_nonneg fun a => norm_nonneg _) : \u2016_\u2016 = \u222b x in s i, \u2016f x\u2016 \u2202\u03bc)]", "annotated_tactic": ["rw [\u2190 (<a>Real.norm_of_nonneg</a> (<a>integral_nonneg</a> fun a => <a>norm_nonneg</a> _) : \u2016_\u2016 = \u222b x in s i, \u2016f x\u2016 \u2202\u03bc)]", [{"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": "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\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : Countable \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nf : C(\u03b1, E)\ns : \u03b2 \u2192 Compacts \u03b1\nhf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))\ni : \u03b2\n\u22a2 \u222b (a : \u03b1) in \u2191(s i), \u2016\u2191f a\u2016 \u2202\u03bc \u2264 \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : Countable \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nf : C(\u03b1, E)\ns : \u03b2 \u2192 Compacts \u03b1\nhf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))\ni : \u03b2\n\u22a2 \u2016\u222b (x : \u03b1) in \u2191(s i), \u2016\u2191f x\u2016 \u2202\u03bc\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))"}, {"tactic": "exact\n  norm_set_integral_le_of_norm_le_const' (s i).isCompact.measure_lt_top\n    (s i).isCompact.isClosed.measurableSet fun x hx =>\n    (norm_norm (f x)).symm \u25b8 (f.restrict (s i : Set \u03b1)).norm_coe_le_norm \u27e8x, hx\u27e9", "annotated_tactic": ["exact\n    <a>norm_set_integral_le_of_norm_le_const'</a> (s i).isCompact.measure_lt_top\n      (s i).isCompact.isClosed.measurableSet fun x hx =>\n      (<a>norm_norm</a> (f x)).<a>symm</a> \u25b8 (f.restrict (s i : <a>Set</a> \u03b1)).<a>norm_coe_le_norm</a> \u27e8x, hx\u27e9", [{"full_name": "MeasureTheory.norm_set_integral_le_of_norm_le_const'", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [580, 9], "def_end_pos": [580, 47]}, {"full_name": "norm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [837, 9], "def_end_pos": [837, 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": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : Countable \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nf : C(\u03b1, E)\ns : \u03b2 \u2192 Compacts \u03b1\nhf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))\ni : \u03b2\n\u22a2 \u2016\u222b (x : \u03b1) in \u2191(s i), \u2016\u2191f x\u2016 \u2202\u03bc\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.bit1_of_bit1", "start": [119, 1], "end": [120, 69], "traced_tactics": [{"tactic": "rw [add_one, bit0_of_bit0]", "annotated_tactic": ["rw [<a>add_one</a>, <a>bit0_of_bit0</a>]", [{"full_name": "PosNum.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [84, 9], "def_end_pos": [84, 16]}, {"full_name": "PosNum.bit0_of_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [113, 9], "def_end_pos": [113, 21]}]], "state_before": "\u03b1 : Type u_1\nn : PosNum\n\u22a2 _root_.bit0 n + 1 = bit1 n", "state_after": "\u03b1 : Type u_1\nn : PosNum\n\u22a2 succ (bit0 n) = bit1 n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nn : PosNum\n\u22a2 succ (bit0 n) = bit1 n", "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_Icc_self", "start": [254, 1], "end": [255, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "full_name": "Set.pairwiseDisjoint_sUnion", "start": [59, 1], "end": [61, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.toMeasure_ofFintype_apply", "start": [229, 1], "end": [231, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.smul_map_volume_mul_left", "start": [295, 1], "end": [305, 74], "traced_tactics": [{"tactic": "refine' (Real.measure_ext_Ioo_rat fun p q => _).symm", "annotated_tactic": ["refine' (<a>Real.measure_ext_Ioo_rat</a> fun p q => _).<a>symm</a>", [{"full_name": "Real.measure_ext_Ioo_rat", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1941, 9], "def_end_pos": [1941, 28]}, {"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 : \u211d\nh : a \u2260 0\n\u22a2 ofReal |a| \u2022 Measure.map (fun x => a * x) volume = volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\np q : \u211a\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)"}, {"tactic": "cases' lt_or_gt_of_ne h with h h", "annotated_tactic": ["cases' <a>lt_or_gt_of_ne</a> h 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": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\np q : \u211a\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)", "state_after": "case inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh\u271d : a \u2260 0\np q : \u211a\nh : a < 0\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)\n\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh\u271d : a \u2260 0\np q : \u211a\nh : a > 0\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)"}, {"tactic": "simp only [Real.volume_Ioo, Measure.smul_apply, \u2190 ENNReal.ofReal_mul (le_of_lt <| neg_pos.2 h),\n  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, neg_sub_neg, neg_mul,\n  preimage_const_mul_Ioo_of_neg _ _ h, abs_of_neg h, mul_sub, smul_eq_mul,\n  mul_div_cancel' _ (ne_of_lt h)]", "annotated_tactic": ["simp only [<a>Real.volume_Ioo</a>, <a>Measure.smul_apply</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>le_of_lt</a> <| <a>neg_pos</a>.2 h),\n      <a>Measure.map_apply</a> (<a>measurable_const_mul</a> a) <a>measurableSet_Ioo</a>, <a>neg_sub_neg</a>, <a>neg_mul</a>,\n      <a>preimage_const_mul_Ioo_of_neg</a> _ _ h, <a>abs_of_neg</a> h, <a>mul_sub</a>, <a>smul_eq_mul</a>,\n      <a>mul_div_cancel'</a> _ (<a>ne_of_lt</a> h)]", [{"full_name": "Real.volume_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "neg_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [677, 24], "def_end_pos": [677, 31]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasurableMul.measurable_const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [82, 3], "def_end_pos": [82, 23]}, {"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"full_name": "neg_sub_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [506, 3], "def_end_pos": [506, 14]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "Set.preimage_const_mul_Ioo_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [662, 9], "def_end_pos": [662, 38]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh\u271d : a \u2260 0\np q : \u211a\nh : a < 0\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)", "state_after": "no goals"}, {"tactic": "simp only [Real.volume_Ioo, Measure.smul_apply, \u2190 ENNReal.ofReal_mul (le_of_lt h),\n  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, preimage_const_mul_Ioo _ _ h,\n  abs_of_pos h, mul_sub, mul_div_cancel' _ (ne_of_gt h), smul_eq_mul]", "annotated_tactic": ["simp only [<a>Real.volume_Ioo</a>, <a>Measure.smul_apply</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>le_of_lt</a> h),\n      <a>Measure.map_apply</a> (<a>measurable_const_mul</a> a) <a>measurableSet_Ioo</a>, <a>preimage_const_mul_Ioo</a> _ _ h,\n      <a>abs_of_pos</a> h, <a>mul_sub</a>, <a>mul_div_cancel'</a> _ (<a>ne_of_gt</a> h), <a>smul_eq_mul</a>]", [{"full_name": "Real.volume_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasurableMul.measurable_const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [82, 3], "def_end_pos": [82, 23]}, {"full_name": "measurableSet_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"full_name": "Set.preimage_const_mul_Ioo", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [618, 9], "def_end_pos": [618, 31]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "case inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh\u271d : a \u2260 0\np q : \u211a\nh : a > 0\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.smul_finite", "start": [3820, 1], "end": [3823, 48], "traced_tactics": [{"tactic": "lift c to \u211d\u22650 using hc", "annotated_tactic": ["lift c to \u211d\u22650 using hc", []], "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\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 IsFiniteMeasure (c \u2022 \u03bc)", "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\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\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\n\u22a2 IsFiniteMeasure (\u2191c \u2022 \u03bc)"}, {"tactic": "exact MeasureTheory.isFiniteMeasureSMulNNReal", "annotated_tactic": ["exact <a>MeasureTheory.isFiniteMeasureSMulNNReal</a>", [{"full_name": "MeasureTheory.isFiniteMeasureSMulNNReal", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2927, 10], "def_end_pos": [2927, 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\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\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\n\u22a2 IsFiniteMeasure (\u2191c \u2022 \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.vars_rename", "start": [868, 1], "end": [873, 60], "traced_tactics": [{"tactic": "classical\nintro i hi\nsimp only [vars_def, exists_prop, Multiset.mem_toFinset, Finset.mem_image] at hi \u22a2\nsimpa only [Multiset.mem_map] using degrees_rename _ _ hi", "annotated_tactic": ["classical\n  intro i hi\n  simp only [<a>vars_def</a>, <a>exists_prop</a>, <a>Multiset.mem_toFinset</a>, <a>Finset.mem_image</a>] at hi \u22a2\n  simpa only [<a>Multiset.mem_map</a>] using <a>degrees_rename</a> _ _ hi", [{"full_name": "MvPolynomial.vars_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3200, 9], "def_end_pos": [3200, 21]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 16]}, {"full_name": "MvPolynomial.degrees_rename", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [235, 9], "def_end_pos": [235, 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\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 vars (\u2191(rename f) \u03c6) \u2286 Finset.image f (vars \u03c6)", "state_after": "no goals"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "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\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 vars (\u2191(rename f) \u03c6) \u2286 Finset.image f (vars \u03c6)", "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\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nhi : i \u2208 vars (\u2191(rename f) \u03c6)\n\u22a2 i \u2208 Finset.image f (vars \u03c6)"}, {"tactic": "simp only [vars_def, exists_prop, Multiset.mem_toFinset, Finset.mem_image] at hi \u22a2", "annotated_tactic": ["simp only [<a>vars_def</a>, <a>exists_prop</a>, <a>Multiset.mem_toFinset</a>, <a>Finset.mem_image</a>] at hi \u22a2", [{"full_name": "MvPolynomial.vars_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3200, 9], "def_end_pos": [3200, 21]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}]], "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\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nhi : i \u2208 vars (\u2191(rename f) \u03c6)\n\u22a2 i \u2208 Finset.image f (vars \u03c6)", "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\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nhi : i \u2208 degrees (\u2191(rename f) \u03c6)\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i"}, {"tactic": "simpa only [Multiset.mem_map] using degrees_rename _ _ hi", "annotated_tactic": ["simpa only [<a>Multiset.mem_map</a>] using <a>degrees_rename</a> _ _ hi", [{"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 16]}, {"full_name": "MvPolynomial.degrees_rename", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [235, 9], "def_end_pos": [235, 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\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nhi : i \u2208 degrees (\u2191(rename f) \u03c6)\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_of_subset_of_ssubset", "start": [433, 1], "end": [435, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.cardinal_measurableSet_le_continuum", "start": [186, 1], "end": [188, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk", "start": [291, 1], "end": [309, 64], "traced_tactics": [{"tactic": "refine' \u27e8fun x => hf.mk f x.2, hf.stronglyMeasurable_mk.comp_measurable measurable_snd, _\u27e9", "annotated_tactic": ["refine' \u27e8fun x => hf.mk f x.2, hf.stronglyMeasurable_mk.comp_measurable <a>measurable_snd</a>, _\u27e9", [{"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\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 (fun x => f x.2) =\u1d50[Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc] fun x => AEStronglyMeasurable.mk f hf x.2"}, {"tactic": "suffices h : Measure.QuasiMeasurePreserving Prod.snd (\u03bc.map fun \u03c9 => (X \u03c9, \u03c9)) \u03bc", "annotated_tactic": ["suffices h : <a>Measure.QuasiMeasurePreserving</a> <a>Prod.snd</a> (\u03bc.map fun \u03c9 => (X \u03c9, \u03c9)) \u03bc", [{"full_name": "MeasureTheory.Measure.QuasiMeasurePreserving", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2197, 11], "def_end_pos": [2197, 33]}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 (fun x => f x.2) =\u1d50[Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc] fun x => AEStronglyMeasurable.mk f hf x.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\nh : Measure.QuasiMeasurePreserving Prod.snd\n\u22a2 (fun x => f x.2) =\u1d50[Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc] fun x => AEStronglyMeasurable.mk f hf x.2\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 Measure.QuasiMeasurePreserving Prod.snd"}, {"tactic": "refine' \u27e8measurable_snd, Measure.AbsolutelyContinuous.mk fun s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>measurable_snd</a>, <a>Measure.AbsolutelyContinuous.mk</a> fun s hs h\u03bcs => _\u27e9", [{"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"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 h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 Measure.QuasiMeasurePreserving Prod.snd", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(Measure.map Prod.snd (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)) s = 0"}, {"tactic": "rw [Measure.map_apply _ hs]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> _ 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\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(Measure.map Prod.snd (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)) s = 0", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 Measurable Prod.snd"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 Measurable Prod.snd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 Measurable Prod.snd\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0"}, {"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": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : \u00acAEMeasurable X\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0"}, {"tactic": "exact Measure.QuasiMeasurePreserving.ae_eq h hf.ae_eq_mk", "annotated_tactic": ["exact <a>Measure.QuasiMeasurePreserving.ae_eq</a> h hf.ae_eq_mk", [{"full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2266, 9], "def_end_pos": [2266, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : 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\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\nh : Measure.QuasiMeasurePreserving Prod.snd\n\u22a2 (fun x => f x.2) =\u1d50[Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc] fun x => AEStronglyMeasurable.mk f hf x.2", "state_after": "no goals"}, {"tactic": "exact measurable_snd", "annotated_tactic": ["exact <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\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 Measurable Prod.snd", "state_after": "no goals"}, {"tactic": "rw [Measure.map_apply_of_aemeasurable]", "annotated_tactic": ["rw [<a>Measure.map_apply_of_aemeasurable</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]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 \u2191\u2191\u03bc ((fun \u03c9 => (X \u03c9, \u03c9)) \u207b\u00b9' (Prod.snd \u207b\u00b9' s)) = 0\n\ncase pos.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 AEMeasurable fun \u03c9 => (X \u03c9, \u03c9)\n\ncase pos.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 MeasurableSet (Prod.snd \u207b\u00b9' s)"}, {"tactic": "rw [\u2190 univ_prod, mk_preimage_prod, preimage_univ, univ_inter, preimage_id']", "annotated_tactic": ["rw [\u2190 <a>univ_prod</a>, <a>mk_preimage_prod</a>, <a>preimage_univ</a>, <a>univ_inter</a>, <a>preimage_id'</a>]", [{"full_name": "Set.univ_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [130, 9], "def_end_pos": [130, 18]}, {"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.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "Set.preimage_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [133, 9], "def_end_pos": [133, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 \u2191\u2191\u03bc ((fun \u03c9 => (X \u03c9, \u03c9)) \u207b\u00b9' (Prod.snd \u207b\u00b9' s)) = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "exact h\u03bcs", "annotated_tactic": ["exact h\u03bcs", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "no goals"}, {"tactic": "exact hX.prod_mk aemeasurable_id", "annotated_tactic": ["exact hX.prod_mk <a>aemeasurable_id</a>", [{"full_name": "aemeasurable_id", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [751, 9], "def_end_pos": [751, 24]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 AEMeasurable fun \u03c9 => (X \u03c9, \u03c9)", "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 pos.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : AEMeasurable X\n\u22a2 MeasurableSet (Prod.snd \u207b\u00b9' s)", "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\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : \u00acAEMeasurable X\n\u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : \u00acAEMeasurable X\n\u22a2 \u2191\u21910 (Prod.snd \u207b\u00b9' s) = 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9\u271d : Type u_3\nF\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\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\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhX : \u00acAEMeasurable X\n\u22a2 \u2191\u21910 (Prod.snd \u207b\u00b9' s) = 0", "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\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\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\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\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\u271d : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\u271d\ninst\u271d\u2076 : MeasurableSpace \u03a9\u271d\ninst\u271d\u2075 : PolishSpace \u03a9\u271d\ninst\u271d\u2074 : BorelSpace \u03a9\u271d\ninst\u271d\u00b3 : Nonempty \u03a9\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d\nX\u271d : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03a9\u271d\nm\u03b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03a9\u271d\nt : Set \u03b2\nf\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d\n\u03a9 : Type u_5\nF : Type u_6\nm\u03a9 : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u03b2\n\u03bc : Measure \u03a9\ninst\u271d : TopologicalSpace F\nf : \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\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/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.lintegral_nnnorm_condexpL2_le", "start": [160, 1], "end": [181, 72], "traced_tactics": [{"tactic": "let h_meas := lpMeas.aeStronglyMeasurable' (condexpL2 \u211d \u211d hm f)", "annotated_tactic": ["let h_meas := <a>lpMeas.aeStronglyMeasurable'</a> (<a>condexpL2</a> \u211d \u211d hm f)", [{"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.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\nf : { x // x \u2208 Lp \u211d 2 }\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "let g := h_meas.choose", "annotated_tactic": ["let g := h_meas.choose", []], "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "have hg_meas : StronglyMeasurable[m] g := h_meas.choose_spec.1", "annotated_tactic": ["have hg_meas : StronglyMeasurable[m] g := h_meas.choose_spec.1", []], "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "have hg_eq : g =\u1d50[\u03bc] condexpL2 \u211d \u211d hm f := h_meas.choose_spec.2.symm", "annotated_tactic": ["have hg_eq : g =\u1d50[\u03bc] <a>condexpL2</a> \u211d \u211d hm f := h_meas.choose_spec.2.<a>symm</a>", [{"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": "\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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "have hg_eq_restrict : g =\u1d50[\u03bc.restrict s] condexpL2 \u211d \u211d hm f := ae_restrict_of_ae hg_eq", "annotated_tactic": ["have hg_eq_restrict : g =\u1d50[\u03bc.restrict s] <a>condexpL2</a> \u211d \u211d hm f := <a>ae_restrict_of_ae</a> hg_eq", [{"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"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": "\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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "have hg_nnnorm_eq : (fun x => (\u2016g x\u2016\u208a : \u211d\u22650\u221e)) =\u1d50[\u03bc.restrict s] fun x =>\n    (\u2016(condexpL2 \u211d \u211d hm f : \u03b1 \u2192 \u211d) x\u2016\u208a : \u211d\u22650\u221e) := by\n  refine' hg_eq_restrict.mono fun x hx => _\n  dsimp only\n  simp_rw [hx]", "annotated_tactic": ["have hg_nnnorm_eq : (fun x => (\u2016g x\u2016\u208a : \u211d\u22650\u221e)) =\u1d50[\u03bc.restrict s] fun x =>\n      (\u2016(<a>condexpL2</a> \u211d \u211d hm f : \u03b1 \u2192 \u211d) x\u2016\u208a : \u211d\u22650\u221e) := by\n    refine' hg_eq_restrict.mono fun x hx => _\n    dsimp only\n    simp_rw [hx]", [{"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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae hg_nnnorm_eq.symm]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> hg_nnnorm_eq.symm]", [{"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\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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016g a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc"}, {"tactic": "refine'\n  lintegral_nnnorm_le_of_forall_fin_meas_integral_eq hm (Lp.stronglyMeasurable f) _ _ _ _ hs h\u03bcs", "annotated_tactic": ["refine'\n    <a>lintegral_nnnorm_le_of_forall_fin_meas_integral_eq</a> hm (<a>Lp.stronglyMeasurable</a> f) _ _ _ _ hs h\u03bcs", [{"full_name": "MeasureTheory.lintegral_nnnorm_le_of_forall_fin_meas_integral_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [203, 9], "def_end_pos": [203, 59]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}]], "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016g a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc", "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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (fun x => \u2191\u2191f x) 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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 StronglyMeasurable fun a => g a\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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (fun a => g a) s\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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \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, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "refine' hg_eq_restrict.mono fun x hx => _", "annotated_tactic": ["refine' hg_eq_restrict.mono fun x hx => _", []], "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a", "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n\u22a2 (fun x => \u2191\u2016g x\u2016\u208a) x = (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) x"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n\u22a2 (fun x => \u2191\u2016g x\u2016\u208a) x = (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) 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\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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n\u22a2 \u2191\u2016Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc) x\u2016\u208a = \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a"}, {"tactic": "simp_rw [hx]", "annotated_tactic": ["simp_rw [hx]", []], "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\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n\u22a2 \u2191\u2016Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc) x\u2016\u208a = \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a", "state_after": "no goals"}, {"tactic": "exact integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim h\u03bcs", "annotated_tactic": ["exact <a>integrableOn_Lp_of_measure_ne_top</a> f 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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (fun x => \u2191\u2191f x) s", "state_after": "no goals"}, {"tactic": "exact hg_meas", "annotated_tactic": ["exact hg_meas", []], "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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 StronglyMeasurable fun a => g a", "state_after": "no goals"}, {"tactic": "rw [IntegrableOn, integrable_congr hg_eq_restrict]", "annotated_tactic": ["rw [<a>IntegrableOn</a>, <a>integrable_congr</a> hg_eq_restrict]", [{"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'_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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (fun a => g a) s", "state_after": "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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)"}, {"tactic": "exact integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs f", "annotated_tactic": ["exact <a>integrableOn_condexpL2_of_measure_ne_top</a> hm h\u03bcs f", [{"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": "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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)", "state_after": "no goals"}, {"tactic": "intro t ht h\u03bct", "annotated_tactic": ["intro t ht h\u03bct", []], "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\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \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, \u2191\u2191f x \u2202\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\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_condexpL2_eq_of_fin_meas_real f ht h\u03bct.ne]", "annotated_tactic": ["rw [\u2190 <a>integral_condexpL2_eq_of_fin_meas_real</a> f ht h\u03bct.ne]", [{"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'_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\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191f x \u2202\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\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d ?m.623979) f) x \u2202\u03bc\n\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\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 m \u2264 m0"}, {"tactic": "exact set_integral_congr_ae (hm t ht) (hg_eq.mono fun x hx _ => hx)", "annotated_tactic": ["exact <a>set_integral_congr_ae</a> (hm t ht) (hg_eq.mono 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]}]], "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\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d ?m.623979) f) x \u2202\u03bc\n\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\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 m \u2264 m0", "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_constL_le", "start": [887, 1], "end": [890, 53], "traced_tactics": [{"tactic": "positivity", "annotated_tactic": ["positivity", []], "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 : IsFiniteMeasure \u03bc\nc : E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p)", "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.reinsertAux_WF", "start": [70, 1], "end": [77, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_disjiUnion", "start": [107, 1], "end": [109, 80], "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_take_ne_nil", "start": [808, 1], "end": [809, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Adapted.lean", "full_name": "MeasureTheory.ProgMeasurable.finset_prod'", "start": [156, 11], "end": [159, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Decidable.not_forall_not", "start": [668, 11], "end": [670, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.mul_pow", "start": [835, 11], "end": [838, 89], "traced_tactics": [{"tactic": "induction n with\n| zero => rw [Nat.pow_zero, Nat.pow_zero, Nat.pow_zero, Nat.mul_one]\n| succ _ ih => rw [Nat.pow_succ, Nat.pow_succ, Nat.pow_succ, Nat.mul_mul_mul_comm, ih]", "annotated_tactic": ["induction n with\n  | <a>zero</a> => rw [<a>Nat.pow_zero</a>, <a>Nat.pow_zero</a>, <a>Nat.pow_zero</a>, <a>Nat.mul_one</a>]\n  | <a>succ</a> _ ih => rw [<a>Nat.pow_succ</a>, <a>Nat.pow_succ</a>, <a>Nat.pow_succ</a>, <a>Nat.mul_mul_mul_comm</a>, ih]", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Nat.pow_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}, {"full_name": "Nat.pow_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}, {"full_name": "Nat.pow_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}, {"full_name": "Nat.mul_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [178, 27], "def_end_pos": [178, 34]}, {"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.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.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.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_mul_mul_comm", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [503, 19], "def_end_pos": [503, 35]}]], "state_before": "a b n : Nat\n\u22a2 (a * b) ^ n = a ^ n * b ^ n", "state_after": "no goals"}, {"tactic": "rw [Nat.pow_zero, Nat.pow_zero, Nat.pow_zero, Nat.mul_one]", "annotated_tactic": ["rw [<a>Nat.pow_zero</a>, <a>Nat.pow_zero</a>, <a>Nat.pow_zero</a>, <a>Nat.mul_one</a>]", [{"full_name": "Nat.pow_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}, {"full_name": "Nat.pow_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}, {"full_name": "Nat.pow_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}, {"full_name": "Nat.mul_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [178, 27], "def_end_pos": [178, 34]}]], "state_before": "case zero\na b : Nat\n\u22a2 (a * b) ^ zero = a ^ zero * b ^ zero", "state_after": "no goals"}, {"tactic": "rw [Nat.pow_succ, Nat.pow_succ, Nat.pow_succ, Nat.mul_mul_mul_comm, ih]", "annotated_tactic": ["rw [<a>Nat.pow_succ</a>, <a>Nat.pow_succ</a>, <a>Nat.pow_succ</a>, <a>Nat.mul_mul_mul_comm</a>, ih]", [{"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.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.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_mul_mul_comm", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [503, 19], "def_end_pos": [503, 35]}]], "state_before": "case succ\na b n\u271d : Nat\nih : (a * b) ^ n\u271d = a ^ n\u271d * b ^ n\u271d\n\u22a2 (a * b) ^ succ n\u271d = a ^ succ n\u271d * b ^ succ n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.piecewise_op", "start": [1520, 1], "end": [1522, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.denseEmbedding", "start": [762, 11], "end": [778, 33], "traced_tactics": [{"tactic": "borelize E", "annotated_tactic": ["borelize E", []], "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u22a2 DenseEmbedding Subtype.val", "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseEmbedding Subtype.val"}, {"tactic": "apply simpleFunc.uniformEmbedding.denseEmbedding", "annotated_tactic": ["apply simpleFunc.uniformEmbedding.denseEmbedding", []], "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseEmbedding Subtype.val", "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseRange Subtype.val"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseRange Subtype.val", "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 closure (Set.range Subtype.val)"}, {"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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 closure (Set.range Subtype.val)", "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)"}, {"tactic": "have hfi' : Mem\u2112p f p \u03bc := Lp.mem\u2112p f", "annotated_tactic": ["have hfi' : <a>Mem\u2112p</a> f p \u03bc := <a>Lp.mem\u2112p</a> f", [{"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]}]], "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd 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\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)"}, {"tactic": "haveI : SeparableSpace (range f \u222a {0} : Set E) :=\n  (Lp.stronglyMeasurable f).separableSpace_range_union_singleton", "annotated_tactic": ["haveI : <a>SeparableSpace</a> (<a>range</a> f \u222a {0} : <a>Set</a> E) :=\n    (<a>Lp.stronglyMeasurable</a> f).<a>separableSpace_range_union_singleton</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": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [207, 19], "def_end_pos": [207, 37]}, {"full_name": "MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [640, 9], "def_end_pos": [640, 45]}]], "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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd 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\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)"}, {"tactic": "refine'\n  \u27e8fun n =>\n    toLp\n      (SimpleFunc.approxOn f (Lp.stronglyMeasurable f).measurable (range f \u222a {0}) 0 _ n)\n      (SimpleFunc.mem\u2112p_approxOn_range (Lp.stronglyMeasurable f).measurable hfi' n),\n    fun n => mem_range_self _, _\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun n =>\n      <a>toLp</a>\n        (<a>SimpleFunc.approxOn</a> f (<a>Lp.stronglyMeasurable</a> f).<a>measurable</a> (<a>range</a> f \u222a {0}) 0 _ n)\n        (<a>SimpleFunc.mem\u2112p_approxOn_range</a> (<a>Lp.stronglyMeasurable</a> f).<a>measurable</a> hfi' n),\n      fun n => <a>mem_range_self</a> _, _\u27e9", [{"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.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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.measurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [341, 19], "def_end_pos": [341, 29]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 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]}, {"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.measurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [341, 19], "def_end_pos": [341, 29]}, {"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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd 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\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      \u2191(toLp (SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n)\n          (_ :\n            Mem\u2112p\n              (\u2191(SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n))\n              p)))\n    atTop (\ud835\udcdd f)"}, {"tactic": "convert SimpleFunc.tendsto_approxOn_range_Lp hp_ne_top (Lp.stronglyMeasurable f).measurable hfi'", "annotated_tactic": ["convert <a>SimpleFunc.tendsto_approxOn_range_Lp</a> hp_ne_top (<a>Lp.stronglyMeasurable</a> f).<a>measurable</a> hfi'", [{"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [189, 9], "def_end_pos": [189, 34]}, {"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.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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      \u2191(toLp (SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n)\n          (_ :\n            Mem\u2112p\n              (\u2191(SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n))\n              p)))\n    atTop (\ud835\udcdd f)", "state_after": "case h.e'_5.h.e'_3\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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 f = Mem\u2112p.toLp (\u2191\u2191f) hfi'"}, {"tactic": "rw [toLp_coeFn f (Lp.mem\u2112p f)]", "annotated_tactic": ["rw [<a>toLp_coeFn</a> f (<a>Lp.mem\u2112p</a> f)]", [{"full_name": "MeasureTheory.Lp.toLp_coeFn", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [193, 9], "def_end_pos": [193, 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": "case h.e'_5.h.e'_3\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\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 f = Mem\u2112p.toLp (\u2191\u2191f) hfi'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Monotone.IicExtend", "start": [321, 11], "end": [322, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.lt_inf'_iff", "start": [1203, 1], "end": [1204, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/SetLike/Basic.lean", "full_name": "SetLike.coe_injective", "start": [134, 1], "end": [135, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ae_essInf_le", "start": [118, 1], "end": [121, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.scanl_get", "start": [390, 1], "end": [401, 59], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with 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\ni : Fin n\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)", "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\ni : Fin Nat.zero\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn : \u2115\nv : Vector \u03b1 (Nat.succ n)\ni : Fin (Nat.succ n)\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)"}, {"tactic": "induction' n with n hn generalizing b", "annotated_tactic": ["induction' n with n hn generalizing b", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn : \u2115\nv : Vector \u03b1 (Nat.succ n)\ni : Fin (Nat.succ n)\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)", "state_after": "case succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)"}, {"tactic": "exact i.elim0", "annotated_tactic": ["exact i.elim0", []], "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\ni : Fin Nat.zero\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)", "state_after": "no goals"}, {"tactic": "have i0 : i = 0 := Fin.eq_zero _", "annotated_tactic": ["have i0 : i = 0 := <a>Fin.eq_zero</a> _", [{"full_name": "Fin.eq_zero", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [113, 9], "def_end_pos": [113, 20]}]], "state_before": "case succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)", "state_after": "case succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)"}, {"tactic": "simp [scanl_singleton, i0, get_zero]", "annotated_tactic": ["simp [<a>scanl_singleton</a>, i0, <a>get_zero</a>]", [{"full_name": "Vector.scanl_singleton", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [364, 9], "def_end_pos": [364, 24]}, {"full_name": "Vector.get_zero", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [264, 9], "def_end_pos": [264, 17]}]], "state_before": "case succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)", "state_after": "case succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n\u22a2 get (b ::\u1d65 f b (head v) ::\u1d65 nil) 1 = f b (head v)"}, {"tactic": "simp [get_eq_get, List.get]", "annotated_tactic": ["simp [<a>get_eq_get</a>, <a>List.get</a>]", [{"full_name": "Vector.get_eq_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 19]}, {"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.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n\u22a2 get (b ::\u1d65 f b (head v) ::\u1d65 nil) 1 = f b (head v)", "state_after": "no goals"}, {"tactic": "rw [\u2190 cons_head_tail v, scanl_cons, get_cons_succ]", "annotated_tactic": ["rw [\u2190 <a>cons_head_tail</a> v, <a>scanl_cons</a>, <a>get_cons_succ</a>]", [{"full_name": "Vector.cons_head_tail", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"full_name": "Vector.scanl_cons", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 19]}, {"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.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)", "state_after": "case succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f (f b (head v)) (tail v)) i =\n    f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc i)) (get (head v ::\u1d65 tail v) i)"}, {"tactic": "refine' Fin.cases _ _ i", "annotated_tactic": ["refine' <a>Fin.cases</a> _ _ i", [{"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": "case succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f (f b (head v)) (tail v)) i =\n    f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc i)) (get (head v ::\u1d65 tail v) i)", "state_after": "case succ.succ.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f (f b (head v)) (tail v)) 0 =\n    f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc 0)) (get (head v ::\u1d65 tail v) 0)\n\ncase succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 \u2200 (i : Fin (n + 1)),\n    get (scanl f (f b (head v)) (tail v)) (Fin.succ i) =\n      f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i))) (get (head v ::\u1d65 tail v) (Fin.succ i))"}, {"tactic": "simp only [get_zero, scanl_head, Fin.castSucc_zero, head_cons]", "annotated_tactic": ["simp only [<a>get_zero</a>, <a>scanl_head</a>, <a>Fin.castSucc_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.scanl_head", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [373, 9], "def_end_pos": [373, 19]}, {"full_name": "Fin.castSucc_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [378, 17], "def_end_pos": [378, 30]}, {"full_name": "Vector.head_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [64, 9], "def_end_pos": [64, 18]}]], "state_before": "case succ.succ.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f (f b (head v)) (tail v)) 0 =\n    f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc 0)) (get (head v ::\u1d65 tail v) 0)", "state_after": "no goals"}, {"tactic": "intro i'", "annotated_tactic": ["intro i'", []], "state_before": "case succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 \u2200 (i : Fin (n + 1)),\n    get (scanl f (f b (head v)) (tail v)) (Fin.succ i) =\n      f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i))) (get (head v ::\u1d65 tail v) (Fin.succ i))", "state_after": "case succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\ni' : Fin (n + 1)\n\u22a2 get (scanl f (f b (head v)) (tail v)) (Fin.succ i') =\n    f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i'))) (get (head v ::\u1d65 tail v) (Fin.succ i'))"}, {"tactic": "simp only [hn, Fin.castSucc_fin_succ, get_cons_succ]", "annotated_tactic": ["simp only [hn, <a>Fin.castSucc_fin_succ</a>, <a>get_cons_succ</a>]", [{"full_name": "Fin.castSucc_fin_succ", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [391, 9], "def_end_pos": [391, 26]}, {"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.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n  \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n    get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\ni' : Fin (n + 1)\n\u22a2 get (scanl f (f b (head v)) (tail v)) (Fin.succ i') =\n    f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i'))) (get (head v ::\u1d65 tail v) (Fin.succ i'))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Snoc.lean", "full_name": "Vector.reverse_snoc", "start": [48, 1], "end": [53, 6], "traced_tactics": [{"tactic": "cases xs", "annotated_tactic": ["cases xs", []], "state_before": "\u03b1 : Type u_1\nn : \u2115\nxs : Vector \u03b1 n\nx : \u03b1\n\u22a2 reverse (snoc xs x) = x ::\u1d65 reverse xs", "state_after": "case mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 reverse (snoc { val := val\u271d, property := property\u271d } x) = x ::\u1d65 reverse { val := val\u271d, property := property\u271d }"}, {"tactic": "simp only [reverse, snoc, cons, toList_mk]", "annotated_tactic": ["simp only [<a>reverse</a>, <a>snoc</a>, <a>cons</a>, <a>toList_mk</a>]", [{"full_name": "Vector.reverse", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [247, 5], "def_end_pos": [247, 12]}, {"full_name": "Vector.snoc", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [24, 5], "def_end_pos": [24, 9]}, {"full_name": "Vector.cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [44, 5], "def_end_pos": [44, 9]}, {"full_name": "Vector.toList_mk", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [245, 9], "def_end_pos": [245, 18]}]], "state_before": "case mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 reverse (snoc { val := val\u271d, property := property\u271d } x) = x ::\u1d65 reverse { val := val\u271d, property := property\u271d }", "state_after": "case mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 {\n      val :=\n        List.reverse\n          (toList\n            (append { val := val\u271d, property := property\u271d }\n              { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })),\n      property :=\n        (_ :\n          (fun l => List.length l = n + 1)\n            (List.reverse\n              (toList\n                (append { val := val\u271d, property := property\u271d }\n                  { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })))) } =\n    { val := x :: List.reverse val\u271d, property := (_ : Nat.succ (List.length (List.reverse val\u271d)) = Nat.succ n) }"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 {\n      val :=\n        List.reverse\n          (toList\n            (append { val := val\u271d, property := property\u271d }\n              { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })),\n      property :=\n        (_ :\n          (fun l => List.length l = n + 1)\n            (List.reverse\n              (toList\n                (append { val := val\u271d, property := property\u271d }\n                  { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })))) } =\n    { val := x :: List.reverse val\u271d, property := (_ : Nat.succ (List.length (List.reverse val\u271d)) = Nat.succ n) }", "state_after": "case mk.e_val\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 List.reverse\n      (toList\n        (append { val := val\u271d, property := property\u271d }\n          { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })) =\n    x :: List.reverse val\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.e_val\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 [] ++ [x] ++ List.reverse val\u271d = x :: List.reverse val\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.not_nontrivial_singleton", "start": [2564, 1], "end": [2567, 37], "traced_tactics": [{"tactic": "rw [nontrivial_iff_exists_ne (mem_singleton x)] at H", "annotated_tactic": ["rw [<a>nontrivial_iff_exists_ne</a> (<a>mem_singleton</a> x)] at H", [{"full_name": "Set.nontrivial_iff_exists_ne", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2520, 9], "def_end_pos": [2520, 33]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 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\nx : \u03b1\nH : Set.Nontrivial {x}\n\u22a2 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 : \u03b1\nH : \u2203 y, y \u2208 {x} \u2227 y \u2260 x\n\u22a2 False"}, {"tactic": "let \u27e8y, hy, hya\u27e9 := H", "annotated_tactic": ["let \u27e8y, hy, hya\u27e9 := H", []], "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 : \u03b1\nH : \u2203 y, y \u2208 {x} \u2227 y \u2260 x\n\u22a2 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 : \u03b1\nH : \u2203 y, y \u2208 {x} \u2227 y \u2260 x\ny : \u03b1\nhy : y \u2208 {x}\nhya : y \u2260 x\n\u22a2 False"}, {"tactic": "exact hya (mem_singleton_iff.1 hy)", "annotated_tactic": ["exact hya (<a>mem_singleton_iff</a>.1 hy)", [{"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 : \u03b1\nH : \u2203 y, y \u2208 {x} \u2227 y \u2260 x\ny : \u03b1\nhy : y \u2208 {x}\nhya : y \u2260 x\n\u22a2 False", "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.sup", "start": [67, 1], "end": [70, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_zmod_eq_zero_iff_of_lt", "start": [584, 1], "end": [587, 55], "traced_tactics": [{"tactic": "rw [\u2190 ZMod.cast_zero (n := m)]", "annotated_tactic": ["rw [\u2190 <a>ZMod.cast_zero</a> (n := m)]", [{"full_name": "ZMod.cast_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 18]}]], "state_before": "m n : \u2115\ninst\u271d : NeZero m\nh : m < n\na : ZMod m\n\u22a2 \u2191a = 0 \u2194 a = 0", "state_after": "m n : \u2115\ninst\u271d : NeZero m\nh : m < n\na : ZMod m\n\u22a2 \u2191a = \u21910 \u2194 a = 0"}, {"tactic": "exact Injective.eq_iff' (cast_injective_of_lt h) rfl", "annotated_tactic": ["exact <a>Injective.eq_iff'</a> (<a>cast_injective_of_lt</a> h) <a>rfl</a>", [{"full_name": "Function.Injective.eq_iff'", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 26]}, {"full_name": "ZMod.cast_injective_of_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [573, 9], "def_end_pos": [573, 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": "m n : \u2115\ninst\u271d : NeZero m\nh : m < n\na : ZMod m\n\u22a2 \u2191a = \u21910 \u2194 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ManyOneDegree.liftOn\u2082_eq", "start": [409, 11], "end": [412, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le_min", "start": [700, 11], "end": [702, 62], "traced_tactics": [{"tactic": "rw [Int.min_def]", "annotated_tactic": ["rw [<a>Int.min_def</a>]", [{"full_name": "Int.min_def", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [686, 19], "def_end_pos": [686, 26]}]], "state_before": "a b c : Int\nx\u271d : a \u2264 b \u2227 a \u2264 c\nh\u2081 : a \u2264 b\nh\u2082 : a \u2264 c\n\u22a2 a \u2264 min b c", "state_after": "a b c : Int\nx\u271d : a \u2264 b \u2227 a \u2264 c\nh\u2081 : a \u2264 b\nh\u2082 : a \u2264 c\n\u22a2 a \u2264 if b \u2264 c then b else c"}, {"tactic": "split <;> assumption", "annotated_tactic": ["split <;> assumption", []], "state_before": "a b c : Int\nx\u271d : a \u2264 b \u2227 a \u2264 c\nh\u2081 : a \u2264 b\nh\u2082 : a \u2264 c\n\u22a2 a \u2264 if b \u2264 c then b else c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Floor.lean", "full_name": "Int.measurable_floor", "start": [25, 1], "end": [27, 70], "traced_tactics": [{"tactic": "simpa only [Int.preimage_floor_singleton] using measurableSet_Ico", "annotated_tactic": ["simpa only [<a>Int.preimage_floor_singleton</a>] using <a>measurableSet_Ico</a>", [{"full_name": "Int.preimage_floor_singleton", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [839, 9], "def_end_pos": [839, 33]}, {"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 : OpensMeasurableSpace R\nx : R\n\u22a2 MeasurableSet (floor \u207b\u00b9' {\u230ax\u230b})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.cast_bitm1", "start": [1142, 1], "end": [1148, 62], "traced_tactics": [{"tactic": "conv =>\n  lhs\n  rw [\u2190 zneg_zneg n]", "annotated_tactic": ["conv =>\n    lhs\n    rw [\u2190 <a>zneg_zneg</a> n]", [{"full_name": "ZNum.zneg_zneg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(ZNum.bitm1 n) = _root_.bit0 \u2191n - 1", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(ZNum.bitm1 (- -n)) = _root_.bit0 \u2191n - 1"}, {"tactic": "rw [\u2190 zneg_bit1, cast_zneg, cast_bit1]", "annotated_tactic": ["rw [\u2190 <a>zneg_bit1</a>, <a>cast_zneg</a>, <a>cast_bit1</a>]", [{"full_name": "ZNum.zneg_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1069, 9], "def_end_pos": [1069, 18]}, {"full_name": "ZNum.cast_zneg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 18]}, {"full_name": "ZNum.cast_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(ZNum.bitm1 (- -n)) = _root_.bit0 \u2191n - 1", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1"}, {"tactic": "have : ((-1 + n + n : \u2124) : \u03b1) = (n + n + -1 : \u2124) := by simp [add_comm, add_left_comm]", "annotated_tactic": ["have : ((-1 + n + n : \u2124) : \u03b1) = (n + n + -1 : \u2124) := by simp [<a>add_comm</a>, <a>add_left_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\nthis : \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1)\n\u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1"}, {"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this", "annotated_tactic": ["simpa [<a>_root_.bit1</a>, <a>_root_.bit0</a>, <a>sub_eq_add_neg</a>] using this", [{"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": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\nthis : \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1)\n\u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1", "state_after": "no goals"}, {"tactic": "simp [add_comm, add_left_comm]", "annotated_tactic": ["simp [<a>add_comm</a>, <a>add_left_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.liftRel_iff_coeFn", "start": [497, 1], "end": [498, 90], "traced_tactics": [{"tactic": "rw [\u2190 liftRel_mk_mk, mk_coeFn, mk_coeFn]", "annotated_tactic": ["rw [\u2190 <a>liftRel_mk_mk</a>, <a>mk_coeFn</a>, <a>mk_coeFn</a>]", [{"full_name": "MeasureTheory.AEEqFun.liftRel_mk_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [492, 9], "def_end_pos": [492, 22]}, {"full_name": "MeasureTheory.AEEqFun.mk_coeFn", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [165, 9], "def_end_pos": [165, 17]}, {"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\nr : \u03b2 \u2192 \u03b3 \u2192 Prop\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\ng : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 LiftRel r f g \u2194 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, r (\u2191f a) (\u2191g a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.mem_condCdfSet_ae", "start": [557, 1], "end": [560, 24], "traced_tactics": [{"tactic": "simp_rw [ae_iff, condCdfSet, not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable]", "annotated_tactic": ["simp_rw [<a>ae_iff</a>, <a>condCdfSet</a>, <a>not_mem_compl_iff</a>, <a>setOf_mem_eq</a>, <a>measure_toMeasurable</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "ProbabilityTheory.condCdfSet", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [541, 5], "def_end_pos": [541, 15]}, {"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\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, a \u2208 condCdfSet \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\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b} = 0"}, {"tactic": "exact hasCondCdf_ae \u03c1", "annotated_tactic": ["exact <a>hasCondCdf_ae</a> \u03c1", [{"full_name": "ProbabilityTheory.hasCondCdf_ae", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [533, 9], "def_end_pos": [533, 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\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b} = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Monotone.measurable", "start": [1242, 11], "end": [1245, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.aefinStronglyMeasurable_of_aemeasurable", "start": [2030, 1], "end": [2032, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.fin_embedding", "start": [1089, 1], "end": [1092, 97], "traced_tactics": [{"tactic": "simp only [Finset.coe_sort_coe, Equiv.asEmbedding_range, Finite.coe_toFinset, setOf_mem_eq]", "annotated_tactic": ["simp only [<a>Finset.coe_sort_coe</a>, <a>Equiv.asEmbedding_range</a>, <a>Finite.coe_toFinset</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Finset.coe_sort_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [285, 9], "def_end_pos": [285, 21]}, {"full_name": "Equiv.asEmbedding_range", "def_path": "Mathlib/Logic/Embedding/Set.lean", "def_pos": [24, 9], "def_end_pos": [24, 32]}, {"full_name": "Set.Finite.coe_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [178, 19], "def_end_pos": [178, 31]}, {"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\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 range \u2191(Equiv.asEmbedding (Fintype.equivFin \u2191\u2191(Finite.toFinset h)).symm) = s", "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_Ico", "start": [696, 1], "end": [701, 17], "traced_tactics": [{"tactic": "simpa only [exists_prop, mem_univ, true_and_iff] using\n  (@dense_univ \u03b1 _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n    mem_univ _", "annotated_tactic": ["simpa only [<a>exists_prop</a>, <a>mem_univ</a>, <a>true_and_iff</a>] using\n    (@<a>dense_univ</a> \u03b1 _).<a>borel_eq_generateFrom_Ico_mem_aux</a> (fun _ _ => <a>mem_univ</a> _) fun _ _ _ _ =>\n      <a>mem_univ</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_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "dense_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [647, 9], "def_end_pos": [647, 19]}, {"full_name": "Dense.borel_eq_generateFrom_Ico_mem_aux", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [651, 9], "def_end_pos": [651, 48]}, {"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": "\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\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 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\n\u22a2 borel \u03b1 = MeasurableSpace.generateFrom {S | \u2203 l u, l < u \u2227 Ico l u = S}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepSets.iInter", "start": [305, 1], "end": [308, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "MeasureTheory.Integrable.integral_norm_condexpKernel", "start": [134, 1], "end": [139, 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\u2075 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NormedAddCommGroup F\nf : \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 Integrable fun \u03c9 => \u222b (y : \u03a9), \u2016f y\u2016 \u2202\u2191(condexpKernel \u03bc m) \u03c9", "state_after": "\u03a9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NormedAddCommGroup F\nf : \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 Integrable fun \u03c9 => \u222b (y : \u03a9), \u2016f y\u2016 \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9"}, {"tactic": "exact Integrable.integral_norm_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.integral_norm_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.integral_norm_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [156, 9], "def_end_pos": [156, 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\u2075 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NormedAddCommGroup F\nf : \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 Integrable fun \u03c9 => \u222b (y : \u03a9), \u2016f y\u2016 \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.cardinal_generateMeasurable_le", "start": [156, 1], "end": [164, 48], "traced_tactics": [{"tactic": "rw [generateMeasurable_eq_rec]", "annotated_tactic": ["rw [<a>generateMeasurable_eq_rec</a>]", [{"full_name": "MeasurableSpace.generateMeasurable_eq_rec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [117, 9], "def_end_pos": [117, 34]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 #\u2191{t | GenerateMeasurable s t} \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 #\u2191(\u22c3 i, generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "apply (mk_iUnion_le _).trans", "annotated_tactic": ["apply (<a>mk_iUnion_le</a> _).<a>trans</a>", [{"full_name": "Cardinal.mk_iUnion_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2171, 9], "def_end_pos": [2171, 21]}, {"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\ns : Set (Set \u03b1)\n\u22a2 #\u2191(\u22c3 i, generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 #(Quotient.out (ord (aleph 1))).\u03b1 * \u2a06 i, #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "rw [(aleph 1).mk_ord_out]", "annotated_tactic": ["rw [(<a>aleph</a> 1).<a>mk_ord_out</a>]", [{"full_name": "Cardinal.aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [245, 5], "def_end_pos": [245, 10]}, {"full_name": "Cardinal.mk_ord_out", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1440, 9], "def_end_pos": [1440, 19]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 #(Quotient.out (ord (aleph 1))).\u03b1 * \u2a06 i, #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 aleph 1 * \u2a06 i, #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "refine le_trans (mul_le_mul' aleph_one_le_continuum\n    (ciSup_le' fun i => cardinal_generateMeasurableRec_le s i)) ?_", "annotated_tactic": ["refine <a>le_trans</a> (<a>mul_le_mul'</a> <a>aleph_one_le_continuum</a>\n      (<a>ciSup_le'</a> fun i => <a>cardinal_generateMeasurableRec_le</a> s i)) ?_", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "Cardinal.aleph_one_le_continuum", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [102, 9], "def_end_pos": [102, 31]}, {"full_name": "ciSup_le'", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1252, 9], "def_end_pos": [1252, 18]}, {"full_name": "MeasurableSpace.cardinal_generateMeasurableRec_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [91, 9], "def_end_pos": [91, 42]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 aleph 1 * \u2a06 i, #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 \ud835\udd20 * max (#\u2191s) 2 ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "refine (mul_le_max_of_aleph0_le_left aleph0_le_continuum).trans (max_le ?_ le_rfl)", "annotated_tactic": ["refine (<a>mul_le_max_of_aleph0_le_left</a> <a>aleph0_le_continuum</a>).<a>trans</a> (<a>max_le</a> ?_ <a>le_rfl</a>)", [{"full_name": "Cardinal.mul_le_max_of_aleph0_le_left", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [604, 9], "def_end_pos": [604, 37]}, {"full_name": "Cardinal.aleph0_le_continuum", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [79, 9], "def_end_pos": [79, 28]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 \ud835\udd20 * max (#\u2191s) 2 ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 \ud835\udd20 \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "exact power_le_power_right (le_max_right _ _)", "annotated_tactic": ["exact <a>power_le_power_right</a> (<a>le_max_right</a> _ _)", [{"full_name": "Cardinal.power_le_power_right", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 29]}, {"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\ns : Set (Set \u03b1)\n\u22a2 \ud835\udd20 \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.move_right_nth", "start": [642, 1], "end": [645, 42], "traced_tactics": [{"tactic": "conv => rhs; rw [\u2190 T.move_right_left]", "annotated_tactic": ["conv => rhs; rw [\u2190 T.move_right_left]", []], "state_before": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n\u22a2 nth (move Dir.right T) i = nth T (i + 1)", "state_after": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n\u22a2 nth (move Dir.right T) i = nth (move Dir.left (move Dir.right T)) (i + 1)"}, {"tactic": "rw [Tape.move_left_nth, add_sub_cancel]", "annotated_tactic": ["rw [<a>Tape.move_left_nth</a>, <a>add_sub_cancel</a>]", [{"full_name": "Turing.Tape.move_left_nth", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [630, 9], "def_end_pos": [630, 27]}, {"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}]], "state_before": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n\u22a2 nth (move Dir.right T) i = nth (move Dir.left (move Dir.right T)) (i + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.notConvergentSeqLTIndex_spec", "start": [122, 1], "end": [127, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.isPure_iff", "start": [232, 1], "end": [233, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_injective", "start": [1549, 1], "end": [1554, 16], "traced_tactics": [{"tactic": "refine' \u27e8fun h y => _, Surjective.preimage_injective\u27e9", "annotated_tactic": ["refine' \u27e8fun h y => _, <a>Surjective.preimage_injective</a>\u27e9", [{"full_name": "Function.Surjective.preimage_injective", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1296, 9], "def_end_pos": [1296, 38]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\n\u22a2 Injective (preimage f) \u2194 Surjective f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\n\u22a2 \u2203 a, f a = y"}, {"tactic": "obtain \u27e8x, hx\u27e9 : (f \u207b\u00b9' {y}).Nonempty := by\n  rw [h.nonempty_apply_iff preimage_empty]\n  apply singleton_nonempty", "annotated_tactic": ["obtain \u27e8x, hx\u27e9 : (f \u207b\u00b9' {y}).<a>Nonempty</a> := by\n    rw [h.nonempty_apply_iff <a>preimage_empty</a>]\n    apply <a>singleton_nonempty</a>", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.preimage_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [59, 9], "def_end_pos": [59, 23]}, {"full_name": "Set.singleton_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1315, 9], "def_end_pos": [1315, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\nx : \u03b1\nhx : x \u2208 f \u207b\u00b9' {y}\n\u22a2 \u2203 a, f a = y"}, {"tactic": "exact \u27e8x, hx\u27e9", "annotated_tactic": ["exact \u27e8x, hx\u27e9", []], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\nx : \u03b1\nhx : x \u2208 f \u207b\u00b9' {y}\n\u22a2 \u2203 a, f a = y", "state_after": "no goals"}, {"tactic": "rw [h.nonempty_apply_iff preimage_empty]", "annotated_tactic": ["rw [h.nonempty_apply_iff <a>preimage_empty</a>]", [{"full_name": "Set.preimage_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [59, 9], "def_end_pos": [59, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\n\u22a2 Set.Nonempty (f \u207b\u00b9' {y})", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\n\u22a2 Set.Nonempty {y}"}, {"tactic": "apply singleton_nonempty", "annotated_tactic": ["apply <a>singleton_nonempty</a>", [{"full_name": "Set.singleton_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1315, 9], "def_end_pos": [1315, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Injective (preimage f)\ny : \u03b2\n\u22a2 Set.Nonempty {y}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.tendsto_lintegral_of_dominated_convergence", "start": [1058, 1], "end": [1072, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.exists_lintegral_le", "start": [743, 1], "end": [745, 61], "traced_tactics": [{"tactic": "simpa only [laverage_eq_lintegral] using\n  exists_laverage_le (IsProbabilityMeasure.ne_zero \u03bc) hint", "annotated_tactic": ["simpa only [<a>laverage_eq_lintegral</a>] using\n    <a>exists_laverage_le</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hint", [{"full_name": "MeasureTheory.laverage_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [95, 9], "def_end_pos": [95, 30]}, {"full_name": "MeasureTheory.exists_laverage_le", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [675, 9], "def_end_pos": [675, 27]}, {"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\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 x, \u222b\u207b (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/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEquiv.measurableSet_image", "start": [1442, 1], "end": [1443, 52], "traced_tactics": [{"tactic": "rw [image_eq_preimage, measurableSet_preimage]", "annotated_tactic": ["rw [<a>image_eq_preimage</a>, <a>measurableSet_preimage</a>]", [{"full_name": "MeasurableEquiv.image_eq_preimage", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 26]}, {"full_name": "MeasurableEquiv.measurableSet_preimage", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1435, 9], "def_end_pos": [1435, 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 \u03b1\n\u22a2 MeasurableSet (\u2191e '' s) \u2194 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.degreeOf_sub_lt", "start": [185, 1], "end": [197, 26], "traced_tactics": [{"tactic": "rw [degreeOf_lt_iff h]", "annotated_tactic": ["rw [<a>degreeOf_lt_iff</a> h]", [{"full_name": "MvPolynomial.degreeOf_lt_iff", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [503, 9], "def_end_pos": [503, 24]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\n\u22a2 degreeOf x (f - g) < k", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\n\u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support (f - g) \u2192 \u2191m x < k"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\n\u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support (f - g) \u2192 \u2191m x < k", "state_after": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\n\u22a2 \u2191m x < k"}, {"tactic": "by_contra' hc", "annotated_tactic": ["by_contra' hc", []], "state_before": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\n\u22a2 \u2191m x < k", "state_after": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\nhc : k \u2264 \u2191m x\n\u22a2 False"}, {"tactic": "have h := support_sub \u03c3 f g hm", "annotated_tactic": ["have h := <a>support_sub</a> \u03c3 f g hm", [{"full_name": "MvPolynomial.support_sub", "def_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "def_pos": [88, 9], "def_end_pos": [88, 20]}]], "state_before": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\nhc : k \u2264 \u2191m x\n\u22a2 False", "state_after": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\n\u22a2 False"}, {"tactic": "simp only [mem_support_iff, Ne.def, coeff_sub, sub_eq_zero] at hm", "annotated_tactic": ["simp only [<a>mem_support_iff</a>, <a>Ne.def</a>, <a>coeff_sub</a>, <a>sub_eq_zero</a>] at hm", [{"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "MvPolynomial.coeff_sub", "def_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\n\u22a2 False", "state_after": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\n\u22a2 False"}, {"tactic": "cases' Finset.mem_union.1 h with cf cg", "annotated_tactic": ["cases' <a>Finset.mem_union</a>.1 h with cf cg", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\n\u22a2 False", "state_after": "case inl\nR : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\ncf : m \u2208 support f\n\u22a2 False\n\ncase inr\nR : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\ncg : m \u2208 support g\n\u22a2 False"}, {"tactic": "exact hm (hf m cf hc)", "annotated_tactic": ["exact hm (hf m cf hc)", []], "state_before": "case inl\nR : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\ncf : m \u2208 support f\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact hm (hg m cg hc)", "annotated_tactic": ["exact hm (hg m cg hc)", []], "state_before": "case inr\nR : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\ncg : m \u2208 support g\n\u22a2 False", "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.Path.insert_toList", "start": [531, 1], "end": [533, 42], "traced_tactics": [{"tactic": "simp [insert]", "annotated_tactic": ["simp [<a>insert</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]}]], "state_before": "\u03b1 : Type u_1\nt : RBNode \u03b1\nv : \u03b1\np : Path \u03b1\n\u22a2 toList (insert p t v) = withList p (toList (setRoot v t))", "state_after": "\u03b1 : Type u_1\nt : RBNode \u03b1\nv : \u03b1\np : Path \u03b1\n\u22a2 toList\n      (match t with\n      | nil => insertNew p v\n      | node c a v_1 b => fill p (node c a v b)) =\n    listL p ++ (toList (setRoot v t) ++ listR p)"}, {"tactic": "split <;> simp [setRoot]", "annotated_tactic": ["split <;> simp [<a>setRoot</a>]", [{"full_name": "Std.RBNode.setRoot", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [32, 5], "def_end_pos": [32, 12]}]], "state_before": "\u03b1 : Type u_1\nt : RBNode \u03b1\nv : \u03b1\np : Path \u03b1\n\u22a2 toList\n      (match t with\n      | nil => insertNew p v\n      | node c a v_1 b => fill p (node c a v b)) =\n    listL p ++ (toList (setRoot v t) ++ listR p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Infinite.exists_subset_ncard_eq", "start": [963, 1], "end": [969, 7], "traced_tactics": [{"tactic": "have := hs.to_subtype", "annotated_tactic": ["have := hs.to_subtype", []], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nk : \u2115\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = k", "state_after": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nk : \u2115\nthis : Infinite \u2191s\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = k"}, {"tactic": "obtain \u27e8t', -, rfl\u27e9 := @Infinite.exists_subset_card_eq s univ infinite_univ k", "annotated_tactic": ["obtain \u27e8t', -, rfl\u27e9 := @<a>Infinite.exists_subset_card_eq</a> s <a>univ</a> <a>infinite_univ</a> k", [{"full_name": "Set.Infinite.exists_subset_card_eq", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1300, 9], "def_end_pos": [1300, 39]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.infinite_univ", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 22]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nk : \u2115\nthis : Infinite \u2191s\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = k", "state_after": "case intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = Finset.card t'"}, {"tactic": "refine' \u27e8Subtype.val '' (t' : Set s), by simp, Finite.image _ (by simp), _\u27e9", "annotated_tactic": ["refine' \u27e8<a>Subtype.val</a> '' (t' : <a>Set</a> s), by simp, <a>Finite.image</a> _ (by simp), _\u27e9", [{"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": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Finite.image", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [862, 9], "def_end_pos": [862, 21]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = Finset.card t'", "state_after": "case intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 Set.ncard (Subtype.val '' \u2191t') = Finset.card t'"}, {"tactic": "rw [ncard_image_of_injective _ Subtype.coe_injective]", "annotated_tactic": ["rw [<a>ncard_image_of_injective</a> _ <a>Subtype.coe_injective</a>]", [{"full_name": "Set.ncard_image_of_injective", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [666, 9], "def_end_pos": [666, 33]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 Set.ncard (Subtype.val '' \u2191t') = Finset.card t'", "state_after": "case intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 Set.ncard \u2191t' = Finset.card t'"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 Set.ncard \u2191t' = Finset.card t'", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 Subtype.val '' \u2191t' \u2286 s", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : Set.Infinite s\nthis : Infinite \u2191s\nt' : Finset \u2191s\n\u22a2 Set.Finite \u2191t'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "full_name": "blimsup_cthickening_mul_ae_eq", "start": [194, 1], "end": [230, 75], "traced_tactics": [{"tactic": "let r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / ((i : \u211d) + 1)", "annotated_tactic": ["let r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / ((i : \u211d) + 1)", []], "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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "have h\u2080 : \u2200 i, p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) := by\n  rintro i \u27e8-, hi\u27e9; congr! 1; change r i = ite (0 < r i) (r i) _; simp [hi]", "annotated_tactic": ["have h\u2080 : \u2200 i, p i \u2227 0 < r i \u2192 <a>cthickening</a> (r i) (s i) = <a>cthickening</a> (r' i) (s i) := by\n    rintro i \u27e8-, hi\u27e9; congr! 1; change r i = <a>ite</a> (0 < r i) (r i) _; simp [hi]", [{"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}]], "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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "have h\u2081 : \u2200 i, p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) := by\n  rintro i \u27e8-, hi\u27e9; simp only [hi, mul_ite, if_true]", "annotated_tactic": ["have h\u2081 : \u2200 i, p i \u2227 0 < r i \u2192 <a>cthickening</a> (M * r i) (s i) = <a>cthickening</a> (M * r' i) (s i) := by\n    rintro i \u27e8-, hi\u27e9; simp only [hi, <a>mul_ite</a>, <a>if_true</a>]", [{"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "mul_ite", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [195, 9], "def_end_pos": [195, 16]}, {"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\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "have h\u2082 : \u2200 i, p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) := by\n  rintro i \u27e8-, hi\u27e9\n  have hi' : M * r i \u2264 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi\n  rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']", "annotated_tactic": ["have h\u2082 : \u2200 i, p i \u2227 r i \u2264 0 \u2192 <a>cthickening</a> (M * r i) (s i) = <a>cthickening</a> (r i) (s i) := by\n    rintro i \u27e8-, hi\u27e9\n    have hi' : M * r i \u2264 0 := <a>mul_nonpos_of_nonneg_of_nonpos</a> hM.le hi\n    rw [<a>cthickening_of_nonpos</a> hi, <a>cthickening_of_nonpos</a> hi']", [{"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1027, 5], "def_end_pos": [1027, 16]}, {"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]}, {"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 30]}, {"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 30]}]], "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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "have hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 := by\n  ext i; simp [\u2190 and_or_left, lt_or_le 0 (r i)]", "annotated_tactic": ["have hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 := by\n    ext i; simp [\u2190 <a>and_or_left</a>, <a>lt_or_le</a> 0 (r i)]", [{"full_name": "and_or_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [321, 9], "def_end_pos": [321, 20]}, {"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "rw [hp, blimsup_or_eq_sup, blimsup_or_eq_sup]", "annotated_tactic": ["rw [hp, <a>blimsup_or_eq_sup</a>, <a>blimsup_or_eq_sup</a>]", [{"full_name": "Filter.blimsup_or_eq_sup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1078, 9], "def_end_pos": [1078, 26]}, {"full_name": "Filter.blimsup_or_eq_sup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1078, 9], "def_end_pos": [1078, 26]}]], "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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u2294\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0) =\u1d50[\u03bc]\n    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u2294\n      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0"}, {"tactic": "simp only [sup_eq_union]", "annotated_tactic": ["simp only [<a>sup_eq_union</a>]", [{"full_name": "Set.sup_eq_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [111, 9], "def_end_pos": [111, 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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u2294\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0) =\u1d50[\u03bc]\n    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u2294\n      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u222a\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0) =\u1d50[\u03bc]\n    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u222a\n      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0"}, {"tactic": "rw [blimsup_congr (eventually_of_forall h\u2080), blimsup_congr (eventually_of_forall h\u2081),\n  blimsup_congr (eventually_of_forall h\u2082)]", "annotated_tactic": ["rw [<a>blimsup_congr</a> (<a>eventually_of_forall</a> h\u2080), <a>blimsup_congr</a> (<a>eventually_of_forall</a> h\u2081),\n    <a>blimsup_congr</a> (<a>eventually_of_forall</a> h\u2082)]", [{"full_name": "Filter.blimsup_congr", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [643, 9], "def_end_pos": [643, 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": "Filter.blimsup_congr", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [643, 9], "def_end_pos": [643, 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": "Filter.blimsup_congr", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [643, 9], "def_end_pos": [643, 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u222a\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0) =\u1d50[\u03bc]\n    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u222a\n      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x \u2227 0 < r x) \u222a\n      blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x \u2227 r x \u2264 0) =\u1d50[\u03bc]\n    (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x \u2227 0 < r x) \u222a\n      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0"}, {"tactic": "exact ae_eq_set_union (this (fun i => p i \u2227 0 < r i) hr') (ae_eq_refl _)", "annotated_tactic": ["exact <a>ae_eq_set_union</a> (this (fun i => p i \u2227 0 < r i) hr') (<a>ae_eq_refl</a> _)", [{"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\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0\n\u22a2 ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x \u2227 0 < r x) \u222a\n      blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x \u2227 r x \u2264 0) =\u1d50[\u03bc]\n    (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x \u2227 0 < r x) \u222a\n      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0", "state_after": "no goals"}, {"tactic": "clear p hr r", "annotated_tactic": ["clear p hr r", []], "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 \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\n\u22a2 \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "intro p r hr", "annotated_tactic": ["intro p r hr", []], "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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\n\u22a2 \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "have hr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[>] 0) := by\n  convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero]", "annotated_tactic": ["have hr' : <a>Tendsto</a> (fun i => M * r i) <a>atTop</a> (\ud835\udcdd[>] 0) := by\n      convert <a>TendstoNhdsWithinIoi.const_mul</a> hM hr <;> simp only [<a>mul_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": "Filter.TendstoNhdsWithinIoi.const_mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [167, 9], "def_end_pos": [167, 46]}, {"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\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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "refine' eventuallyLE_antisymm_iff.mpr \u27e8_, _\u27e9", "annotated_tactic": ["refine' eventuallyLE_antisymm_iff.mpr \u27e8_, _\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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (M * r i) (s i)) atTop p"}, {"tactic": "convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero]", "annotated_tactic": ["convert <a>TendstoNhdsWithinIoi.const_mul</a> hM hr <;> simp only [<a>mul_zero</a>]", [{"full_name": "Filter.TendstoNhdsWithinIoi.const_mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [167, 9], "def_end_pos": [167, 46]}, {"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\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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)", "state_after": "no goals"}, {"tactic": "exact blimsup_cthickening_ae_le_of_eventually_mul_le \u03bc p (inv_pos.mpr hM) hr'\n  (eventually_of_forall fun i => by rw [inv_mul_cancel_left\u2080 hM.ne' (r i)])", "annotated_tactic": ["exact <a>blimsup_cthickening_ae_le_of_eventually_mul_le</a> \u03bc p (inv_pos.mpr hM) hr'\n        (<a>eventually_of_forall</a> fun i => by rw [<a>inv_mul_cancel_left\u2080</a> hM.ne' (r i)])", [{"full_name": "blimsup_cthickening_ae_le_of_eventually_mul_le", "def_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "def_pos": [158, 9], "def_end_pos": [158, 55]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"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 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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p", "state_after": "no goals"}, {"tactic": "rw [inv_mul_cancel_left\u2080 hM.ne' (r i)]", "annotated_tactic": ["rw [<a>inv_mul_cancel_left\u2080</a> hM.ne' (r i)]", [{"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": "\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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\n\u22a2 M\u207b\u00b9 * (M * r i) \u2264 r i", "state_after": "no goals"}, {"tactic": "exact blimsup_cthickening_ae_le_of_eventually_mul_le \u03bc p hM hr\n  (eventually_of_forall fun i => le_refl _)", "annotated_tactic": ["exact <a>blimsup_cthickening_ae_le_of_eventually_mul_le</a> \u03bc p hM hr\n        (<a>eventually_of_forall</a> fun i => <a>le_refl</a> _)", [{"full_name": "blimsup_cthickening_ae_le_of_eventually_mul_le", "def_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "def_pos": [158, 9], "def_end_pos": [158, 55]}, {"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_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "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\ns : \u2115 \u2192 Set \u03b1\nM : \u211d\nhM : 0 < M\np : \u2115 \u2192 Prop\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (M * r i) (s i)) atTop p", "state_after": "no goals"}, {"tactic": "refine' tendsto_nhdsWithin_iff.mpr\n  \u27e8Tendsto.if' hr tendsto_one_div_add_atTop_nhds_0_nat, eventually_of_forall fun i => _\u27e9", "annotated_tactic": ["refine' tendsto_nhdsWithin_iff.mpr\n      \u27e8<a>Tendsto.if'</a> hr <a>tendsto_one_div_add_atTop_nhds_0_nat</a>, <a>eventually_of_forall</a> fun i => _\u27e9", [{"full_name": "Filter.Tendsto.if'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3214, 19], "def_end_pos": [3214, 30]}, {"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]}, {"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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\n\u22a2 Tendsto r' atTop (\ud835\udcdd[Ioi 0] 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\n\u22a2 r' i \u2208 Ioi 0"}, {"tactic": "by_cases hi : 0 < r i", "annotated_tactic": ["by_cases 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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\n\u22a2 r' i \u2208 Ioi 0", "state_after": "case pos\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\nhi : 0 < r i\n\u22a2 r' i \u2208 Ioi 0\n\ncase neg\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\nhi : \u00ac0 < r i\n\u22a2 r' i \u2208 Ioi 0"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "case pos\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\nhi : 0 < r i\n\u22a2 r' i \u2208 Ioi 0", "state_after": "no goals"}, {"tactic": "simp only [hi, one_div, mem_Ioi, if_false, inv_pos]", "annotated_tactic": ["simp only [hi, <a>one_div</a>, <a>mem_Ioi</a>, <a>if_false</a>, <a>inv_pos</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 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": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "case neg\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\nhi : \u00ac0 < r i\n\u22a2 r' i \u2208 Ioi 0", "state_after": "case neg\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\nhi : \u00ac0 < r i\n\u22a2 0 < \u2191i + 1"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case neg\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\ni : \u2115\nhi : \u00ac0 < r i\n\u22a2 0 < \u2191i + 1", "state_after": "no goals"}, {"tactic": "rintro i \u27e8-, hi\u27e9", "annotated_tactic": ["rintro i \u27e8-, hi\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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\n\u22a2 \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\nhi : 0 < r i\n\u22a2 cthickening (r i) (s i) = cthickening (r' i) (s i)"}, {"tactic": "congr! 1", "annotated_tactic": ["congr! 1", []], "state_before": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\nhi : 0 < r i\n\u22a2 cthickening (r i) (s i) = cthickening (r' i) (s i)", "state_after": "case intro.h.e'_3\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\nhi : 0 < r i\n\u22a2 r i = r' i"}, {"tactic": "change r i = ite (0 < r i) (r i) _", "annotated_tactic": ["change r i = <a>ite</a> (0 < r i) (r i) _", [{"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}]], "state_before": "case intro.h.e'_3\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\nhi : 0 < r i\n\u22a2 r i = r' i", "state_after": "case intro.h.e'_3\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\nhi : 0 < r i\n\u22a2 r i = if 0 < r i then r i else 1 / (\u2191i + 1)"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "case intro.h.e'_3\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\ni : \u2115\nhi : 0 < r i\n\u22a2 r i = if 0 < r i then r i else 1 / (\u2191i + 1)", "state_after": "no goals"}, {"tactic": "rintro i \u27e8-, hi\u27e9", "annotated_tactic": ["rintro i \u27e8-, hi\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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\n\u22a2 \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\ni : \u2115\nhi : 0 < r i\n\u22a2 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)"}, {"tactic": "simp only [hi, mul_ite, if_true]", "annotated_tactic": ["simp only [hi, <a>mul_ite</a>, <a>if_true</a>]", [{"full_name": "mul_ite", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [195, 9], "def_end_pos": [195, 16]}, {"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}]], "state_before": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\ni : \u2115\nhi : 0 < r i\n\u22a2 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)", "state_after": "no goals"}, {"tactic": "rintro i \u27e8-, hi\u27e9", "annotated_tactic": ["rintro i \u27e8-, hi\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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\n\u22a2 \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : \u2115\nhi : r i \u2264 0\n\u22a2 cthickening (M * r i) (s i) = cthickening (r i) (s i)"}, {"tactic": "have hi' : M * r i \u2264 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi", "annotated_tactic": ["have hi' : M * r i \u2264 0 := <a>mul_nonpos_of_nonneg_of_nonpos</a> hM.le hi", [{"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 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : \u2115\nhi : r i \u2264 0\n\u22a2 cthickening (M * r i) (s i) = cthickening (r i) (s i)", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : \u2115\nhi : r i \u2264 0\nhi' : M * r i \u2264 0\n\u22a2 cthickening (M * r i) (s i) = cthickening (r i) (s i)"}, {"tactic": "rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']", "annotated_tactic": ["rw [<a>cthickening_of_nonpos</a> hi, <a>cthickening_of_nonpos</a> hi']", [{"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 30]}, {"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 30]}]], "state_before": "case 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\nM : \u211d\nhM : 0 < M\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : \u2115\nhi : r i \u2264 0\nhi' : M * r i \u2264 0\n\u22a2 cthickening (M * r i) (s i) = cthickening (r i) (s i)", "state_after": "no goals"}, {"tactic": "ext i", "annotated_tactic": ["ext 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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\n\u22a2 p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0", "state_after": "case h.a\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\ni : \u2115\n\u22a2 p i \u2194 p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0"}, {"tactic": "simp [\u2190 and_or_left, lt_or_le 0 (r i)]", "annotated_tactic": ["simp [\u2190 <a>and_or_left</a>, <a>lt_or_le</a> 0 (r i)]", [{"full_name": "and_or_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [321, 9], "def_end_pos": [321, 20]}, {"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case h.a\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)\nthis :\n  \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},\n    Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192\n      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1)\nhr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0)\nh\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i)\nh\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i)\ni : \u2115\n\u22a2 p i \u2194 p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.insert_union", "start": [610, 1], "end": [611, 64], "traced_tactics": [{"tactic": "simp [AList.insert_union]", "annotated_tactic": ["simp [<a>AList.insert_union</a>]", [{"full_name": "AList.insert_union", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [488, 9], "def_end_pos": [488, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u2081 s\u2082 : Finmap \u03b2\na\u2081 a\u2082 : AList \u03b2\n\u22a2 insert a b (\u27e6a\u2081\u27e7 \u222a \u27e6a\u2082\u27e7) = insert a b \u27e6a\u2081\u27e7 \u222a \u27e6a\u2082\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.fixed_point\u2082", "start": [1188, 1], "end": [1191, 55], "traced_tactics": [{"tactic": "simp [e.symm, ef, Part.map_id']", "annotated_tactic": ["simp [e.symm, ef, <a>Part.map_id'</a>]", [{"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "state_before": "f : Code \u2192 \u2115 \u2192. \u2115\nhf : Partrec\u2082 f\ncf : Code\nef : eval cf = fun n => Part.bind \u2191(decode n) fun a => Part.map encode ((fun p => f p.1 p.2) a)\nc : Code\ne : eval (curry cf (encode c)) = eval c\nn : \u2115\n\u22a2 eval c n = f c n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.cprankMax_1", "start": [329, 1], "end": [332, 33], "traced_tactics": [{"tactic": "have h' := CPRankMax.succ 0 x 0 h CPRankMax.zero", "annotated_tactic": ["have h' := <a>CPRankMax.succ</a> 0 x 0 h <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.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 ds\nh : CPRankMax1 x\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 ds\nh : CPRankMax1 x\nh' : CPRankMax (0 + 1) (x + 0)\n\u22a2 CPRankMax 1 x"}, {"tactic": "rwa [zero_add, add_zero] at h'", "annotated_tactic": ["rwa [<a>zero_add</a>, <a>add_zero</a>] at h'", [{"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]}]], "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 ds\nh : CPRankMax1 x\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/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "ContinuousOn.intervalIntegrable", "start": [354, 1], "end": [356, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.norm_indicatorConstLp'", "start": [774, 1], "end": [779, 46], "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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\n\u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\nhp_top : p = \u22a4\n\u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc s) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [hp_top, ENNReal.top_toReal, _root_.div_zero, Real.rpow_zero, mul_one]", "annotated_tactic": ["rw [hp_top, <a>ENNReal.top_toReal</a>, <a>_root_.div_zero</a>, <a>Real.rpow_zero</a>, <a>mul_one</a>]", [{"full_name": "ENNReal.top_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [227, 17], "def_end_pos": [227, 27]}, {"full_name": "div_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "Real.rpow_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [97, 9], "def_end_pos": [97, 18]}, {"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\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\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\nhp_top : p = \u22a4\n\u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\nhp_top : p = \u22a4\n\u22a2 \u2016indicatorConstLp \u22a4 hs h\u03bcs c\u2016 = \u2016c\u2016"}, {"tactic": "exact norm_indicatorConstLp_top h\u03bcs_pos", "annotated_tactic": ["exact <a>norm_indicatorConstLp_top</a> h\u03bcs_pos", [{"full_name": "MeasureTheory.norm_indicatorConstLp_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [767, 9], "def_end_pos": [767, 34]}]], "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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\nhp_top : p = \u22a4\n\u22a2 \u2016indicatorConstLp \u22a4 hs h\u03bcs c\u2016 = \u2016c\u2016", "state_after": "no goals"}, {"tactic": "exact norm_indicatorConstLp hp_pos hp_top", "annotated_tactic": ["exact <a>norm_indicatorConstLp</a> hp_pos hp_top", [{"full_name": "MeasureTheory.norm_indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [760, 9], "def_end_pos": [760, 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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nhp_pos : p \u2260 0\nh\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc s) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.univ_finite_iff_nonempty_fintype", "start": [1058, 1], "end": [1059, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.map_bind\u2082", "start": [255, 1], "end": [259, 41], "traced_tactics": [{"tactic": "simp only [bind\u2082, eval\u2082_comp_right, coe_eval\u2082Hom, eval\u2082_map]", "annotated_tactic": ["simp only [<a>bind\u2082</a>, <a>eval\u2082_comp_right</a>, <a>coe_eval\u2082Hom</a>, <a>eval\u2082_map</a>]", [{"full_name": "MvPolynomial.bind\u2082", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [81, 5], "def_end_pos": [81, 10]}, {"full_name": "MvPolynomial.eval\u2082_comp_right", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 25]}, {"full_name": "MvPolynomial.coe_eval\u2082Hom", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1051, 9], "def_end_pos": [1051, 21]}, {"full_name": "MvPolynomial.eval\u2082_map", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1357, 9], "def_end_pos": [1357, 18]}]], "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\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(map g) (\u2191(bind\u2082 f) \u03c6) = \u2191(bind\u2082 (RingHom.comp (map g) 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+* MvPolynomial \u03c3 S\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 (RingHom.comp (map g) f) (\u2191(map g) \u2218 X) \u03c6 = eval\u2082 (RingHom.comp (map g) f) X \u03c6"}, {"tactic": "congr 1 with : 1", "annotated_tactic": ["congr 1 with : 1", []], "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\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 (RingHom.comp (map g) f) (\u2191(map g) \u2218 X) \u03c6 = eval\u2082 (RingHom.comp (map g) f) X \u03c6", "state_after": "case e_g.h\n\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\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\nx\u271d : \u03c3\n\u22a2 (\u2191(map g) \u2218 X) x\u271d = X x\u271d"}, {"tactic": "simp only [Function.comp_apply, map_X]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>map_X</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": "MvPolynomial.map_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1243, 9], "def_end_pos": [1243, 14]}]], "state_before": "case e_g.h\n\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\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\nx\u271d : \u03c3\n\u22a2 (\u2191(map g) \u2218 X) x\u271d = X x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.hrecOn\u2082'_mk''", "start": [737, 1], "end": [741, 6], "traced_tactics": []}, {"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": [301, 1], "end": [312, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.core_res", "start": [492, 1], "end": [495, 34], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "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 : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 core (res f s) t = s\u1d9c \u222a f \u207b\u00b9' 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\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\u1d9c \u222a f \u207b\u00b9' t"}, {"tactic": "rw [mem_core_res]", "annotated_tactic": ["rw [<a>mem_core_res</a>]", [{"full_name": "PFun.mem_core_res", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [484, 9], "def_end_pos": [484, 21]}]], "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\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\u1d9c \u222a f \u207b\u00b9' 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\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 s \u2192 f x \u2208 t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t"}, {"tactic": "by_cases h : x \u2208 s <;> simp [h]", "annotated_tactic": ["by_cases h : x \u2208 s <;> simp [h]", []], "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\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 s \u2192 f x \u2208 t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.sym_nonempty", "start": [198, 1], "end": [201, 45], "traced_tactics": [{"tactic": "simp_rw [nonempty_iff_ne_empty, Ne.def]", "annotated_tactic": ["simp_rw [<a>nonempty_iff_ne_empty</a>, <a>Ne.def</a>]", [{"full_name": "Finset.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [604, 9], "def_end_pos": [604, 30]}, {"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 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 Finset.Nonempty (Finset.sym s n) \u2194 n = 0 \u2228 Finset.Nonempty s", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 \u00acFinset.sym s n = \u2205 \u2194 n = 0 \u2228 \u00acs = \u2205"}, {"tactic": "rwa [sym_eq_empty, not_and_or, not_ne_iff]", "annotated_tactic": ["rwa [<a>sym_eq_empty</a>, 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"commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize.reduced'", "start": [20, 1], "end": [23, 35], "traced_tactics": [{"tactic": "rw [\u2190 Int.div_eq_ediv_of_dvd (e \u25b8 Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]", "annotated_tactic": ["rw [\u2190 <a>Int.div_eq_ediv_of_dvd</a> (e \u25b8 <a>Int.ofNat_dvd_left</a>.2 (<a>Nat.gcd_dvd_left</a> ..))]", [{"full_name": "Int.div_eq_ediv_of_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [794, 9], "def_end_pos": [794, 27]}, {"full_name": "Int.ofNat_dvd_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [662, 9], "def_end_pos": [662, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}]], "state_before": "num : Int\nden g : Nat\nden_nz : den \u2260 0\ne : g = Nat.gcd (Int.natAbs num) den\n\u22a2 Nat.Coprime 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"annotated_tactic": ["simp only [<a>ENNReal.tsum_eq_iSup_sum</a>]", [{"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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u03b2), f i a \u2202\u03bc = \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1), f i 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 s, \u2211 i in s, f i a \u2202\u03bc = \u2a06 s, \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc"}, {"tactic": "rw [lintegral_iSup_directed]", "annotated_tactic": ["rw [<a>lintegral_iSup_directed</a>]", [{"full_name": "MeasureTheory.lintegral_iSup_directed", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1154, 9], "def_end_pos": [1154, 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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 s, \u2211 i in s, f i a \u2202\u03bc = \u2a06 s, \u2211 i in s, \u222b\u207b (a : \u03b1), f i 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u2a06 b, \u222b\u207b (a : \u03b1), \u2211 i in b, f i a \u2202\u03bc = \u2a06 s, \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u2200 (b : Finset \u03b2), AEMeasurable fun a => \u2211 i in b, f i a\n\ncase h_directed\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun s a => \u2211 i in s, f i a"}, {"tactic": "simp [lintegral_finset_sum' _ fun i _ => hf i]", "annotated_tactic": ["simp [<a>lintegral_finset_sum'</a> _ fun i _ => hf i]", [{"full_name": "MeasureTheory.lintegral_finset_sum'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [655, 9], "def_end_pos": [655, 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u2a06 b, \u222b\u207b (a : \u03b1), \u2211 i in b, f i a \u2202\u03bc = \u2a06 s, \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u2200 (b : Finset \u03b2), AEMeasurable fun a => \u2211 i in b, f i a", "state_after": "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\nb : Finset \u03b2\n\u22a2 AEMeasurable fun a => \u2211 i in b, f i a"}, {"tactic": "exact Finset.aemeasurable_sum _ fun i _ => hf i", "annotated_tactic": ["exact <a>Finset.aemeasurable_sum</a> _ fun i _ => hf i", [{"full_name": "Finset.aemeasurable_sum", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [951, 3], "def_end_pos": [951, 14]}]], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\nb : Finset \u03b2\n\u22a2 AEMeasurable fun a => \u2211 i in b, f i a", "state_after": "no goals"}, {"tactic": "intro s t", "annotated_tactic": ["intro s t", []], "state_before": "case h_directed\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun s a => \u2211 i in s, f i a", "state_after": "case h_directed\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) z) \u2227\n      (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) z)"}, {"tactic": "use s \u222a t", "annotated_tactic": ["use s \u222a t", []], "state_before": "case h_directed\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) z) \u2227\n      (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) z)", "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) \u2227\n    (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t))"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) \u2227\n    (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t))", "state_after": "case h.left\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t))\n\ncase h.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t))"}, {"tactic": "exact fun a => Finset.sum_le_sum_of_subset (Finset.subset_union_left _ _)", "annotated_tactic": ["exact fun a => <a>Finset.sum_le_sum_of_subset</a> (<a>Finset.subset_union_left</a> _ _)", [{"full_name": "Finset.sum_le_sum_of_subset", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [403, 15], "def_end_pos": [403, 35]}, {"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 h.left\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t))", "state_after": "no goals"}, {"tactic": "exact fun a => Finset.sum_le_sum_of_subset (Finset.subset_union_right _ _)", "annotated_tactic": ["exact fun a => <a>Finset.sum_le_sum_of_subset</a> (<a>Finset.subset_union_right</a> _ _)", [{"full_name": "Finset.sum_le_sum_of_subset", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [403, 15], "def_end_pos": [403, 35]}, {"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 h.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.hasSum_intervalIntegral_of_summable_norm", "start": [1044, 1], "end": [1059, 48], "traced_tactics": [{"tactic": "apply hasSum_integral_of_dominated_convergence\n  (fun i (x : \u211d) => \u2016(f i).restrict \u2191(\u27e8uIcc a b, isCompact_uIcc\u27e9 : Compacts \u211d)\u2016)\n  (fun i => (map_continuous <| f i).aestronglyMeasurable)", "annotated_tactic": ["apply <a>hasSum_integral_of_dominated_convergence</a>\n    (fun i (x : \u211d) => \u2016(f i).<a>restrict</a> \u2191(\u27e8<a>uIcc</a> a b, <a>isCompact_uIcc</a>\u27e9 : <a>Compacts</a> \u211d)\u2016)\n    (fun i => (<a>map_continuous</a> <| f i).<a>aestronglyMeasurable</a>)", [{"full_name": "intervalIntegral.hasSum_integral_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1025, 16], "def_end_pos": [1025, 56]}, {"full_name": "ContinuousMap.restrict", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [371, 5], "def_end_pos": [371, 13]}, {"full_name": "Set.uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [54, 5], "def_end_pos": [54, 9]}, {"full_name": "isCompact_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [133, 9], "def_end_pos": [133, 23]}, {"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}, {"full_name": "ContinuousMapClass.map_continuous", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 17]}, {"full_name": "Continuous.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1271, 9], "def_end_pos": [1271, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 HasSum (fun i => \u222b (x : \u211d) in a..b, \u2191(f i) x) (\u222b (x : \u211d) in a..b, \u2211' (i : \u03b9), \u2191(f i) x)", "state_after": "case h_bound\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 \u2200 (n : \u03b9),\n    \u2200\u1d50 (t : \u211d),\n      t \u2208 \u0399 a b \u2192\n        \u2016\u2191(f n) t\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016\n\ncase bound_summable\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 \u2200\u1d50 (t : \u211d),\n    t \u2208 \u0399 a b \u2192\n      Summable fun n =>\n        \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016\n\ncase bound_integrable\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 IntervalIntegrable\n    (fun t =>\n      \u2211' (n : \u03b9), \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016)\n    volume a b\n\ncase h_lim\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 \u2200\u1d50 (t : \u211d), t \u2208 \u0399 a b \u2192 HasSum (fun n => \u2191(f n) t) (\u2211' (i : \u03b9), \u2191(f i) t)"}, {"tactic": "refine fun i => ae_of_all _ fun x hx => ?_", "annotated_tactic": ["refine fun i => <a>ae_of_all</a> _ fun x hx => ?_", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}]], "state_before": "case h_bound\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 \u2200 (n : \u03b9),\n    \u2200\u1d50 (t : \u211d),\n      t \u2208 \u0399 a b \u2192\n        \u2016\u2191(f n) t\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016", "state_after": "case h_bound\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\ni : \u03b9\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 \u2016\u2191(f i) x\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016"}, {"tactic": "apply ContinuousMap.norm_coe_le_norm ((f i).restrict _) \u27e8x, _\u27e9", "annotated_tactic": ["apply <a>ContinuousMap.norm_coe_le_norm</a> ((f i).<a>restrict</a> _) \u27e8x, _\u27e9", [{"full_name": "ContinuousMap.norm_coe_le_norm", "def_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "def_pos": [202, 9], "def_end_pos": [202, 25]}, {"full_name": "ContinuousMap.restrict", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [371, 5], "def_end_pos": [371, 13]}]], "state_before": "case h_bound\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\ni : \u03b9\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 \u2016\u2191(f i) x\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\ni : \u03b9\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 x \u2208 \u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }"}, {"tactic": "exact \u27e8hx.1.le, hx.2\u27e9", "annotated_tactic": ["exact \u27e8hx.1.<a>le</a>, hx.2\u27e9", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\ni : \u03b9\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 x \u2208 \u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }", "state_after": "no goals"}, {"tactic": "exact ae_of_all _ fun x _ => hf_sum", "annotated_tactic": ["exact <a>ae_of_all</a> _ fun x _ => hf_sum", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}]], "state_before": "case bound_summable\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 \u2200\u1d50 (t : \u211d),\n    t \u2208 \u0399 a b \u2192\n      Summable fun n =>\n        \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016", "state_after": "no goals"}, {"tactic": "exact intervalIntegrable_const", "annotated_tactic": ["exact <a>intervalIntegrable_const</a>", [{"full_name": "intervalIntegrable_const", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [125, 9], "def_end_pos": [125, 33]}]], "state_before": "case bound_integrable\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 IntervalIntegrable\n    (fun t =>\n      \u2211' (n : \u03b9), \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016)\n    volume a b", "state_after": "no goals"}, {"tactic": "refine ae_of_all _ fun x hx => Summable.hasSum ?_", "annotated_tactic": ["refine <a>ae_of_all</a> _ fun x hx => <a>Summable.hasSum</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": "Summable.hasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 24]}]], "state_before": "case h_lim\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\n\u22a2 \u2200\u1d50 (t : \u211d), t \u2208 \u0399 a b \u2192 HasSum (fun n => \u2191(f n) t) (\u2211' (i : \u03b9), \u2191(f i) t)", "state_after": "case h_lim\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 Summable fun n => \u2191(f n) x"}, {"tactic": "let x : (\u27e8uIcc a b, isCompact_uIcc\u27e9 : Compacts \u211d) := \u27e8x, ?_\u27e9", "annotated_tactic": ["let x : (\u27e8<a>uIcc</a> a b, <a>isCompact_uIcc</a>\u27e9 : <a>Compacts</a> \u211d) := \u27e8x, ?_\u27e9", [{"full_name": "Set.uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [54, 5], "def_end_pos": [54, 9]}, {"full_name": "isCompact_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [133, 9], "def_end_pos": [133, 23]}, {"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}]], "state_before": "case h_lim\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 Summable fun n => \u2191(f n) x", "state_after": "case h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := ?h_lim.refine_1 }\n\u22a2 Summable fun n => \u2191(f n) x\u271d\n\ncase h_lim.refine_1\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := ?h_lim.refine_1 }\n\u22a2 Summable fun n => \u2191(f n) x\u271d\n\ncase h_lim.refine_1\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }", "state_after": "case h_lim.refine_1\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }\n\ncase h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := ?h_lim.refine_1 }\n\u22a2 Summable fun n => \u2191(f n) x\u271d"}, {"tactic": "exact \u27e8hx.1.le, hx.2\u27e9", "annotated_tactic": ["exact \u27e8hx.1.<a>le</a>, hx.2\u27e9", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case h_lim.refine_1\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx : \u211d\nhx : x \u2208 \u0399 a b\n\u22a2 x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }\n\ncase h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := ?h_lim.refine_1 }\n\u22a2 Summable fun n => \u2191(f n) x\u271d", "state_after": "case h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := (_ : a \u2293 b \u2264 x\u271d \u2227 x\u271d \u2264 a \u2294 b) }\n\u22a2 Summable fun n => \u2191(f n) x\u271d"}, {"tactic": "have := summable_of_summable_norm hf_sum", "annotated_tactic": ["have := <a>summable_of_summable_norm</a> hf_sum", [{"full_name": "summable_of_summable_norm", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [173, 9], "def_end_pos": [173, 34]}]], "state_before": "case h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := (_ : a \u2293 b \u2264 x\u271d \u2227 x\u271d \u2264 a \u2294 b) }\n\u22a2 Summable fun n => \u2191(f n) x\u271d", "state_after": "case h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := (_ : a \u2293 b \u2264 x\u271d \u2227 x\u271d \u2264 a \u2294 b) }\nthis :\n  Summable fun a_1 => ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f a_1)\n\u22a2 Summable fun n => \u2191(f n) x\u271d"}, {"tactic": "simpa only [Compacts.coe_mk, ContinuousMap.restrict_apply]\n  using ContinuousMap.summable_apply this x", "annotated_tactic": ["simpa only [<a>Compacts.coe_mk</a>, <a>ContinuousMap.restrict_apply</a>]\n      using <a>ContinuousMap.summable_apply</a> this x", [{"full_name": "TopologicalSpace.Compacts.coe_mk", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [67, 9], "def_end_pos": [67, 15]}, {"full_name": "ContinuousMap.restrict_apply", "def_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 23]}, {"full_name": "ContinuousMap.summable_apply", "def_path": "Mathlib/Topology/ContinuousFunction/Algebra.lean", "def_pos": [432, 9], "def_end_pos": [432, 23]}]], "state_before": "case h_lim.refine_2\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 c d : \u211d\nf\u271d g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 C(\u211d, E)\nhf_sum :\n  Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016\nx\u271d : \u211d\nhx : x\u271d \u2208 \u0399 a b\nx : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=\n  { val := x\u271d, property := (_ : a \u2293 b \u2264 x\u271d \u2227 x\u271d \u2264 a \u2294 b) }\nthis :\n  Summable fun a_1 => ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f a_1)\n\u22a2 Summable fun n => \u2191(f n) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Ico.coe_ne_zero", "start": [205, 1], "end": [206, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.Measure.ext", "start": [135, 1], "end": [137, 57], "traced_tactics": [{"tactic": "rw [\u2190 trimmed, OuterMeasure.trim_congr (h _), trimmed]", "annotated_tactic": ["rw [\u2190 <a>trimmed</a>, <a>OuterMeasure.trim_congr</a> (h _), <a>trimmed</a>]", [{"full_name": "MeasureTheory.Measure.trimmed", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [78, 3], "def_end_pos": [78, 10]}, {"full_name": "MeasureTheory.OuterMeasure.trim_congr", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1613, 9], "def_end_pos": [1613, 19]}, {"full_name": "MeasureTheory.Measure.trimmed", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [78, 3], "def_end_pos": [78, 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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc\u2081 s = \u2191\u2191\u03bc\u2082 s\n\u22a2 \u2191\u03bc\u2081 = \u2191\u03bc\u2082", "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.Ordered.erase", "start": [457, 11], "end": [458, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "indicator_ae_eq_restrict_compl", "start": [4489, 1], "end": [4491, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.preimage_asSubtype", "start": [523, 1], "end": [531, 36], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "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 asSubtype f \u207b\u00b9' s = Subtype.val \u207b\u00b9' preimage f 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\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 x \u2208 asSubtype f \u207b\u00b9' s \u2194 x \u2208 Subtype.val \u207b\u00b9' preimage f s"}, {"tactic": "simp only [Set.mem_preimage, Set.mem_setOf_eq, PFun.asSubtype, PFun.mem_preimage]", "annotated_tactic": ["simp only [<a>Set.mem_preimage</a>, <a>Set.mem_setOf_eq</a>, <a>PFun.asSubtype</a>, <a>PFun.mem_preimage</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_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "PFun.asSubtype", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [118, 5], "def_end_pos": [118, 14]}, {"full_name": "PFun.mem_preimage", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [431, 9], "def_end_pos": [431, 21]}]], "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\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 x \u2208 asSubtype f \u207b\u00b9' s \u2194 x \u2208 Subtype.val \u207b\u00b9' preimage f 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\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 fn f \u2191x (_ : \u2191x \u2208 Dom f) \u2208 s \u2194 \u2203 y, y \u2208 s \u2227 y \u2208 f \u2191x"}, {"tactic": "exact\n  Iff.intro (fun h => \u27e8_, h, Part.get_mem _\u27e9) fun \u27e8y, ys, fxy\u27e9 =>\n    have : f.fn x.val x.property \u2208 f x.val := Part.get_mem _\n    Part.mem_unique fxy this \u25b8 ys", "annotated_tactic": ["exact\n    <a>Iff.intro</a> (fun h => \u27e8_, h, <a>Part.get_mem</a> _\u27e9) fun \u27e8y, ys, fxy\u27e9 =>\n      have : f.fn x.val x.property \u2208 f x.val := <a>Part.get_mem</a> _\n      <a>Part.mem_unique</a> fxy this \u25b8 ys", [{"full_name": "Iff.intro", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [88, 3], "def_end_pos": [88, 8]}, {"full_name": "Part.get_mem", "def_path": "Mathlib/Data/Part.lean", "def_pos": [105, 9], "def_end_pos": [105, 16]}, {"full_name": "Part.get_mem", "def_path": "Mathlib/Data/Part.lean", "def_pos": [105, 9], "def_end_pos": [105, 16]}, {"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}]], "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\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 fn f \u2191x (_ : \u2191x \u2208 Dom f) \u2208 s \u2194 \u2203 y, y \u2208 s \u2227 y \u2208 f \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_mem_image_coe", "start": [1619, 1], "end": [1621, 53], "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_le_le", "start": [2744, 1], "end": [2749, 72], "traced_tactics": [{"tactic": "simp [monotoneOn_iff_monotone, antitoneOn_iff_antitone, and_assoc, exists_and_left,\n  not_monotone_not_antitone_iff_exists_le_le, @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_le_le</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_le_le", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [928, 7], "def_end_pos": [928, 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 \u2264 b \u2227 b \u2264 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/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_eq_zero_of_hasDerivWithinAt'", "start": [469, 1], "end": [477, 28], "traced_tactics": [{"tactic": "by_cases hi : CircleIntegrable f' c R", "annotated_tactic": ["by_cases hi : <a>CircleIntegrable</a> f' c R", [{"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\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0", "state_after": "case pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0\n\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : \u00acCircleIntegrable f' c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0"}, {"tactic": "rw [\u2190 sub_eq_zero.2 ((periodic_circleMap c R).comp f).eq]", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>.2 ((<a>periodic_circleMap</a> c R).<a>comp</a> f).<a>eq</a>]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "periodic_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [93, 9], "def_end_pos": [93, 27]}, {"full_name": "Function.Periodic.comp", "def_path": "Mathlib/Algebra/Periodic.lean", "def_pos": [57, 19], "def_end_pos": [57, 32]}, {"full_name": "Function.Periodic.eq", "def_path": "Mathlib/Algebra/Periodic.lean", "def_pos": [258, 19], "def_end_pos": [258, 30]}]], "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0", "state_after": "case pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = (f \u2218 circleMap c R) (2 * \u03c0) - (f \u2218 circleMap c R) 0"}, {"tactic": "refine' intervalIntegral.integral_eq_sub_of_hasDerivAt (fun \u03b8 _ => _) hi.out", "annotated_tactic": ["refine' <a>intervalIntegral.integral_eq_sub_of_hasDerivAt</a> (fun \u03b8 _ => _) hi.out", [{"full_name": "intervalIntegral.integral_eq_sub_of_hasDerivAt", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 38]}]], "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = (f \u2218 circleMap c R) (2 * \u03c0) - (f \u2218 circleMap c R) 0", "state_after": "case pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 HasDerivAt (f \u2218 circleMap c R) (deriv (circleMap c R) \u03b8 \u2022 (fun z => f' z) (circleMap c R \u03b8)) \u03b8"}, {"tactic": "exact (h _ (circleMap_mem_sphere' _ _ _)).scomp_hasDerivAt \u03b8\n  (differentiable_circleMap _ _ _).hasDerivAt (circleMap_mem_sphere' _ _)", "annotated_tactic": ["exact (h _ (<a>circleMap_mem_sphere'</a> _ _ _)).<a>scomp_hasDerivAt</a> \u03b8\n      (<a>differentiable_circleMap</a> _ _ _).<a>hasDerivAt</a> (<a>circleMap_mem_sphere'</a> _ _)", [{"full_name": "circleMap_mem_sphere'", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [119, 9], "def_end_pos": [119, 30]}, {"full_name": "HasDerivWithinAt.scomp_hasDerivAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Comp.lean", "def_pos": [82, 9], "def_end_pos": [82, 42]}, {"full_name": "differentiable_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [180, 9], "def_end_pos": [180, 33]}, {"full_name": "DifferentiableAt.hasDerivAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 36]}, {"full_name": "circleMap_mem_sphere'", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [119, 9], "def_end_pos": [119, 30]}]], "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 HasDerivAt (f \u2218 circleMap c R) (deriv (circleMap c R) \u03b8 \u2022 (fun z => f' z) (circleMap c R \u03b8)) \u03b8", "state_after": "no goals"}, {"tactic": "exact integral_undef hi", "annotated_tactic": ["exact <a>integral_undef</a> hi", [{"full_name": "circleIntegral.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [377, 9], "def_end_pos": [377, 23]}]], "state_before": "case neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf f' : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nh : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : \u00acCircleIntegrable f' c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.cprankMax_upper_bound", "start": [377, 1], "end": [397, 26], "traced_tactics": [{"tactic": "have h_summands :\n  \u2200 i : { x // x \u2208 Finset.range d },\n    CPRankMax ds.prod (unitVec d i.1 \u2297 slice x i.1 (mem_range.1 i.2)) :=\n  fun i => cprankMax_mul _ _ _ (cprankMax_upper_bound (slice x i.1 (mem_range.1 i.2)))", "annotated_tactic": ["have h_summands :\n      \u2200 i : { x // x \u2208 <a>Finset.range</a> d },\n        <a>CPRankMax</a> ds.prod (<a>unitVec</a> d i.1 \u2297 <a>slice</a> x i.1 (<a>mem_range</a>.1 i.2)) :=\n      fun i => <a>cprankMax_mul</a> _ _ _ (cprankMax_upper_bound (<a>slice</a> x i.1 (<a>mem_range</a>.1 i.2)))", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Holor.CPRankMax", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [318, 11], "def_end_pos": [318, 20]}, {"full_name": "Holor.unitVec", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [238, 5], "def_end_pos": [238, 12]}, {"full_name": "Holor.slice", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [233, 5], "def_end_pos": [233, 10]}, {"full_name": "List.mem_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 18]}, {"full_name": "Holor.cprankMax_mul", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [350, 9], "def_end_pos": [350, 22]}, {"full_name": "Holor.slice", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [233, 5], "def_end_pos": [233, 10]}, {"full_name": "List.mem_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 18]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\n\u22a2 CPRankMax (prod (d :: ds)) x", "state_after": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) x"}, {"tactic": "have h_dds_prod : (List.cons d ds).prod = Finset.card (Finset.range d) * prod ds := by\n  simp [Finset.card_range]", "annotated_tactic": ["have h_dds_prod : (<a>List.cons</a> d ds).<a>prod</a> = <a>Finset.card</a> (<a>Finset.range</a> d) * <a>prod</a> ds := by\n      simp [<a>Finset.card_range</a>]", [{"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.prod", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "List.prod", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) x", "state_after": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\n\u22a2 CPRankMax (prod (d :: ds)) x"}, {"tactic": "have :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d),\n      unitVec d i.val \u2297 slice x i.val (mem_range.1 i.2)) :=\n  cprankMax_sum (Finset.range d).attach _ fun i _ => h_summands i", "annotated_tactic": ["have :\n      <a>CPRankMax</a> (<a>Finset.card</a> (<a>Finset.attach</a> (<a>Finset.range</a> d)) * <a>prod</a> ds)\n        (\u2211 i in <a>Finset.attach</a> (<a>Finset.range</a> d),\n          <a>unitVec</a> d i.val \u2297 <a>slice</a> x i.val (<a>mem_range</a>.1 i.2)) :=\n      <a>cprankMax_sum</a> (<a>Finset.range</a> d).<a>attach</a> _ fun i _ => h_summands i", [{"full_name": "Holor.CPRankMax", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [318, 11], "def_end_pos": [318, 20]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2440, 5], "def_end_pos": [2440, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "List.prod", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "Finset.attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2440, 5], "def_end_pos": [2440, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Holor.unitVec", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [238, 5], "def_end_pos": [238, 12]}, {"full_name": "Holor.slice", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [233, 5], "def_end_pos": [233, 10]}, {"full_name": "List.mem_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 18]}, {"full_name": "Holor.cprankMax_sum", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [361, 9], "def_end_pos": [361, 22]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2440, 5], "def_end_pos": [2440, 11]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\n\u22a2 CPRankMax (prod (d :: ds)) x", "state_after": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) x"}, {"tactic": "have h_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d),\n      unitVec d i.val \u2297 slice x i.val (mem_range.1 i.2)) := by rwa [Finset.card_attach] at this", "annotated_tactic": ["have h_cprankMax_sum :\n      <a>CPRankMax</a> (<a>Finset.card</a> (<a>Finset.range</a> d) * <a>prod</a> ds)\n        (\u2211 i in <a>Finset.attach</a> (<a>Finset.range</a> d),\n          <a>unitVec</a> d i.val \u2297 <a>slice</a> x i.val (<a>mem_range</a>.1 i.2)) := by rwa [<a>Finset.card_attach</a>] at this", [{"full_name": "Holor.CPRankMax", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [318, 11], "def_end_pos": [318, 20]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "List.prod", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [81, 5], "def_end_pos": [81, 9]}, {"full_name": "Finset.attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2440, 5], "def_end_pos": [2440, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Holor.unitVec", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [238, 5], "def_end_pos": [238, 12]}, {"full_name": "Holor.slice", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [233, 5], "def_end_pos": [233, 10]}, {"full_name": "List.mem_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2087, 9], "def_end_pos": [2087, 18]}, {"full_name": "Finset.card_attach", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) x", "state_after": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) x"}, {"tactic": "rw [\u2190 sum_unitVec_mul_slice x]", "annotated_tactic": ["rw [\u2190 <a>sum_unitVec_mul_slice</a> x]", [{"full_name": "Holor.sum_unitVec_mul_slice", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [290, 9], "def_end_pos": [290, 30]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) x", "state_after": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d))"}, {"tactic": "rw [h_dds_prod]", "annotated_tactic": ["rw [h_dds_prod]", []], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (prod (d :: ds)) (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d))", "state_after": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d))"}, {"tactic": "exact h_cprankMax_sum", "annotated_tactic": ["exact h_cprankMax_sum", []], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d))", "state_after": "no goals"}, {"tactic": "simp [Finset.card_range]", "annotated_tactic": ["simp [<a>Finset.card_range</a>]", [{"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 prod (d :: ds) = Finset.card (Finset.range d) * prod ds", "state_after": "no goals"}, {"tactic": "rwa [Finset.card_attach] at this", "annotated_tactic": ["rwa [<a>Finset.card_attach</a>] at this", [{"full_name": "Finset.card_attach", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "\u03b1 : Type\nd\u271d : \u2115\nds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nd : \u2115\nds : List \u2115\nx : Holor \u03b1 (d :: ds)\nh_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\nh_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds\nthis :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))\n\u22a2 CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d))", "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.add_measure", "start": [1731, 1], "end": [1734, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.take_subset", "start": [1901, 1], "end": [1902, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.support_smul_eq", "start": [751, 1], "end": [753, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "full_name": "MeasureTheory.L2.inner_indicatorConstLp_eq_set_integral_inner", "start": [234, 1], "end": [258, 33], "traced_tactics": [{"tactic": "rw [inner_def, \u2190 integral_add_compl hs (L2.integrable_inner _ f)]", "annotated_tactic": ["rw [<a>inner_def</a>, \u2190 <a>integral_add_compl</a> hs (<a>L2.integrable_inner</a> _ f)]", [{"full_name": "MeasureTheory.L2.inner_def", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [153, 9], "def_end_pos": [153, 18]}, {"full_name": "MeasureTheory.integral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [162, 9], "def_end_pos": [162, 27]}, {"full_name": "MeasureTheory.L2.integrable_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [191, 9], "def_end_pos": [191, 25]}]], "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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 inner (indicatorConstLp 2 hs h\u03bcs c) f = \u222b (x : \u03b1) in s, inner c (\u2191\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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc +\n      \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc =\n    \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc"}, {"tactic": "have h_left : (\u222b x in s, \u27ea(indicatorConstLp 2 hs h\u03bcs c) x, f x\u27eb \u2202\u03bc) = \u222b x in s, \u27eac, f x\u27eb \u2202\u03bc := by\n  suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u27eaindicatorConstLp 2 hs h\u03bcs c x, f x\u27eb = \u27eac, f x\u27eb\n  exact set_integral_congr_ae hs h_ae_eq\n  have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2208 s \u2192 indicatorConstLp 2 hs h\u03bcs c x = c :=\n    indicatorConstLp_coeFn_mem\n  refine' h_indicator.mono fun x hx hxs => _\n  congr\n  exact hx hxs", "annotated_tactic": ["have h_left : (\u222b x in s, \u27ea(<a>indicatorConstLp</a> 2 hs h\u03bcs c) x, f x\u27eb \u2202\u03bc) = \u222b x in s, \u27eac, f x\u27eb \u2202\u03bc := by\n    suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u27ea<a>indicatorConstLp</a> 2 hs h\u03bcs c x, f x\u27eb = \u27eac, f x\u27eb\n    exact <a>set_integral_congr_ae</a> hs h_ae_eq\n    have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2208 s \u2192 <a>indicatorConstLp</a> 2 hs h\u03bcs c x = c :=\n      <a>indicatorConstLp_coeFn_mem</a>\n    refine' h_indicator.mono fun x hx hxs => _\n    congr\n    exact hx hxs", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"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.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.indicatorConstLp_coeFn_mem", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [752, 9], "def_end_pos": [752, 35]}]], "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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc +\n      \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc =\n    \u222b (x : \u03b1) in s, inner c (\u2191\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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc +\n      \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc =\n    \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc"}, {"tactic": "rw [h_left, h_right, add_zero]", "annotated_tactic": ["rw [h_left, h_right, <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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_right : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc +\n      \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc =\n    \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u27eaindicatorConstLp 2 hs h\u03bcs c x, f x\u27eb = \u27eac, f x\u27eb", "annotated_tactic": ["suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u27ea<a>indicatorConstLp</a> 2 hs h\u03bcs c x, f x\u27eb = \u27eac, f x\u27eb", [{"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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\n\ncase h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)"}, {"tactic": "exact set_integral_congr_ae hs h_ae_eq", "annotated_tactic": ["exact <a>set_integral_congr_ae</a> hs h_ae_eq", [{"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": "\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)\n\u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\n\ncase h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)", "state_after": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)"}, {"tactic": "have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2208 s \u2192 indicatorConstLp 2 hs h\u03bcs c x = c :=\n  indicatorConstLp_coeFn_mem", "annotated_tactic": ["have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2208 s \u2192 <a>indicatorConstLp</a> 2 hs h\u03bcs c x = c :=\n      <a>indicatorConstLp_coeFn_mem</a>", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.indicatorConstLp_coeFn_mem", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [752, 9], "def_end_pos": [752, 35]}]], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)", "state_after": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)"}, {"tactic": "refine' h_indicator.mono fun x hx hxs => _", "annotated_tactic": ["refine' h_indicator.mono fun x hx hxs => _", []], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)", "state_after": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nx : \u03b1\nhx : x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nhxs : x \u2208 s\n\u22a2 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nx : \u03b1\nhx : x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nhxs : x \u2208 s\n\u22a2 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x)", "state_after": "case h_ae_eq.e_a\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nx : \u03b1\nhx : x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nhxs : x \u2208 s\n\u22a2 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c"}, {"tactic": "exact hx hxs", "annotated_tactic": ["exact hx hxs", []], "state_before": "case h_ae_eq.e_a\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nx : \u03b1\nhx : x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c\nhxs : x \u2208 s\n\u22a2 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c", "state_after": "no goals"}, {"tactic": "suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2209 s \u2192 \u27eaindicatorConstLp 2 hs h\u03bcs c x, f x\u27eb = 0", "annotated_tactic": ["suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2209 s \u2192 \u27ea<a>indicatorConstLp</a> 2 hs h\u03bcs c x, f x\u27eb = 0", [{"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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f 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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0\n\ncase h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0"}, {"tactic": "have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2209 s \u2192 indicatorConstLp 2 hs h\u03bcs c x = 0 :=\n  indicatorConstLp_coeFn_nmem", "annotated_tactic": ["have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2209 s \u2192 <a>indicatorConstLp</a> 2 hs h\u03bcs c x = 0 :=\n      <a>indicatorConstLp_coeFn_nmem</a>", [{"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.indicatorConstLp_coeFn_nmem", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [756, 9], "def_end_pos": [756, 36]}]], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0", "state_after": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0"}, {"tactic": "refine' h_indicator.mono fun x hx hxs => _", "annotated_tactic": ["refine' h_indicator.mono fun x hx hxs => _", []], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0", "state_after": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nx : \u03b1\nhx : \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nhxs : \u00acx \u2208 s\n\u22a2 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0"}, {"tactic": "rw [hx hxs]", "annotated_tactic": ["rw [hx hxs]", []], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nx : \u03b1\nhx : \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nhxs : \u00acx \u2208 s\n\u22a2 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0", "state_after": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nx : \u03b1\nhx : \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nhxs : \u00acx \u2208 s\n\u22a2 inner 0 (\u2191\u2191f x) = 0"}, {"tactic": "exact inner_zero_left _", "annotated_tactic": ["exact <a>inner_zero_left</a> _", [{"full_name": "inner_zero_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 24]}]], "state_before": "case h_ae_eq\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nx : \u03b1\nhx : \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0\nhxs : \u00acx \u2208 s\n\u22a2 inner 0 (\u2191\u2191f x) = 0", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 Set.mem_compl_iff] at h_ae_eq", "annotated_tactic": ["simp_rw [\u2190 <a>Set.mem_compl_iff</a>] at h_ae_eq", [{"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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f 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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0"}, {"tactic": "suffices h_int_zero :\n  (\u222b x in s\u1d9c, inner (indicatorConstLp 2 hs h\u03bcs c x) (f x) \u2202\u03bc) = \u222b _ in s\u1d9c, (0 : \ud835\udd5c) \u2202\u03bc", "annotated_tactic": ["suffices h_int_zero :\n        (\u222b x in s\u1d9c, <a>inner</a> (<a>indicatorConstLp</a> 2 hs h\u03bcs c x) (f x) \u2202\u03bc) = \u222b _ in s\u1d9c, (0 : \ud835\udd5c) \u2202\u03bc", [{"full_name": "Inner.inner", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [79, 3], "def_end_pos": [79, 8]}, {"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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f 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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\nh_int_zero : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0\n\ncase h_int_zero\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc"}, {"tactic": "exact set_integral_congr_ae hs.compl h_ae_eq", "annotated_tactic": ["exact <a>set_integral_congr_ae</a> hs.compl h_ae_eq", [{"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 h_int_zero\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [h_int_zero]", "annotated_tactic": ["rw [h_int_zero]", []], "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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\nh_int_zero : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f 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\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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\nh_int_zero : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\ns : Set \u03b1\nf : { x // x \u2208 Lp E 2 }\nhs : MeasurableSet s\nc : E\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0\nh_int_zero : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc = 0", "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_Ioc", "start": [725, 1], "end": [730, 17], "traced_tactics": [{"tactic": "simpa only [exists_prop, mem_univ, true_and_iff] using\n  (@dense_univ \u03b1 _).borel_eq_generateFrom_Ioc_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n    mem_univ _", "annotated_tactic": ["simpa only [<a>exists_prop</a>, <a>mem_univ</a>, <a>true_and_iff</a>] using\n    (@<a>dense_univ</a> \u03b1 _).<a>borel_eq_generateFrom_Ioc_mem_aux</a> (fun _ _ => <a>mem_univ</a> _) fun _ _ _ _ =>\n      <a>mem_univ</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_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "dense_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [647, 9], "def_end_pos": [647, 19]}, {"full_name": "Dense.borel_eq_generateFrom_Ioc_mem_aux", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [704, 9], "def_end_pos": [704, 48]}, {"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": "\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\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 : SecondCountableTopology \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\n\u22a2 borel \u03b1 = MeasurableSpace.generateFrom {S | \u2203 l u, l < u \u2227 Ioc l u = S}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "full_name": "Option.join_ne_none'", "start": [113, 1], "end": [114, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Prime.lean", "full_name": "PosNum.minFacAux_to_nat", "start": [44, 1], "end": [54, 43], "traced_tactics": [{"tactic": "induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux]", "annotated_tactic": ["induction' fuel with fuel ih generalizing k <;> rw [<a>minFacAux</a>, <a>Nat.minFacAux</a>]", [{"full_name": "PosNum.minFacAux", "def_path": "Mathlib/Data/Num/Prime.lean", "def_pos": [37, 5], "def_end_pos": [37, 14]}, {"full_name": "Nat.minFacAux", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [250, 5], "def_end_pos": [250, 14]}]], "state_before": "fuel : \u2115\nn k : PosNum\nh : Nat.sqrt \u2191n < fuel + \u2191(bit1 k)\n\u22a2 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)", "state_after": "case zero\nfuel : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel + \u2191(bit1 k\u271d)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.zero + \u2191(bit1 k)\n\u22a2 \u2191n =\n    if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n\n    else\n      if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)\n      else\n        let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));\n        Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)\n\ncase succ\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\n\u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =\n    if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n\n    else\n      if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)\n      else\n        let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));\n        Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)"}, {"tactic": "simp_rw [\u2190 mul_to_nat]", "annotated_tactic": ["simp_rw [\u2190 <a>mul_to_nat</a>]", [{"full_name": "PosNum.mul_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [124, 9], "def_end_pos": [124, 19]}]], "state_before": "case succ\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\n\u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =\n    if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n\n    else\n      if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)\n      else\n        let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));\n        Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)", "state_after": "case succ\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\n\u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =\n    if h : \u2191n < \u2191(bit1 k * bit1 k) then \u2191n else if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k) else Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)"}, {"tactic": "simp only [cast_lt, dvd_to_nat]", "annotated_tactic": ["simp only [<a>cast_lt</a>, <a>dvd_to_nat</a>]", [{"full_name": "PosNum.cast_lt", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [699, 9], "def_end_pos": [699, 16]}, {"full_name": "PosNum.dvd_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [562, 9], "def_end_pos": [562, 19]}]], "state_before": "case succ\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\n\u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =\n    if h : \u2191n < \u2191(bit1 k * bit1 k) then \u2191n else if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k) else Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)", "state_after": "case succ\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\n\u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =\n    if h : n < bit1 k * bit1 k then \u2191n else if bit1 k \u2223 n then \u2191(bit1 k) else Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)"}, {"tactic": "split_ifs <;> try rfl", "annotated_tactic": ["split_ifs <;> try rfl", []], "state_before": "case succ\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\n\u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =\n    if h : n < bit1 k * bit1 k then \u2191n else if bit1 k \u2223 n then \u2191(bit1 k) else Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)", "state_after": "case neg\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d\u00b2 : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\nh\u271d\u00b9 : \u00acn < bit1 k * bit1 k\nh\u271d : \u00acbit1 k \u2223 n\n\u22a2 \u2191(minFacAux n fuel (succ k)) = Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)"}, {"tactic": "rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>\n  simp only [_root_.bit1, _root_.bit0, cast_bit1, cast_succ, Nat.succ_eq_add_one, add_assoc,\n    add_left_comm, \u2190 one_add_one_eq_two]", "annotated_tactic": ["rw [ih] <;> [congr; convert <a>Nat.lt_succ_of_lt</a> h using 1] <;>\n    simp only [<a>_root_.bit1</a>, <a>_root_.bit0</a>, <a>cast_bit1</a>, <a>cast_succ</a>, <a>Nat.succ_eq_add_one</a>, <a>add_assoc</a>,\n      <a>add_left_comm</a>, \u2190 <a>one_add_one_eq_two</a>]", [{"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": "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": "PosNum.cast_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [52, 9], "def_end_pos": [52, 18]}, {"full_name": "PosNum.cast_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"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]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "one_add_one_eq_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [218, 9], "def_end_pos": [218, 27]}]], "state_before": "case neg\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d\u00b2 : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\nh\u271d\u00b9 : \u00acn < bit1 k * bit1 k\nh\u271d : \u00acbit1 k \u2223 n\n\u22a2 \u2191(minFacAux n fuel (succ k)) = Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)", "state_after": "no goals"}, {"tactic": "rw [Nat.zero_add, Nat.sqrt_lt] at h", "annotated_tactic": ["rw [<a>Nat.zero_add</a>, <a>Nat.sqrt_lt</a>] at h", [{"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]}, {"full_name": "Nat.sqrt_lt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}]], "state_before": "case zero\nfuel : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel + \u2191(bit1 k\u271d)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.zero + \u2191(bit1 k)\n\u22a2 \u2191n =\n    if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n\n    else\n      if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)\n      else\n        let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));\n        Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)", "state_after": "case zero\nfuel : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel + \u2191(bit1 k\u271d)\nk : PosNum\nh : \u2191n < \u2191(bit1 k) * \u2191(bit1 k)\n\u22a2 \u2191n =\n    if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n\n    else\n      if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)\n      else\n        let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));\n        Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)"}, {"tactic": "simp only [h, dite_true]", "annotated_tactic": ["simp only [h, <a>dite_true</a>]", [{"full_name": "dite_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [79, 17], "def_end_pos": [79, 26]}]], "state_before": "case zero\nfuel : \u2115\nn k\u271d : PosNum\nh\u271d : Nat.sqrt \u2191n < fuel + \u2191(bit1 k\u271d)\nk : PosNum\nh : \u2191n < \u2191(bit1 k) * \u2191(bit1 k)\n\u22a2 \u2191n =\n    if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n\n    else\n      if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)\n      else\n        let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));\n        Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\nfuel\u271d : \u2115\nn k\u271d : PosNum\nh\u271d\u00b2 : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d)\nfuel : \u2115\nih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k)\nk : PosNum\nh : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k)\nh\u271d\u00b9 : \u00acn < bit1 k * bit1 k\nh\u271d : bit1 k \u2223 n\n\u22a2 \u2191(bit1 k) = \u2191(bit1 k)", "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", "start": [1014, 1], "end": [1016, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.submartingale_iff_condexp_sub_nonneg", "start": [324, 1], "end": [328, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_union_eq_left_of_ae", "start": [302, 1], "end": [318, 91], "traced_tactics": [{"tactic": "have ht : IntegrableOn f t \u03bc := by apply integrableOn_zero.congr_fun_ae; symm; exact ht_eq", "annotated_tactic": ["have ht : <a>IntegrableOn</a> f t \u03bc := by apply integrableOn_zero.congr_fun_ae; symm; exact ht_eq", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "by_cases H : IntegrableOn f (s \u222a t) \u03bc", "annotated_tactic": ["by_cases H : <a>IntegrableOn</a> f (s \u222a t) \u03bc", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : \u00acIntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : \u00acIntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : \u00acIntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "let f' := H.1.mk f", "annotated_tactic": ["let f' := H.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": "case pos\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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "calc\n  \u222b x : \u03b1 in s \u222a t, f x \u2202\u03bc = \u222b x : \u03b1 in s \u222a t, f' x \u2202\u03bc := integral_congr_ae H.1.ae_eq_mk\n  _ = \u222b x in s, f' x \u2202\u03bc := by\n    apply\n      integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk)\n    filter_upwards [ht_eq,\n      ae_mono (Measure.restrict_mono (subset_union_right s t) le_rfl) H.1.ae_eq_mk] with x hx h'x\n    rw [\u2190 h'x, hx]\n  _ = \u222b x in s, f x \u2202\u03bc :=\n    integral_congr_ae\n      (ae_mono (Measure.restrict_mono (subset_union_left s t) le_rfl) H.1.ae_eq_mk.symm)", "annotated_tactic": ["calc\n    \u222b x : \u03b1 in s \u222a t, f x \u2202\u03bc = \u222b x : \u03b1 in s \u222a t, f' x \u2202\u03bc := <a>integral_congr_ae</a> H.1.<a>ae_eq_mk</a>\n    _ = \u222b x in s, f' x \u2202\u03bc := by\n      apply\n        <a>integral_union_eq_left_of_ae_aux</a> _ H.1.<a>stronglyMeasurable_mk</a> (H.congr_fun_ae H.1.<a>ae_eq_mk</a>)\n      filter_upwards [ht_eq,\n        <a>ae_mono</a> (<a>Measure.restrict_mono</a> (<a>subset_union_right</a> s t) <a>le_rfl</a>) H.1.<a>ae_eq_mk</a>] with x hx h'x\n      rw [\u2190 h'x, hx]\n    _ = \u222b x in s, f x \u2202\u03bc :=\n      <a>integral_congr_ae</a>\n        (<a>ae_mono</a> (<a>Measure.restrict_mono</a> (<a>subset_union_left</a> s t) <a>le_rfl</a>) H.1.ae_eq_mk.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.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "MeasureTheory.integral_union_eq_left_of_ae_aux", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [285, 9], "def_end_pos": [285, 41]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"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.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [833, 9], "def_end_pos": [833, 27]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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": "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.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply integrableOn_zero.congr_fun_ae", "annotated_tactic": ["apply integrableOn_zero.congr_fun_ae", []], "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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\n\u22a2 IntegrableOn f t", "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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\n\u22a2 (fun x => 0) =\u1d50[Measure.restrict \u03bc t] f"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\n\u22a2 (fun x => 0) =\u1d50[Measure.restrict \u03bc t] 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] fun x => 0"}, {"tactic": "exact ht_eq", "annotated_tactic": ["exact ht_eq", []], "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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] fun x => 0", "state_after": "no goals"}, {"tactic": "rw [integral_undef H, integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a> H, <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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : \u00acIntegrableOn f (s \u222a t)\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : \u00acIntegrableOn f (s \u222a t)\n\u22a2 \u00acIntegrable fun x => f x"}, {"tactic": "simpa [integrableOn_union, ht] using H", "annotated_tactic": ["simpa [<a>integrableOn_union</a>, ht] using H", [{"full_name": "MeasureTheory.integrableOn_union", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [186, 9], "def_end_pos": [186, 27]}]], "state_before": "case neg\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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : \u00acIntegrableOn f (s \u222a t)\n\u22a2 \u00acIntegrable fun x => f x", "state_after": "no goals"}, {"tactic": "apply\n  integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk)", "annotated_tactic": ["apply\n        <a>integral_union_eq_left_of_ae_aux</a> _ H.1.<a>stronglyMeasurable_mk</a> (H.congr_fun_ae H.1.<a>ae_eq_mk</a>)", [{"full_name": "MeasureTheory.integral_union_eq_left_of_ae_aux", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [285, 9], "def_end_pos": [285, 41]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\n\u22a2 \u222b (x : \u03b1) in s \u222a t, f' x \u2202\u03bc = \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\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\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t,\n    AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x = 0"}, {"tactic": "filter_upwards [ht_eq,\n  ae_mono (Measure.restrict_mono (subset_union_right s t) le_rfl) H.1.ae_eq_mk] with x hx h'x", "annotated_tactic": ["filter_upwards [ht_eq,\n        <a>ae_mono</a> (<a>Measure.restrict_mono</a> (<a>subset_union_right</a> s t) <a>le_rfl</a>) H.1.<a>ae_eq_mk</a>] with x hx h'x", [{"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.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [833, 9], "def_end_pos": [833, 27]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t,\n    AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x = 0", "state_after": "case 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\nx : \u03b1\nhx : f x = 0\nh'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x\n\u22a2 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x = 0"}, {"tactic": "rw [\u2190 h'x, hx]", "annotated_tactic": ["rw [\u2190 h'x, hx]", []], "state_before": "case 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0\nht : IntegrableOn f t\nH : IntegrableOn f (s \u222a t)\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t)))\nx : \u03b1\nhx : f x = 0\nh'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x\n\u22a2 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.Nontrivial.ne_singleton", "start": [2570, 1], "end": [2572, 36], "traced_tactics": [{"tactic": "rw [H] at hs", "annotated_tactic": ["rw [H] at hs", []], "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 : \u03b1\nhs : Set.Nontrivial s\nH : s = {x}\n\u22a2 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 : \u03b1\nhs : Set.Nontrivial {x}\nH : s = {x}\n\u22a2 False"}, {"tactic": "exact not_nontrivial_singleton hs", "annotated_tactic": ["exact <a>not_nontrivial_singleton</a> hs", [{"full_name": "Set.not_nontrivial_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2564, 9], "def_end_pos": [2564, 33]}]], "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 : \u03b1\nhs : Set.Nontrivial {x}\nH : s = {x}\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.apply_le_one", "start": [182, 9], "end": [183, 43], "traced_tactics": [{"tactic": "simpa using apply_mono \u03bc (subset_univ s)", "annotated_tactic": ["simpa using <a>apply_mono</a> \u03bc (<a>subset_univ</a> s)", [{"full_name": "MeasureTheory.ProbabilityMeasure.apply_mono", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "Function.Periodic.intervalIntegral_add_zsmul_eq", "start": [276, 1], "end": [295, 60], "traced_tactics": [{"tactic": "suffices (\u222b x in (0)..(n \u2022 T), f x) = n \u2022 \u222b x in (0)..T, f x by\n  simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).intervalIntegral_add_eq t 0, zero_add,\n    this]", "annotated_tactic": ["suffices (\u222b x in (0)..(n \u2022 T), f x) = n \u2022 \u222b x in (0)..T, f x by\n    simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).<a>intervalIntegral_add_eq</a> t 0, <a>zero_add</a>,\n      this]", [{"full_name": "Function.Periodic.intervalIntegral_add_eq", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [256, 9], "def_end_pos": [256, 32]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\n\u22a2 \u222b (x : \u211d) in t..t + n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in t..t + T, f x", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\n\u22a2 \u222b (x : \u211d) in 0 ..n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in 0 ..T, f x"}, {"tactic": "cases' n with n n", "annotated_tactic": ["cases' n with n n", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\n\u22a2 \u222b (x : \u211d) in 0 ..n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in 0 ..T, f x", "state_after": "case ofNat\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..Int.ofNat n \u2022 T, f x = Int.ofNat n \u2022 \u222b (x : \u211d) in 0 ..T, f x\n\ncase negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = Int.negSucc n \u2022 \u222b (x : \u211d) in 0 ..T, f x"}, {"tactic": "simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).intervalIntegral_add_eq t 0, zero_add,\n  this]", "annotated_tactic": ["simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).<a>intervalIntegral_add_eq</a> t 0, <a>zero_add</a>,\n      this]", [{"full_name": "Function.Periodic.intervalIntegral_add_eq", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [256, 9], "def_end_pos": [256, 32]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u222b (x : \u211d) in 0 ..n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in 0 ..T, f x\n\u22a2 \u222b (x : \u211d) in t..t + n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in t..t + T, f x", "state_after": "no goals"}, {"tactic": "induction' m with m ih", "annotated_tactic": ["induction' m with m ih", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nm : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x", "state_after": "case zero\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\n\u22a2 \u222b (x : \u211d) in 0 ..Nat.zero \u2022 T, f x = Nat.zero \u2022 \u222b (x : \u211d) in 0 ..T, f x\n\ncase succ\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nm : \u2115\nih : \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\n\u22a2 \u222b (x : \u211d) in 0 ..Nat.succ m \u2022 T, f x = Nat.succ m \u2022 \u222b (x : \u211d) in 0 ..T, f x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\n\u22a2 \u222b (x : \u211d) in 0 ..Nat.zero \u2022 T, f x = Nat.zero \u2022 \u222b (x : \u211d) in 0 ..T, f x", "state_after": "no goals"}, {"tactic": "simp only [succ_nsmul', hf.intervalIntegral_add_eq_add 0 (m \u2022 T) h_int, ih, zero_add]", "annotated_tactic": ["simp only [<a>succ_nsmul'</a>, hf.intervalIntegral_add_eq_add 0 (m \u2022 T) h_int, ih, <a>zero_add</a>]", [{"full_name": "succ_nsmul'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [212, 15], "def_end_pos": [212, 26]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case succ\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nn : \u2124\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nm : \u2115\nih : \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\n\u22a2 \u222b (x : \u211d) in 0 ..Nat.succ m \u2022 T, f x = Nat.succ m \u2022 \u222b (x : \u211d) in 0 ..T, f x", "state_after": "no goals"}, {"tactic": "simp [\u2190 this n]", "annotated_tactic": ["simp [\u2190 this n]", []], "state_before": "case ofNat\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..Int.ofNat n \u2022 T, f x = Int.ofNat n \u2022 \u222b (x : \u211d) in 0 ..T, f x", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [negSucc_zsmul]", "annotated_tactic": ["conv_rhs => rw [<a>negSucc_zsmul</a>]", [{"full_name": "negSucc_zsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [963, 9], "def_end_pos": [963, 22]}]], "state_before": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = Int.negSucc n \u2022 \u222b (x : \u211d) in 0 ..T, f x", "state_after": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = -((n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, f x)"}, {"tactic": "have h\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0 := by simp; linarith", "annotated_tactic": ["have h\u2080 : <a>Int.negSucc</a> n \u2022 T + (n + 1) \u2022 T = 0 := by simp; linarith", [{"full_name": "Int.negSucc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 12]}]], "state_before": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = -((n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, f x)", "state_after": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\nh\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0\n\u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = -((n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, f x)"}, {"tactic": "rw [integral_symm, \u2190 (hf.nsmul (n + 1)).funext, neg_inj]", "annotated_tactic": ["rw [<a>integral_symm</a>, \u2190 (hf.nsmul (n + 1)).<a>funext</a>, <a>neg_inj</a>]", [{"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}, {"full_name": "Function.Periodic.funext", "def_path": "Mathlib/Algebra/Periodic.lean", "def_pos": [53, 19], "def_end_pos": [53, 34]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [256, 3], "def_end_pos": [256, 14]}]], "state_before": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\nh\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0\n\u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = -((n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, f x)", "state_after": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\nh\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0\n\u22a2 \u222b (x : \u211d) in Int.negSucc n \u2022 T..0, (fun x => f (x + (n + 1) \u2022 T)) x =\n    (n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, (fun x => f (x + (n + 1) \u2022 T)) x"}, {"tactic": "simp_rw [integral_comp_add_right, h\u2080, zero_add, this (n + 1), add_comm T,\n  hf.intervalIntegral_add_eq ((n + 1) \u2022 T) 0, zero_add]", "annotated_tactic": ["simp_rw [<a>integral_comp_add_right</a>, h\u2080, <a>zero_add</a>, this (n + 1), <a>add_comm</a> T,\n      hf.intervalIntegral_add_eq ((n + 1) \u2022 T) 0, <a>zero_add</a>]", [{"full_name": "intervalIntegral.integral_comp_add_right", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [735, 9], "def_end_pos": [735, 32]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case negSucc\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\nh\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0\n\u22a2 \u222b (x : \u211d) in Int.negSucc n \u2022 T..0, (fun x => f (x + (n + 1) \u2022 T)) x =\n    (n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, (fun x => f (x + (n + 1) \u2022 T)) x", "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 : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 (-1 + -\u2191n) * T + (\u2191n + 1) * T = 0"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nt : \u211d\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082\nthis : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x\nn : \u2115\n\u22a2 (-1 + -\u2191n) * T + (\u2191n + 1) * T = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_findGreatest", "start": [487, 1], "end": [492, 44], "traced_tactics": [{"tactic": "refine' measurable_findGreatest' fun k hk => _", "annotated_tactic": ["refine' <a>measurable_findGreatest'</a> fun k hk => _", [{"full_name": "measurable_findGreatest'", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 33]}]], "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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\n\u22a2 Measurable fun x => Nat.findGreatest (p x) N", "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 t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\n\u22a2 MeasurableSet {x | Nat.findGreatest (p x) N = k}"}, {"tactic": "simp only [Nat.findGreatest_eq_iff, setOf_and, setOf_forall, \u2190 compl_setOf]", "annotated_tactic": ["simp only [<a>Nat.findGreatest_eq_iff</a>, <a>setOf_and</a>, <a>setOf_forall</a>, \u2190 <a>compl_setOf</a>]", [{"full_name": "Nat.findGreatest_eq_iff", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 28]}, {"full_name": "Set.setOf_and", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 18]}, {"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}, {"full_name": "Set.compl_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1649, 9], "def_end_pos": [1649, 20]}]], "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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\n\u22a2 MeasurableSet {x | Nat.findGreatest (p x) N = k}", "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 t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\n\u22a2 MeasurableSet ({a | k \u2264 N} \u2229 ((\u22c2 (_ : k \u2260 0), {x | p x k}) \u2229 \u22c2 i, \u22c2 (_ : k < i), \u22c2 (_ : i \u2264 N), {a | p a i}\u1d9c))"}, {"tactic": "repeat' apply_rules [MeasurableSet.inter, MeasurableSet.const, MeasurableSet.iInter,\n  MeasurableSet.compl, hN] <;> try intros", "annotated_tactic": ["repeat' apply_rules [<a>MeasurableSet.inter</a>, <a>MeasurableSet.const</a>, <a>MeasurableSet.iInter</a>,\n    <a>MeasurableSet.compl</a>, hN] <;> try intros", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSet.const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [241, 19], "def_end_pos": [241, 38]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"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\u03b9\ns t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\n\u22a2 MeasurableSet ({a | k \u2264 N} \u2229 ((\u22c2 (_ : k \u2260 0), {x | p x k}) \u2229 \u22c2 i, \u22c2 (_ : k < i), \u22c2 (_ : i \u2264 N), {a | p a i}\u1d9c))", "state_after": "no goals"}, {"tactic": "apply_rules [MeasurableSet.inter, MeasurableSet.const, MeasurableSet.iInter,\nMeasurableSet.compl, hN] <;> try intros", "annotated_tactic": ["apply_rules [<a>MeasurableSet.inter</a>, <a>MeasurableSet.const</a>, <a>MeasurableSet.iInter</a>,\n    <a>MeasurableSet.compl</a>, hN] <;> try intros", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSet.const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [241, 19], "def_end_pos": [241, 38]}, {"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "state_before": "case h\u2082.h\u2082\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\nb\u271d\u00b2 : \u2115\nb\u271d\u00b9 : k < b\u271d\u00b2\nb\u271d : b\u271d\u00b2 \u2264 N\n\u22a2 MeasurableSet {a | p a b\u271d\u00b2}\u1d9c", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case h\u2082.h\u2082\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\nb\u271d\u00b9 : \u2115\nb\u271d : k < b\u271d\u00b9\n\u22a2 b\u271d\u00b9 \u2264 N \u2192 MeasurableSet {a | p a b\u271d\u00b9}\u1d9c", "state_after": "case h\u2082.h\u2082\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : \u03b1 \u2192 \u2115 \u2192 Prop\ninst\u271d : (x : \u03b1) \u2192 DecidablePred (p x)\nN : \u2115\nhN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k}\nk : \u2115\nhk : k \u2264 N\nb\u271d\u00b2 : \u2115\nb\u271d\u00b9 : k < b\u271d\u00b2\nb\u271d : b\u271d\u00b2 \u2264 N\n\u22a2 MeasurableSet {a | p a b\u271d\u00b2}\u1d9c"}]}, {"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_replaceF", "start": [1505, 1], "end": [1513, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpInd_disjoint_union", "start": [321, 1], "end": [324, 79], "traced_tactics": [{"tactic": "ext1 x", "annotated_tactic": ["ext1 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\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\n\u22a2 condexpInd G hm \u03bc (s \u222a t) = condexpInd G hm \u03bc s + condexpInd G hm \u03bc t", "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 \u2191(condexpInd G hm \u03bc (s \u222a t)) x = \u2191(condexpInd G hm \u03bc s + condexpInd G hm \u03bc t) x"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 \u2191(condexpInd G hm \u03bc (s \u222a t)) x = \u2191(condexpInd G hm \u03bc s + condexpInd G hm \u03bc t) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 \u2191(condexpInd G hm \u03bc (s \u222a t)) x = (\u2191(condexpInd G hm \u03bc s) + \u2191(condexpInd G hm \u03bc t)) x"}, {"tactic": "exact condexpInd_disjoint_union_apply hs ht h\u03bcs h\u03bct hst x", "annotated_tactic": ["exact <a>condexpInd_disjoint_union_apply</a> hs ht h\u03bcs h\u03bct hst x", [{"full_name": "MeasureTheory.condexpInd_disjoint_union_apply", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [315, 9], "def_end_pos": [315, 40]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 \u2191(condexpInd G hm \u03bc (s \u222a t)) x = (\u2191(condexpInd G hm \u03bc s) + \u2191(condexpInd G hm \u03bc t)) x", "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_of_lt", "start": [133, 1], "end": [134, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "ProbabilityTheory.condexp_ae_eq_integral_condexpKernel", "start": [199, 1], "end": [202, 95], "traced_tactics": [{"tactic": "rw [inf_of_le_left hm]", "annotated_tactic": ["rw [<a>inf_of_le_left</a> hm]", [{"full_name": "inf_of_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [454, 22], "def_end_pos": [454, 36]}]], "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\nhm : m \u2264 m\u03a9\nhf_int : Integrable f\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] \u03bc[f|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.disjoint_of_disjoint_cons_left", "start": [1537, 1], "end": [1538, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.singularPart_smul", "start": [268, 1], "end": [282, 64], "traced_tactics": [{"tactic": "by_cases hr : r = 0", "annotated_tactic": ["by_cases hr : r = 0", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : r = 0\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd"}, {"tactic": "by_cases hl : HaveLebesgueDecomposition \u03bc \u03bd", "annotated_tactic": ["by_cases hl : <a>HaveLebesgueDecomposition</a> \u03bc \u03bd", [{"full_name": "MeasureTheory.Measure.HaveLebesgueDecomposition", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [79, 7], "def_end_pos": [79, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : HaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd"}, {"tactic": "rw [hr, zero_smul, zero_smul, singularPart_zero]", "annotated_tactic": ["rw [hr, <a>zero_smul</a>, <a>zero_smul</a>, <a>singularPart_zero</a>]", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "MeasureTheory.Measure.singularPart_zero", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [263, 9], "def_end_pos": [263, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : r = 0\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd", "state_after": "no goals"}, {"tactic": "haveI := hl", "annotated_tactic": ["haveI := hl", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : HaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl this : HaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd"}, {"tactic": "refine'\n  (eq_singularPart ((measurable_rnDeriv \u03bc \u03bd).const_smul (r : \u211d\u22650\u221e))\n      (MutuallySingular.smul r (haveLebesgueDecomposition_spec _ _).2.1) _).symm", "annotated_tactic": ["refine'\n      (<a>eq_singularPart</a> ((<a>measurable_rnDeriv</a> \u03bc \u03bd).<a>const_smul</a> (r : \u211d\u22650\u221e))\n          (<a>MutuallySingular.smul</a> r (<a>haveLebesgueDecomposition_spec</a> _ _).2.1) _).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.eq_singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [228, 9], "def_end_pos": [228, 24]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}, {"full_name": "Measurable.const_smul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [650, 9], "def_end_pos": [650, 30]}, {"full_name": "MeasureTheory.Measure.MutuallySingular.smul", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [118, 9], "def_end_pos": [118, 13]}, {"full_name": "MeasureTheory.Measure.haveLebesgueDecomposition_spec", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 39]}, {"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\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl this : HaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl this : HaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 r \u2022 \u03bc = \u2191r \u2022 singularPart \u03bc \u03bd + withDensity \u03bd (\u2191r \u2022 rnDeriv \u03bc \u03bd)"}, {"tactic": "rw [withDensity_smul _ (measurable_rnDeriv _ _), \u2190 smul_add,\n  \u2190 haveLebesgueDecomposition_add \u03bc \u03bd, ENNReal.smul_def]", "annotated_tactic": ["rw [<a>withDensity_smul</a> _ (<a>measurable_rnDeriv</a> _ _), \u2190 <a>smul_add</a>,\n      \u2190 <a>haveLebesgueDecomposition_add</a> \u03bc \u03bd, <a>ENNReal.smul_def</a>]", [{"full_name": "MeasureTheory.withDensity_smul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [89, 9], "def_end_pos": [89, 25]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}, {"full_name": "MeasureTheory.Measure.haveLebesgueDecomposition_add", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [106, 9], "def_end_pos": [106, 38]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl this : HaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 r \u2022 \u03bc = \u2191r \u2022 singularPart \u03bc \u03bd + withDensity \u03bd (\u2191r \u2022 rnDeriv \u03bc \u03bd)", "state_after": "no goals"}, {"tactic": "rw [singularPart, singularPart, dif_neg hl, dif_neg, smul_zero]", "annotated_tactic": ["rw [<a>singularPart</a>, <a>singularPart</a>, <a>dif_neg</a> hl, <a>dif_neg</a>, <a>smul_zero</a>]", [{"full_name": "MeasureTheory.Measure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [88, 17], "def_end_pos": [88, 29]}, {"full_name": "MeasureTheory.Measure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [88, 17], "def_end_pos": [88, 29]}, {"full_name": "dif_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [812, 9], "def_end_pos": [812, 16]}, {"full_name": "dif_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [812, 9], "def_end_pos": [812, 16]}, {"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\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd", "state_after": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 \u00acHaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd"}, {"tactic": "refine' fun hl' => hl _", "annotated_tactic": ["refine' fun hl' => hl _", []], "state_before": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\n\u22a2 \u00acHaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd", "state_after": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\nhl' : HaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd\n\u22a2 HaveLebesgueDecomposition \u03bc \u03bd"}, {"tactic": "rw [\u2190 inv_smul_smul\u2080 hr \u03bc]", "annotated_tactic": ["rw [\u2190 <a>inv_smul_smul\u2080</a> hr \u03bc]", [{"full_name": "inv_smul_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [192, 9], "def_end_pos": [192, 23]}]], "state_before": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\nhl' : HaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd\n\u22a2 HaveLebesgueDecomposition \u03bc \u03bd", "state_after": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\nhl' : HaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd\n\u22a2 HaveLebesgueDecomposition (r\u207b\u00b9 \u2022 r \u2022 \u03bc) \u03bd"}, {"tactic": "exact @Measure.haveLebesgueDecomposition_smul _ _ _ _ hl' _", "annotated_tactic": ["exact @<a>Measure.haveLebesgueDecomposition_smul</a> _ _ _ _ hl' _", [{"full_name": "MeasureTheory.Measure.haveLebesgueDecomposition_smul", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [111, 10], "def_end_pos": [111, 40]}]], "state_before": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\nhr : \u00acr = 0\nhl : \u00acHaveLebesgueDecomposition \u03bc \u03bd\nhl' : HaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd\n\u22a2 HaveLebesgueDecomposition (r\u207b\u00b9 \u2022 r \u2022 \u03bc) \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.norm_setToL1_le_mul_norm", "start": [1219, 1], "end": [1224, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Option.lean", "full_name": "Finset.eraseNone_image_some", "start": [107, 1], "end": [108, 92], "traced_tactics": [{"tactic": "simpa only [map_eq_image] using eraseNone_map_some s", "annotated_tactic": ["simpa only [<a>map_eq_image</a>] using <a>eraseNone_map_some</a> s", [{"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [342, 9], "def_end_pos": [342, 21]}, {"full_name": "Finset.eraseNone_map_some", "def_path": "Mathlib/Data/Finset/Option.lean", "def_pos": [101, 9], "def_end_pos": [101, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq (Option \u03b1)\ns : Finset \u03b1\n\u22a2 \u2191eraseNone (image some s) = s", "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.split_of_valid", "start": [478, 1], "end": [479, 55], "traced_tactics": [{"tactic": "simpa [split] using splitAux_of_valid p [] [] s.1 []", "annotated_tactic": ["simpa [<a>split</a>] using <a>splitAux_of_valid</a> p [] [] s.1 []", [{"full_name": "String.split", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [245, 19], "def_end_pos": [245, 24]}, {"full_name": "String.splitAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [465, 9], "def_end_pos": [465, 26]}]], "state_before": "s : String\np : Char \u2192 Bool\n\u22a2 split s p = List.map mk (List.splitOnP p s.data)", "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.natAbs_sign_of_nonzero", "start": [204, 1], "end": [205, 34], "traced_tactics": [{"tactic": "rw [Int.natAbs_sign, if_neg hz]", "annotated_tactic": ["rw [<a>Int.natAbs_sign</a>, <a>if_neg</a> hz]", [{"full_name": "Int.natAbs_sign", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [201, 9], "def_end_pos": [201, 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": "z : Int\nhz : z \u2260 0\n\u22a2 natAbs (sign z) = 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_limsup", "start": [1527, 1], "end": [1529, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.indep_biSup_limsup", "start": [75, 1], "end": [80, 45], "traced_tactics": [{"tactic": "refine' indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) _", "annotated_tactic": ["refine' <a>indep_of_indep_of_le_right</a> (<a>indep_biSup_compl</a> h_le h_indep t) _", [{"full_name": "ProbabilityTheory.indep_of_indep_of_le_right", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [270, 9], "def_end_pos": [270, 35]}, {"full_name": "ProbabilityTheory.indep_biSup_compl", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [55, 9], "def_end_pos": [55, 26]}]], "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\nt : Set \u03b9\nht : p t\n\u22a2 Indep (\u2a06 n \u2208 t, 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\nt : Set \u03b9\nht : p t\n\u22a2 limsup s f \u2264 \u2a06 n \u2208 t\u1d9c, s n"}, {"tactic": "refine' limsSup_le_of_le (by isBoundedDefault) _", "annotated_tactic": ["refine' <a>limsSup_le_of_le</a> (by isBoundedDefault) _", [{"full_name": "Filter.limsSup_le_of_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [491, 9], "def_end_pos": [491, 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\nt : Set \u03b9\nht : p t\n\u22a2 limsup s f \u2264 \u2a06 n \u2208 t\u1d9c, 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\nt : Set \u03b9\nht : p t\n\u22a2 \u2200\u1da0 (n : MeasurableSpace \u03a9) in map s f, n \u2264 \u2a06 n \u2208 t\u1d9c, s n"}, {"tactic": "simp only [Set.mem_compl_iff, eventually_map]", "annotated_tactic": ["simp only [<a>Set.mem_compl_iff</a>, <a>eventually_map</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": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}]], "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\nt : Set \u03b9\nht : p t\n\u22a2 \u2200\u1da0 (n : MeasurableSpace \u03a9) in map s f, n \u2264 \u2a06 n \u2208 t\u1d9c, 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\nt : Set \u03b9\nht : p t\n\u22a2 \u2200\u1da0 (a : \u03b9) in f, s a \u2264 \u2a06 n, \u2a06 (_ : \u00acn \u2208 t), s n"}, {"tactic": "exact eventually_of_mem (hf t ht) le_iSup\u2082", "annotated_tactic": ["exact <a>eventually_of_mem</a> (hf t ht) <a>le_iSup\u2082</a>", [{"full_name": "Filter.eventually_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1098, 9], "def_end_pos": [1098, 26]}, {"full_name": "le_iSup\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [857, 9], "def_end_pos": [857, 17]}]], "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\nt : Set \u03b9\nht : p t\n\u22a2 \u2200\u1da0 (a : \u03b9) in f, s a \u2264 \u2a06 n, \u2a06 (_ : \u00acn \u2208 t), s n", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "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\nt : Set \u03b9\nht : p t\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map s f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.move_right_n_head", "start": [649, 1], "end": [653, 90], "traced_tactics": [{"tactic": "induction i generalizing T", "annotated_tactic": ["induction i generalizing T", []], "state_before": "\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2115\n\u22a2 ((move Dir.right)^[i] T).head = nth T \u2191i", "state_after": "case zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 ((move Dir.right)^[Nat.zero] T).head = nth T \u2191Nat.zero\n\ncase succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nn\u271d : \u2115\nn_ih\u271d : \u2200 (T : Tape \u0393), ((move Dir.right)^[n\u271d] T).head = nth T \u2191n\u271d\nT : Tape \u0393\n\u22a2 ((move Dir.right)^[Nat.succ n\u271d] T).head = nth T \u2191(Nat.succ n\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 ((move Dir.right)^[Nat.zero] T).head = nth T \u2191Nat.zero", "state_after": "no goals"}, {"tactic": "simp only [*, Tape.move_right_nth, Int.ofNat_succ, iterate_succ, Function.comp_apply]", "annotated_tactic": ["simp only [*, <a>Tape.move_right_nth</a>, <a>Int.ofNat_succ</a>, <a>iterate_succ</a>, <a>Function.comp_apply</a>]", [{"full_name": "Turing.Tape.move_right_nth", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [642, 9], "def_end_pos": [642, 28]}, {"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": "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\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nn\u271d : \u2115\nn_ih\u271d : \u2200 (T : Tape \u0393), ((move Dir.right)^[n\u271d] T).head = nth T \u2191n\u271d\nT : Tape \u0393\n\u22a2 ((move Dir.right)^[Nat.succ n\u271d] T).head = nth T \u2191(Nat.succ n\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "parallelepiped_orthonormalBasis_one_dim", "start": [84, 1], "end": [111, 96], "traced_tactics": [{"tactic": "have e : \u03b9 \u2243 Fin 1 := by\n  apply Fintype.equivFinOfCardEq\n  simp only [\u2190 finrank_eq_card_basis b.toBasis, finrank_self]", "annotated_tactic": ["have e : \u03b9 \u2243 <a>Fin</a> 1 := by\n    apply <a>Fintype.equivFinOfCardEq</a>\n    simp only [\u2190 <a>finrank_eq_card_basis</a> b.toBasis, <a>finrank_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": "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]}, {"full_name": "FiniteDimensional.finrank_self", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [151, 9], "def_end_pos": [151, 21]}]], "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\n\u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0", "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\n\u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0"}, {"tactic": "have B : parallelepiped (b.reindex e) = parallelepiped b := by\n  convert parallelepiped_comp_equiv b e.symm\n  ext i\n  simp only [OrthonormalBasis.coe_reindex]", "annotated_tactic": ["have B : <a>parallelepiped</a> (b.reindex e) = <a>parallelepiped</a> b := by\n    convert <a>parallelepiped_comp_equiv</a> b e.symm\n    ext i\n    simp only [<a>OrthonormalBasis.coe_reindex</a>]", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}, {"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}, {"full_name": "parallelepiped_comp_equiv", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [62, 9], "def_end_pos": [62, 34]}, {"full_name": "OrthonormalBasis.coe_reindex", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [597, 19], "def_end_pos": [597, 30]}]], "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\n\u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0", "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\n\u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0"}, {"tactic": "rw [\u2190 B]", "annotated_tactic": ["rw [\u2190 B]", []], "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\n\u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0", "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0"}, {"tactic": "let F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t => fun _i => t", "annotated_tactic": ["let F : \u211d \u2192 <a>Fin</a> 1 \u2192 \u211d := fun t => fun _i => t", [{"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 : 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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0"}, {"tactic": "rcases orthonormalBasis_one_dim (b.reindex e) with (H | H)", "annotated_tactic": ["rcases <a>orthonormalBasis_one_dim</a> (b.reindex e) with (H | H)", [{"full_name": "orthonormalBasis_one_dim", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [840, 9], "def_end_pos": [840, 33]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0", "state_after": "case inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => 1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0\n\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => -1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0"}, {"tactic": "apply Fintype.equivFinOfCardEq", "annotated_tactic": ["apply <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\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\nb : OrthonormalBasis \u03b9 \u211d \u211d\n\u22a2 \u03b9 \u2243 Fin 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\nb : OrthonormalBasis \u03b9 \u211d \u211d\n\u22a2 Fintype.card \u03b9 = 1"}, {"tactic": "simp only [\u2190 finrank_eq_card_basis b.toBasis, finrank_self]", "annotated_tactic": ["simp only [\u2190 <a>finrank_eq_card_basis</a> b.toBasis, <a>finrank_self</a>]", [{"full_name": "FiniteDimensional.finrank_eq_card_basis", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [135, 9], "def_end_pos": [135, 30]}, {"full_name": "FiniteDimensional.finrank_self", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [151, 9], "def_end_pos": [151, 21]}]], "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\n\u22a2 Fintype.card \u03b9 = 1", "state_after": "no goals"}, {"tactic": "convert parallelepiped_comp_equiv b e.symm", "annotated_tactic": ["convert <a>parallelepiped_comp_equiv</a> b e.symm", [{"full_name": "parallelepiped_comp_equiv", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [62, 9], "def_end_pos": [62, 34]}]], "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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b", "state_after": "case h.e'_2.h.e'_6\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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\n\u22a2 \u2191(OrthonormalBasis.reindex b e) = \u2191b \u2218 \u2191e.symm"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "case h.e'_2.h.e'_6\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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\n\u22a2 \u2191(OrthonormalBasis.reindex b e) = \u2191b \u2218 \u2191e.symm", "state_after": "case h.e'_2.h.e'_6.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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\ni : Fin 1\n\u22a2 \u2191(OrthonormalBasis.reindex b e) i = (\u2191b \u2218 \u2191e.symm) i"}, {"tactic": "simp only [OrthonormalBasis.coe_reindex]", "annotated_tactic": ["simp only [<a>OrthonormalBasis.coe_reindex</a>]", [{"full_name": "OrthonormalBasis.coe_reindex", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [597, 19], "def_end_pos": [597, 30]}]], "state_before": "case h.e'_2.h.e'_6.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\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\ni : Fin 1\n\u22a2 \u2191(OrthonormalBasis.reindex b e) i = (\u2191b \u2218 \u2191e.symm) i", "state_after": "no goals"}, {"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": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\n\u22a2 Icc 0 1 = F '' Icc 0 1", "state_after": "case h\u2081\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\n\u22a2 Icc 0 1 \u2286 F '' Icc 0 1\n\ncase h\u2082\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\n\u22a2 F '' Icc 0 1 \u2286 Icc 0 1"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\u2081\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\n\u22a2 Icc 0 1 \u2286 F '' Icc 0 1", "state_after": "case h\u2081\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nx : Fin 1 \u2192 \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 x \u2208 F '' Icc 0 1"}, {"tactic": "refine' \u27e8x 0, \u27e8hx.1 0, hx.2 0\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8x 0, \u27e8hx.1 0, hx.2 0\u27e9, _\u27e9", []], "state_before": "case h\u2081\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nx : Fin 1 \u2192 \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 x \u2208 F '' Icc 0 1", "state_after": "case h\u2081\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nx : Fin 1 \u2192 \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 F (x 0) = x"}, {"tactic": "ext j", "annotated_tactic": ["ext j", []], "state_before": "case h\u2081\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nx : Fin 1 \u2192 \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 F (x 0) = x", "state_after": "case h\u2081.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nx : Fin 1 \u2192 \u211d\nhx : x \u2208 Icc 0 1\nj : Fin 1\n\u22a2 F (x 0) j = x j"}, {"tactic": "simp only [Subsingleton.elim j 0]", "annotated_tactic": ["simp only [<a>Subsingleton.elim</a> j 0]", [{"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\u2081.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nx : Fin 1 \u2192 \u211d\nhx : x \u2208 Icc 0 1\nj : Fin 1\n\u22a2 F (x 0) j = x j", "state_after": "no goals"}, {"tactic": "rintro x \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8y, hy, rfl\u27e9", []], "state_before": "case h\u2082\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\n\u22a2 F '' Icc 0 1 \u2286 Icc 0 1", "state_after": "case h\u2082.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\ny : \u211d\nhy : y \u2208 Icc 0 1\n\u22a2 F y \u2208 Icc 0 1"}, {"tactic": "exact \u27e8fun _j => hy.1, fun _j => hy.2\u27e9", "annotated_tactic": ["exact \u27e8fun _j => hy.1, fun _j => hy.2\u27e9", []], "state_before": "case h\u2082.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\ny : \u211d\nhy : y \u2208 Icc 0 1\n\u22a2 F y \u2208 Icc 0 1", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => 1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0", "state_after": "case inl.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => 1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1"}, {"tactic": "simp_rw [parallelepiped, H, A, Algebra.id.smul_eq_mul, mul_one]", "annotated_tactic": ["simp_rw [<a>parallelepiped</a>, H, A, <a>Algebra.id.smul_eq_mul</a>, <a>mul_one</a>]", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 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_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case inl.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => 1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1", "state_after": "case inl.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => 1\n\u22a2 (fun a => \u2211 x : Fin 1, a x) '' ((fun a _i => a) '' Icc 0 1) = Icc 0 1"}, {"tactic": "simp only [Finset.univ_unique, Fin.default_eq_zero, smul_eq_mul, mul_one, Finset.sum_singleton,\n  \u2190 image_comp, Function.comp_apply, image_id', ge_iff_le, zero_le_one, not_true, gt_iff_lt]", "annotated_tactic": ["simp only [<a>Finset.univ_unique</a>, <a>Fin.default_eq_zero</a>, <a>smul_eq_mul</a>, <a>mul_one</a>, <a>Finset.sum_singleton</a>,\n      \u2190 <a>image_comp</a>, <a>Function.comp_apply</a>, <a>image_id'</a>, <a>ge_iff_le</a>, <a>zero_le_one</a>, <a>not_true</a>, <a>gt_iff_lt</a>]", [{"full_name": "Finset.univ_unique", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [119, 9], "def_end_pos": [119, 20]}, {"full_name": "Fin.default_eq_zero", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [124, 9], "def_end_pos": [124, 28]}, {"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": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 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": "Set.image_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [396, 9], "def_end_pos": [396, 18]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}]], "state_before": "case inl.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => 1\n\u22a2 (fun a => \u2211 x : Fin 1, a x) '' ((fun a _i => a) '' Icc 0 1) = Icc 0 1", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => -1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0", "state_after": "case inr.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => -1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0"}, {"tactic": "simp_rw [H, parallelepiped, Algebra.id.smul_eq_mul, A]", "annotated_tactic": ["simp_rw [H, <a>parallelepiped</a>, <a>Algebra.id.smul_eq_mul</a>, A]", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}]], "state_before": "case inr.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => -1\n\u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0", "state_after": "case inr.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => -1\n\u22a2 (fun a => \u2211 x : Fin 1, a x * -1) '' ((fun a _i => a) '' Icc 0 1) = Icc (-1) 0"}, {"tactic": "simp only [Finset.univ_unique, Fin.default_eq_zero, mul_neg, mul_one, Finset.sum_neg_distrib,\n  Finset.sum_singleton, \u2190 image_comp, Function.comp, image_neg, preimage_neg_Icc, neg_zero]", "annotated_tactic": ["simp only [<a>Finset.univ_unique</a>, <a>Fin.default_eq_zero</a>, <a>mul_neg</a>, <a>mul_one</a>, <a>Finset.sum_neg_distrib</a>,\n      <a>Finset.sum_singleton</a>, \u2190 <a>image_comp</a>, <a>Function.comp</a>, <a>image_neg</a>, <a>preimage_neg_Icc</a>, <a>neg_zero</a>]", [{"full_name": "Finset.univ_unique", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [119, 9], "def_end_pos": [119, 20]}, {"full_name": "Fin.default_eq_zero", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [124, 9], "def_end_pos": [124, 28]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [294, 9], "def_end_pos": [294, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "Finset.sum_neg_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1814, 3], "def_end_pos": [1814, 14]}, {"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}, {"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": "Set.image_neg", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [256, 3], "def_end_pos": [256, 14]}, {"full_name": "Set.preimage_neg_Icc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [150, 9], "def_end_pos": [150, 25]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1014, 3], "def_end_pos": [1014, 14]}]], "state_before": "case inr.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF\u271d : 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\u271d\ninst\u271d : Module \u211d F\u271d\nb : OrthonormalBasis \u03b9 \u211d \u211d\ne : \u03b9 \u2243 Fin 1\nB : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b\nF : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t\nA : Icc 0 1 = F '' Icc 0 1\nH : \u2191(OrthonormalBasis.reindex b e) = fun x => -1\n\u22a2 (fun a => \u2211 x : Fin 1, a x * -1) '' ((fun a _i => a) '' Icc 0 1) = Icc (-1) 0", "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.sub_le_sub_iff_right", "start": [364, 11], "end": [365, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_fintype", "start": [1821, 1], "end": [1826, 57], "traced_tactics": [{"tactic": "rw [\u2190 integral_finset .univ , Finset.coe_univ, Measure.restrict_univ]", "annotated_tactic": ["rw [\u2190 <a>integral_finset</a> .univ , <a>Finset.coe_univ</a>, <a>Measure.restrict_univ</a>]", [{"full_name": "MeasureTheory.integral_finset", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1816, 9], "def_end_pos": [1816, 24]}, {"full_name": "Finset.coe_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 17]}, {"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\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : FirstCountableTopology X\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc = \u2211 x : \u03b1, ENNReal.toReal (\u2191\u2191\u03bc {x}) \u2022 f x", "state_after": "case hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : FirstCountableTopology X\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 Integrable fun x => f x"}, {"tactic": "simp only [Finset.coe_univ, Measure.restrict_univ, hf]", "annotated_tactic": ["simp only [<a>Finset.coe_univ</a>, <a>Measure.restrict_univ</a>, hf]", [{"full_name": "Finset.coe_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 17]}, {"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 hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\nG : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : FirstCountableTopology X\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 Integrable fun x => f x", "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", "start": [1688, 1], "end": [1699, 53], "traced_tactics": [{"tactic": "have : \u2200 a b : Num, (a * b).natSize \u2264 a.natSize + b.natSize := by\n  intros\n  simp only [natSize_to_nat, cast_mul]\n  rw [Nat.size_le, pow_add]\n  exact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _)", "annotated_tactic": ["have : \u2200 a b : <a>Num</a>, (a * b).<a>natSize</a> \u2264 a.natSize + b.natSize := by\n    intros\n    simp only [<a>natSize_to_nat</a>, <a>cast_mul</a>]\n    rw [<a>Nat.size_le</a>, <a>pow_add</a>]\n    exact <a>mul_lt_mul''</a> (<a>Nat.lt_size_self</a> _) (<a>Nat.lt_size_self</a> _) (<a>Nat.zero_le</a> _) (<a>Nat.zero_le</a> _)", [{"full_name": "Num", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [43, 11], "def_end_pos": [43, 14]}, {"full_name": "Num.natSize", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [281, 5], "def_end_pos": [281, 12]}, {"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.cast_mul", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [743, 9], "def_end_pos": [743, 17]}, {"full_name": "Nat.size_le", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "mul_lt_mul''", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [559, 9], "def_end_pos": [559, 21]}, {"full_name": "Nat.lt_size_self", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [112, 9], "def_end_pos": [112, 21]}, {"full_name": "Nat.lt_size_self", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [112, 9], "def_end_pos": [112, 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": "\u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b", "state_after": "this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\n\u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\n\u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b", "state_after": "this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\n\u22a2 \u2191(gcd a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d"}, {"tactic": "unfold gcd", "annotated_tactic": ["unfold <a>gcd</a>", [{"full_name": "Num.gcd", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [633, 5], "def_end_pos": [633, 8]}]], "state_before": "this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\n\u22a2 \u2191(gcd a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d", "state_after": "this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\n\u22a2 \u2191(if a\u271d \u2264 b\u271d then gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d else gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) =\n    Nat.gcd \u2191a\u271d \u2191b\u271d"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\n\u22a2 \u2191(if a\u271d \u2264 b\u271d then gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d else gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) =\n    Nat.gcd \u2191a\u271d \u2191b\u271d", "state_after": "case pos\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : a\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d\n\ncase neg\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : \u00aca\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "\u22a2 \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b", "state_after": "a\u271d b\u271d : Num\n\u22a2 natSize (a\u271d * b\u271d) \u2264 natSize a\u271d + natSize b\u271d"}, {"tactic": "simp only [natSize_to_nat, cast_mul]", "annotated_tactic": ["simp only [<a>natSize_to_nat</a>, <a>cast_mul</a>]", [{"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.cast_mul", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [743, 9], "def_end_pos": [743, 17]}]], "state_before": "a\u271d b\u271d : Num\n\u22a2 natSize (a\u271d * b\u271d) \u2264 natSize a\u271d + natSize b\u271d", "state_after": "a\u271d b\u271d : Num\n\u22a2 Nat.size (\u2191a\u271d * \u2191b\u271d) \u2264 Nat.size \u2191a\u271d + Nat.size \u2191b\u271d"}, {"tactic": "rw [Nat.size_le, pow_add]", "annotated_tactic": ["rw [<a>Nat.size_le</a>, <a>pow_add</a>]", [{"full_name": "Nat.size_le", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "a\u271d b\u271d : Num\n\u22a2 Nat.size (\u2191a\u271d * \u2191b\u271d) \u2264 Nat.size \u2191a\u271d + Nat.size \u2191b\u271d", "state_after": "a\u271d b\u271d : Num\n\u22a2 \u2191a\u271d * \u2191b\u271d < 2 ^ Nat.size \u2191a\u271d * 2 ^ Nat.size \u2191b\u271d"}, {"tactic": "exact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _)", "annotated_tactic": ["exact <a>mul_lt_mul''</a> (<a>Nat.lt_size_self</a> _) (<a>Nat.lt_size_self</a> _) (<a>Nat.zero_le</a> _) (<a>Nat.zero_le</a> _)", [{"full_name": "mul_lt_mul''", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [559, 9], "def_end_pos": [559, 21]}, {"full_name": "Nat.lt_size_self", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [112, 9], "def_end_pos": [112, 21]}, {"full_name": "Nat.lt_size_self", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [112, 9], "def_end_pos": [112, 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": "a\u271d b\u271d : Num\n\u22a2 \u2191a\u271d * \u2191b\u271d < 2 ^ Nat.size \u2191a\u271d * 2 ^ Nat.size \u2191b\u271d", "state_after": "no goals"}, {"tactic": "exact gcd_to_nat_aux h (this _ _)", "annotated_tactic": ["exact <a>gcd_to_nat_aux</a> h (this _ _)", [{"full_name": "Num.gcd_to_nat_aux", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1662, 9], "def_end_pos": [1662, 23]}]], "state_before": "case pos\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : a\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d", "state_after": "no goals"}, {"tactic": "rw [Nat.gcd_comm]", "annotated_tactic": ["rw [<a>Nat.gcd_comm</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]}]], "state_before": "case neg\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : \u00aca\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d", "state_after": "case neg\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : \u00aca\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191b\u271d \u2191a\u271d"}, {"tactic": "exact gcd_to_nat_aux (le_of_not_le h) (this _ _)", "annotated_tactic": ["exact <a>gcd_to_nat_aux</a> (<a>le_of_not_le</a> h) (this _ _)", [{"full_name": "Num.gcd_to_nat_aux", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1662, 9], "def_end_pos": [1662, 23]}, {"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 neg\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : \u00aca\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191b\u271d \u2191a\u271d", "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.setCurr'", "start": [592, 1], "end": [599, 50], "traced_tactics": [{"tactic": "cases h.out'", "annotated_tactic": ["cases h.out'", []], "state_before": "l r : List Char\nc : Char\nit : Iterator\nh : ValidFor l r it\n\u22a2 ValidFor l (List.modifyHead (fun x => c) r) (setCurr it c)", "state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor l (List.modifyHead (fun x => c) r)\n    (setCurr { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } c)"}, {"tactic": "simp [Iterator.setCurr]", "annotated_tactic": ["simp [<a>Iterator.setCurr</a>]", [{"full_name": "String.Iterator.setCurr", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [338, 5], "def_end_pos": [338, 12]}]], "state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor l (List.modifyHead (fun x => c) r)\n    (setCurr { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } c)", "state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor l\n    (match r with\n    | [] => []\n    | a :: l => c :: l)\n    { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }"}, {"tactic": "refine .of_eq _ ?_ (by simp)", "annotated_tactic": ["refine .of_eq _ ?_ (by simp)", []], "state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor l\n    (match r with\n    | [] => []\n    | a :: l => c :: l)\n    { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }", "state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n    List.reverseAux l\n      (match r with\n      | [] => []\n      | a :: l => c :: l)"}, {"tactic": "have := set_of_valid l.reverse r c", "annotated_tactic": ["have := <a>set_of_valid</a> l.reverse r c", [{"full_name": "String.set_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 21]}]], "state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n    List.reverseAux l\n      (match r with\n      | [] => []\n      | a :: l => c :: l)", "state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n  set { data := List.reverse l ++ r } { byteIdx := utf8Len (List.reverse l) } c =\n    { data := List.reverse l ++ List.modifyHead (fun x => c) r }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n    List.reverseAux l\n      (match r with\n      | [] => []\n      | a :: l => c :: l)"}, {"tactic": "simp at this", "annotated_tactic": ["simp at this", []], "state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n  set { data := List.reverse l ++ r } { byteIdx := utf8Len (List.reverse l) } c =\n    { data := List.reverse l ++ List.modifyHead (fun x => c) r }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n    List.reverseAux l\n      (match r with\n      | [] => []\n      | a :: l => c :: l)", "state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n  set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c =\n    {\n      data :=\n        List.reverse l ++\n          match r with\n          | [] => []\n          | a :: l => c :: l }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n    List.reverseAux l\n      (match r with\n      | [] => []\n      | a :: l => c :: l)"}, {"tactic": "simp [List.reverseAux_eq, this]", "annotated_tactic": ["simp [<a>List.reverseAux_eq</a>, this]", [{"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}]], "state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n  set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c =\n    {\n      data :=\n        List.reverse l ++\n          match r with\n          | [] => []\n          | a :: l => c :: l }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n    List.reverseAux l\n      (match r with\n      | [] => []\n      | a :: l => c :: l)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.i.byteIdx =\n    utf8Len l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.head_stack_ok", "start": [1496, 1], "end": [1527, 30], "traced_tactics": [{"tactic": "cases' L\u2082 with a L\u2082", "annotated_tactic": ["cases' L\u2082 with a L\u2082", []], "state_before": "q : \u039b'\ns : Option \u0393'\nL\u2081 L\u2082 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trList (List.headI L\u2082 :: L\u2081)) [] [] L\u2083 }", "state_after": "case nil\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trList (List.headI [] :: L\u2081)) [] [] L\u2083 }\n\ncase cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trList (List.headI (a :: L\u2082) :: L\u2081)) [] [] L\u2083 }"}, {"tactic": "refine'\n  TransGen.trans\n    (move_ok (by decide)\n      (splitAtPred_eq _ _ [] (some \u0393'.cons\u2097) L\u2083 (by rintro _ \u27e8\u27e9) \u27e8rfl, rfl\u27e9))\n    (TransGen.head rfl (TransGen.head rfl _))", "annotated_tactic": ["refine'\n      <a>TransGen.trans</a>\n        (<a>move_ok</a> (by decide)\n          (<a>splitAtPred_eq</a> _ _ [] (<a>some</a> <a>\u0393'.cons\u2097</a>) L\u2083 (by rintro _ \u27e8\u27e9) \u27e8<a>rfl</a>, <a>rfl</a>\u27e9))\n        (<a>TransGen.head</a> <a>rfl</a> (<a>TransGen.head</a> <a>rfl</a> _))", [{"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}, {"full_name": "Turing.PartrecToTM2.move_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 16]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"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.PartrecToTM2.\u0393'.cons\u2097", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [868, 5], "def_end_pos": [868, 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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case nil\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trList (List.headI [] :: L\u2081)) [] [] L\u2083 }", "state_after": "case nil\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux\n      (tr\n        ((fun x =>\n            \u039b'.read fun s => ite (s = some \u0393'.cons\u2097) id (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack) (unrev q))\n          (some \u0393'.cons\u2097)))\n      (some \u0393'.cons\u2097)\n      (update\n        (update (update (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083)) stack L\u2083) rev\n          (List.reverseAux [] (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) rev)))\n        rev\n        ((fun s => Option.iget ((fun x => some \u0393'.cons) s)) (some \u0393'.cons\u2097) ::\n          update (update (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083)) stack L\u2083) rev\n            (List.reverseAux [] (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) rev)) rev)))\n    { l := some q, var := none, stk := elim (trList (List.headI [] :: L\u2081)) [] [] L\u2083 }"}, {"tactic": "simp only [TM2.step, Option.mem_def, TM2.stepAux, ite_true, id_eq, trList, List.nil_append,\n  elim_update_stack, elim_rev, List.reverseAux_nil, elim_update_rev, Function.update_same,\n  List.headI_nil, trNat_default]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>ite_true</a>, <a>id_eq</a>, <a>trList</a>, <a>List.nil_append</a>,\n      <a>elim_update_stack</a>, <a>elim_rev</a>, <a>List.reverseAux_nil</a>, <a>elim_update_rev</a>, <a>Function.update_same</a>,\n      <a>List.headI_nil</a>, <a>trNat_default</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"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": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"full_name": "List.nil_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [92, 17], "def_end_pos": [92, 27]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}, {"full_name": "Turing.PartrecToTM2.K'.elim_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 20]}, {"full_name": "List.reverseAux_nil", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [127, 17], "def_end_pos": [127, 31]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1284, 9], "def_end_pos": [1284, 27]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "List.headI_nil", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [44, 17], "def_end_pos": [44, 26]}, {"full_name": "Turing.PartrecToTM2.trNat_default", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1203, 9], "def_end_pos": [1203, 22]}]], "state_before": "case nil\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux\n      (tr\n        ((fun x =>\n            \u039b'.read fun s => ite (s = some \u0393'.cons\u2097) id (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack) (unrev q))\n          (some \u0393'.cons\u2097)))\n      (some \u0393'.cons\u2097)\n      (update\n        (update (update (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083)) stack L\u2083) rev\n          (List.reverseAux [] (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) rev)))\n        rev\n        ((fun s => Option.iget ((fun x => some \u0393'.cons) s)) (some \u0393'.cons\u2097) ::\n          update (update (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083)) stack L\u2083) rev\n            (List.reverseAux [] (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) rev)) rev)))\n    { l := some q, var := none, stk := elim (trList (List.headI [] :: L\u2081)) [] [] L\u2083 }", "state_after": "case nil\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (unrev q), var := some \u0393'.cons\u2097, stk := elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 }\n    { l := some q, var := none, stk := elim (\u0393'.cons :: trList L\u2081) [] [] L\u2083 }"}, {"tactic": "convert unrev_ok using 2", "annotated_tactic": ["convert <a>unrev_ok</a> using 2", [{"full_name": "Turing.PartrecToTM2.unrev_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1384, 9], "def_end_pos": [1384, 17]}]], "state_before": "case nil\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (unrev q), var := some \u0393'.cons\u2097, stk := elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 }\n    { l := some q, var := none, stk := elim (\u0393'.cons :: trList L\u2081) [] [] L\u2083 }", "state_after": "case h.e'_2.h.e'_7\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 elim (\u0393'.cons :: trList L\u2081) [] [] L\u2083 =\n    update (update (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083) rev []) main\n      (List.reverseAux (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 rev)\n        (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 main))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_7\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 elim (\u0393'.cons :: trList L\u2081) [] [] L\u2083 =\n    update (update (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083) rev []) main\n      (List.reverseAux (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 rev)\n        (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 main))", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "q : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 stack \u2260 rev", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8\u27e9", "annotated_tactic": ["rintro _ \u27e8\u27e9", []], "state_before": "q : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\n\u22a2 \u2200 (x : \u0393'), x \u2208 [] \u2192 natEnd x = false", "state_after": "no goals"}, {"tactic": "refine'\n  TransGen.trans\n    (move_ok (by decide)\n      (splitAtPred_eq _ _ (trNat a) (some \u0393'.cons) (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083)\n        (trNat_natEnd _) \u27e8rfl, by simp\u27e9))\n    (TransGen.head rfl (TransGen.head rfl _))", "annotated_tactic": ["refine'\n      <a>TransGen.trans</a>\n        (<a>move_ok</a> (by decide)\n          (<a>splitAtPred_eq</a> _ _ (<a>trNat</a> a) (<a>some</a> <a>\u0393'.cons</a>) (<a>trList</a> L\u2082 ++ <a>\u0393'.cons\u2097</a> :: L\u2083)\n            (<a>trNat_natEnd</a> _) \u27e8<a>rfl</a>, by simp\u27e9))\n        (<a>TransGen.head</a> <a>rfl</a> (<a>TransGen.head</a> <a>rfl</a> _))", [{"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}, {"full_name": "Turing.PartrecToTM2.move_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 16]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"full_name": "Turing.PartrecToTM2.trNat", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1194, 5], "def_end_pos": [1194, 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.PartrecToTM2.\u0393'.cons", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [869, 5], "def_end_pos": [869, 9]}, {"full_name": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"full_name": "Turing.PartrecToTM2.\u0393'.cons\u2097", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [868, 5], "def_end_pos": [868, 10]}, {"full_name": "Turing.PartrecToTM2.trNat_natEnd", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1465, 9], "def_end_pos": [1465, 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": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trList (List.headI (a :: L\u2082) :: L\u2081)) [] [] L\u2083 }", "state_after": "case cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux\n      (tr\n        ((fun x =>\n            \u039b'.read fun s => ite (s = some \u0393'.cons\u2097) id (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack) (unrev q))\n          (some \u0393'.cons)))\n      (some \u0393'.cons)\n      (update\n        (update\n          (update (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083)) stack (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083)) rev\n          (List.reverseAux (trNat a) (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) rev)))\n        rev\n        ((fun s => Option.iget ((fun x => some \u0393'.cons) s)) (some \u0393'.cons) ::\n          update\n            (update (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083)) stack (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n            rev (List.reverseAux (trNat a) (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) rev)) rev)))\n    { l := some q, var := none, stk := elim (trList (List.headI (a :: L\u2082) :: L\u2081)) [] [] L\u2083 }"}, {"tactic": "simp only [TM2.step, Option.mem_def, TM2.stepAux, ite_false, trList, List.append_assoc,\n  List.cons_append, elim_update_stack, elim_rev, elim_update_rev, Function.update_same,\n  List.headI_cons]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>ite_false</a>, <a>trList</a>, <a>List.append_assoc</a>,\n      <a>List.cons_append</a>, <a>elim_update_stack</a>, <a>elim_rev</a>, <a>elim_update_rev</a>, <a>Function.update_same</a>,\n      <a>List.headI_cons</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "ite_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [78, 17], "def_end_pos": [78, 26]}, {"full_name": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"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.cons_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [99, 17], "def_end_pos": [99, 28]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}, {"full_name": "Turing.PartrecToTM2.K'.elim_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 20]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1284, 9], "def_end_pos": [1284, 27]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "List.headI_cons", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [45, 17], "def_end_pos": [45, 27]}]], "state_before": "case cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux\n      (tr\n        ((fun x =>\n            \u039b'.read fun s => ite (s = some \u0393'.cons\u2097) id (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack) (unrev q))\n          (some \u0393'.cons)))\n      (some \u0393'.cons)\n      (update\n        (update\n          (update (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083)) stack (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083)) rev\n          (List.reverseAux (trNat a) (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) rev)))\n        rev\n        ((fun s => Option.iget ((fun x => some \u0393'.cons) s)) (some \u0393'.cons) ::\n          update\n            (update (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083)) stack (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n            rev (List.reverseAux (trNat a) (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) rev)) rev)))\n    { l := some q, var := none, stk := elim (trList (List.headI (a :: L\u2082) :: L\u2081)) [] [] L\u2083 }", "state_after": "case cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack (unrev q)), var := some \u0393'.cons,\n      stk :=\n        elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 }"}, {"tactic": "refine'\n  TransGen.trans\n    (clear_ok\n      (splitAtPred_eq _ _ (trList L\u2082) (some \u0393'.cons\u2097) L\u2083\n        (fun x h => Bool.decide_false (trList_ne_cons\u2097 _ _ h)) \u27e8rfl, by simp\u27e9))\n    _", "annotated_tactic": ["refine'\n      <a>TransGen.trans</a>\n        (<a>clear_ok</a>\n          (<a>splitAtPred_eq</a> _ _ (<a>trList</a> L\u2082) (<a>some</a> <a>\u0393'.cons\u2097</a>) L\u2083\n            (fun x h => <a>Bool.decide_false</a> (<a>trList_ne_cons\u2097</a> _ _ h)) \u27e8<a>rfl</a>, by simp\u27e9))\n        _", [{"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}, {"full_name": "Turing.PartrecToTM2.clear_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 17]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"full_name": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"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.PartrecToTM2.\u0393'.cons\u2097", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [868, 5], "def_end_pos": [868, 10]}, {"full_name": "Bool.decide_false", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [149, 9], "def_end_pos": [149, 21]}, {"full_name": "Turing.PartrecToTM2.trList_ne_cons\u2097", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1469, 9], "def_end_pos": [1469, 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 cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack (unrev q)), var := some \u0393'.cons,\n      stk :=\n        elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) }\n    { l := some q, var := none, stk := elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 }", "state_after": "case cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (unrev q), var := some \u0393'.cons\u2097,\n      stk :=\n        update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083 }\n    { l := some q, var := none, stk := elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 }"}, {"tactic": "convert unrev_ok using 2", "annotated_tactic": ["convert <a>unrev_ok</a> using 2", [{"full_name": "Turing.PartrecToTM2.unrev_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1384, 9], "def_end_pos": [1384, 17]}]], "state_before": "case cons\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (unrev q), var := some \u0393'.cons\u2097,\n      stk :=\n        update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083 }\n    { l := some q, var := none, stk := elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 }", "state_after": "case h.e'_2.h.e'_7\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 =\n    update\n      (update\n        (update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083)\n        rev [])\n      main\n      (List.reverseAux\n        (update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083 rev)\n        (update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083 main))"}, {"tactic": "simp [List.reverseAux_eq]", "annotated_tactic": ["simp [<a>List.reverseAux_eq</a>]", [{"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}]], "state_before": "case h.e'_2.h.e'_7\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 =\n    update\n      (update\n        (update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083)\n        rev [])\n      main\n      (List.reverseAux\n        (update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083 rev)\n        (update\n          (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []\n            (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))\n          stack L\u2083 main))", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "q : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 stack \u2260 rev", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "q : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) stack =\n    trNat a ++ \u0393'.cons :: (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "q : \u039b'\ns : Option \u0393'\nL\u2081 : List \u2115\nL\u2083 : List \u0393'\na : \u2115\nL\u2082 : List \u2115\n\u22a2 elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) stack =\n    trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083", "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_left_identity", "start": [507, 1], "end": [509, 81], "traced_tactics": [{"tactic": "rw [coe_image\u2082, coe_singleton, Set.image2_left_identity h]", "annotated_tactic": ["rw [<a>coe_image\u2082</a>, <a>coe_singleton</a>, <a>Set.image2_left_identity</a> h]", [{"full_name": "Finset.coe_image\u2082", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [49, 9], "def_end_pos": [49, 19]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [725, 9], "def_end_pos": [725, 22]}, {"full_name": "Set.image2_left_identity", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [438, 7], "def_end_pos": [438, 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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\nu u' : Finset \u03b3\na\u271d a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b3\na : \u03b1\nh : \u2200 (b : \u03b3), f a b = b\nt : Finset \u03b3\n\u22a2 \u2191(image\u2082 f {a} t) = \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt", "start": [521, 1], "end": [532, 47], "traced_tactics": [{"tactic": "have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM\n  (lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2", "annotated_tactic": ["have := (<a>crossing_eq_crossing_of_lowerCrossingTime_lt</a> hNM\n    (<a>lt_of_le_of_lt</a> <a>lowerCrossingTime_le_upperCrossingTime_succ</a> h)).2", [{"full_name": "MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [492, 9], "def_end_pos": [492, 53]}, {"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.lowerCrossingTime_le_upperCrossingTime_succ", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [207, 9], "def_end_pos": [207, 52]}]], "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 + 1) \u03c9 < N\n\u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9 \u2227\n    lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime 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\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N (n + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9 \u2227\n    lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9"}, {"tactic": "refine' \u27e8_, this\u27e9", "annotated_tactic": ["refine' \u27e8_, this\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 : \u03a9\n\u2131 : Filtration \u2115 m0\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N (n + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9 \u2227\n    lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime 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\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N (n + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9"}, {"tactic": "rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]", "annotated_tactic": ["rw [<a>upperCrossingTime_succ_eq</a>, <a>upperCrossingTime_succ_eq</a>, <a>eq_comm</a>, this]", [{"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.upperCrossingTime_succ_eq", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [173, 9], "def_end_pos": [173, 34]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "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 + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \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\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N (n + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 = hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) M \u03c9"}, {"tactic": "refine' hitting_eq_hitting_of_exists hNM _", "annotated_tactic": ["refine' <a>hitting_eq_hitting_of_exists</a> hNM _", [{"full_name": "MeasureTheory.hitting_eq_hitting_of_exists", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [195, 9], "def_end_pos": [195, 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\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N (n + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 = hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) M \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\nM : \u2115\nhNM : N \u2264 M\nh : upperCrossingTime a b f N (n + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b"}, {"tactic": "rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h", "annotated_tactic": ["rw [<a>upperCrossingTime_succ_eq</a>, <a>hitting_lt_iff</a>] at h", [{"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_lt_iff", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [182, 9], "def_end_pos": [182, 23]}]], "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 + 1) \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b", "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\nM : \u2115\nhNM : N \u2264 M\nh : \u2203 j, j \u2208 Set.Ico (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\ncase hi\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\nM : \u2115\nhNM : N \u2264 M\nh : hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 N \u2264 N"}, {"tactic": "obtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h", "annotated_tactic": ["obtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h", []], "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 : \u2203 j, j \u2208 Set.Ico (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\ncase hi\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\nM : \u2115\nhNM : N \u2264 M\nh : hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 N \u2264 N", "state_after": "case intro.intro\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\nM : \u2115\nhNM : N \u2264 M\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\nj : \u2115\nhj\u2081 : j \u2208 Set.Ico (lowerCrossingTime a b f N n \u03c9) N\nhj\u2082 : f j \u03c9 \u2208 Set.Ici b\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\ncase hi\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\nM : \u2115\nhNM : N \u2264 M\nh : hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 N \u2264 N"}, {"tactic": "exacts [\u27e8j, \u27e8hj\u2081.1, hj\u2081.2.le\u27e9, hj\u2082\u27e9, le_rfl]", "annotated_tactic": ["exacts [\u27e8j, \u27e8hj\u2081.1, hj\u2081.2.<a>le</a>\u27e9, hj\u2082\u27e9, <a>le_rfl</a>]", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro.intro\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\nM : \u2115\nhNM : N \u2264 M\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\nj : \u2115\nhj\u2081 : j \u2208 Set.Ico (lowerCrossingTime a b f N n \u03c9) N\nhj\u2082 : f j \u03c9 \u2208 Set.Ici b\n\u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b\n\ncase hi\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\nM : \u2115\nhNM : N \u2264 M\nh : hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 < N\nthis : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9\n\u22a2 N \u2264 N", "state_after": "no goals"}]}, {"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": [56, 1], "end": [58, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasureTheory.Measure.ext_of_Ici", "start": [835, 1], "end": [838, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mkRBSet_eq", "start": [618, 9], "end": [618, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.Mem\u2112p.induction_stronglyMeasurable", "start": [687, 1], "end": [712, 92], "traced_tactics": [{"tactic": "intro f hf hfm", "annotated_tactic": ["intro f hf hfm", []], "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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 F\u2984, Mem\u2112p f p \u2192 AEStronglyMeasurable' m f \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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\n\u22a2 P f"}, {"tactic": "let f_Lp := hf.toLp f", "annotated_tactic": ["let f_Lp := hf.toLp f", []], "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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\n\u22a2 P f"}, {"tactic": "have hfm_Lp : AEStronglyMeasurable' m f_Lp \u03bc := hfm.congr hf.coeFn_toLp.symm", "annotated_tactic": ["have hfm_Lp : <a>AEStronglyMeasurable'</a> m f_Lp \u03bc := hfm.congr hf.coeFn_toLp.symm", [{"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P f"}, {"tactic": "refine' h_ae hf.coeFn_toLp (Lp.mem\u2112p _) _", "annotated_tactic": ["refine' h_ae hf.coeFn_toLp (<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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P \u2191\u2191(toLp f hf)"}, {"tactic": "change P f_Lp", "annotated_tactic": ["change P f_Lp", []], "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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P \u2191\u2191(toLp f hf)", "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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P \u2191\u2191f_Lp"}, {"tactic": "refine' Lp.induction_stronglyMeasurable hm hp_ne_top (P := fun f => P f) _ _ h_closed f_Lp hfm_Lp", "annotated_tactic": ["refine' <a>Lp.induction_stronglyMeasurable</a> hm hp_ne_top (P := fun f => P f) _ _ h_closed f_Lp hfm_Lp", [{"full_name": "MeasureTheory.Lp.induction_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [621, 9], "def_end_pos": [621, 40]}]], "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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P \u2191\u2191f_Lp", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    (fun f => P \u2191\u2191f) \u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n\ncase 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 \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          (fun f => P \u2191\u2191f) (toLp f hf) \u2192 (fun f => P \u2191\u2191f) (toLp g hg) \u2192 (fun f => P \u2191\u2191f) (toLp f hf + toLp g hg)"}, {"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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n    (fun f => P \u2191\u2191f) \u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 P \u2191\u2191\u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)"}, {"tactic": "rw [Lp.simpleFunc.coe_indicatorConst]", "annotated_tactic": ["rw [<a>Lp.simpleFunc.coe_indicatorConst</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.coe_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [700, 9], "def_end_pos": [700, 27]}]], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 P \u2191\u2191\u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 P \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)"}, {"tactic": "refine' h_ae indicatorConstLp_coeFn.symm _ (h_ind c hs h\u03bcs)", "annotated_tactic": ["refine' h_ae indicatorConstLp_coeFn.symm _ (h_ind c hs h\u03bcs)", []], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 P \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Mem\u2112p (Set.indicator s fun x => c) p"}, {"tactic": "exact mem\u2112p_indicator_const p (hm s hs) c (Or.inr h\u03bcs.ne)", "annotated_tactic": ["exact <a>mem\u2112p_indicator_const</a> p (hm s hs) c (<a>Or.inr</a> h\u03bcs.ne)", [{"full_name": "MeasureTheory.mem\u2112p_indicator_const", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [696, 9], "def_end_pos": [696, 30]}, {"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 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Mem\u2112p (Set.indicator s fun x => c) p", "state_after": "no goals"}, {"tactic": "intro f g hf_mem hg_mem hfm hgm h_disj hfP hgP", "annotated_tactic": ["intro f g hf_mem hg_mem hfm hgm h_disj hfP hgP", []], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 \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          (fun f => P \u2191\u2191f) (toLp f hf) \u2192 (fun f => P \u2191\u2191f) (toLp g hg) \u2192 (fun f => P \u2191\u2191f) (toLp f hf + toLp g hg)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)"}, {"tactic": "have hfP' : P f := h_ae hf_mem.coeFn_toLp (Lp.mem\u2112p _) hfP", "annotated_tactic": ["have hfP' : P f := h_ae hf_mem.coeFn_toLp (<a>Lp.mem\u2112p</a> _) hfP", [{"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}]], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)"}, {"tactic": "have hgP' : P g := h_ae hg_mem.coeFn_toLp (Lp.mem\u2112p _) hgP", "annotated_tactic": ["have hgP' : P g := h_ae hg_mem.coeFn_toLp (<a>Lp.mem\u2112p</a> _) hgP", [{"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}]], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)"}, {"tactic": "specialize h_add h_disj hf_mem hg_mem hfm hgm hfP' hgP'", "annotated_tactic": ["specialize h_add h_disj hf_mem hg_mem hfm hgm hfP' hgP'", []], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\nh_add : P (f + g)\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)"}, {"tactic": "refine' h_ae _ (hf_mem.add hg_mem) h_add", "annotated_tactic": ["refine' h_ae _ (hf_mem.add hg_mem) h_add", []], "state_before": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\nh_add : P (f + g)\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)", "state_after": "case 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\nh_add : P (f + g)\n\u22a2 f + g =\u1d50[\u03bc] \u2191\u2191(toLp f hf_mem + toLp g hg_mem)"}, {"tactic": "exact (hf_mem.coeFn_toLp.symm.add hg_mem.coeFn_toLp.symm).trans (Lp.coeFn_add _ _).symm", "annotated_tactic": ["exact (hf_mem.coeFn_toLp.symm.add hg_mem.coeFn_toLp.symm).<a>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 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\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 : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\nh_add : P (f + g)\n\u22a2 f + g =\u1d50[\u03bc] \u2191\u2191(toLp f hf_mem + toLp g hg_mem)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.sigma_nonempty_iff", "start": [215, 1], "end": [216, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.restrict_const_lintegral", "start": [1085, 1], "end": [1087, 79], "traced_tactics": [{"tactic": "rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict]", "annotated_tactic": ["rw [<a>restrict_lintegral_eq_lintegral_restrict</a> _ hs, <a>const_lintegral_restrict</a>]", [{"full_name": "MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1068, 9], "def_end_pos": [1068, 49]}, {"full_name": "MeasureTheory.SimpleFunc.const_lintegral_restrict", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 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\nc : \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 lintegral (restrict (const \u03b1 c) s) \u03bc = c * \u2191\u2191\u03bc 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_sInf_eq_sInf_restrict", "start": [1881, 1], "end": [1888, 33], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "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\u271d : 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\nm0 : MeasurableSpace \u03b1\nm : Set (Measure \u03b1)\nhm : Set.Nonempty m\nht : MeasurableSet t\n\u22a2 restrict (sInf m) t = sInf ((fun \u03bc => restrict \u03bc t) '' m)", "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\u271d : 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\nm0 : MeasurableSpace \u03b1\nm : Set (Measure \u03b1)\nhm : Set.Nonempty m\nht : MeasurableSet t\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict (sInf m) t) s = \u2191\u2191(sInf ((fun \u03bc => restrict \u03bc t) '' m)) s"}, {"tactic": "simp_rw [sInf_apply hs, restrict_apply hs, sInf_apply (MeasurableSet.inter hs ht),\n  Set.image_image, restrict_toOuterMeasure_eq_toOuterMeasure_restrict ht, \u2190\n  Set.image_image _ toOuterMeasure, \u2190 OuterMeasure.restrict_sInf_eq_sInf_restrict _ (hm.image _),\n  OuterMeasure.restrict_apply]", "annotated_tactic": ["simp_rw [<a>sInf_apply</a> hs, <a>restrict_apply</a> hs, <a>sInf_apply</a> (<a>MeasurableSet.inter</a> hs ht),\n    <a>Set.image_image</a>, <a>restrict_toOuterMeasure_eq_toOuterMeasure_restrict</a> ht, \u2190\n    <a>Set.image_image</a> _ toOuterMeasure, \u2190 <a>OuterMeasure.restrict_sInf_eq_sInf_restrict</a> _ (hm.image _),\n    <a>OuterMeasure.restrict_apply</a>]", [{"full_name": "MeasureTheory.Measure.sInf_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 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.Measure.sInf_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 19]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}, {"full_name": "MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 59]}, {"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}, {"full_name": "MeasureTheory.OuterMeasure.restrict_sInf_eq_sInf_restrict", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1281, 9], "def_end_pos": [1281, 39]}, {"full_name": "MeasureTheory.OuterMeasure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [561, 9], "def_end_pos": [561, 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\u271d : 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\nm0 : MeasurableSpace \u03b1\nm : Set (Measure \u03b1)\nhm : Set.Nonempty m\nht : MeasurableSet t\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict (sInf m) t) s = \u2191\u2191(sInf ((fun \u03bc => restrict \u03bc t) '' m)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.exists_ne_of_one_lt_card", "start": [607, 1], "end": [611, 22], "traced_tactics": [{"tactic": "obtain \u27e8x, hx, y, hy, hxy\u27e9 := Finset.one_lt_card.mp hs", "annotated_tactic": ["obtain \u27e8x, hx, y, hy, hxy\u27e9 := Finset.one_lt_card.mp hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na : \u03b1\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a"}, {"tactic": "by_cases ha : y = a", "annotated_tactic": ["by_cases ha : y = a", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nha : y = a\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nha : \u00acy = a\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a"}, {"tactic": "exact \u27e8x, hx, ne_of_ne_of_eq hxy ha\u27e9", "annotated_tactic": ["exact \u27e8x, hx, <a>ne_of_ne_of_eq</a> hxy ha\u27e9", [{"full_name": "ne_of_ne_of_eq", "def_path": "Mathlib/Init/CCLemmas.lean", "def_pos": [124, 9], "def_end_pos": [124, 23]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nha : y = a\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a", "state_after": "no goals"}, {"tactic": "exact \u27e8y, hy, ha\u27e9", "annotated_tactic": ["exact \u27e8y, hy, ha\u27e9", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nha : \u00acy = a\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.laverage_mem_openSegment_compl_self", "start": [191, 1], "end": [196, 27], "traced_tactics": [{"tactic": "simpa only [union_compl_self, restrict_univ] using\n  laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs\u2080 hsc\u2080 (measure_ne_top _ _)\n    (measure_ne_top _ _)", "annotated_tactic": ["simpa only [<a>union_compl_self</a>, <a>restrict_univ</a>] using\n    <a>laverage_union_mem_openSegment</a> <a>aedisjoint_compl_right</a> hs.compl hs\u2080 hsc\u2080 (<a>measure_ne_top</a> _ _)\n      (<a>measure_ne_top</a> _ _)", [{"full_name": "Set.union_compl_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1739, 9], "def_end_pos": [1739, 25]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "MeasureTheory.laverage_union_mem_openSegment", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [168, 9], "def_end_pos": [168, 39]}, {"full_name": "MeasureTheory.aedisjoint_compl_right", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [161, 9], "def_end_pos": [161, 31]}, {"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": "\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 g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhs : NullMeasurableSet s\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nhsc\u2080 : \u2191\u2191\u03bc s\u1d9c \u2260 0\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc \u2208 openSegment \u211d\u22650\u221e (\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d\u207b (x : \u03b1) in s\u1d9c, f x \u2202\u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.prod_map_id", "start": [762, 1], "end": [767, 6], "traced_tactics": [{"tactic": "ext i x : 2", "annotated_tactic": ["ext i x : 2", []], "state_before": "n : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} n\n\u22a2 (id \u2297' id) = id", "state_after": "case a.h\nn : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} n\ni : Fin2 n\nx : (\u03b1 \u2297 \u03b2) i\n\u22a2 (id \u2297' id) i x = id i x"}, {"tactic": "induction i <;> simp only [TypeVec.prod.map, *, dropFun_id]", "annotated_tactic": ["induction i <;> simp only [<a>TypeVec.prod.map</a>, *, <a>dropFun_id</a>]", [{"full_name": "TypeVec.prod.map", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [547, 15], "def_end_pos": [547, 23]}, {"full_name": "TypeVec.dropFun_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [757, 9], "def_end_pos": [757, 19]}]], "state_before": "case a.h\nn : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} n\ni : Fin2 n\nx : (\u03b1 \u2297 \u03b2) i\n\u22a2 (id \u2297' id) i x = id i x", "state_after": "case a.h.fz\nn n\u271d : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) Fin2.fz\n\u22a2 (id Fin2.fz x.fst, id Fin2.fz x.snd) = id Fin2.fz x\n\ncase a.h.fs\nn n\u271d : \u2115\na\u271d : Fin2 n\u271d\na_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d)\n\u22a2 id a\u271d x = id (Fin2.fs a\u271d) x"}, {"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "case a.h.fz\nn n\u271d : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) Fin2.fz\n\u22a2 (id Fin2.fz x.fst, id Fin2.fz x.snd) = id Fin2.fz x\n\ncase a.h.fs\nn n\u271d : \u2115\na\u271d : Fin2 n\u271d\na_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d)\n\u22a2 id a\u271d x = id (Fin2.fs a\u271d) x", "state_after": "case a.h.fz.mk\nn n\u271d : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nfst\u271d : last \u03b1\nsnd\u271d : last \u03b2\n\u22a2 (id Fin2.fz (fst\u271d, snd\u271d).fst, id Fin2.fz (fst\u271d, snd\u271d).snd) = id Fin2.fz (fst\u271d, snd\u271d)\n\ncase a.h.fs\nn n\u271d : \u2115\na\u271d : Fin2 n\u271d\na_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d)\n\u22a2 id a\u271d x = id (Fin2.fs a\u271d) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a.h.fz.mk\nn n\u271d : \u2115\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nfst\u271d : last \u03b1\nsnd\u271d : last \u03b2\n\u22a2 (id Fin2.fz (fst\u271d, snd\u271d).fst, id Fin2.fz (fst\u271d, snd\u271d).snd) = id Fin2.fz (fst\u271d, snd\u271d)\n\ncase a.h.fs\nn n\u271d : \u2115\na\u271d : Fin2 n\u271d\na_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d)\n\u22a2 id a\u271d x = id (Fin2.fs a\u271d) x", "state_after": "case a.h.fs\nn n\u271d : \u2115\na\u271d : Fin2 n\u271d\na_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d)\n\u22a2 id a\u271d x = id (Fin2.fs a\u271d) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a.h.fs\nn n\u271d : \u2115\na\u271d : Fin2 n\u271d\na_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x\n\u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d)\nx : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d)\n\u22a2 id a\u271d x = id (Fin2.fs a\u271d) x", "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", "start": [2701, 1], "end": [2720, 48], "traced_tactics": [{"tactic": "intro c\u2081 c\u2082 h", "annotated_tactic": ["intro c\u2081 c\u2082 h", []], "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\n\u22a2 Respects (TM2.step M) (TM1.step (tr M)) TrCfg", "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\nc\u2081 : Cfg\u2082\nc\u2082 : TM1.Cfg \u0393' \u039b' \u03c3\nh : TrCfg c\u2081 c\u2082\n\u22a2 match TM2.step M c\u2081 with\n  | some b\u2081 => \u2203 b\u2082, TrCfg b\u2081 b\u2082 \u2227 Reaches\u2081 (TM1.step (tr M)) c\u2082 b\u2082\n  | none => TM1.step (tr M) c\u2082 = none"}, {"tactic": "cases' h with l v S L hT", "annotated_tactic": ["cases' h with l v S L hT", []], "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\nc\u2081 : Cfg\u2082\nc\u2082 : TM1.Cfg \u0393' \u039b' \u03c3\nh : TrCfg c\u2081 c\u2082\n\u22a2 match TM2.step M c\u2081 with\n  | some b\u2081 => \u2203 b\u2082, TrCfg b\u2081 b\u2082 \u2227 Reaches\u2081 (TM1.step (tr M)) c\u2082 b\u2082\n  | none => TM1.step (tr M) c\u2082 = none", "state_after": "case mk\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\nl : Option \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 match TM2.step M { l := l, var := v, stk := S } with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      TrCfg b\u2081 b\u2082 \u2227\n        Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n  | none => TM1.step (tr M) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none"}, {"tactic": "cases' l with l", "annotated_tactic": ["cases' l with l", []], "state_before": "case mk\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\nl : Option \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 match TM2.step M { l := l, var := v, stk := S } with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      TrCfg b\u2081 b\u2082 \u2227\n        Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n  | none => TM1.step (tr M) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none", "state_after": "case mk.none\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 match TM2.step M { l := none, var := v, stk := S } with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      TrCfg b\u2081 b\u2082 \u2227\n        Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n  | none => TM1.step (tr M) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none\n\ncase mk.some\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 match TM2.step M { l := some l, var := v, stk := S } with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      TrCfg b\u2081 b\u2082 \u2227\n        Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n  | none => TM1.step (tr M) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none"}, {"tactic": "simp only [TM2.step, Respects, Option.map_some']", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Respects</a>, <a>Option.map_some'</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"full_name": "Turing.Respects", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [877, 5], "def_end_pos": [877, 13]}, {"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 mk.some\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 match TM2.step M { l := some l, var := v, stk := S } with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      TrCfg b\u2081 b\u2082 \u2227\n        Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n  | none => TM1.step (tr M) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none", "state_after": "case mk.some\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b\u2082,\n    TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227\n      Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082"}, {"tactic": "rsuffices \u27e8b, c, r\u27e9 : \u2203 b, _ \u2227 Reaches (TM1.step (tr M)) _ _", "annotated_tactic": ["rsuffices \u27e8b, c, r\u27e9 : \u2203 b, _ \u2227 <a>Reaches</a> (<a>TM1.step</a> (<a>tr</a> M)) _ _", [{"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.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}]], "state_before": "case mk.some\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b\u2082,\n    TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227\n      Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082", "state_after": "case mk.some.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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nb : ?m.732398\nc : ?m.732726 b\nr : Reaches (TM1.step (tr M)) (?m.732727 b) (?m.732728 b)\n\u22a2 \u2203 b\u2082,\n    TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227\n      Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b, ?m.732726 b \u2227 Reaches (TM1.step (tr M)) (?m.732727 b) (?m.732728 b)"}, {"tactic": "simp only [tr]", "annotated_tactic": ["simp only [<a>tr</a>]", [{"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}]], "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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (M l) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (tr M (normal l)) v (Tape.mk' \u2205 (addBottom L))) 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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (M l) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (M l)) v (Tape.mk' \u2205 (addBottom L))) b"}, {"tactic": "generalize M l = N", "annotated_tactic": ["generalize M l = N", []], "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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (M l) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (M l)) v (Tape.mk' \u2205 (addBottom L))) 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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nN : Stmt\u2082\n\u22a2 \u2203 b, TrCfg (TM2.stepAux N v S) b \u2227 Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' \u2205 (addBottom L))) b"}, {"tactic": "induction N using stmtStRec generalizing v S L hT with\n| H\u2081 k s q IH => exact tr_respects_aux M hT s @IH\n| H\u2082 a _ IH => exact IH _ hT\n| H\u2083 p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n  unfold TM2.stepAux trNormal TM1.stepAux\n  simp only []\n  cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n| H\u2084 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n| H\u2085 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9", "annotated_tactic": ["induction N using <a>stmtStRec</a> generalizing v S L hT with\n  | H\u2081 k s q IH => exact <a>tr_respects_aux</a> M hT s @IH\n  | H\u2082 a _ IH => exact IH _ hT\n  | H\u2083 p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n    unfold <a>TM2.stepAux</a> <a>trNormal</a> <a>TM1.stepAux</a>\n    simp only []\n    cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n  | H\u2084 => exact \u27e8_, \u27e8_, hT\u27e9, <a>ReflTransGen.refl</a>\u27e9\n  | H\u2085 => exact \u27e8_, \u27e8_, hT\u27e9, <a>ReflTransGen.refl</a>\u27e9", [{"full_name": "Turing.TM2to1.stmtStRec", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2445, 5], "def_end_pos": [2445, 14]}, {"full_name": "Turing.TM2to1.tr_respects_aux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2678, 9], "def_end_pos": [2678, 24]}, {"full_name": "Turing.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "Turing.TM2to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2516, 5], "def_end_pos": [2516, 13]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nN : Stmt\u2082\n\u22a2 \u2203 b, TrCfg (TM2.stepAux N v S) b \u2227 Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' \u2205 (addBottom L))) b", "state_after": "no goals"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case mk.none\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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 match TM2.step M { l := none, var := v, stk := S } with\n  | some b\u2081 =>\n    \u2203 b\u2082,\n      TrCfg b\u2081 b\u2082 \u2227\n        Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n  | none => TM1.step (tr M) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none", "state_after": "no goals"}, {"tactic": "exact \u27e8b, c, TransGen.head' rfl r\u27e9", "annotated_tactic": ["exact \u27e8b, c, <a>TransGen.head'</a> <a>rfl</a> r\u27e9", [{"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": "case mk.some.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\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nb : ?m.732398\nc : ?m.732726 b\nr : Reaches (TM1.step (tr M)) (?m.732727 b) (?m.732728 b)\n\u22a2 \u2203 b\u2082,\n    TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227\n      Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082", "state_after": "no goals"}, {"tactic": "exact tr_respects_aux M hT s @IH", "annotated_tactic": ["exact <a>tr_respects_aux</a> M hT s @IH", [{"full_name": "Turing.TM2to1.tr_respects_aux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2678, 9], "def_end_pos": [2678, 24]}]], "state_before": "case H\u2081\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\nl : \u039b\nk : K\ns : StAct k\nq : Stmt\u2082\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = 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 L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (stRun s q) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun s q)) v (Tape.mk' \u2205 (addBottom L))) b", "state_after": "no goals"}, {"tactic": "exact IH _ hT", "annotated_tactic": ["exact IH _ hT", []], "state_before": "case H\u2082\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\nl : \u039b\na : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u271d v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u271d) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (TM2.Stmt.load a q\u271d) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.load a q\u271d)) v (Tape.mk' \u2205 (addBottom L))) b", "state_after": "no goals"}, {"tactic": "unfold TM2.stepAux trNormal TM1.stepAux", "annotated_tactic": ["unfold <a>TM2.stepAux</a> <a>trNormal</a> <a>TM1.stepAux</a>", [{"full_name": "Turing.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "Turing.TM2to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2516, 5], "def_end_pos": [2516, 13]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}]], "state_before": "case H\u2083\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\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (TM2.Stmt.branch p q\u2081 q\u2082) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) v (Tape.mk' \u2205 (addBottom L))) b", "state_after": "case H\u2083\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\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n      Reaches (TM1.step (tr M))\n        (bif (fun x => p) (Tape.mk' \u2205 (addBottom L)).head v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n        else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n        b"}, {"tactic": "simp only []", "annotated_tactic": ["simp only []", []], "state_before": "case H\u2083\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\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n      Reaches (TM1.step (tr M))\n        (bif (fun x => p) (Tape.mk' \u2205 (addBottom L)).head v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n        else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n        b", "state_after": "case H\u2083\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\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n      Reaches (TM1.step (tr M))\n        (bif p v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n        else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n        b"}, {"tactic": "cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]", "annotated_tactic": ["cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]", []], "state_before": "case H\u2083\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\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n  \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n    (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n      \u2203 b,\n        TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n      Reaches (TM1.step (tr M))\n        (bif p v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n        else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n        b", "state_after": "no goals"}, {"tactic": "exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9", "annotated_tactic": ["exact \u27e8_, \u27e8_, hT\u27e9, <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 H\u2084\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\nl : \u039b\nl\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux (TM2.Stmt.goto l\u271d) v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.goto l\u271d)) v (Tape.mk' \u2205 (addBottom L))) b", "state_after": "no goals"}, {"tactic": "exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9", "annotated_tactic": ["exact \u27e8_, \u27e8_, hT\u27e9, <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 H\u2085\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\nl : \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n    TrCfg (TM2.stepAux TM2.Stmt.halt v S) b \u2227\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal TM2.Stmt.halt) v (Tape.mk' \u2205 (addBottom L))) 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.FinMeasSupp.iff_lintegral_lt_top", "start": [1249, 1], "end": [1251, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt", "start": [476, 1], "end": [479, 69], "traced_tactics": [{"tactic": "refine' le_antisymm upperCrossingTime_le (not_lt.1 _)", "annotated_tactic": ["refine' <a>le_antisymm</a> <a>upperCrossingTime_le</a> (<a>not_lt</a>.1 _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.upperCrossingTime_le", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [187, 9], "def_end_pos": [187, 29]}, {"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 : upcrossingsBefore a b f N \u03c9 < n\n\u22a2 upperCrossingTime a b f N n \u03c9 = 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\nhn : upcrossingsBefore a b f N \u03c9 < n\n\u22a2 \u00acupperCrossingTime a b f N n \u03c9 < N"}, {"tactic": "convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)", "annotated_tactic": ["convert <a>not_mem_of_csSup_lt</a> hn (<a>upperCrossingTime_lt_bddAbove</a> hab)", [{"full_name": "not_mem_of_csSup_lt", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [617, 9], "def_end_pos": [617, 28]}, {"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 \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/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.mem\u2112p_approxOn", "start": [139, 1], "end": [167, 38], "traced_tactics": [{"tactic": "refine' \u27e8(approxOn f fmeas s y\u2080 h\u2080 n).aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8(<a>approxOn</a> f fmeas s y\u2080 h\u2080 n).<a>aestronglyMeasurable</a>, _\u27e9", [{"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Mem\u2112p (\u2191(approxOn f fmeas s y\u2080 h\u2080 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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4"}, {"tactic": "suffices snorm (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc < \u22a4 by\n  have : Mem\u2112p (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n    \u27e8(approxOn f fmeas s y\u2080 h\u2080 n - const \u03b2 y\u2080).aestronglyMeasurable, this\u27e9\n  convert snorm_add_lt_top this hi\u2080\n  ext x\n  simp", "annotated_tactic": ["suffices <a>snorm</a> (fun x => <a>approxOn</a> f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc < \u22a4 by\n    have : <a>Mem\u2112p</a> (fun x => <a>approxOn</a> f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n      \u27e8(<a>approxOn</a> f fmeas s y\u2080 h\u2080 n - <a>const</a> \u03b2 y\u2080).<a>aestronglyMeasurable</a>, this\u27e9\n    convert <a>snorm_add_lt_top</a> this hi\u2080\n    ext x\n    simp", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 40]}, {"full_name": "MeasureTheory.snorm_add_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [893, 9], "def_end_pos": [893, 25]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4"}, {"tactic": "have hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p \u03bc := by\n  have h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 := by\n    simp only [\u2190 dist_eq_norm]\n    exact (continuous_id.dist continuous_const).measurable.comp fmeas\n  refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9\n  rw [snorm_norm]\n  convert snorm_add_lt_top hf hi\u2080.neg with x\n  simp [sub_eq_add_neg]", "annotated_tactic": ["have hf' : <a>Mem\u2112p</a> (fun x => \u2016f x - y\u2080\u2016) p \u03bc := by\n    have h_meas : <a>Measurable</a> fun x => \u2016f x - y\u2080\u2016 := by\n      simp only [\u2190 <a>dist_eq_norm</a>]\n      exact (continuous_id.dist <a>continuous_const</a>).measurable.comp fmeas\n    refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9\n    rw [<a>snorm_norm</a>]\n    convert <a>snorm_add_lt_top</a> hf hi\u2080.neg with x\n    simp [<a>sub_eq_add_neg</a>]", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}, {"full_name": "MeasureTheory.snorm_norm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [501, 9], "def_end_pos": [501, 19]}, {"full_name": "MeasureTheory.snorm_add_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [893, 9], "def_end_pos": [893, 25]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4"}, {"tactic": "have : \u2200\u1d50 x \u2202\u03bc, \u2016approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 := by\n  refine' eventually_of_forall _\n  intro x\n  convert norm_approxOn_y\u2080_le fmeas h\u2080 x n using 1\n  rw [Real.norm_eq_abs, abs_of_nonneg]\n  exact add_nonneg (norm_nonneg _) (norm_nonneg _)", "annotated_tactic": ["have : \u2200\u1d50 x \u2202\u03bc, \u2016<a>approxOn</a> f fmeas s y\u2080 h\u2080 n x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 := by\n    refine' <a>eventually_of_forall</a> _\n    intro x\n    convert <a>norm_approxOn_y\u2080_le</a> fmeas h\u2080 x n using 1\n    rw [<a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a>]\n    exact <a>add_nonneg</a> (<a>norm_nonneg</a> _) (<a>norm_nonneg</a> _)", [{"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 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": "MeasureTheory.SimpleFunc.norm_approxOn_y\u2080_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [78, 9], "def_end_pos": [78, 28]}, {"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": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nthis : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4"}, {"tactic": "calc\n  snorm (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc \u2264\n      snorm (fun x => \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016) p \u03bc :=\n    snorm_mono_ae this\n  _ < \u22a4 := snorm_add_lt_top hf' hf'", "annotated_tactic": ["calc\n    <a>snorm</a> (fun x => <a>approxOn</a> f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc \u2264\n        <a>snorm</a> (fun x => \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016) p \u03bc :=\n      <a>snorm_mono_ae</a> this\n    _ < \u22a4 := <a>snorm_add_lt_top</a> hf' hf'", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"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.snorm_add_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [893, 9], "def_end_pos": [893, 25]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nthis : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "have : Mem\u2112p (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n  \u27e8(approxOn f fmeas s y\u2080 h\u2080 n - const \u03b2 y\u2080).aestronglyMeasurable, this\u27e9", "annotated_tactic": ["have : <a>Mem\u2112p</a> (fun x => <a>approxOn</a> f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n      \u27e8(<a>approxOn</a> f fmeas s y\u2080 h\u2080 n - <a>const</a> \u03b2 y\u2080).<a>aestronglyMeasurable</a>, this\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.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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.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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4"}, {"tactic": "convert snorm_add_lt_top this hi\u2080", "annotated_tactic": ["convert <a>snorm_add_lt_top</a> this hi\u2080", [{"full_name": "MeasureTheory.snorm_add_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [893, 9], "def_end_pos": [893, 25]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4", "state_after": "case h.e'_3.h.e'_5\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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\n\u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) = (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_3.h.e'_5\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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\n\u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) = (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080", "state_after": "case h.e'_3.h.e'_5.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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\nx : \u03b2\n\u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x = ((fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3.h.e'_5.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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\nx : \u03b2\n\u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x = ((fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080) x", "state_after": "no goals"}, {"tactic": "have h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 := by\n  simp only [\u2190 dist_eq_norm]\n  exact (continuous_id.dist continuous_const).measurable.comp fmeas", "annotated_tactic": ["have h_meas : <a>Measurable</a> fun x => \u2016f x - y\u2080\u2016 := by\n      simp only [\u2190 <a>dist_eq_norm</a>]\n      exact (continuous_id.dist <a>continuous_const</a>).measurable.comp fmeas", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) 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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p"}, {"tactic": "refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9", []], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) 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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 snorm (fun x => \u2016f x - y\u2080\u2016) p \u03bc < \u22a4"}, {"tactic": "rw [snorm_norm]", "annotated_tactic": ["rw [<a>snorm_norm</a>]", [{"full_name": "MeasureTheory.snorm_norm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [501, 9], "def_end_pos": [501, 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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 snorm (fun x => \u2016f x - y\u2080\u2016) p \u03bc < \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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 snorm (fun x => f x - y\u2080) p \u03bc < \u22a4"}, {"tactic": "convert snorm_add_lt_top hf hi\u2080.neg with x", "annotated_tactic": ["convert <a>snorm_add_lt_top</a> hf hi\u2080.neg with x", [{"full_name": "MeasureTheory.snorm_add_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [893, 9], "def_end_pos": [893, 25]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 snorm (fun x => f x - y\u2080) p \u03bc < \u22a4", "state_after": "case h.e'_3.h.e'_5.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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\nx : \u03b2\n\u22a2 f x - y\u2080 = (f + -fun x => y\u2080) x"}, {"tactic": "simp [sub_eq_add_neg]", "annotated_tactic": ["simp [<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'_3.h.e'_5.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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\nx : \u03b2\n\u22a2 f x - y\u2080 = (f + -fun x => y\u2080) x", "state_after": "no goals"}, {"tactic": "simp only [\u2190 dist_eq_norm]", "annotated_tactic": ["simp only [\u2190 <a>dist_eq_norm</a>]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Measurable fun x => \u2016f x - y\u2080\u2016", "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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Measurable fun x => dist (f x) y\u2080"}, {"tactic": "exact (continuous_id.dist continuous_const).measurable.comp fmeas", "annotated_tactic": ["exact (continuous_id.dist <a>continuous_const</a>).measurable.comp fmeas", [{"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Measurable fun x => dist (f x) y\u2080", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall _", "annotated_tactic": ["refine' <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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016", "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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 \u2200 (x : \u03b2), \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 \u2200 (x : \u03b2), \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016", "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\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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016"}, {"tactic": "convert norm_approxOn_y\u2080_le fmeas h\u2080 x n using 1", "annotated_tactic": ["convert <a>norm_approxOn_y\u2080_le</a> fmeas h\u2080 x n using 1", [{"full_name": "MeasureTheory.SimpleFunc.norm_approxOn_y\u2080_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [78, 9], "def_end_pos": [78, 28]}]], "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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016", "state_after": "case h.e'_4\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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 = \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016"}, {"tactic": "rw [Real.norm_eq_abs, abs_of_nonneg]", "annotated_tactic": ["rw [<a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</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": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}]], "state_before": "case h.e'_4\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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 = \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016", "state_after": "case h.e'_4\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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 0 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016"}, {"tactic": "exact add_nonneg (norm_nonneg _) (norm_nonneg _)", "annotated_tactic": ["exact <a>add_nonneg</a> (<a>norm_nonneg</a> _) (<a>norm_nonneg</a> _)", [{"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"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'_4\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 \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\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 0 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurableSet_prod", "start": [784, 1], "end": [788, 97], "traced_tactics": [{"tactic": "cases' (s \u00d7\u02e2 t).eq_empty_or_nonempty with h h", "annotated_tactic": ["cases' (s \u00d7\u02e2 t).<a>eq_empty_or_nonempty</a> 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\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\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "case inl\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\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ns : Set \u03b1\nt : Set \u03b2\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205\n\ncase inr\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\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205"}, {"tactic": "simp [h, prod_eq_empty_iff.mp h]", "annotated_tactic": ["simp [h, prod_eq_empty_iff.mp h]", []], "state_before": "case inl\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\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ns : Set \u03b1\nt : Set \u03b2\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "no goals"}, {"tactic": "simp [\u2190 not_nonempty_iff_eq_empty, prod_nonempty_iff.mp h, measurableSet_prod_of_nonempty h]", "annotated_tactic": ["simp [\u2190 <a>not_nonempty_iff_eq_empty</a>, prod_nonempty_iff.mp h, <a>measurableSet_prod_of_nonempty</a> h]", [{"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": "measurableSet_prod_of_nonempty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [775, 9], "def_end_pos": [775, 39]}]], "state_before": "case inr\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\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ns : Set \u03b1\nt : Set \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.map_decode_iff", "start": [659, 1], "end": [662, 27], "traced_tactics": [{"tactic": "convert (bind_decode_iff (f := fun a => Option.some \u2218 f a)).trans option_some_iff", "annotated_tactic": ["convert (<a>bind_decode_iff</a> (f := fun a => <a>Option.some</a> \u2218 f a)).<a>trans</a> <a>option_some_iff</a>", [{"full_name": "Computable.bind_decode_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [635, 9], "def_end_pos": [635, 24]}, {"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": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "Computable.option_some_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [631, 9], "def_end_pos": [631, 24]}]], "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 \u2192 \u03c3\n\u22a2 (Computable\u2082 fun a n => Option.map (f a) (decode n)) \u2194 Computable\u2082 f", "state_after": "case h.e'_1.h.e'_7.h.h\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 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nx\u271d\u00b9 : \u03b1\nx\u271d : \u2115\n\u22a2 Option.map (f x\u271d\u00b9) (decode x\u271d) = Option.bind (decode x\u271d) (Option.some \u2218 f x\u271d\u00b9)"}, {"tactic": "apply Option.map_eq_bind", "annotated_tactic": ["apply <a>Option.map_eq_bind</a>", [{"full_name": "Option.map_eq_bind", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 20]}]], "state_before": "case h.e'_1.h.e'_7.h.h\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 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nx\u271d\u00b9 : \u03b1\nx\u271d : \u2115\n\u22a2 Option.map (f x\u271d\u00b9) (decode x\u271d) = Option.bind (decode x\u271d) (Option.some \u2218 f x\u271d\u00b9)", "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_sub_div", "start": [843, 1], "end": [845, 59], "traced_tactics": [{"tactic": "by_cases hc : c = 0 <;> simp [hc, integral_comp_sub_div]", "annotated_tactic": ["by_cases hc : c = 0 <;> simp [hc, <a>integral_comp_sub_div</a>]", [{"full_name": "intervalIntegral.integral_comp_sub_div", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [837, 9], "def_end_pos": [837, 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 - b / c..d - a / c, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bitwise_bit", "start": [293, 1], "end": [301, 42], "traced_tactics": [{"tactic": "cases' m with m m <;> cases' n with n n <;>\nsimp only [bitwise, ofNat_eq_coe, bit_coe_nat, natBitwise, Bool.not_false, Bool.not_eq_false',\n  bit_negSucc]", "annotated_tactic": ["cases' m with m m <;> cases' n with n n <;>\n  simp only [<a>bitwise</a>, <a>ofNat_eq_coe</a>, <a>bit_coe_nat</a>, <a>natBitwise</a>, <a>Bool.not_false</a>, <a>Bool.not_eq_false'</a>,\n    <a>bit_negSucc</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.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.bit_coe_nat", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [153, 9], "def_end_pos": [153, 20]}, {"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.not_eq_false'", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [129, 17], "def_end_pos": [129, 35]}, {"full_name": "Int.bit_negSucc", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [159, 9], "def_end_pos": [159, 20]}]], "state_before": "f : Bool \u2192 Bool \u2192 Bool\na : Bool\nm : \u2124\nb : Bool\nn : \u2124\n\u22a2 bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n)", "state_after": "case ofNat.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false false then -[Nat.bitwise (fun x y => !f x y) (Nat.bit a m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise f (Nat.bit a m) (Nat.bit b n))) =\n    bit (f a b) (bif f false false then -[Nat.bitwise (fun x y => !f x y) m n+1] else \u2191(Nat.bitwise f m n))\n\ncase ofNat.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false true then -[Nat.bitwise (fun x y => !f x !y) (Nat.bit a m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f x !y) (Nat.bit a m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f false true then -[Nat.bitwise (fun x y => !f x !y) m n+1] else \u2191(Nat.bitwise (fun x y => f x !y) m n))\n\ncase negSucc.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) (Nat.bit (!a) m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) y) (Nat.bit (!a) m) (Nat.bit b n))) =\n    bit (f a b)\n      (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) y) m n))\n\ncase negSucc.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) !y) m n))"}, {"tactic": "by_cases h : f false false <;> simp [h]", "annotated_tactic": ["by_cases h : f <a>false</a> <a>false</a> <;> simp [h]", [{"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": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "state_before": "case ofNat.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false false then -[Nat.bitwise (fun x y => !f x y) (Nat.bit a m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise f (Nat.bit a m) (Nat.bit b n))) =\n    bit (f a b) (bif f false false then -[Nat.bitwise (fun x y => !f x y) m n+1] else \u2191(Nat.bitwise f m n))", "state_after": "no goals"}, {"tactic": "by_cases h : f false true <;> simp [h]", "annotated_tactic": ["by_cases h : f <a>false</a> <a>true</a> <;> simp [h]", [{"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": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "case ofNat.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false true then -[Nat.bitwise (fun x y => !f x !y) (Nat.bit a m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f x !y) (Nat.bit a m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f false true then -[Nat.bitwise (fun x y => !f x !y) m n+1] else \u2191(Nat.bitwise (fun x y => f x !y) m n))", "state_after": "no goals"}, {"tactic": "by_cases h : f true false <;> simp [h]", "annotated_tactic": ["by_cases h : f <a>true</a> <a>false</a> <;> simp [h]", [{"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": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "state_before": "case negSucc.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) (Nat.bit (!a) m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) y) (Nat.bit (!a) m) (Nat.bit b n))) =\n    bit (f a b)\n      (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) y) m n))", "state_after": "no goals"}, {"tactic": "by_cases h : f true true <;> simp [h]", "annotated_tactic": ["by_cases h : f <a>true</a> <a>true</a> <;> simp [h]", [{"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": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "case negSucc.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) !y) m n))", "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_smul_left", "start": [932, 1], "end": [934, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_indexOf\u2081", "start": [778, 1], "end": [779, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.restrict_iUnion_congr", "start": [1858, 1], "end": [1865, 97], "traced_tactics": [{"tactic": "refine' \u27e8fun h i => restrict_congr_mono (subset_iUnion _ _) h, fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h i => <a>restrict_congr_mono</a> (<a>subset_iUnion</a> _ _) h, fun h => _\u27e9", [{"full_name": "MeasureTheory.Measure.restrict_congr_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 28]}, {"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\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\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i, s i) = restrict \u03bd (\u22c3 i, s i) \u2194 \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (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\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\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\n\u22a2 restrict \u03bc (\u22c3 i, s i) = restrict \u03bd (\u22c3 i, s i)"}, {"tactic": "ext1 t ht", "annotated_tactic": ["ext1 t ht", []], "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\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\n\u22a2 restrict \u03bc (\u22c3 i, s i) = restrict \u03bd (\u22c3 i, s i)", "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\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\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 i, s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 i, s i)) t"}, {"tactic": "rw [iUnion_eq_iUnion_finset]", "annotated_tactic": ["rw [<a>iUnion_eq_iUnion_finset</a>]", [{"full_name": "Set.iUnion_eq_iUnion_finset", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1943, 9], "def_end_pos": [1943, 32]}]], "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\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\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\nD : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 i \u2208 t, s i\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 i, s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 i, s i)) 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\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\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\nD : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 i \u2208 t, s i\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 t, \u22c3 i \u2208 t, s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 t, \u22c3 i \u2208 t, s i)) t"}, {"tactic": "simp only [restrict_iUnion_apply_eq_iSup D ht, restrict_finset_biUnion_congr.2 fun i _ => h i]", "annotated_tactic": ["simp only [<a>restrict_iUnion_apply_eq_iSup</a> D ht, <a>restrict_finset_biUnion_congr</a>.2 fun i _ => h i]", [{"full_name": "MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1781, 9], "def_end_pos": [1781, 38]}, {"full_name": "MeasureTheory.Measure.restrict_finset_biUnion_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1850, 9], "def_end_pos": [1850, 38]}]], "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\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\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\nD : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 i \u2208 t, s i\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 t, \u22c3 i \u2208 t, s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 t, \u22c3 i \u2208 t, s i)) 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_univ", "start": [526, 1], "end": [531, 22], "traced_tactics": [{"tactic": "cases' finite_or_infinite \u03b1 with h h", "annotated_tactic": ["cases' <a>finite_or_infinite</a> \u03b1 with h h", [{"full_name": "finite_or_infinite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}]], "state_before": "\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u22a2 ncard univ = Nat.card \u03b1", "state_after": "case inl\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Finite \u03b1\n\u22a2 ncard univ = Nat.card \u03b1\n\ncase inr\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Infinite \u03b1\n\u22a2 ncard univ = Nat.card \u03b1"}, {"tactic": "rw [Nat.card_eq_zero_of_infinite, Infinite.ncard]", "annotated_tactic": ["rw [<a>Nat.card_eq_zero_of_infinite</a>, <a>Infinite.ncard</a>]", [{"full_name": "Nat.card_eq_zero_of_infinite", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [47, 9], "def_end_pos": [47, 33]}, {"full_name": "Set.Infinite.ncard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [504, 9], "def_end_pos": [504, 23]}]], "state_before": "case inr\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Infinite \u03b1\n\u22a2 ncard univ = Nat.card \u03b1", "state_after": "case inr\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Infinite \u03b1\n\u22a2 Set.Infinite univ"}, {"tactic": "exact infinite_univ", "annotated_tactic": ["exact <a>infinite_univ</a>", [{"full_name": "Set.infinite_univ", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 22]}]], "state_before": "case inr\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Infinite \u03b1\n\u22a2 Set.Infinite univ", "state_after": "no goals"}, {"tactic": "have hft := Fintype.ofFinite \u03b1", "annotated_tactic": ["have hft := <a>Fintype.ofFinite</a> \u03b1", [{"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [448, 19], "def_end_pos": [448, 35]}]], "state_before": "case inl\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Finite \u03b1\n\u22a2 ncard univ = Nat.card \u03b1", "state_after": "case inl\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Finite \u03b1\nhft : Fintype \u03b1\n\u22a2 ncard univ = Nat.card \u03b1"}, {"tactic": "rw [ncard_eq_toFinset_card, Finite.toFinset_univ, Finset.card_univ, Nat.card_eq_fintype_card]", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a>, <a>Finite.toFinset_univ</a>, <a>Finset.card_univ</a>, <a>Nat.card_eq_fintype_card</a>]", [{"full_name": "Set.toFinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [82, 9], "def_end_pos": [82, 17]}, {"full_name": "Set.Finite.toFinset_univ", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [287, 19], "def_end_pos": [287, 32]}, {"full_name": "Finset.card_univ", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [247, 9], "def_end_pos": [247, 25]}, {"full_name": "Nat.card_eq_fintype_card", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [42, 9], "def_end_pos": [42, 29]}]], "state_before": "case inl\n\u03b1\u271d : Type ?u.175153\ns t : Set \u03b1\u271d\n\u03b1 : Type u_1\nh : Finite \u03b1\nhft : Fintype \u03b1\n\u22a2 ncard univ = Nat.card \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/FinEnum.lean", "full_name": "FinEnum.Finset.mem_enum", "start": [137, 1], "end": [168, 26], "traced_tactics": [{"tactic": "induction' xs with xs_hd generalizing s <;> simp [*, Finset.enum]", "annotated_tactic": ["induction' xs with xs_hd generalizing s <;> simp [*, <a>Finset.enum</a>]", [{"full_name": "FinEnum.Finset.enum", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [129, 5], "def_end_pos": [129, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nxs : List \u03b1\n\u22a2 s \u2208 enum xs \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 xs", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d s : Finset \u03b1\n\u22a2 s = \u2205 \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 False\n\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\n\u22a2 (\u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d"}, {"tactic": "simp [Finset.eq_empty_iff_forall_not_mem]", "annotated_tactic": ["simp [<a>Finset.eq_empty_iff_forall_not_mem</a>]", [{"full_name": "Finset.eq_empty_iff_forall_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 36]}]], "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d s : Finset \u03b1\n\u22a2 s = \u2205 \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 False", "state_after": "no goals"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\n\u22a2 (\u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "case cons.mp\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\n\u22a2 (\u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)) \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d) \u2192 \u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)"}, {"tactic": "rintro \u27e8a, h, h'\u27e9 x hx", "annotated_tactic": ["rintro \u27e8a, h, h'\u27e9 x hx", []], "state_before": "case cons.mp\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\n\u22a2 (\u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)) \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "case cons.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nh' : s = a \u2228 s = {xs_hd} \u222a a\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d"}, {"tactic": "cases' h' with _ h' a b", "annotated_tactic": ["cases' h' with _ h' a b", []], "state_before": "case cons.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nh' : s = a \u2228 s = {xs_hd} \u222a a\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "case cons.mp.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh\u271d : s = a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d\n\ncase cons.mp.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh' : s = {xs_hd} \u222a a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case cons.mp.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh\u271d : s = a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "case cons.mp.intro.intro.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh\u271d : s = a\n\u22a2 x \u2208 tail\u271d"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case cons.mp.intro.intro.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh\u271d : s = a\n\u22a2 x \u2208 tail\u271d", "state_after": "case cons.mp.intro.intro.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh\u271d : s = a\n\u22a2 x \u2208 a"}, {"tactic": "subst a", "annotated_tactic": ["subst a", []], "state_before": "case cons.mp.intro.intro.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh\u271d : s = a\n\u22a2 x \u2208 a", "state_after": "case cons.mp.intro.intro.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\n\u22a2 x \u2208 s"}, {"tactic": "exact hx", "annotated_tactic": ["exact hx", []], "state_before": "case cons.mp.intro.intro.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "simp only [h', mem_union, mem_singleton] at hx \u22a2", "annotated_tactic": ["simp only [h', <a>mem_union</a>, <a>mem_singleton</a>] at hx \u22a2", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case cons.mp.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nhx : x \u2208 s\nh' : s = {xs_hd} \u222a a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "case cons.mp.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nh' : s = {xs_hd} \u222a a\nhx : x = xs_hd \u2228 x \u2208 a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d"}, {"tactic": "cases' hx with hx hx'", "annotated_tactic": ["cases' hx with hx hx'", []], "state_before": "case cons.mp.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nh' : s = {xs_hd} \u222a a\nhx : x = xs_hd \u2228 x \u2208 a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "case cons.mp.intro.intro.inr.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nh' : s = {xs_hd} \u222a a\nhx : x = xs_hd\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d\n\ncase cons.mp.intro.intro.inr.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nh' : s = {xs_hd} \u222a a\nhx' : x \u2208 a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d"}, {"tactic": "exact Or.inl hx", "annotated_tactic": ["exact <a>Or.inl</a> hx", [{"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 cons.mp.intro.intro.inr.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nh' : s = {xs_hd} \u222a a\nhx : x = xs_hd\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "no goals"}, {"tactic": "exact Or.inr (h _ hx')", "annotated_tactic": ["exact <a>Or.inr</a> (h _ hx')", [{"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 cons.mp.intro.intro.inr.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns a : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d\nx : \u03b1\nh' : s = {xs_hd} \u222a a\nhx' : x \u2208 a\n\u22a2 x = xs_hd \u2228 x \u2208 tail\u271d", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d) \u2192 \u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)", "state_after": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\u22a2 \u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)"}, {"tactic": "exists s \\ ({xs_hd} : Finset \u03b1)", "annotated_tactic": ["exists s \\ ({xs_hd} : <a>Finset</a> \u03b1)", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}]], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\u22a2 \u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)", "state_after": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \\ {xs_hd} \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd})"}, {"tactic": "simp only [and_imp, mem_sdiff, mem_singleton]", "annotated_tactic": ["simp only [<a>and_imp</a>, <a>mem_sdiff</a>, <a>mem_singleton</a>]", [{"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "Finset.mem_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2077, 9], "def_end_pos": [2077, 18]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \\ {xs_hd} \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd})", "state_after": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd})"}, {"tactic": "simp only [or_iff_not_imp_left] at h", "annotated_tactic": ["simp only [<a>or_iff_not_imp_left</a>] at h", [{"full_name": "or_iff_not_imp_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 28]}]], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd})", "state_after": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd})"}, {"tactic": "exists h", "annotated_tactic": ["exists h", []], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd})", "state_after": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}"}, {"tactic": "by_cases h : xs_hd \u2208 s", "annotated_tactic": ["by_cases h : xs_hd \u2208 s", []], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : xs_hd \u2208 s\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}\n\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}"}, {"tactic": "have : {xs_hd} \u2286 s := by\n  simp only [HasSubset.Subset, *, forall_eq, mem_singleton]", "annotated_tactic": ["have : {xs_hd} \u2286 s := by\n          simp only [<a>HasSubset.Subset</a>, *, <a>forall_eq</a>, <a>mem_singleton</a>]", [{"full_name": "HasSubset.Subset", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [11, 3], "def_end_pos": [11, 9]}, {"full_name": "forall_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [450, 17], "def_end_pos": [450, 26]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : xs_hd \u2208 s\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : xs_hd \u2208 s\nthis : {xs_hd} \u2286 s\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}"}, {"tactic": "simp only [union_sdiff_of_subset this, or_true_iff, Finset.union_sdiff_of_subset,\n  eq_self_iff_true]", "annotated_tactic": ["simp only [<a>union_sdiff_of_subset</a> this, <a>or_true_iff</a>, <a>Finset.union_sdiff_of_subset</a>,\n          <a>eq_self_iff_true</a>]", [{"full_name": "Finset.union_sdiff_of_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2102, 9], "def_end_pos": [2102, 30]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "Finset.union_sdiff_of_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2102, 9], "def_end_pos": [2102, 30]}, {"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\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : xs_hd \u2208 s\nthis : {xs_hd} \u2286 s\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}", "state_after": "no goals"}, {"tactic": "simp only [HasSubset.Subset, *, forall_eq, mem_singleton]", "annotated_tactic": ["simp only [<a>HasSubset.Subset</a>, *, <a>forall_eq</a>, <a>mem_singleton</a>]", [{"full_name": "HasSubset.Subset", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [11, 3], "def_end_pos": [11, 9]}, {"full_name": "forall_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [450, 17], "def_end_pos": [450, 26]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : xs_hd \u2208 s\n\u22a2 {xs_hd} \u2286 s", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}", "state_after": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s = s \\ {xs_hd}"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s = s \\ {xs_hd}", "state_after": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s \\ {xs_hd} = s"}, {"tactic": "simp only [sdiff_eq_self]", "annotated_tactic": ["simp only [<a>sdiff_eq_self</a>]", [{"full_name": "Finset.sdiff_eq_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2924, 9], "def_end_pos": [2924, 22]}]], "state_before": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s \\ {xs_hd} = s", "state_after": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s \u2229 {xs_hd} \u2286 \u2205"}, {"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\n\u22a2 s \u2229 {xs_hd} \u2286 \u2205", "state_after": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\na : \u03b1\n\u22a2 a \u2208 s \u2229 {xs_hd} \u2192 a \u2208 \u2205"}, {"tactic": "simp only [and_imp, mem_inter, mem_singleton]", "annotated_tactic": ["simp only [<a>and_imp</a>, <a>mem_inter</a>, <a>mem_singleton</a>]", [{"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "Finset.mem_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1601, 9], "def_end_pos": [1601, 18]}, {"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.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\na : \u03b1\n\u22a2 a \u2208 s \u2229 {xs_hd} \u2192 a \u2208 \u2205", "state_after": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\na : \u03b1\n\u22a2 a \u2208 s \u2192 a = xs_hd \u2192 a \u2208 \u2205"}, {"tactic": "rintro h\u2080 rfl", "annotated_tactic": ["rintro h\u2080 rfl", []], "state_before": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nxs_hd : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d\nh : \u00acxs_hd \u2208 s\na : \u03b1\n\u22a2 a \u2208 s \u2192 a = xs_hd \u2192 a \u2208 \u2205", "state_after": "case neg.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\na : \u03b1\nh\u2080 : a \u2208 s\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 x \u2208 tail\u271d\nh : \u00aca \u2208 s\n\u22a2 a \u2208 \u2205"}, {"tactic": "exact (h h\u2080).elim", "annotated_tactic": ["exact (h h\u2080).<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.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d\ns : Finset \u03b1\na : \u03b1\nh\u2080 : a \u2208 s\nh\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 x \u2208 tail\u271d\nh : \u00aca \u2208 s\n\u22a2 a \u2208 \u2205", "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", "start": [173, 1], "end": [183, 12], "traced_tactics": [{"tactic": "induction xs using Vector.revInductionOn generalizing s\u2081 s\u2082", "annotated_tactic": ["induction xs using <a>Vector.revInductionOn</a> generalizing s\u2081 s\u2082", [{"full_name": "Vector.revInductionOn", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [82, 5], "def_end_pos": [82, 19]}]], "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\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 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2", "state_after": "case nil\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 nil s\u2081).1 (mapAccumr f\u2082 nil s\u2082).1 \u2227 (mapAccumr f\u2081 nil s\u2081).2 = (mapAccumr f\u2082 nil s\u2082).2\n\ncase snoc\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs\u271d s\u2081).1 (mapAccumr f\u2082 xs\u271d s\u2082).1 \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).2 = (mapAccumr f\u2082 xs\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).1 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).2 = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).2"}, {"tactic": "next => exact \u27e8h\u2080, rfl\u27e9", "annotated_tactic": ["next => exact \u27e8h\u2080, <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 nil\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 nil s\u2081).1 (mapAccumr f\u2082 nil s\u2082).1 \u2227 (mapAccumr f\u2081 nil s\u2081).2 = (mapAccumr f\u2082 nil s\u2082).2\n\ncase snoc\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs\u271d s\u2081).1 (mapAccumr f\u2082 xs\u271d s\u2082).1 \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).2 = (mapAccumr f\u2082 xs\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).1 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).2 = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).2", "state_after": "case snoc\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs\u271d s\u2081).1 (mapAccumr f\u2082 xs\u271d s\u2082).1 \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).2 = (mapAccumr f\u2082 xs\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).1 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).2 = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).2"}, {"tactic": "next xs x ih =>\n  rcases (hR x h\u2080) with \u27e8hR, _\u27e9\n  simp only [mapAccumr_snoc, ih hR, true_and]\n  congr 1", "annotated_tactic": ["next xs x ih =>\n    rcases (hR x h\u2080) with \u27e8hR, _\u27e9\n    simp only [<a>mapAccumr_snoc</a>, ih hR, <a>true_and</a>]\n    congr 1", [{"full_name": "Vector.mapAccumr_snoc", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [137, 9], "def_end_pos": [137, 23]}, {"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 snoc\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs\u271d s\u2081).1 (mapAccumr f\u2082 xs\u271d s\u2082).1 \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).2 = (mapAccumr f\u2082 xs\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).1 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).2 = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).2", "state_after": "no goals"}, {"tactic": "exact \u27e8h\u2080, rfl\u27e9", "annotated_tactic": ["exact \u27e8h\u2080, <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": "\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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 nil s\u2081).1 (mapAccumr f\u2082 nil s\u2082).1 \u2227 (mapAccumr f\u2081 nil s\u2081).2 = (mapAccumr f\u2082 nil s\u2082).2", "state_after": "no goals"}, {"tactic": "rcases (hR x h\u2080) with \u27e8hR, _\u27e9", "annotated_tactic": ["rcases (hR x h\u2080) with \u27e8hR, _\u27e9", []], "state_before": "\u03b1 : Type u_2\nn : \u2115\nxs\u271d : 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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\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\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).1 (mapAccumr f\u2082 (snoc xs x) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs x) s\u2081).2 = (mapAccumr f\u2082 (snoc xs x) s\u2082).2", "state_after": "case intro\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : 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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \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\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x s\u2081).1 (f\u2082 x s\u2082).1\nright\u271d : (f\u2081 x s\u2081).2 = (f\u2082 x s\u2082).2\n\u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).1 (mapAccumr f\u2082 (snoc xs x) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs x) s\u2081).2 = (mapAccumr f\u2082 (snoc xs x) s\u2082).2"}, {"tactic": "simp only [mapAccumr_snoc, ih hR, true_and]", "annotated_tactic": ["simp only [<a>mapAccumr_snoc</a>, ih hR, <a>true_and</a>]", [{"full_name": "Vector.mapAccumr_snoc", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [137, 9], "def_end_pos": [137, 23]}, {"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\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : 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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \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\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x s\u2081).1 (f\u2082 x s\u2082).1\nright\u271d : (f\u2081 x s\u2081).2 = (f\u2082 x s\u2082).2\n\u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).1 (mapAccumr f\u2082 (snoc xs x) s\u2082).1 \u2227\n    (mapAccumr f\u2081 (snoc xs x) s\u2081).2 = (mapAccumr f\u2082 (snoc xs x) s\u2082).2", "state_after": "case intro\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : 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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \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\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x s\u2081).1 (f\u2082 x s\u2082).1\nright\u271d : (f\u2081 x s\u2081).2 = (f\u2082 x s\u2082).2\n\u22a2 snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).1).2 (f\u2081 x s\u2081).2 = snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).1).2 (f\u2082 x s\u2082).2"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case intro\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : 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\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \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\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x s\u2081).1 (f\u2082 x s\u2082).1\nright\u271d : (f\u2081 x s\u2081).2 = (f\u2082 x s\u2082).2\n\u22a2 snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).1).2 (f\u2081 x s\u2081).2 = snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).1).2 (f\u2082 x s\u2082).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_def", "start": [1069, 1], "end": [1070, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.pos_iff_ne_zero", "start": [115, 1], "end": [116, 64], "traced_tactics": [{"tactic": "rw [lt_def, val_zero, Nat.pos_iff_ne_zero, \u2190 val_ne_iff]", "annotated_tactic": ["rw [<a>lt_def</a>, <a>val_zero</a>, <a>Nat.pos_iff_ne_zero</a>, \u2190 <a>val_ne_iff</a>]", [{"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.val_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"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": "Fin.val_ne_iff", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [40, 9], "def_end_pos": [40, 19]}]], "state_before": "n : Nat\na : Fin (n + 1)\n\u22a2 0 < a \u2194 a \u2260 0", "state_after": "n : Nat\na : Fin (n + 1)\n\u22a2 \u2191a \u2260 0 \u2194 \u2191a \u2260 \u21910"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : Nat\na : Fin (n + 1)\n\u22a2 \u2191a \u2260 0 \u2194 \u2191a \u2260 \u21910", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Kernel.lean", "full_name": "ProbabilityTheory.kernel.iIndepSets.indepSets", "start": [246, 1], "end": [269, 37], "traced_tactics": [{"tactic": "intro t\u2081 t\u2082 ht\u2081 ht\u2082", "annotated_tactic": ["intro t\u2081 t\u2082 ht\u2081 ht\u2082", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 IndepSets (s i) (s j) \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "have h1 : t\u2081 = ite (i = i) t\u2081 t\u2082 := by simp only [if_true, eq_self_iff_true]", "annotated_tactic": ["have h1 : t\u2081 = <a>ite</a> (i = i) t\u2081 t\u2082 := by simp only [<a>if_true</a>, <a>eq_self_iff_true</a>]", [{"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"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]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "have h2 : t\u2082 = ite (j = i) t\u2081 t\u2082 := by simp only [hij.symm, if_false]", "annotated_tactic": ["have h2 : t\u2082 = <a>ite</a> (j = i) t\u2081 t\u2082 := by simp only [hij.symm, <a>if_false</a>]", [{"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "have h_inter : \u22c2 (t : \u03b9) (_ : t \u2208 ({i, j} : Finset \u03b9)), ite (t = i) t\u2081 t\u2082 =\n    ite (i = i) t\u2081 t\u2082 \u2229 ite (j = i) t\u2081 t\u2082 := by\n  simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]", "annotated_tactic": ["have h_inter : \u22c2 (t : \u03b9) (_ : t \u2208 ({i, j} : <a>Finset</a> \u03b9)), <a>ite</a> (t = i) t\u2081 t\u2082 =\n      <a>ite</a> (i = i) t\u2081 t\u2082 \u2229 <a>ite</a> (j = i) t\u2081 t\u2082 := by\n    simp only [<a>Finset.set_biInter_singleton</a>, <a>Finset.set_biInter_insert</a>]", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 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": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "Finset.set_biInter_singleton", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2099, 9], "def_end_pos": [2099, 30]}, {"full_name": "Finset.set_biInter_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2141, 9], "def_end_pos": [2141, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "filter_upwards [h_indep {i, j} hf_m] with a h_indep'", "annotated_tactic": ["filter_upwards [h_indep {i, j} hf_m] with a h_indep'", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "have h_prod : (\u220f t : \u03b9 in ({i, j} : Finset \u03b9), \u03ba a (ite (t = i) t\u2081 t\u2082))\n    = \u03ba a (ite (i = i) t\u2081 t\u2082) * \u03ba a (ite (j = i) t\u2081 t\u2082) := by\n  simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff,\n    Finset.mem_singleton]", "annotated_tactic": ["have h_prod : (\u220f t : \u03b9 in ({i, j} : <a>Finset</a> \u03b9), \u03ba a (<a>ite</a> (t = i) t\u2081 t\u2082))\n      = \u03ba a (<a>ite</a> (i = i) t\u2081 t\u2082) * \u03ba a (<a>ite</a> (j = i) t\u2081 t\u2082) := by\n    simp only [hij, <a>Finset.prod_singleton</a>, <a>Finset.prod_insert</a>, <a>not_false_iff</a>,\n      <a>Finset.mem_singleton</a>]", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 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": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "Finset.prod_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 23]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "rw [h1]", "annotated_tactic": ["rw [h1]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "nth_rw 2 [h2]", "annotated_tactic": ["nth_rw 2 [h2]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082) = \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) t\u2082"}, {"tactic": "nth_rw 4 [h2]", "annotated_tactic": ["nth_rw 4 [h2]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082) = \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) t\u2082", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)"}, {"tactic": "rw [\u2190 h_inter, \u2190 h_prod, h_indep']", "annotated_tactic": ["rw [\u2190 h_inter, \u2190 h_prod, h_indep']", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\nh_prod :\n  \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\n\u22a2 \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nx : \u03b9\nhx : x \u2208 {i, j}\n\u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x"}, {"tactic": "cases' Finset.mem_insert.mp hx with hx hx", "annotated_tactic": ["cases' Finset.mem_insert.mp hx with hx hx", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nx : \u03b9\nhx : x \u2208 {i, j}\n\u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x", "state_after": "case inl\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nx : \u03b9\nhx\u271d : x \u2208 {i, j}\nhx : x = i\n\u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x\n\ncase inr\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nx : \u03b9\nhx\u271d : x \u2208 {i, j}\nhx : x \u2208 {j}\n\u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x"}, {"tactic": "simp [hx, ht\u2081]", "annotated_tactic": ["simp [hx, ht\u2081]", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nx : \u03b9\nhx\u271d : x \u2208 {i, j}\nhx : x = i\n\u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x", "state_after": "no goals"}, {"tactic": "simp [Finset.mem_singleton.mp hx, hij.symm, ht\u2082]", "annotated_tactic": ["simp [Finset.mem_singleton.mp hx, hij.symm, ht\u2082]", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nx : \u03b9\nhx\u271d : x \u2208 {i, j}\nhx : x \u2208 {j}\n\u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x", "state_after": "no goals"}, {"tactic": "simp only [if_true, eq_self_iff_true]", "annotated_tactic": ["simp only [<a>if_true</a>, <a>eq_self_iff_true</a>]", [{"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]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\n\u22a2 t\u2081 = if i = i then t\u2081 else t\u2082", "state_after": "no goals"}, {"tactic": "simp only [hij.symm, if_false]", "annotated_tactic": ["simp only [hij.symm, <a>if_false</a>]", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\n\u22a2 t\u2082 = if j = i then t\u2081 else t\u2082", "state_after": "no goals"}, {"tactic": "simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]", "annotated_tactic": ["simp only [<a>Finset.set_biInter_singleton</a>, <a>Finset.set_biInter_insert</a>]", [{"full_name": "Finset.set_biInter_singleton", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2099, 9], "def_end_pos": [2099, 30]}, {"full_name": "Finset.set_biInter_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2141, 9], "def_end_pos": [2141, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\n\u22a2 (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082", "state_after": "no goals"}, {"tactic": "simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff,\n  Finset.mem_singleton]", "annotated_tactic": ["simp only [hij, <a>Finset.prod_singleton</a>, <a>Finset.prod_insert</a>, <a>not_false_iff</a>,\n      <a>Finset.mem_singleton</a>]", [{"full_name": "Finset.prod_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 23]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\ns : \u03b9 \u2192 Set (Set \u03a9)\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\nh_indep : iIndepSets s \u03ba\ni j : \u03b9\nhij : i \u2260 j\nt\u2081 t\u2082 : Set \u03a9\nht\u2081 : t\u2081 \u2208 s i\nht\u2082 : t\u2082 \u2208 s j\nhf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x\nh1 : t\u2081 = if i = i then t\u2081 else t\u2082\nh2 : t\u2082 = if j = i then t\u2081 else t\u2082\nh_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082\na : \u03b1\nh_indep' :\n  \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082)\n\u22a2 \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =\n    \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.RightInvOn.surjOn", "start": [1141, 1], "end": [1142, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_nonneg", "start": [1181, 1], "end": [1194, 83], "traced_tactics": [{"tactic": "suffices : \u2200 f : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g }, 0 \u2264 setToL1 hT f", "annotated_tactic": ["suffices : \u2200 f : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g }, 0 \u2264 <a>setToL1</a> hT f", [{"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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 0 \u2264 \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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\nthis : \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f\n\u22a2 0 \u2264 \u2191(setToL1 hT) f\n\ncase this\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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f"}, {"tactic": "exact this (\u27e8f, hf\u27e9 : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g })", "annotated_tactic": ["exact this (\u27e8f, hf\u27e9 : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g })", []], "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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\nthis : \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f\n\u22a2 0 \u2264 \u2191(setToL1 hT) f\n\ncase this\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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f", "state_after": "case this\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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f"}, {"tactic": "refine' fun g =>\n  @isClosed_property { g : \u03b1 \u2192\u2081\u209b[\u03bc] G' // 0 \u2264 g } { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g } _ _\n    (fun g => 0 \u2264 setToL1 hT g)\n    (denseRange_coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' one_ne_top) _ _ g", "annotated_tactic": ["refine' fun g =>\n    @<a>isClosed_property</a> { g : \u03b1 \u2192\u2081\u209b[\u03bc] G' // 0 \u2264 g } { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g } _ _\n      (fun g => 0 \u2264 <a>setToL1</a> hT g)\n      (<a>denseRange_coeSimpleFuncNonnegToLpNonneg</a> 1 \u03bc G' <a>one_ne_top</a>) _ _ g", [{"full_name": "isClosed_property", "def_path": "Mathlib/Topology/DenseEmbedding.lean", "def_pos": [308, 9], "def_end_pos": [308, 26]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [852, 9], "def_end_pos": [852, 49]}, {"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 this\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\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f", "state_after": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 IsClosed {x | (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) x}\n\ncase this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 \u2200 (a : { g // 0 \u2264 g }), (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' a)"}, {"tactic": "exact isClosed_le continuous_zero ((setToL1 hT).continuous.comp continuous_induced_dom)", "annotated_tactic": ["exact <a>isClosed_le</a> <a>continuous_zero</a> ((<a>setToL1</a> hT).continuous.comp <a>continuous_induced_dom</a>)", [{"full_name": "isClosed_le", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 20]}, {"full_name": "continuous_zero", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [31, 3], "def_end_pos": [31, 14]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "continuous_induced_dom", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [730, 9], "def_end_pos": [730, 31]}]], "state_before": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 IsClosed {x | (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) x}", "state_after": "no goals"}, {"tactic": "intro g", "annotated_tactic": ["intro g", []], "state_before": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 \u2200 (a : { g // 0 \u2264 g }), (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' a)", "state_after": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)"}, {"tactic": "have : (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g : \u03b1 \u2192\u2081[\u03bc] G') = (g : \u03b1 \u2192\u2081\u209b[\u03bc] G') := rfl", "annotated_tactic": ["have : (<a>coeSimpleFuncNonnegToLpNonneg</a> 1 \u03bc G' g : \u03b1 \u2192\u2081[\u03bc] G') = (g : \u03b1 \u2192\u2081\u209b[\u03bc] G') := <a>rfl</a>", [{"full_name": "MeasureTheory.Lp.simpleFunc.coeSimpleFuncNonnegToLpNonneg", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [848, 5], "def_end_pos": [848, 34]}, {"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 this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)", "state_after": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)"}, {"tactic": "rw [this, setToL1_eq_setToL1SCLM]", "annotated_tactic": ["rw [this, <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 this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)", "state_after": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1SCLM \u03b1 G' \u03bc hT) \u2191g"}, {"tactic": "exact setToL1S_nonneg (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2", "annotated_tactic": ["exact <a>setToL1S_nonneg</a> (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2", [{"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": "case this.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\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\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 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1SCLM \u03b1 G' \u03bc hT) \u2191g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/UInt.lean", "full_name": "UInt8.toChar_aux", "start": [121, 1], "end": [124, 35], "traced_tactics": [{"tactic": "rw [UInt32.val_eq_of_lt]", "annotated_tactic": ["rw [<a>UInt32.val_eq_of_lt</a>]", [{"full_name": "UInt32.val_eq_of_lt", "def_path": "Mathlib/Data/UInt.lean", "def_pos": [12, 7], "def_end_pos": [12, 26]}]], "state_before": "n : \u2115\nh : n < size\n\u22a2 Nat.isValidChar \u2191(UInt32.ofNat n).val", "state_after": "n : \u2115\nh : n < size\n\u22a2 Nat.isValidChar n\n\nn : \u2115\nh : n < size\n\u22a2 n < UInt32.size"}, {"tactic": "exact Or.inl $ Nat.lt_trans h $ by decide", "annotated_tactic": ["exact <a>Or.inl</a> $ <a>Nat.lt_trans</a> h $ by decide", [{"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.lt_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1596, 19], "def_end_pos": [1596, 31]}]], "state_before": "n : \u2115\nh : n < size\n\u22a2 Nat.isValidChar n\n\nn : \u2115\nh : n < size\n\u22a2 n < UInt32.size", "state_after": "n : \u2115\nh : n < size\n\u22a2 n < UInt32.size"}, {"tactic": "exact Nat.lt_trans h $ by decide", "annotated_tactic": ["exact <a>Nat.lt_trans</a> h $ by decide", [{"full_name": "Nat.lt_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1596, 19], "def_end_pos": [1596, 31]}]], "state_before": "n : \u2115\nh : n < size\n\u22a2 n < UInt32.size", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n : \u2115\nh : n < size\n\u22a2 size < 55296", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n : \u2115\nh : n < size\n\u22a2 size < UInt32.size", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.measure_le_le_exp_cgf", "start": [373, 1], "end": [378, 67], "traced_tactics": [{"tactic": "refine' (measure_le_le_exp_mul_mgf \u03b5 ht h_int).trans _", "annotated_tactic": ["refine' (<a>measure_le_le_exp_mul_mgf</a> \u03b5 ht h_int).<a>trans</a> _", [{"full_name": "ProbabilityTheory.measure_le_le_exp_mul_mgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [352, 9], "def_end_pos": [352, 34]}, {"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\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : t \u2264 0\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}) \u2264 rexp (-t * \u03b5 + cgf X \u03bc t)", "state_after": "\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : t \u2264 0\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 rexp (-t * \u03b5) * mgf (fun \u03c9 => X \u03c9) \u03bc t \u2264 rexp (-t * \u03b5 + cgf X \u03bc t)"}, {"tactic": "rw [exp_add]", "annotated_tactic": ["rw [<a>exp_add</a>]", [{"full_name": "Real.exp_add", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1138, 16], "def_end_pos": [1138, 23]}]], "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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : t \u2264 0\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 rexp (-t * \u03b5) * mgf (fun \u03c9 => X \u03c9) \u03bc t \u2264 rexp (-t * \u03b5 + cgf X \u03bc t)", "state_after": "\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : t \u2264 0\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 rexp (-t * \u03b5) * mgf (fun \u03c9 => X \u03c9) \u03bc t \u2264 rexp (-t * \u03b5) * rexp (cgf X \u03bc t)"}, {"tactic": "exact mul_le_mul le_rfl (le_exp_log _) mgf_nonneg (exp_pos _).le", "annotated_tactic": ["exact <a>mul_le_mul</a> <a>le_rfl</a> (<a>le_exp_log</a> _) <a>mgf_nonneg</a> (<a>exp_pos</a> _).<a>le</a>", [{"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_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Real.le_exp_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "ProbabilityTheory.mgf_nonneg", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [182, 9], "def_end_pos": [182, 19]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : t \u2264 0\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 rexp (-t * \u03b5) * mgf (fun \u03c9 => X \u03c9) \u03bc t \u2264 rexp (-t * \u03b5) * rexp (cgf X \u03bc t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.stoppedValue_stoppedValue_leastGE", "start": [89, 1], "end": [92, 80], "traced_tactics": [{"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u03c9 : \u03a9\n\u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 \u03c9 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0 \u03c9"}, {"tactic": "simp_rw [stoppedProcess, stoppedValue]", "annotated_tactic": ["simp_rw [<a>stoppedProcess</a>, <a>stoppedValue</a>]", [{"full_name": "MeasureTheory.stoppedProcess", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [781, 5], "def_end_pos": [781, 19]}, {"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}]], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u03c9 : \u03a9\n\u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 \u03c9 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0 \u03c9", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u03c9 : \u03a9\n\u22a2 f (leastGE f r (\u03c0 \u03c9) \u03c9) \u03c9 = f (min (\u03c0 \u03c9) (leastGE f r n \u03c9)) \u03c9"}, {"tactic": "rw [leastGE_eq_min _ _ _ h\u03c0n]", "annotated_tactic": ["rw [<a>leastGE_eq_min</a> _ _ _ h\u03c0n]", [{"full_name": "MeasureTheory.leastGE_eq_min", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}]], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u03c9 : \u03a9\n\u22a2 f (leastGE f r (\u03c0 \u03c9) \u03c9) \u03c9 = f (min (\u03c0 \u03c9) (leastGE f r n \u03c9)) \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.finset_prod_mem_finset_prod", "start": [154, 1], "end": [156, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat", "start": [440, 1], "end": [443, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.SimpleFunc.integral_smul", "start": [542, 1], "end": [543, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_eq_zero", "start": [92, 9], "end": [95, 33], "traced_tactics": [{"tactic": "rw [encard, \u2190PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply,\n  PartENat.withTopEquiv_symm_zero, PartENat.card_eq_zero_iff_empty, isEmpty_subtype,\n  eq_empty_iff_forall_not_mem]", "annotated_tactic": ["rw [<a>encard</a>, \u2190PartENat.withTopEquiv.symm.injective.eq_iff, <a>Equiv.symm_apply_apply</a>,\n    <a>PartENat.withTopEquiv_symm_zero</a>, <a>PartENat.card_eq_zero_iff_empty</a>, <a>isEmpty_subtype</a>,\n    <a>eq_empty_iff_forall_not_mem</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_zero", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [728, 9], "def_end_pos": [728, 31]}, {"full_name": "PartENat.card_eq_zero_iff_empty", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [223, 9], "def_end_pos": [223, 31]}, {"full_name": "isEmpty_subtype", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [167, 9], "def_end_pos": [167, 24]}, {"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": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 encard s = 0 \u2194 s = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.divMonomial_add_modMonomial_single", "start": [200, 1], "end": [202, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.cond_mul_eq_inter", "start": [145, 1], "end": [147, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.rec_computable", "start": [499, 1], "end": [601, 45], "traced_tactics": [{"tactic": "intros _ _ _ _ F", "annotated_tactic": ["intros _ _ _ _ F", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\n\u22a2 let PR := fun a cf cg hf hg => pr a (cf, cg, hf, hg);\n  let CO := fun a cf cg hf hg => co a (cf, cg, hf, hg);\n  let PC := fun a cf cg hf hg => pc a (cf, cg, hf, hg);\n  let RF := fun a cf hf => rf a (cf, hf);\n  let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a);\n  Computable fun a => F a (c a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\n\u22a2 Computable fun a => F a (c a)"}, {"tactic": "let G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p =>\n  let a := p.1.1\n  let IH := p.1.2\n  let n := p.2.1\n  let m := p.2.2\n  (IH.get? m).bind fun s =>\n    (IH.get? m.unpair.1).bind fun s\u2081 =>\n      (IH.get? m.unpair.2).map fun s\u2082 =>\n        cond n.bodd\n          (cond n.div2.bodd (rf a (ofNat Code m, s))\n            (pc a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n          (cond n.div2.bodd (co a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))\n            (pr a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))", "annotated_tactic": ["let G\u2081 : (\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 <a>Option</a> \u03c3 := fun p =>\n    let a := p.1.1\n    let IH := p.1.2\n    let n := p.2.1\n    let m := p.2.2\n    (IH.get? m).<a>bind</a> fun s =>\n      (IH.get? m.unpair.1).<a>bind</a> fun s\u2081 =>\n        (IH.get? m.unpair.2).<a>map</a> fun s\u2082 =>\n          <a>cond</a> n.bodd\n            (<a>cond</a> n.div2.bodd (rf a (<a>ofNat</a> <a>Code</a> m, s))\n              (pc a (<a>ofNat</a> <a>Code</a> m.unpair.1, <a>ofNat</a> <a>Code</a> m.unpair.2, s\u2081, s\u2082)))\n            (<a>cond</a> n.div2.bodd (co a (<a>ofNat</a> <a>Code</a> m.unpair.1, <a>ofNat</a> <a>Code</a> m.unpair.2, s\u2081, s\u2082))\n              (pr a (<a>ofNat</a> <a>Code</a> m.unpair.1, <a>ofNat</a> <a>Code</a> m.unpair.2, s\u2081, s\u2082)))", [{"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": "Option.bind", "def_path": "lake-packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [33, 25], "def_end_pos": [33, 29]}, {"full_name": "Option.bind", "def_path": "lake-packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [33, 25], "def_end_pos": [33, 29]}, {"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": "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]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\n\u22a2 Computable fun a => F a (c a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable fun a => F a (c a)"}, {"tactic": "have : Computable G\u2081 := by\n  refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n  unfold Computable\u2082\n  refine'\n    option_bind\n      ((list_get?.comp (snd.comp fst)\n        (fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _\n  unfold Computable\u2082\n  refine'\n    option_map\n      ((list_get?.comp (snd.comp fst)\n        (snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n  have a : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n    fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n  have n : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n    fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n    snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m\u2081 := fst.comp (Computable.unpair.comp m)\n  have m\u2082 := snd.comp (Computable.unpair.comp m)\n  have s : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n    snd.comp (fst.comp fst)\n  have s\u2081 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n    snd.comp fst\n  have s\u2082 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n    snd\n  exact\n    (nat_bodd.comp n).cond\n      ((nat_bodd.comp <| nat_div2.comp n).cond\n        (hrf.comp a (((Computable.ofNat Code).comp m).pair s))\n        (hpc.comp a\n          (((Computable.ofNat Code).comp m\u2081).pair <|\n            ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n      (Computable.cond (nat_bodd.comp <| nat_div2.comp n)\n        (hco.comp a\n          (((Computable.ofNat Code).comp m\u2081).pair <|\n            ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n        (hpr.comp a\n          (((Computable.ofNat Code).comp m\u2081).pair <|\n            ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))", "annotated_tactic": ["have : <a>Computable</a> G\u2081 := by\n    refine' <a>option_bind</a> (list_get?.comp (snd.comp <a>fst</a>) (snd.comp <a>snd</a>)) _\n    unfold <a>Computable\u2082</a>\n    refine'\n      <a>option_bind</a>\n        ((list_get?.comp (snd.comp <a>fst</a>)\n          (fst.comp <| Computable.unpair.comp (snd.comp <a>snd</a>))).<a>comp</a> <a>fst</a>) _\n    unfold <a>Computable\u2082</a>\n    refine'\n      <a>option_map</a>\n        ((list_get?.comp (snd.comp <a>fst</a>)\n          (snd.comp <| Computable.unpair.comp (snd.comp <a>snd</a>))).<a>comp</a> <| fst.comp <a>fst</a>) _\n    have a : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n      fst.comp (fst.comp <| fst.comp <| fst.comp <a>fst</a>)\n    have n : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n      fst.comp (snd.comp <| fst.comp <| fst.comp <a>fst</a>)\n    have m : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n      snd.comp (snd.comp <| fst.comp <| fst.comp <a>fst</a>)\n    have m\u2081 := fst.comp (Computable.unpair.comp m)\n    have m\u2082 := snd.comp (Computable.unpair.comp m)\n    have s : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n      snd.comp (fst.comp <a>fst</a>)\n    have s\u2081 : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n      snd.comp <a>fst</a>\n    have s\u2082 : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n      <a>snd</a>\n    exact\n      (nat_bodd.comp n).<a>cond</a>\n        ((nat_bodd.comp <| nat_div2.comp n).<a>cond</a>\n          (hrf.comp a (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m).<a>pair</a> s))\n          (hpc.comp a\n            (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2081).<a>pair</a> <|\n              ((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2082).<a>pair</a> <| s\u2081.pair s\u2082)))\n        (<a>Computable.cond</a> (nat_bodd.comp <| nat_div2.comp n)\n          (hco.comp a\n            (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2081).<a>pair</a> <|\n              ((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2082).<a>pair</a> <| s\u2081.pair s\u2082))\n          (hpr.comp a\n            (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2081).<a>pair</a> <|\n              ((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2082).<a>pair</a> <| s\u2081.pair s\u2082)))", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 15]}, {"full_name": "Computable.option_bind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [690, 9], "def_end_pos": [690, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [249, 5], "def_end_pos": [249, 16]}, {"full_name": "Computable.option_bind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [690, 9], "def_end_pos": [690, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [249, 5], "def_end_pos": [249, 16]}, {"full_name": "Computable.option_map", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.cond", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [677, 9], "def_end_pos": [677, 13]}, {"full_name": "Computable.cond", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [677, 9], "def_end_pos": [677, 13]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.cond", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [677, 9], "def_end_pos": [677, 13]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable fun a => F a (c a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis : Computable G\u2081\n\u22a2 Computable fun a => F a (c a)"}, {"tactic": "let G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH =>\n  IH.length.casesOn (some (z a)) fun n =>\n    n.casesOn (some (s a)) fun n =>\n      n.casesOn (some (l a)) fun n =>\n        n.casesOn (some (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2)", "annotated_tactic": ["let G : \u03b1 \u2192 <a>List</a> \u03c3 \u2192 <a>Option</a> \u03c3 := fun a IH =>\n    IH.length.casesOn (<a>some</a> (z a)) fun n =>\n      n.casesOn (<a>some</a> (s a)) fun n =>\n        n.casesOn (<a>some</a> (l a)) fun n =>\n          n.casesOn (<a>some</a> (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2)", [{"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": "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": "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]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis : Computable G\u2081\n\u22a2 Computable fun a => F a (c a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Computable fun a => F a (c a)"}, {"tactic": "have : Computable\u2082 G :=\n  Computable.nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) <|\n    Computable.nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) <|\n      Computable.nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) <|\n        Computable.nat_casesOn snd\n          (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst)))\n          (this.comp <|\n            ((Computable.fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n              snd.pair <| nat_div2.comp <| nat_div2.comp snd)", "annotated_tactic": ["have : <a>Computable\u2082</a> G :=\n    <a>Computable.nat_casesOn</a> (list_length.comp <a>snd</a>) (<a>option_some_iff</a>.2 (hz.comp <a>fst</a>)) <|\n      <a>Computable.nat_casesOn</a> <a>snd</a> (<a>option_some_iff</a>.2 (hs.comp (fst.comp <a>fst</a>))) <|\n        <a>Computable.nat_casesOn</a> <a>snd</a> (<a>option_some_iff</a>.2 (hl.comp (fst.comp <| fst.comp <a>fst</a>))) <|\n          <a>Computable.nat_casesOn</a> <a>snd</a>\n            (<a>option_some_iff</a>.2 (hr.comp (fst.comp <| fst.comp <| fst.comp <a>fst</a>)))\n            (this.comp <|\n              ((Computable.fst.pair <a>snd</a>).<a>comp</a> <| fst.comp <| fst.comp <| fst.comp <| <a>fst</a>).<a>pair</a> <|\n                snd.pair <| nat_div2.comp <| nat_div2.comp <a>snd</a>)", [{"full_name": "Computable\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [249, 5], "def_end_pos": [249, 16]}, {"full_name": "Computable.nat_casesOn", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [671, 9], "def_end_pos": [671, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.option_some_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [631, 9], "def_end_pos": [631, 24]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.nat_casesOn", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [671, 9], "def_end_pos": [671, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.option_some_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [631, 9], "def_end_pos": [631, 24]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.nat_casesOn", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [671, 9], "def_end_pos": [671, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.option_some_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [631, 9], "def_end_pos": [631, 24]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.nat_casesOn", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [671, 9], "def_end_pos": [671, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.option_some_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [631, 9], "def_end_pos": [631, 24]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Computable fun a => F a (c a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\n\u22a2 Computable fun a => F a (c a)"}, {"tactic": "refine'\n  ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to\u2082 fun a n => _).comp Computable.id <|\n    encode_iff.2 hc).of_eq fun a => by simp", "annotated_tactic": ["refine'\n    ((<a>nat_strong_rec</a> (fun a n => F a (<a>ofNat</a> <a>Code</a> n)) this.to\u2082 fun a n => _).<a>comp</a> <a>Computable.id</a> <|\n      <a>encode_iff</a>.2 hc).<a>of_eq</a> fun a => by simp", [{"full_name": "Computable.nat_strong_rec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [724, 9], "def_end_pos": [724, 23]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable\u2082.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [545, 16], "def_end_pos": [545, 20]}, {"full_name": "Computable.id", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [298, 19], "def_end_pos": [298, 21]}, {"full_name": "Computable.encode_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}, {"full_name": "Computable.of_eq", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [280, 9], "def_end_pos": [280, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\n\u22a2 Computable fun a => F a (c a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).1\n      (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).2 =\n    some ((fun a n => F a (ofNat Code n)) a n)"}, {"tactic": "simp (config := { zeta := false })", "annotated_tactic": ["simp (config := { zeta := <a>false</a> })", [{"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).1\n      (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).2 =\n    some ((fun a n => F a (ofNat Code n)) a n)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))"}, {"tactic": "simp only []", "annotated_tactic": ["simp only []", []], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).1)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).2))\n                  n_3)\n              n_2)\n          n_1)\n      (List.length\n        (List.map\n          (fun n =>\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))"}, {"tactic": "rw [List.length_map, List.length_range]", "annotated_tactic": ["rw [<a>List.length_map</a>, <a>List.length_range</a>]", [{"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": "List.length_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2074, 17], "def_end_pos": [2074, 29]}]], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).1)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).2))\n                  n_3)\n              n_2)\n          n_1)\n      (List.length\n        (List.map\n          (fun n =>\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).1)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).2))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))"}, {"tactic": "let m := n.div2.div2", "annotated_tactic": ["let m := n.div2.div2", []], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).1)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).2))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).1)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).2))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))"}, {"tactic": "show\n  G\u2081 ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) =\n    some (F a (ofNat Code (n + 4)))", "annotated_tactic": ["show\n    G\u2081 ((a, (<a>List.range</a> (n + 4)).<a>map</a> fun n => F a (<a>ofNat</a> <a>Code</a> n)), n, m) =\n      <a>some</a> (F a (<a>ofNat</a> <a>Code</a> (n + 4)))", [{"full_name": "List.range", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [572, 5], "def_end_pos": [572, 10]}, {"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": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"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": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}]], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).1)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,\n                                    s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).2))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (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": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (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": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (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": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))"}, {"tactic": "simp [List.get?_map, List.get?_range, hm, m1, m2]", "annotated_tactic": ["simp [<a>List.get?_map</a>, <a>List.get?_range</a>, hm, m1, m2]", [{"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.get?_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2094, 9], "def_end_pos": [2094, 19]}]], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (n + 4))"}, {"tactic": "rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl]", "annotated_tactic": ["rw [show <a>ofNat</a> <a>Code</a> (n + 4) = <a>ofNatCode</a> (n + 4) from <a>rfl</a>]", [{"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"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.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 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 succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (n + 4))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNatCode (n + 4))"}, {"tactic": "simp [ofNatCode]", "annotated_tactic": ["simp [<a>ofNatCode</a>]", [{"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}]], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNatCode (n + 4))", "state_after": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\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))))"}, {"tactic": "cases n.bodd <;> cases n.div2.bodd <;> rfl", "annotated_tactic": ["cases n.bodd <;> cases n.div2.bodd <;> rfl", []], "state_before": "case succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).1 < n + 4\nm2 : (unpair m).2 < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\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))))", "state_after": "no goals"}, {"tactic": "refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _", "annotated_tactic": ["refine' <a>option_bind</a> (list_get?.comp (snd.comp <a>fst</a>) (snd.comp <a>snd</a>)) _", [{"full_name": "Computable.option_bind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [690, 9], "def_end_pos": [690, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable G\u2081", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable\u2082 fun p s =>\n    Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 =>\n      Option.map\n        (fun s\u2082 =>\n          bif bodd p.2.1 then\n            bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s)\n            else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n          else\n            bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n            else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082))\n        (List.get? p.1.2 (unpair p.2.2).2)"}, {"tactic": "unfold Computable\u2082", "annotated_tactic": ["unfold <a>Computable\u2082</a>", [{"full_name": "Computable\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [249, 5], "def_end_pos": [249, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable\u2082 fun p s =>\n    Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 =>\n      Option.map\n        (fun s\u2082 =>\n          bif bodd p.2.1 then\n            bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s)\n            else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n          else\n            bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n            else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082))\n        (List.get? p.1.2 (unpair p.2.2).2)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable fun p =>\n    (fun p s =>\n        Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 =>\n          Option.map\n            (fun s\u2082 =>\n              bif bodd p.2.1 then\n                bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s)\n                else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n              else\n                bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n                else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082))\n            (List.get? p.1.2 (unpair p.2.2).2))\n      p.1 p.2"}, {"tactic": "refine'\n  option_bind\n    ((list_get?.comp (snd.comp fst)\n      (fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _", "annotated_tactic": ["refine'\n      <a>option_bind</a>\n        ((list_get?.comp (snd.comp <a>fst</a>)\n          (fst.comp <| Computable.unpair.comp (snd.comp <a>snd</a>))).<a>comp</a> <a>fst</a>) _", [{"full_name": "Computable.option_bind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [690, 9], "def_end_pos": [690, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable fun p =>\n    (fun p s =>\n        Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 =>\n          Option.map\n            (fun s\u2082 =>\n              bif bodd p.2.1 then\n                bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s)\n                else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n              else\n                bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)\n                else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082))\n            (List.get? p.1.2 (unpair p.2.2).2))\n      p.1 p.2", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable\u2082 fun p s\u2081 =>\n    Option.map\n      (fun s\u2082 =>\n        bif bodd p.1.2.1 then\n          bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2)\n          else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n        else\n          bif bodd (div2 p.1.2.1) then co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n          else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082))\n      (List.get? p.1.1.2 (unpair p.1.2.2).2)"}, {"tactic": "unfold Computable\u2082", "annotated_tactic": ["unfold <a>Computable\u2082</a>", [{"full_name": "Computable\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [249, 5], "def_end_pos": [249, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable\u2082 fun p s\u2081 =>\n    Option.map\n      (fun s\u2082 =>\n        bif bodd p.1.2.1 then\n          bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2)\n          else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n        else\n          bif bodd (div2 p.1.2.1) then co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n          else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082))\n      (List.get? p.1.1.2 (unpair p.1.2.2).2)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable fun p =>\n    (fun p s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd p.1.2.1 then\n              bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2)\n              else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 p.1.2.1) then\n                co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n              else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082))\n          (List.get? p.1.1.2 (unpair p.1.2.2).2))\n      p.1 p.2"}, {"tactic": "refine'\n  option_map\n    ((list_get?.comp (snd.comp fst)\n      (snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _", "annotated_tactic": ["refine'\n      <a>option_map</a>\n        ((list_get?.comp (snd.comp <a>fst</a>)\n          (snd.comp <| Computable.unpair.comp (snd.comp <a>snd</a>))).<a>comp</a> <| fst.comp <a>fst</a>) _", [{"full_name": "Computable.option_map", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable fun p =>\n    (fun p s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd p.1.2.1 then\n              bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2)\n              else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 p.1.2.1) then\n                co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)\n              else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082))\n          (List.get? p.1.1.2 (unpair p.1.2.2).2))\n      p.1 p.2", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have a : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n  fst.comp (fst.comp <| fst.comp <| fst.comp fst)", "annotated_tactic": ["have a : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n      fst.comp (fst.comp <| fst.comp <| fst.comp <a>fst</a>)", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have n : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n  fst.comp (snd.comp <| fst.comp <| fst.comp fst)", "annotated_tactic": ["have n : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n      fst.comp (snd.comp <| fst.comp <| fst.comp <a>fst</a>)", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have m : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n  snd.comp (snd.comp <| fst.comp <| fst.comp fst)", "annotated_tactic": ["have m : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n      snd.comp (snd.comp <| fst.comp <| fst.comp <a>fst</a>)", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have m\u2081 := fst.comp (Computable.unpair.comp m)", "annotated_tactic": ["have m\u2081 := fst.comp (Computable.unpair.comp m)", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have m\u2082 := snd.comp (Computable.unpair.comp m)", "annotated_tactic": ["have m\u2082 := snd.comp (Computable.unpair.comp m)", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have s : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n  snd.comp (fst.comp fst)", "annotated_tactic": ["have s : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n      snd.comp (fst.comp <a>fst</a>)", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\ns : Computable fun p => p.1.1.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have s\u2081 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n  snd.comp fst", "annotated_tactic": ["have s\u2081 : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n      snd.comp <a>fst</a>", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\ns : Computable fun p => p.1.1.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\ns : Computable fun p => p.1.1.2\ns\u2081 : Computable fun p => p.1.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "have s\u2082 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n  snd", "annotated_tactic": ["have s\u2082 : <a>Computable</a> (fun p : ((((\u03b1 \u00d7 <a>List</a> \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n      <a>snd</a>", [{"full_name": "Computable", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [245, 5], "def_end_pos": [245, 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": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\ns : Computable fun p => p.1.1.2\ns\u2081 : Computable fun p => p.1.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\ns : Computable fun p => p.1.1.2\ns\u2081 : Computable fun p => p.1.2\ns\u2082 : Computable fun p => p.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)"}, {"tactic": "exact\n  (nat_bodd.comp n).cond\n    ((nat_bodd.comp <| nat_div2.comp n).cond\n      (hrf.comp a (((Computable.ofNat Code).comp m).pair s))\n      (hpc.comp a\n        (((Computable.ofNat Code).comp m\u2081).pair <|\n          ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n    (Computable.cond (nat_bodd.comp <| nat_div2.comp n)\n      (hco.comp a\n        (((Computable.ofNat Code).comp m\u2081).pair <|\n          ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n      (hpr.comp a\n        (((Computable.ofNat Code).comp m\u2081).pair <|\n          ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))", "annotated_tactic": ["exact\n      (nat_bodd.comp n).<a>cond</a>\n        ((nat_bodd.comp <| nat_div2.comp n).<a>cond</a>\n          (hrf.comp a (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m).<a>pair</a> s))\n          (hpc.comp a\n            (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2081).<a>pair</a> <|\n              ((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2082).<a>pair</a> <| s\u2081.pair s\u2082)))\n        (<a>Computable.cond</a> (nat_bodd.comp <| nat_div2.comp n)\n          (hco.comp a\n            (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2081).<a>pair</a> <|\n              ((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2082).<a>pair</a> <| s\u2081.pair s\u2082))\n          (hpr.comp a\n            (((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2081).<a>pair</a> <|\n              ((<a>Computable.ofNat</a> <a>Code</a>).<a>comp</a> m\u2082).<a>pair</a> <| s\u2081.pair s\u2082)))", [{"full_name": "Computable.cond", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [677, 9], "def_end_pos": [677, 13]}, {"full_name": "Computable.cond", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [677, 9], "def_end_pos": [677, 13]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.cond", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [677, 9], "def_end_pos": [677, 13]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}, {"full_name": "Computable.ofNat", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [408, 19], "def_end_pos": [408, 24]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}, {"full_name": "Computable.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [527, 16], "def_end_pos": [527, 20]}, {"full_name": "Computable.pair", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [310, 16], "def_end_pos": [310, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\na : Computable fun p => p.1.1.1.1.1\nn : Computable fun p => p.1.1.1.2.1\nm : Computable fun p => p.1.1.1.2.2\nm\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1\nm\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2\ns : Computable fun p => p.1.1.2\ns\u2081 : Computable fun p => p.1.2\ns\u2082 : Computable fun p => p.2\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.1.1.2.1 then\n      bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)\n      else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n    else\n      bif bodd (div2 p.1.1.2.1) then\n        co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)\n      else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 F (id a) (ofNat Code (encode (c a))) = F a (c a)", "state_after": "no goals"}, {"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "case succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n)))))", "state_after": "case succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))\n\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))"}, {"tactic": "simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]", "annotated_tactic": ["simp (config := { zeta := <a>false</a> }) [<a>ofNatCode_eq</a>, <a>ofNatCode</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": "Nat.Partrec.Code.ofNatCode_eq", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [198, 9], "def_end_pos": [198, 21]}, {"full_name": "Nat.Partrec.Code.ofNatCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [143, 5], "def_end_pos": [143, 14]}]], "state_before": "case succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))", "state_after": "case succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNatCode n)) (List.range (Nat.succ (Nat.succ (Nat.succ 0))))) = some (F a right)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNatCode n)) (List.range (Nat.succ (Nat.succ (Nat.succ 0))))) = some (F a right)", "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": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \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": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.1.1;\n    let IH := p.1.2;\n    let n := p.2.1;\n    let m := p.2.2;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))\n          (List.get? IH (unpair m).2)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \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/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.self_eq_mass_smul_normalize", "start": [354, 1], "end": [358, 60], "traced_tactics": [{"tactic": "apply eq_of_forall_apply_eq", "annotated_tactic": ["apply <a>eq_of_forall_apply_eq</a>", [{"full_name": "MeasureTheory.FiniteMeasure.eq_of_forall_apply_eq", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [204, 9], "def_end_pos": [204, 30]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 \u03bc = mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 \u2200 (s : Set \u03a9),\n    MeasurableSet s \u2192\n      (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =\n        (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \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 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 \u2200 (s : Set \u03a9),\n    MeasurableSet s \u2192\n      (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =\n        (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =\n    (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s"}, {"tactic": "rw [\u03bc.self_eq_mass_mul_normalize s, coeFn_smul_apply, smul_eq_mul,\n  ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn]", "annotated_tactic": ["rw [\u03bc.self_eq_mass_mul_normalize s, <a>coeFn_smul_apply</a>, <a>smul_eq_mul</a>,\n    <a>ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.coeFn_smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [276, 9], "def_end_pos": [276, 25]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "MeasureTheory.ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [165, 9], "def_end_pos": [165, 44]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =\n    (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s", "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.suffix_or_suffix_of_suffix", "start": [1850, 1], "end": [1852, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "LipschitzWith.norm_compLp_sub_le", "start": [1056, 1], "end": [1062, 36], "traced_tactics": [{"tactic": "apply Lp.norm_le_mul_norm_of_ae_le_mul", "annotated_tactic": ["apply <a>Lp.norm_le_mul_norm_of_ae_le_mul</a>", [{"full_name": "MeasureTheory.Lp.norm_le_mul_norm_of_ae_le_mul", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [377, 9], "def_end_pos": [377, 38]}]], "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\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\n\u22a2 \u2016compLp hg g0 f - compLp hg g0 f'\u2016 \u2264 \u2191c * \u2016f - f'\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191(compLp hg g0 f - compLp hg g0 f') x\u2016 \u2264 \u2191c * \u2016\u2191\u2191(f - f') x\u2016"}, {"tactic": "filter_upwards [hg.coeFn_compLp g0 f, hg.coeFn_compLp g0 f',\n  Lp.coeFn_sub (hg.compLp g0 f) (hg.compLp g0 f'), Lp.coeFn_sub f f'] with a ha1 ha2 ha3 ha4", "annotated_tactic": ["filter_upwards [hg.coeFn_compLp g0 f, hg.coeFn_compLp g0 f',\n    <a>Lp.coeFn_sub</a> (hg.compLp g0 f) (hg.compLp g0 f'), <a>Lp.coeFn_sub</a> f f'] with a ha1 ha2 ha3 ha4", [{"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.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191(compLp hg g0 f - compLp hg g0 f') x\u2016 \u2264 \u2191c * \u2016\u2191\u2191(f - f') x\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\na : \u03b1\nha1 : \u2191\u2191(compLp hg g0 f) a = (g \u2218 \u2191\u2191f) a\nha2 : \u2191\u2191(compLp hg g0 f') a = (g \u2218 \u2191\u2191f') a\nha3 : \u2191\u2191(compLp hg g0 f - compLp hg g0 f') a = (\u2191\u2191(compLp hg g0 f) - \u2191\u2191(compLp hg g0 f')) a\nha4 : \u2191\u2191(f - f') a = (\u2191\u2191f - \u2191\u2191f') a\n\u22a2 \u2016\u2191\u2191(compLp hg g0 f - compLp hg g0 f') a\u2016 \u2264 \u2191c * \u2016\u2191\u2191(f - f') a\u2016"}, {"tactic": "simp only [ha1, ha2, ha3, ha4, \u2190 dist_eq_norm, Pi.sub_apply, Function.comp_apply]", "annotated_tactic": ["simp only [ha1, ha2, ha3, ha4, \u2190 <a>dist_eq_norm</a>, <a>Pi.sub_apply</a>, <a>Function.comp_apply</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": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 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]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\na : \u03b1\nha1 : \u2191\u2191(compLp hg g0 f) a = (g \u2218 \u2191\u2191f) a\nha2 : \u2191\u2191(compLp hg g0 f') a = (g \u2218 \u2191\u2191f') a\nha3 : \u2191\u2191(compLp hg g0 f - compLp hg g0 f') a = (\u2191\u2191(compLp hg g0 f) - \u2191\u2191(compLp hg g0 f')) a\nha4 : \u2191\u2191(f - f') a = (\u2191\u2191f - \u2191\u2191f') a\n\u22a2 \u2016\u2191\u2191(compLp hg g0 f - compLp hg g0 f') a\u2016 \u2264 \u2191c * \u2016\u2191\u2191(f - f') a\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\na : \u03b1\nha1 : \u2191\u2191(compLp hg g0 f) a = (g \u2218 \u2191\u2191f) a\nha2 : \u2191\u2191(compLp hg g0 f') a = (g \u2218 \u2191\u2191f') a\nha3 : \u2191\u2191(compLp hg g0 f - compLp hg g0 f') a = (\u2191\u2191(compLp hg g0 f) - \u2191\u2191(compLp hg g0 f')) a\nha4 : \u2191\u2191(f - f') a = (\u2191\u2191f - \u2191\u2191f') a\n\u22a2 dist (g (\u2191\u2191f a)) (g (\u2191\u2191f' a)) \u2264 \u2191c * dist (\u2191\u2191f a) (\u2191\u2191f' a)"}, {"tactic": "exact hg.dist_le_mul (f a) (f' a)", "annotated_tactic": ["exact hg.dist_le_mul (f a) (f' a)", []], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\ng : E \u2192 F\nc : \u211d\u22650\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : { x // x \u2208 Lp E p }\na : \u03b1\nha1 : \u2191\u2191(compLp hg g0 f) a = (g \u2218 \u2191\u2191f) a\nha2 : \u2191\u2191(compLp hg g0 f') a = (g \u2218 \u2191\u2191f') a\nha3 : \u2191\u2191(compLp hg g0 f - compLp hg g0 f') a = (\u2191\u2191(compLp hg g0 f) - \u2191\u2191(compLp hg g0 f')) a\nha4 : \u2191\u2191(f - f') a = (\u2191\u2191f - \u2191\u2191f') a\n\u22a2 dist (g (\u2191\u2191f a)) (g (\u2191\u2191f' a)) \u2264 \u2191c * dist (\u2191\u2191f a) (\u2191\u2191f' 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_Ioi_of_neg", "start": [644, 1], "end": [646, 64], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Ioi_of_neg a h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Ioi_of_neg</a> a 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_Ioi_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [558, 9], "def_end_pos": [558, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a c : \u03b1\nh : c < 0\n\u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Ioi a = Iio (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepSets.iIndep", "start": [469, 1], "end": [473, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.exists_not_mem_null_le_lintegral", "start": [750, 1], "end": [753, 75], "traced_tactics": [{"tactic": "simpa only [laverage_eq_lintegral] using\n  exists_not_mem_null_le_laverage (IsProbabilityMeasure.ne_zero \u03bc) hf hN", "annotated_tactic": ["simpa only [<a>laverage_eq_lintegral</a>] using\n    <a>exists_not_mem_null_le_laverage</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hf hN", [{"full_name": "MeasureTheory.laverage_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [95, 9], "def_end_pos": [95, 30]}, {"full_name": "MeasureTheory.exists_not_mem_null_le_laverage", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [709, 9], "def_end_pos": [709, 40]}, {"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\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : AEMeasurable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.compl_inter_self", "start": [1672, 1], "end": [1673, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inter_subset_ite", "start": [2327, 1], "end": [2328, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.image_fintype_prod_pi", "start": [186, 1], "end": [188, 72], "traced_tactics": [{"tactic": "simpa only [Finset.coe_univ] using image_finset_prod_pi Finset.univ S", "annotated_tactic": ["simpa only [<a>Finset.coe_univ</a>] using <a>image_finset_prod_pi</a> <a>Finset.univ</a> S", [{"full_name": "Finset.coe_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 17]}, {"full_name": "Set.image_finset_prod_pi", "def_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "def_pos": [177, 9], "def_end_pos": [177, 29]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b3 : CommMonoid \u03b1\ninst\u271d\u00b2 : CommMonoid \u03b2\ninst\u271d\u00b9 : MonoidHomClass F \u03b1 \u03b2\ninst\u271d : Fintype \u03b9\nS : \u03b9 \u2192 Set \u03b1\n\u22a2 (fun f => \u220f i : \u03b9, f i) '' pi univ S = \u220f i : \u03b9, S i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.fderiv_integral", "start": [739, 1], "end": [744, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.mul_sign", "start": [840, 1], "end": [843, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_ssubset_Icc_left", "start": [275, 1], "end": [278, 42], "traced_tactics": [{"tactic": "rw [\u2190 coe_ssubset, coe_Icc, coe_Icc]", "annotated_tactic": ["rw [\u2190 <a>coe_ssubset</a>, <a>coe_Icc</a>, <a>coe_Icc</a>]", [{"full_name": "Finset.coe_ssubset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [411, 9], "def_end_pos": [411, 20]}, {"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\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 < a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Icc a\u2081 b\u2081 \u2282 Icc a\u2082 b\u2082", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 < a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082"}, {"tactic": "exact Set.Icc_ssubset_Icc_left hI ha hb", "annotated_tactic": ["exact <a>Set.Icc_ssubset_Icc_left</a> hI ha hb", [{"full_name": "Set.Icc_ssubset_Icc_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [585, 9], "def_end_pos": [585, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 < a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082", "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_biUnion", "start": [2171, 1], "end": [2173, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.orderEmbOfFin_unique'", "start": [225, 1], "end": [227, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.tr_supports", "start": [2768, 1], "end": [2813, 43], "traced_tactics": [{"tactic": "suffices \u2200 (q) (_ : TM2.SupportsStmt S q) (_ : \u2200 x \u2208 trStmts\u2081 q, x \u2208 trSupp M S),\n    TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n    \u2200 l' \u2208 trStmts\u2081 q, TM1.SupportsStmt (trSupp M S) (tr M l') by\n  rcases Finset.mem_biUnion.1 h with \u27e8l, lS, h\u27e9\n  have :=\n    this _ (ss.2 l lS) fun x hx \u21a6 Finset.mem_biUnion.2 \u27e8_, lS, Finset.mem_insert_of_mem hx\u27e9\n  rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]", "annotated_tactic": ["suffices \u2200 (q) (_ : <a>TM2.SupportsStmt</a> S q) (_ : \u2200 x \u2208 <a>trStmts\u2081</a> q, x \u2208 <a>trSupp</a> M S),\n        <a>TM1.SupportsStmt</a> (<a>trSupp</a> M S) (<a>trNormal</a> q) \u2227\n        \u2200 l' \u2208 <a>trStmts\u2081</a> q, <a>TM1.SupportsStmt</a> (<a>trSupp</a> M S) (<a>tr</a> M l') by\n      rcases <a>Finset.mem_biUnion</a>.1 h with \u27e8l, lS, h\u27e9\n      have :=\n        this _ (ss.2 l lS) fun x hx \u21a6 <a>Finset.mem_biUnion</a>.2 \u27e8_, lS, <a>Finset.mem_insert_of_mem</a> hx\u27e9\n      rcases <a>Finset.mem_insert</a>.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]", [{"full_name": "Turing.TM2.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2165, 5], "def_end_pos": [2165, 17]}, {"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}, {"full_name": "Turing.TM2to1.trSupp", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2764, 19], "def_end_pos": [2764, 25]}, {"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM2to1.trSupp", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2764, 19], "def_end_pos": [2764, 25]}, {"full_name": "Turing.TM2to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2516, 5], "def_end_pos": [2516, 13]}, {"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}, {"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM2to1.trSupp", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2764, 19], "def_end_pos": [2764, 25]}, {"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"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", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}]], "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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M 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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\n\u22a2 \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "clear h l'", "annotated_tactic": ["clear h l'", []], "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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\n\u22a2 \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M 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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "refine' stmtStRec _ _ _ _ _", "annotated_tactic": ["refine' <a>stmtStRec</a> _ _ _ _ _", [{"full_name": "Turing.TM2to1.stmtStRec", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2445, 5], "def_end_pos": [2445, 14]}]], "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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (k : K) (s : StAct k) (q : Stmt\u2082),\n    (TM2.SupportsStmt S q \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n      TM2.SupportsStmt S (stRun s q) \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q) \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q)) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_2\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (a : \u03c3 \u2192 \u03c3) (q : Stmt\u2082),\n    (TM2.SupportsStmt S q \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n      TM2.SupportsStmt S (TM2.Stmt.load a q) \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a q)) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_3\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (p : \u03c3 \u2192 Bool) (q\u2081 q\u2082 : Stmt\u2082),\n    (TM2.SupportsStmt S q\u2081 \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081 \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal q\u2081) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n      (TM2.SupportsStmt S q\u2082 \u2192\n          (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082 \u2192 x \u2208 trSupp M S) \u2192\n            TM1.SupportsStmt (trSupp M S) (trNormal q\u2082) \u2227\n              \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n        TM2.SupportsStmt S (TM2.Stmt.branch p q\u2081 q\u2082) \u2192\n          (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 x \u2208 trSupp M S) \u2192\n            TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) \u2227\n              \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (l : \u03c3 \u2192 \u039b),\n    TM2.SupportsStmt S (TM2.Stmt.goto l) \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_5\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 TM2.SupportsStmt S TM2.Stmt.halt \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "rcases Finset.mem_biUnion.1 h with \u27e8l, lS, h\u27e9", "annotated_tactic": ["rcases <a>Finset.mem_biUnion</a>.1 h with \u27e8l, lS, h\u27e9", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}]], "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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\nthis :\n  \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')", "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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis :\n  \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "have :=\n  this _ (ss.2 l lS) fun x hx \u21a6 Finset.mem_biUnion.2 \u27e8_, lS, Finset.mem_insert_of_mem hx\u27e9", "annotated_tactic": ["have :=\n        this _ (ss.2 l lS) fun x hx \u21a6 <a>Finset.mem_biUnion</a>.2 \u27e8_, lS, <a>Finset.mem_insert_of_mem</a> hx\u27e9", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"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 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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis :\n  \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')", "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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis\u271d :\n  \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\nthis :\n  TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]", "annotated_tactic": ["rcases <a>Finset.mem_insert</a>.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]", [{"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}]], "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\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis\u271d :\n  \u2200 (q : Stmt\u2082),\n    TM2.SupportsStmt S q \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\nthis :\n  TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "no goals"}, {"tactic": "intro _ s _ IH ss' sub", "annotated_tactic": ["intro _ s _ IH ss' sub", []], "state_before": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (k : K) (s : StAct k) (q : Stmt\u2082),\n    (TM2.SupportsStmt S q \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n      TM2.SupportsStmt S (stRun s q) \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q) \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q)) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (stRun s q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "rw [TM2to1.supports_run] at ss'", "annotated_tactic": ["rw [<a>TM2to1.supports_run</a>] at ss'", [{"full_name": "Turing.TM2to1.supports_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2458, 9], "def_end_pos": [2458, 21]}]], "state_before": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (stRun s q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "simp only [TM2to1.trStmts\u2081_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton]\n  at sub", "annotated_tactic": ["simp only [<a>TM2to1.trStmts\u2081_run</a>, <a>Finset.mem_union</a>, <a>Finset.mem_insert</a>, <a>Finset.mem_singleton</a>]\n        at sub", [{"full_name": "Turing.TM2to1.trStmts\u2081_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2543, 9], "def_end_pos": [2543, 21]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "have hgo := sub _ (Or.inl <| Or.inl rfl)", "annotated_tactic": ["have hgo := sub _ (<a>Or.inl</a> <| <a>Or.inl</a> <a>rfl</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": "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 refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "have hret := sub _ (Or.inl <| Or.inr rfl)", "annotated_tactic": ["have hret := sub _ (<a>Or.inl</a> <| <a>Or.inr</a> <a>rfl</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": "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 refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "cases' IH ss' fun x hx \u21a6 sub x <| Or.inr hx with IH\u2081 IH\u2082", "annotated_tactic": ["cases' IH ss' fun x hx \u21a6 sub x <| <a>Or.inr</a> hx with IH\u2081 IH\u2082", [{"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 refine'_1\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "refine' \u27e8by simp only [trNormal_run, TM1.SupportsStmt]; intros; exact hgo, fun l h \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8by simp only [<a>trNormal_run</a>, <a>TM1.SupportsStmt</a>]; intros; exact hgo, fun l h \u21a6 _\u27e9", [{"full_name": "Turing.TM2to1.trNormal_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2526, 9], "def_end_pos": [2526, 21]}, {"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}]], "state_before": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 (stRun s q\u271d)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)"}, {"tactic": "rw [trStmts\u2081_run] at h", "annotated_tactic": ["rw [<a>trStmts\u2081_run</a>] at h", [{"full_name": "Turing.TM2to1.trStmts\u2081_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2543, 9], "def_end_pos": [2543, 21]}]], "state_before": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 (stRun s q\u271d)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)", "state_after": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 {go k\u271d s q\u271d, ret q\u271d} \u222a trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)"}, {"tactic": "simp only [TM2to1.trStmts\u2081_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton]\n  at h", "annotated_tactic": ["simp only [<a>TM2to1.trStmts\u2081_run</a>, <a>Finset.mem_union</a>, <a>Finset.mem_insert</a>, <a>Finset.mem_singleton</a>]\n        at h", [{"full_name": "Turing.TM2to1.trStmts\u2081_run", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2543, 9], "def_end_pos": [2543, 21]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 {go k\u271d s q\u271d, ret q\u271d} \u222a trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)", "state_after": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : (l = go k\u271d s q\u271d \u2228 l = ret q\u271d) \u2228 l \u2208 trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)"}, {"tactic": "rcases h with (\u27e8rfl | rfl\u27e9 | h)", "annotated_tactic": ["rcases h with (\u27e8rfl | rfl\u27e9 | h)", []], "state_before": "case refine'_1.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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : (l = go k\u271d s q\u271d \u2228 l = ret q\u271d) \u2228 l \u2208 trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)", "state_after": "case refine'_1.intro.inl.inl\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d s q\u271d))\n\ncase refine'_1.intro.inl.inr\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (ret q\u271d))\n\ncase refine'_1.intro.inr\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)"}, {"tactic": "simp only [trNormal_run, TM1.SupportsStmt]", "annotated_tactic": ["simp only [<a>trNormal_run</a>, <a>TM1.SupportsStmt</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.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d))", "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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 \u0393' \u2192 \u03c3 \u2192 go k\u271d s q\u271d \u2208 trSupp M S"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 \u0393' \u2192 \u03c3 \u2192 go k\u271d s q\u271d \u2208 trSupp M S", "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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u0393'\nv\u271d : \u03c3\n\u22a2 go k\u271d s q\u271d \u2208 trSupp M S"}, {"tactic": "exact hgo", "annotated_tactic": ["exact hgo", []], "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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u0393'\nv\u271d : \u03c3\n\u22a2 go k\u271d s q\u271d \u2208 trSupp M S", "state_after": "no goals"}, {"tactic": "cases s", "annotated_tactic": ["cases s", []], "state_before": "case refine'_1.intro.inl.inl\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d s q\u271d))", "state_after": "case refine'_1.intro.inl.inl.push\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.push a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.push a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.push a\u271d) q\u271d))\n\ncase refine'_1.intro.inl.inl.peek\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.peek a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.peek a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.peek a\u271d) q\u271d))\n\ncase refine'_1.intro.inl.inl.pop\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.pop a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.pop a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.pop a\u271d) q\u271d))"}, {"tactic": "exact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9", "annotated_tactic": ["exact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9", []], "state_before": "case refine'_1.intro.inl.inl.push\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.push a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.push a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.push a\u271d) q\u271d))", "state_after": "no goals"}, {"tactic": "exact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9", "annotated_tactic": ["exact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9", []], "state_before": "case refine'_1.intro.inl.inl.peek\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.peek a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.peek a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.peek a\u271d) q\u271d))", "state_after": "no goals"}, {"tactic": "exact \u27e8\u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hret\u27e9, fun _ _ \u21a6 hgo\u27e9", "annotated_tactic": ["exact \u27e8\u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hret\u27e9, fun _ _ \u21a6 hgo\u27e9", []], "state_before": "case refine'_1.intro.inl.inl.pop\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.pop a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.pop a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.pop a\u271d) q\u271d))", "state_after": "no goals"}, {"tactic": "unfold TM1.SupportsStmt TM2to1.tr", "annotated_tactic": ["unfold <a>TM1.SupportsStmt</a> <a>TM2to1.tr</a>", [{"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}]], "state_before": "case refine'_1.intro.inl.inr\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (ret q\u271d))", "state_after": "case refine'_1.intro.inl.inr\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 match\n    match ret q\u271d with\n    | normal q => trNormal (M q)\n    | go k s q =>\n      branch (fun a x => Option.isNone (a.2 k)) (trStAct (goto fun x x => ret q) s)\n        (move Dir.right (goto fun x x => go k s q))\n    | ret q => branch (fun a x => a.1) (trNormal q) (move Dir.left (goto fun x x => ret q)) with\n  | move a q => TM1.SupportsStmt (trSupp M S) q\n  | write a q => TM1.SupportsStmt (trSupp M S) q\n  | load a q => TM1.SupportsStmt (trSupp M S) q\n  | branch a q\u2081 q\u2082 => TM1.SupportsStmt (trSupp M S) q\u2081 \u2227 TM1.SupportsStmt (trSupp M S) q\u2082\n  | goto l => \u2200 (a : \u0393') (v : \u03c3), l a v \u2208 trSupp M S\n  | halt => True"}, {"tactic": "exact \u27e8IH\u2081, fun _ _ \u21a6 hret\u27e9", "annotated_tactic": ["exact \u27e8IH\u2081, fun _ _ \u21a6 hret\u27e9", []], "state_before": "case refine'_1.intro.inl.inr\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 match\n    match ret q\u271d with\n    | normal q => trNormal (M q)\n    | go k s q =>\n      branch (fun a x => Option.isNone (a.2 k)) (trStAct (goto fun x x => ret q) s)\n        (move Dir.right (goto fun x x => go k s q))\n    | ret q => branch (fun a x => a.1) (trNormal q) (move Dir.left (goto fun x x => ret q)) with\n  | move a q => TM1.SupportsStmt (trSupp M S) q\n  | write a q => TM1.SupportsStmt (trSupp M S) q\n  | load a q => TM1.SupportsStmt (trSupp M S) q\n  | branch a q\u2081 q\u2082 => TM1.SupportsStmt (trSupp M S) q\u2081 \u2227 TM1.SupportsStmt (trSupp M S) q\u2082\n  | goto l => \u2200 (a : \u0393') (v : \u03c3), l a v \u2208 trSupp M S\n  | halt => True", "state_after": "no goals"}, {"tactic": "exact IH\u2082 _ h", "annotated_tactic": ["exact IH\u2082 _ h", []], "state_before": "case refine'_1.intro.inr\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\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)", "state_after": "no goals"}, {"tactic": "intro _ _ IH ss' sub", "annotated_tactic": ["intro _ _ IH ss' sub", []], "state_before": "case refine'_2\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (a : \u03c3 \u2192 \u03c3) (q : Stmt\u2082),\n    (TM2.SupportsStmt S q \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n      TM2.SupportsStmt S (TM2.Stmt.load a q) \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a q)) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_2\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\nS : Finset \u039b\nss : TM2.Supports M S\na\u271d : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "unfold TM2to1.trStmts\u2081 at ss' sub \u22a2", "annotated_tactic": ["unfold <a>TM2to1.trStmts\u2081</a> at ss' sub \u22a2", [{"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}]], "state_before": "case refine'_2\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\nS : Finset \u039b\nss : TM2.Supports M S\na\u271d : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_2\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\nS : Finset \u039b\nss : TM2.Supports M S\na\u271d : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "exact IH ss' sub", "annotated_tactic": ["exact IH ss' sub", []], "state_before": "case refine'_2\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\nS : Finset \u039b\nss : TM2.Supports M S\na\u271d : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n  TM2.SupportsStmt S q\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "no goals"}, {"tactic": "intro _ _ _ IH\u2081 IH\u2082 ss' sub", "annotated_tactic": ["intro _ _ _ IH\u2081 IH\u2082 ss' sub", []], "state_before": "case refine'_3\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (p : \u03c3 \u2192 Bool) (q\u2081 q\u2082 : Stmt\u2082),\n    (TM2.SupportsStmt S q\u2081 \u2192\n        (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081 \u2192 x \u2208 trSupp M S) \u2192\n          TM1.SupportsStmt (trSupp M S) (trNormal q\u2081) \u2227\n            \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n      (TM2.SupportsStmt S q\u2082 \u2192\n          (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082 \u2192 x \u2208 trSupp M S) \u2192\n            TM1.SupportsStmt (trSupp M S) (trNormal q\u2082) \u2227\n              \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n        TM2.SupportsStmt S (TM2.Stmt.branch p q\u2081 q\u2082) \u2192\n          (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 x \u2208 trSupp M S) \u2192\n            TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) \u2227\n              \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_3\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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "unfold TM2to1.trStmts\u2081 at sub", "annotated_tactic": ["unfold <a>TM2to1.trStmts\u2081</a> at sub", [{"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}]], "state_before": "case refine'_3\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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_3\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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "cases' IH\u2081 ss'.1 fun x hx \u21a6 sub x <| Finset.mem_union_left _ hx with IH\u2081\u2081 IH\u2081\u2082", "annotated_tactic": ["cases' IH\u2081 ss'.1 fun x hx \u21a6 sub x <| <a>Finset.mem_union_left</a> _ hx with IH\u2081\u2081 IH\u2081\u2082", [{"full_name": "Finset.mem_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1382, 9], "def_end_pos": [1382, 23]}]], "state_before": "case refine'_3\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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "cases' IH\u2082 ss'.2 fun x hx \u21a6 sub x <| Finset.mem_union_right _ hx with IH\u2082\u2081 IH\u2082\u2082", "annotated_tactic": ["cases' IH\u2082 ss'.2 fun x hx \u21a6 sub x <| <a>Finset.mem_union_right</a> _ hx with IH\u2082\u2081 IH\u2082\u2082", [{"full_name": "Finset.mem_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}]], "state_before": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "refine' \u27e8\u27e8IH\u2081\u2081, IH\u2082\u2081\u27e9, fun l h \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8IH\u2081\u2081, IH\u2082\u2081\u27e9, fun l h \u21a6 _\u27e9", []], "state_before": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)"}, {"tactic": "rw [trStmts\u2081] at h", "annotated_tactic": ["rw [<a>trStmts\u2081</a>] at h", [{"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}]], "state_before": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)", "state_after": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)"}, {"tactic": "rcases Finset.mem_union.1 h with (h | h) <;> [exact IH\u2081\u2082 _ h; exact IH\u2082\u2082 _ h]", "annotated_tactic": ["rcases <a>Finset.mem_union</a>.1 h with (h | h) <;> [exact IH\u2081\u2082 _ h; exact IH\u2082\u2082 _ h]", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "case refine'_3.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\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n  TM2.SupportsStmt S q\u2081\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n  TM2.SupportsStmt S q\u2082\u271d \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)", "state_after": "no goals"}, {"tactic": "intro _ ss' _", "annotated_tactic": ["intro _ ss' _", []], "state_before": "case refine'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (l : \u03c3 \u2192 \u039b),\n    TM2.SupportsStmt S (TM2.Stmt.goto l) \u2192\n      (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 x \u2208 trSupp M S) \u2192\n        TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) \u2227\n          \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "simp only [trStmts\u2081, Finset.not_mem_empty]", "annotated_tactic": ["simp only [<a>trStmts\u2081</a>, <a>Finset.not_mem_empty</a>]", [{"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}, {"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 refine'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227\n    \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "refine' \u27e8_, fun _ \u21a6 False.elim\u27e9", "annotated_tactic": ["refine' \u27e8_, fun _ \u21a6 <a>False.elim</a>\u27e9", [{"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'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227\n    \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d))"}, {"tactic": "exact fun _ v \u21a6 Finset.mem_biUnion.2 \u27e8_, ss' v, Finset.mem_insert_self _ _\u27e9", "annotated_tactic": ["exact fun _ v \u21a6 <a>Finset.mem_biUnion</a>.2 \u27e8_, ss' v, <a>Finset.mem_insert_self</a> _ _\u27e9", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"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'_4\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\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d))", "state_after": "no goals"}, {"tactic": "intro _ _", "annotated_tactic": ["intro _ _", []], "state_before": "case refine'_5\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\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 TM2.SupportsStmt S TM2.Stmt.halt \u2192\n    (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S) \u2192\n      TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227\n        \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_5\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\nS : Finset \u039b\nss : TM2.Supports M S\nx\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "simp only [trStmts\u2081, Finset.not_mem_empty]", "annotated_tactic": ["simp only [<a>trStmts\u2081</a>, <a>Finset.not_mem_empty</a>]", [{"full_name": "Turing.TM2to1.trStmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2534, 19], "def_end_pos": [2534, 27]}, {"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 refine'_5\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\nS : Finset \u039b\nss : TM2.Supports M S\nx\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227\n    \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "case refine'_5\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\nS : Finset \u039b\nss : TM2.Supports M S\nx\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227 \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')"}, {"tactic": "exact \u27e8trivial, fun _ \u21a6 False.elim\u27e9", "annotated_tactic": ["exact \u27e8<a>trivial</a>, fun _ \u21a6 <a>False.elim</a>\u27e9", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"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'_5\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\nS : Finset \u039b\nss : TM2.Supports M S\nx\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227 \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.card_disjSum", "start": [50, 1], "end": [51, 28], "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_drop", "start": [1925, 1], "end": [1925, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.mapsTo_prod_map_diagonal", "start": [400, 1], "end": [401, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion\u2080", "start": [3489, 1], "end": [3505, 56], "traced_tactics": [{"tactic": "have obs : { i : \u03b9 | 0 < \u03bc (As i) } \u2286 \u22c3 n, { i : \u03b9 | 0 < \u03bc (As i \u2229 spanningSets \u03bc n) } := by\n  intro i i_in_nonzeroes\n  by_contra con\n  simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *\n  simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes", "annotated_tactic": ["have obs : { i : \u03b9 | 0 < \u03bc (As i) } \u2286 \u22c3 n, { i : \u03b9 | 0 < \u03bc (As i \u2229 <a>spanningSets</a> \u03bc n) } := by\n    intro i i_in_nonzeroes\n    by_contra con\n    simp only [<a>mem_iUnion</a>, <a>mem_setOf_eq</a>, <a>not_exists</a>, <a>not_lt</a>, <a>nonpos_iff_eq_zero</a>] at *\n    simp [(<a>forall_measure_inter_spanningSets_eq_zero</a> _).<a>mp</a> con] at i_in_nonzeroes", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 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_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"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": "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.forall_measure_inter_spanningSets_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3409, 9], "def_end_pos": [3409, 50]}, {"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\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\n\u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)}"}, {"tactic": "apply Countable.mono obs", "annotated_tactic": ["apply <a>Countable.mono</a> obs", [{"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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 Set.Countable (\u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)})"}, {"tactic": "refine' countable_iUnion fun n => countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080 \u03bc _ _ _", "annotated_tactic": ["refine' <a>countable_iUnion</a> fun n => <a>countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080</a> \u03bc _ _ _", [{"full_name": "Set.countable_iUnion", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [185, 9], "def_end_pos": [185, 25]}, {"full_name": "MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3459, 9], "def_end_pos": [3459, 62]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 Set.Countable (\u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)})", "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\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u2200 (i : \u03b9), NullMeasurableSet (As i \u2229 spanningSets \u03bc n)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 Pairwise (AEDisjoint \u03bc on fun i => As i \u2229 spanningSets \u03bc n)\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, As i \u2229 spanningSets \u03bc n) \u2260 \u22a4"}, {"tactic": "intro i i_in_nonzeroes", "annotated_tactic": ["intro i i_in_nonzeroes", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\n\u22a2 {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\ni : \u03b9\ni_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)}\n\u22a2 i \u2208 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}"}, {"tactic": "by_contra con", "annotated_tactic": ["by_contra con", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\ni : \u03b9\ni_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)}\n\u22a2 i \u2208 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\ni : \u03b9\ni_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)}\ncon : \u00aci \u2208 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 False"}, {"tactic": "simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_setOf_eq</a>, <a>not_exists</a>, <a>not_lt</a>, <a>nonpos_iff_eq_zero</a>] at *", [{"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": "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": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\ni : \u03b9\ni_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)}\ncon : \u00aci \u2208 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\ni : \u03b9\ni_in_nonzeroes : 0 < \u2191\u2191\u03bc (As i)\ncon : \u2200 (x : \u2115), \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc x) = 0\n\u22a2 False"}, {"tactic": "simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes", "annotated_tactic": ["simp [(<a>forall_measure_inter_spanningSets_eq_zero</a> _).<a>mp</a> con] at i_in_nonzeroes", [{"full_name": "MeasureTheory.Measure.forall_measure_inter_spanningSets_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3409, 9], "def_end_pos": [3409, 50]}, {"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\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\ni : \u03b9\ni_in_nonzeroes : 0 < \u2191\u2191\u03bc (As i)\ncon : \u2200 (x : \u2115), \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc x) = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact fun i \u21a6 NullMeasurableSet.inter (As_mble i)\n  (measurable_spanningSets \u03bc n).nullMeasurableSet", "annotated_tactic": ["exact fun i \u21a6 <a>NullMeasurableSet.inter</a> (As_mble i)\n      (<a>measurable_spanningSets</a> \u03bc n).<a>nullMeasurableSet</a>", [{"full_name": "MeasureTheory.NullMeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [195, 19], "def_end_pos": [195, 24]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}, {"full_name": "MeasurableSet.nullMeasurableSet", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [103, 9], "def_end_pos": [103, 47]}]], "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\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u2200 (i : \u03b9), NullMeasurableSet (As i \u2229 spanningSets \u03bc n)", "state_after": "no goals"}, {"tactic": "exact fun i j i_ne_j \u21a6 (As_disj i_ne_j).mono\n  (inter_subset_left (As i) (spanningSets \u03bc n)) (inter_subset_left (As j) (spanningSets \u03bc n))", "annotated_tactic": ["exact fun i j i_ne_j \u21a6 (As_disj i_ne_j).<a>mono</a>\n      (<a>inter_subset_left</a> (As i) (<a>spanningSets</a> \u03bc n)) (<a>inter_subset_left</a> (As j) (<a>spanningSets</a> \u03bc n))", [{"full_name": "MeasureTheory.AEDisjoint.mono", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [83, 19], "def_end_pos": [83, 23]}, {"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.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"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.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 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\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 Pairwise (AEDisjoint \u03bc on fun i => As i \u2229 spanningSets \u03bc n)", "state_after": "no goals"}, {"tactic": "refine' (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top \u03bc n)).ne", "annotated_tactic": ["refine' (<a>lt_of_le_of_lt</a> (<a>measure_mono</a> _) (<a>measure_spanningSets_lt_top</a> \u03bc n)).<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": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 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": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "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\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, As i \u2229 spanningSets \u03bc n) \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\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u22c3 i, As i \u2229 spanningSets \u03bc n \u2286 spanningSets \u03bc n"}, {"tactic": "exact iUnion_subset fun i => inter_subset_right _ _", "annotated_tactic": ["exact <a>iUnion_subset</a> fun i => <a>inter_subset_right</a> _ _", [{"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_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "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\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u22c3 i, As i \u2229 spanningSets \u03bc n \u2286 spanningSets \u03bc n", "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_Icc_left", "start": [223, 1], "end": [225, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.eq_insert_of_ncard_eq_succ", "start": [1081, 1], "end": [1091, 7], "traced_tactics": [{"tactic": "have hsf := finite_of_ncard_pos (n.zero_lt_succ.trans_eq h.symm)", "annotated_tactic": ["have hsf := <a>finite_of_ncard_pos</a> (n.zero_lt_succ.trans_eq h.symm)", [{"full_name": "Set.finite_of_ncard_pos", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [550, 9], "def_end_pos": [550, 28]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nh : ncard s = n + 1\n\u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nh : ncard s = n + 1\nhsf : Set.Finite s\n\u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n"}, {"tactic": "rw [ncard_eq_toFinset_card _ hsf, Finset.card_eq_succ] at h", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a> _ hsf, <a>Finset.card_eq_succ</a>] at h", [{"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": "Finset.card_eq_succ", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [627, 9], "def_end_pos": [627, 21]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nh : ncard s = n + 1\nhsf : Set.Finite s\n\u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhsf : Set.Finite s\nh\u271d : Finset.card (Finite.toFinset hsf) = n + 1\nh : \u2203 a t, \u00aca \u2208 t \u2227 insert a t = Finite.toFinset hsf \u2227 Finset.card t = n\n\u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n"}, {"tactic": "obtain \u27e8a, t, hat, hts, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8a, t, hat, hts, rfl\u27e9 := h", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhsf : Set.Finite s\nh\u271d : Finset.card (Finite.toFinset hsf) = n + 1\nh : \u2203 a t, \u00aca \u2208 t \u2227 insert a t = Finite.toFinset hsf \u2227 Finset.card t = n\n\u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nhts : insert a t = Finite.toFinset hsf\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\n\u22a2 \u2203 a t_1, \u00aca \u2208 t_1 \u2227 insert a t_1 = s \u2227 ncard t_1 = Finset.card t"}, {"tactic": "simp only [Finset.ext_iff, Finset.mem_insert, Finite.mem_toFinset] at hts", "annotated_tactic": ["simp only [<a>Finset.ext_iff</a>, <a>Finset.mem_insert</a>, <a>Finite.mem_toFinset</a>] at hts", [{"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [173, 19], "def_end_pos": [173, 31]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nhts : insert a t = Finite.toFinset hsf\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\n\u22a2 \u2203 a t_1, \u00aca \u2208 t_1 \u2227 insert a t_1 = s \u2227 ncard t_1 = Finset.card t", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 \u2203 a t_1, \u00aca \u2208 t_1 \u2227 insert a t_1 = s \u2227 ncard t_1 = Finset.card t"}, {"tactic": "refine' \u27e8a, t, hat, _, _\u27e9", "annotated_tactic": ["refine' \u27e8a, t, hat, _, _\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 \u2203 a t_1, \u00aca \u2208 t_1 \u2227 insert a t_1 = s \u2227 ncard t_1 = Finset.card t", "state_after": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 insert a \u2191t = s\n\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 ncard \u2191t = Finset.card t"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 ncard \u2191t = Finset.card t", "state_after": "no goals"}, {"tactic": "simp only [Finset.mem_coe, ext_iff, mem_insert_iff]", "annotated_tactic": ["simp only [<a>Finset.mem_coe</a>, <a>ext_iff</a>, <a>mem_insert_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": "Set.ext_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 16]}, {"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 intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 insert a \u2191t = s", "state_after": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 \u2200 (x : \u03b1), x = a \u2228 x \u2208 t \u2194 x \u2208 s"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhsf : Set.Finite s\na : \u03b1\nt : Finset \u03b1\nhat : \u00aca \u2208 t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s\n\u22a2 \u2200 (x : \u03b1), x = a \u2228 x \u2208 t \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.FiniteSpanningSetsIn.ext", "start": [3652, 11], "end": [3654, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.sup_def", "start": [221, 1], "end": [222, 26], "traced_tactics": [{"tactic": "rw [sup_eq_eqvGen]", "annotated_tactic": ["rw [<a>sup_eq_eqvGen</a>]", [{"full_name": "Setoid.sup_eq_eqvGen", "def_path": "Mathlib/Data/Setoid/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr s : Setoid \u03b1\n\u22a2 r \u2294 s = EqvGen.Setoid (Rel r \u2294 Rel s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr s : Setoid \u03b1\n\u22a2 (EqvGen.Setoid fun x y => Rel r x y \u2228 Rel s x y) = EqvGen.Setoid (Rel r \u2294 Rel s)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr s : Setoid \u03b1\n\u22a2 (EqvGen.Setoid fun x y => Rel r x y \u2228 Rel s x y) = EqvGen.Setoid (Rel r \u2294 Rel s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "full_name": "MeasureTheory.Integrable.comp_div", "start": [183, 1], "end": [185, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar", "start": [371, 1], "end": [384, 85], "traced_tactics": [{"tactic": "have : IsCompact (haarProduct (K\u2080 : Set G)) := by\n  apply isCompact_univ_pi; intro K; apply isCompact_Icc", "annotated_tactic": ["have : <a>IsCompact</a> (<a>haarProduct</a> (K\u2080 : <a>Set</a> G)) := by\n    apply <a>isCompact_univ_pi</a>; intro K; apply <a>isCompact_Icc</a>", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "MeasureTheory.Measure.haar.haarProduct", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [136, 5], "def_end_pos": [136, 16]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "isCompact_univ_pi", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [1007, 9], "def_end_pos": [1007, 26]}, {"full_name": "CompactIccSpace.isCompact_Icc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [53, 3], "def_end_pos": [53, 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\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 V, clPrehaar (\u2191K\u2080) V)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 V, clPrehaar (\u2191K\u2080) V)"}, {"tactic": "refine' this.inter_iInter_nonempty (clPrehaar K\u2080) (fun s => isClosed_closure) fun t => _", "annotated_tactic": ["refine' this.inter_iInter_nonempty (<a>clPrehaar</a> K\u2080) (fun s => <a>isClosed_closure</a>) fun t => _", [{"full_name": "MeasureTheory.Measure.haar.clPrehaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [154, 5], "def_end_pos": [154, 14]}, {"full_name": "isClosed_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 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\nthis : IsCompact (haarProduct \u2191K\u2080)\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 V, clPrehaar (\u2191K\u2080) V)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)"}, {"tactic": "let V\u2080 := \u22c2 V \u2208 t, (V : OpenNhdsOf (1 : G)).carrier", "annotated_tactic": ["let V\u2080 := \u22c2 V \u2208 t, (V : <a>OpenNhdsOf</a> (1 : G)).<a>carrier</a>", [{"full_name": "TopologicalSpace.OpenNhdsOf", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [422, 11], "def_end_pos": [422, 21]}, {"full_name": "TopologicalSpace.Opens.carrier", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [65, 3], "def_end_pos": [65, 10]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)"}, {"tactic": "have h1V\u2080 : IsOpen V\u2080 := isOpen_biInter_finset $ by rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2081", "annotated_tactic": ["have h1V\u2080 : <a>IsOpen</a> V\u2080 := <a>isOpen_biInter_finset</a> $ by rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2081", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "isOpen_biInter_finset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 30]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)"}, {"tactic": "have h2V\u2080 : (1 : G) \u2208 V\u2080 := by simp only [mem_iInter]; rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2082", "annotated_tactic": ["have h2V\u2080 : (1 : G) \u2208 V\u2080 := by simp only [<a>mem_iInter</a>]; rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2082", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 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\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)"}, {"tactic": "refine' \u27e8prehaar K\u2080 V\u2080, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>prehaar</a> K\u2080 V\u2080, _\u27e9", [{"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\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i", "state_after": "case left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080\n\ncase right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i"}, {"tactic": "apply isCompact_univ_pi", "annotated_tactic": ["apply <a>isCompact_univ_pi</a>", [{"full_name": "isCompact_univ_pi", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [1007, 9], "def_end_pos": [1007, 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\n\u22a2 IsCompact (haarProduct \u2191K\u2080)", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2200 (i : Compacts G), IsCompact (Icc 0 \u2191(index \u2191i \u2191K\u2080))"}, {"tactic": "intro K", "annotated_tactic": ["intro K", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2200 (i : Compacts G), IsCompact (Icc 0 \u2191(index \u2191i \u2191K\u2080))", "state_after": "case 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\n\u22a2 IsCompact (Icc 0 \u2191(index \u2191K \u2191K\u2080))"}, {"tactic": "apply isCompact_Icc", "annotated_tactic": ["apply <a>isCompact_Icc</a>", [{"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 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\n\u22a2 IsCompact (Icc 0 \u2191(index \u2191K \u2191K\u2080))", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _", "annotated_tactic": ["rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\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\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 IsOpen i.carrier", "state_after": "case mk.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nV : Set G\nhV\u2081 : IsOpen V\nhV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier\na\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t\n\u22a2 IsOpen { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier"}, {"tactic": "exact hV\u2081", "annotated_tactic": ["exact hV\u2081", []], "state_before": "case mk.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nV : Set G\nhV\u2081 : IsOpen V\nhV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier\na\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t\n\u22a2 IsOpen { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier", "state_after": "no goals"}, {"tactic": "simp only [mem_iInter]", "annotated_tactic": ["simp only [<a>mem_iInter</a>]", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 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\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 1 \u2208 V\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\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 1 \u2208 i.carrier"}, {"tactic": "rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _", "annotated_tactic": ["rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\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\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 1 \u2208 i.carrier", "state_after": "case mk.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nV : Set G\nhV\u2081 : IsOpen V\nhV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier\ni\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t\n\u22a2 1 \u2208 { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier"}, {"tactic": "exact hV\u2082", "annotated_tactic": ["exact hV\u2082", []], "state_before": "case mk.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nV : Set G\nhV\u2081 : IsOpen V\nhV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier\ni\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t\n\u22a2 1 \u2208 { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier", "state_after": "no goals"}, {"tactic": "apply prehaar_mem_haarProduct K\u2080", "annotated_tactic": ["apply <a>prehaar_mem_haarProduct</a> K\u2080", [{"full_name": "MeasureTheory.Measure.haar.prehaar_mem_haarProduct", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [364, 9], "def_end_pos": [364, 32]}]], "state_before": "case left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080", "state_after": "case left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 Set.Nonempty (interior V\u2080)"}, {"tactic": "use 1", "annotated_tactic": ["use 1", []], "state_before": "case left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 Set.Nonempty (interior V\u2080)", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 1 \u2208 interior V\u2080"}, {"tactic": "rwa [h1V\u2080.interior_eq]", "annotated_tactic": ["rwa [h1V\u2080.interior_eq]", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 1 \u2208 interior V\u2080", "state_after": "no goals"}, {"tactic": "simp only [mem_iInter]", "annotated_tactic": ["simp only [<a>mem_iInter</a>]", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "case right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i", "state_after": "case right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 clPrehaar (\u2191K\u2080) i"}, {"tactic": "rintro \u27e8V, hV\u27e9 h2V", "annotated_tactic": ["rintro \u27e8V, hV\u27e9 h2V", []], "state_before": "case right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 clPrehaar (\u2191K\u2080) i", "state_after": "case right.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 clPrehaar \u2191K\u2080 { toOpens := V, mem' := hV }"}, {"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 right.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 clPrehaar \u2191K\u2080 { toOpens := V, mem' := hV }", "state_after": "case right.mk.a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}"}, {"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": "case right.mk.a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}", "state_after": "case right.mk.a.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 \u22c2 V \u2208 t, V.carrier \u2208 {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}"}, {"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 right.mk.a.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 \u22c2 V \u2208 t, V.carrier \u2208 {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}", "state_after": "case right.mk.a.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 \u22c2 V \u2208 t, V.carrier \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen (\u22c2 V \u2208 t, V.carrier) \u2227 1 \u2208 \u22c2 V \u2208 t, V.carrier"}, {"tactic": "exact \u27e8Subset.trans (iInter_subset _ \u27e8V, hV\u27e9) (iInter_subset _ h2V), h1V\u2080, h2V\u2080\u27e9", "annotated_tactic": ["exact \u27e8<a>Subset.trans</a> (<a>iInter_subset</a> _ \u27e8V, hV\u27e9) (<a>iInter_subset</a> _ h2V), h1V\u2080, h2V\u2080\u27e9", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.iInter_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [435, 9], "def_end_pos": [435, 22]}, {"full_name": "Set.iInter_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [435, 9], "def_end_pos": [435, 22]}]], "state_before": "case right.mk.a.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 V \u2208 t, V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 \u22c2 V \u2208 t, V.carrier \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen (\u22c2 V \u2208 t, V.carrier) \u2227 1 \u2208 \u22c2 V \u2208 t, V.carrier", "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_le_prod_diam", "start": [270, 1], "end": [278, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.biUnion_null_iff", "start": [135, 1], "end": [138, 58], "traced_tactics": [{"tactic": "haveI := hs.toEncodable", "annotated_tactic": ["haveI := hs.toEncodable", []], "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 : Set \u03b2\nhs : Set.Countable s\nt : \u03b2 \u2192 Set \u03b1\n\u22a2 \u2191m (\u22c3 i \u2208 s, t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 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 : Set \u03b2\nhs : Set.Countable s\nt : \u03b2 \u2192 Set \u03b1\nthis : Encodable \u2191s\n\u22a2 \u2191m (\u22c3 i \u2208 s, t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 0"}, {"tactic": "rw [biUnion_eq_iUnion, iUnion_null_iff, SetCoe.forall']", "annotated_tactic": ["rw [<a>biUnion_eq_iUnion</a>, <a>iUnion_null_iff</a>, <a>SetCoe.forall'</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.OuterMeasure.iUnion_null_iff", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "SetCoe.forall'", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 23]}]], "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 : Set \u03b2\nhs : Set.Countable s\nt : \u03b2 \u2192 Set \u03b1\nthis : Encodable \u2191s\n\u22a2 \u2191m (\u22c3 i \u2208 s, t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.mem_disjointed_spanningSetsIndex", "start": [3363, 1], "end": [3365, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.move_ok", "start": [1349, 1], "end": [1381, 40], "traced_tactics": [{"tactic": "induction' L\u2081 with a L\u2081 IH generalizing S s", "annotated_tactic": ["induction' L\u2081 with a L\u2081 IH generalizing S s", []], "state_before": "p : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne : splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }", "state_after": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux [] (S k\u2082)) }\n\ncase cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux (a :: L\u2081) (S k\u2082)) }"}, {"tactic": "rw [(_ : [].reverseAux _ = _), Function.update_eq_self]", "annotated_tactic": ["rw [(_ : [].<a>reverseAux</a> _ = _), <a>Function.update_eq_self</a>]", [{"full_name": "List.reverseAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [42, 5], "def_end_pos": [42, 15]}, {"full_name": "Function.update_eq_self", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 23]}]], "state_before": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux [] (S k\u2082)) }", "state_after": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }\n\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 List.reverseAux [] (S k\u2082) = update S k\u2081 L\u2082 k\u2082"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }\n\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 List.reverseAux [] (S k\u2082) = update S k\u2081 L\u2082 k\u2082", "state_after": "p : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 List.reverseAux [] (S k\u2082) = update S k\u2081 L\u2082 k\u2082\n\ncase nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"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": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 ReflTransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr (\u039b'.move p k\u2081 k\u2082 q)) s S)\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>Option.elim</a>, <a>ne_eq</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "Option.elim", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [13, 31], "def_end_pos": [13, 35]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}]], "state_before": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 ReflTransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr (\u039b'.move p k\u2081 k\u2082 q)) s S)\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (S k\u2081), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),\n        stk :=\n          update (update S k\u2081 (List.tail (S k\u2081))) k\u2082\n            (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "revert e", "annotated_tactic": ["revert e", []], "state_before": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (S k\u2081), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),\n        stk :=\n          update (update S k\u2081 (List.tail (S k\u2081))) k\u2082\n            (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\n\u22a2 splitAtPred p (S k\u2081) = ([], o, L\u2082) \u2192\n    ReflTransGen\n      (fun a b =>\n        (match a with\n          | { l := none, var := var, stk := stk } => none\n          | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n          some b)\n      (bif\n          match List.head? (S k\u2081), true, p with\n          | some x, x_1, f => f x\n          | none, y, x => y then\n        { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }\n      else\n        { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),\n          stk :=\n            update (update S k\u2081 (List.tail (S k\u2081))) k\u2082\n              (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })\n      { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "cases' S k\u2081 with a Sk <;> intro e", "annotated_tactic": ["cases' S k\u2081 with a Sk <;> intro e", []], "state_before": "case nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\n\u22a2 splitAtPred p (S k\u2081) = ([], o, L\u2082) \u2192\n    ReflTransGen\n      (fun a b =>\n        (match a with\n          | { l := none, var := var, stk := stk } => none\n          | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n          some b)\n      (bif\n          match List.head? (S k\u2081), true, p with\n          | some x, x_1, f => f x\n          | none, y, x => y then\n        { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }\n      else\n        { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),\n          stk :=\n            update (update S k\u2081 (List.tail (S k\u2081))) k\u2082\n              (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })\n      { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil.nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p [] = ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? [], true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],\n        stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }\n\ncase nil.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne : splitAtPred p (a :: Sk) = ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a :: Sk), stk := update S k\u2081 (List.tail (a :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a :: Sk))) k\u2082\n            (Option.iget (List.head? (a :: Sk)) :: update S k\u2081 (List.tail (a :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "simp only [splitAtPred, Option.elim, List.head?, List.tail_cons, Option.iget_some] at e \u22a2", "annotated_tactic": ["simp only [<a>splitAtPred</a>, <a>Option.elim</a>, <a>List.head?</a>, <a>List.tail_cons</a>, <a>Option.iget_some</a>] at e \u22a2", [{"full_name": "Turing.PartrecToTM2.splitAtPred", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1317, 5], "def_end_pos": [1317, 16]}, {"full_name": "Option.elim", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [13, 31], "def_end_pos": [13, 35]}, {"full_name": "List.head?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/BasicAux.lean", "def_pos": [34, 5], "def_end_pos": [34, 10]}, {"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": "Option.iget_some", "def_path": "Mathlib/Data/Option/Defs.lean", "def_pos": [111, 9], "def_end_pos": [111, 18]}]], "state_before": "case nil.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne : splitAtPred p (a :: Sk) = ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a :: Sk), stk := update S k\u2081 (List.tail (a :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a :: Sk))) k\u2082\n            (Option.iget (List.head? (a :: Sk)) :: update S k\u2081 (List.tail (a :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne :\n  (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =\n    ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif p a then { l := some q, var := some a, stk := update S k\u2081 Sk }\n    else { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a, stk := update (update S k\u2081 Sk) k\u2082 (a :: update S k\u2081 Sk k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "revert e", "annotated_tactic": ["revert e", []], "state_before": "case nil.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne :\n  (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =\n    ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif p a then { l := some q, var := some a, stk := update S k\u2081 Sk }\n    else { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a, stk := update (update S k\u2081 Sk) k\u2082 (a :: update S k\u2081 Sk k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\n\u22a2 (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =\n      ([], o, L\u2082) \u2192\n    ReflTransGen\n      (fun a b =>\n        (match a with\n          | { l := none, var := var, stk := stk } => none\n          | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n          some b)\n      (bif p a then { l := some q, var := some a, stk := update S k\u2081 Sk }\n      else { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a, stk := update (update S k\u2081 Sk) k\u2082 (a :: update S k\u2081 Sk k\u2082) })\n      { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "cases p a <;> intro e <;>\nsimp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e \u22a2", "annotated_tactic": ["cases p a <;> intro e <;>\n      simp only [<a>cond_false</a>, <a>cond_true</a>, Prod.mk.injEq, <a>true_and</a>, <a>false_and</a>] at e \u22a2", [{"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": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"full_name": "false_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [86, 17], "def_end_pos": [86, 26]}]], "state_before": "case nil.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\n\u22a2 (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =\n      ([], o, L\u2082) \u2192\n    ReflTransGen\n      (fun a b =>\n        (match a with\n          | { l := none, var := var, stk := stk } => none\n          | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n          some b)\n      (bif p a then { l := some q, var := some a, stk := update S k\u2081 Sk }\n      else { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a, stk := update (update S k\u2081 Sk) k\u2082 (a :: update S k\u2081 Sk k\u2082) })\n      { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne : some a = o \u2227 Sk = L\u2082\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some q, var := some a, stk := update S k\u2081 Sk } { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "rw [Function.update_noteq h\u2081.symm]", "annotated_tactic": ["rw [<a>Function.update_noteq</a> h\u2081.symm]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}]], "state_before": "p : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 List.reverseAux [] (S k\u2082) = update S k\u2081 L\u2082 k\u2082", "state_after": "p : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 List.reverseAux [] (S k\u2082) = S k\u2082"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "p : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = ([], o, L\u2082)\n\u22a2 List.reverseAux [] (S k\u2082) = S k\u2082", "state_after": "no goals"}, {"tactic": "cases e", "annotated_tactic": ["cases e", []], "state_before": "case nil.nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p [] = ([], o, L\u2082)\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? [], true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],\n        stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil.nil.refl\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, none, [])\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? [], true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],\n        stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })\n    { l := some q, var := none, stk := update S k\u2081 [] }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil.nil.refl\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, none, [])\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? [], true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],\n        stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })\n    { l := some q, var := none, stk := update S k\u2081 [] }", "state_after": "no goals"}, {"tactic": "simp only [e]", "annotated_tactic": ["simp only [e]", []], "state_before": "case nil.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne : some a = o \u2227 Sk = L\u2082\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some q, var := some a, stk := update S k\u2081 Sk } { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "state_after": "case nil.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne : some a = o \u2227 Sk = L\u2082\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 } { l := some q, var := o, stk := update S k\u2081 L\u2082 }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082)\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na : \u0393'\nSk : List \u0393'\ne : some a = o \u2227 Sk = L\u2082\n\u22a2 ReflTransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some q, var := o, stk := update S k\u2081 L\u2082 } { l := some q, var := o, stk := update S k\u2081 L\u2082 }", "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": [385, 9], "def_end_pos": [385, 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": "case cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082)\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux (a :: L\u2081) (S k\u2082)) }", "state_after": "case cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr (\u039b'.move p k\u2081 k\u2082 q)) s S)\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux (a :: L\u2081) (S k\u2082)) }"}, {"tactic": "simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq, List.reverseAux_cons]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>Option.elim</a>, <a>ne_eq</a>, <a>List.reverseAux_cons</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "Option.elim", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [13, 31], "def_end_pos": [13, 35]}, {"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": "List.reverseAux_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [128, 17], "def_end_pos": [128, 32]}]], "state_before": "case cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr (\u039b'.move p k\u2081 k\u2082 q)) s S)\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux (a :: L\u2081) (S k\u2082)) }", "state_after": "case cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (S k\u2081), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),\n        stk :=\n          update (update S k\u2081 (List.tail (S k\u2081))) k\u2082\n            (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }"}, {"tactic": "cases' e\u2081 : S k\u2081 with a' Sk <;> rw [e\u2081, splitAtPred] at e", "annotated_tactic": ["cases' e\u2081 : S k\u2081 with a' Sk <;> rw [e\u2081, <a>splitAtPred</a>] at e", [{"full_name": "Turing.PartrecToTM2.splitAtPred", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1317, 5], "def_end_pos": [1317, 16]}]], "state_before": "case cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (S k\u2081), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),\n        stk :=\n          update (update S k\u2081 (List.tail (S k\u2081))) k\u2082\n            (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "case cons.nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : ([], none, []) = (a :: L\u2081, o, L\u2082)\ne\u2081 : S k\u2081 = []\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? [], true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],\n        stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }\n\ncase cons.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne :\n  (bif p a' then ([], some a', Sk)\n    else\n      match splitAtPred p Sk with\n      | (l\u2081, o, l\u2082) => (a' :: l\u2081, o, l\u2082)) =\n    (a :: L\u2081, o, L\u2082)\ne\u2081 : S k\u2081 = a' :: Sk\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }"}, {"tactic": "cases e\u2082 : p a' <;> simp only [e\u2082, cond] at e", "annotated_tactic": ["cases e\u2082 : p a' <;> simp only [e\u2082, <a>cond</a>] at e", [{"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "case cons.cons\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne :\n  (bif p a' then ([], some a', Sk)\n    else\n      match splitAtPred p Sk with\n      | (l\u2081, o, l\u2082) => (a' :: l\u2081, o, l\u2082)) =\n    (a :: L\u2081, o, L\u2082)\ne\u2081 : S k\u2081 = a' :: Sk\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "case cons.cons.false\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\ne : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }\n\ncase cons.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = true\ne : ([], some a', Sk) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case cons.cons.false\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\ne : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }\n\ncase cons.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = true\ne : ([], some a', Sk) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "case cons.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = true\ne : ([], some a', Sk) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }\n\ncase cons.cons.false\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\ne : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }"}, {"tactic": "rcases e\u2083 : splitAtPred p Sk with \u27e8_, _, _\u27e9", "annotated_tactic": ["rcases e\u2083 : <a>splitAtPred</a> p Sk with \u27e8_, _, _\u27e9", [{"full_name": "Turing.PartrecToTM2.splitAtPred", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1317, 5], "def_end_pos": [1317, 16]}]], "state_before": "case cons.cons.false\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\ne : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "case cons.cons.false.mk.mk\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\ne : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082)\nfst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }"}, {"tactic": "rw [e\u2083] at e", "annotated_tactic": ["rw [e\u2083] at e", []], "state_before": "case cons.cons.false.mk.mk\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\ne : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082)\nfst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "case cons.cons.false.mk.mk\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\nfst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne : (a' :: (fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) = (a :: L\u2081, o, L\u2082)\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }"}, {"tactic": "cases e", "annotated_tactic": ["cases e", []], "state_before": "case cons.cons.false.mk.mk\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = false\nfst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne : (a' :: (fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) = (a :: L\u2081, o, L\u2082)\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "case cons.cons.false.mk.mk.refl\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\na : \u0393'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\nSk fst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\ne\u2081 : S k\u2081 = a :: Sk\ne\u2082 : p a = false\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2)\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n          stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) }\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a :: Sk), stk := update S k\u2081 (List.tail (a :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a :: Sk))) k\u2082\n            (Option.iget (List.head? (a :: Sk)) :: update S k\u2081 (List.tail (a :: Sk)) k\u2082) })\n    { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n      stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (a :: S k\u2082)) }"}, {"tactic": "simp only [List.head?_cons, e\u2082, List.tail_cons, ne_eq, cond_false]", "annotated_tactic": ["simp only [<a>List.head?_cons</a>, e\u2082, <a>List.tail_cons</a>, <a>ne_eq</a>, <a>cond_false</a>]", [{"full_name": "List.head?_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [26, 17], "def_end_pos": [26, 27]}, {"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": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "cond_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [146, 17], "def_end_pos": [146, 27]}]], "state_before": "case cons.cons.false.mk.mk.refl\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\na : \u0393'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\nSk fst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\ne\u2081 : S k\u2081 = a :: Sk\ne\u2082 : p a = false\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2)\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n          stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) }\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a :: Sk), stk := update S k\u2081 (List.tail (a :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a :: Sk))) k\u2082\n            (Option.iget (List.head? (a :: Sk)) :: update S k\u2081 (List.tail (a :: Sk)) k\u2082) })\n    { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n      stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (a :: S k\u2082)) }", "state_after": "case cons.cons.false.mk.mk.refl\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\na : \u0393'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\nSk fst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\ne\u2081 : S k\u2081 = a :: Sk\ne\u2082 : p a = false\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2)\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n          stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) }\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a,\n      stk := update (update S k\u2081 Sk) k\u2082 (Option.iget (some a) :: update S k\u2081 Sk k\u2082) }\n    { l := some q, var := fst\u271d, stk := update (update S k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) }"}, {"tactic": "convert @IH _ (update (update S k\u2081 Sk) k\u2082 (a :: S k\u2082)) _ using 2 <;>\n  simp [Function.update_noteq, h\u2081, h\u2081.symm, e\u2083, List.reverseAux]", "annotated_tactic": ["convert @IH _ (<a>update</a> (<a>update</a> S k\u2081 Sk) k\u2082 (a :: S k\u2082)) _ using 2 <;>\n      simp [<a>Function.update_noteq</a>, h\u2081, h\u2081.symm, e\u2083, <a>List.reverseAux</a>]", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "List.reverseAux", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [42, 5], "def_end_pos": [42, 15]}]], "state_before": "case cons.cons.false.mk.mk.refl\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\na : \u0393'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\nSk fst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\ne\u2081 : S k\u2081 = a :: Sk\ne\u2082 : p a = false\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2)\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n          stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) }\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a,\n      stk := update (update S k\u2081 Sk) k\u2082 (Option.iget (some a) :: update S k\u2081 Sk k\u2082) }\n    { l := some q, var := fst\u271d, stk := update (update S k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) }", "state_after": "case h.e'_2.h.e'_7\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\na : \u0393'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\nSk fst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\ne\u2081 : S k\u2081 = a :: Sk\ne\u2082 : p a = false\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2)\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n          stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) }\n\u22a2 update (update S k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) =\n    update (update (update (update S k\u2081 Sk) k\u2082 (a :: S k\u2082)) k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082))"}, {"tactic": "simp [Function.update_comm h\u2081.symm]", "annotated_tactic": ["simp [<a>Function.update_comm</a> h\u2081.symm]", [{"full_name": "Function.update_comm", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [672, 9], "def_end_pos": [672, 20]}]], "state_before": "case h.e'_2.h.e'_7\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\na : \u0393'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\nSk fst\u271d\u00b9 : List \u0393'\nfst\u271d : Option \u0393'\nsnd\u271d : List \u0393'\ne\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d)\ne\u2081 : S k\u2081 = a :: Sk\ne\u2082 : p a = false\ne : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2)\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,\n          stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) }\n\u22a2 update (update S k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) =\n    update (update (update (update S k\u2081 Sk) k\u2082 (a :: S k\u2082)) k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082))", "state_after": "no goals"}, {"tactic": "cases e", "annotated_tactic": ["cases e", []], "state_before": "case cons.nil\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\ne : ([], none, []) = (a :: L\u2081, o, L\u2082)\ne\u2081 : S k\u2081 = []\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? [], true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],\n        stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "state_after": "no goals"}, {"tactic": "cases e", "annotated_tactic": ["cases e", []], "state_before": "case cons.cons.true\np : \u0393' \u2192 Bool\nk\u2081 k\u2082 : K'\nq : \u039b'\ns\u271d : Option \u0393'\nL\u2081\u271d : List \u0393'\no : Option \u0393'\nL\u2082 : List \u0393'\nS\u271d : K' \u2192 List \u0393'\nh\u2081 : k\u2081 \u2260 k\u2082\ne\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082)\na : \u0393'\nL\u2081 : List \u0393'\nIH :\n  \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},\n    splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }\n        { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) }\ns : Option \u0393'\nS : K' \u2192 List \u0393'\na' : \u0393'\nSk : List \u0393'\ne\u2081 : S k\u2081 = a' :: Sk\ne\u2082 : p a' = true\ne : ([], some a', Sk) = (a :: L\u2081, o, L\u2082)\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    (bif\n        match List.head? (a' :: Sk), true, p with\n        | some x, x_1, f => f x\n        | none, y, x => y then\n      { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }\n    else\n      { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),\n        stk :=\n          update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082\n            (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })\n    { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }", "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": [1499, 1], "end": [1501, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_insert_le", "start": [578, 1], "end": [582, 22], "traced_tactics": [{"tactic": "obtain hs | hs := s.finite_or_infinite", "annotated_tactic": ["obtain hs | hs := s.finite_or_infinite", []], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 ncard (insert a s) \u2264 ncard s + 1", "state_after": "case inl\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 ncard (insert a s) \u2264 ncard s + 1\n\ncase inr\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Infinite s\n\u22a2 ncard (insert a s) \u2264 ncard s + 1"}, {"tactic": "rw [(hs.mono (subset_insert a s)).ncard]", "annotated_tactic": ["rw [(hs.mono (<a>subset_insert</a> a s)).<a>ncard</a>]", [{"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 22]}, {"full_name": "Set.Infinite.ncard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [504, 9], "def_end_pos": [504, 23]}]], "state_before": "case inr\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Infinite s\n\u22a2 ncard (insert a s) \u2264 ncard s + 1", "state_after": "case inr\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Infinite s\n\u22a2 0 \u2264 ncard s + 1"}, {"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": "case inr\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Infinite s\n\u22a2 0 \u2264 ncard s + 1", "state_after": "no goals"}, {"tactic": "to_encard_tac", "annotated_tactic": ["to_encard_tac", []], "state_before": "case inl\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 ncard (insert a s) \u2264 ncard s + 1", "state_after": "case inl\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 \u2191(ncard (insert a s)) \u2264 \u2191(ncard s) + 1"}, {"tactic": "rw [hs.cast_ncard_eq, (hs.insert _).cast_ncard_eq]", "annotated_tactic": ["rw [hs.cast_ncard_eq, (hs.insert _).<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": "case inl\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 \u2191(ncard (insert a s)) \u2264 \u2191(ncard s) + 1", "state_after": "case inl\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 encard (insert a s) \u2264 encard s + 1"}, {"tactic": "apply encard_insert_le", "annotated_tactic": ["apply <a>encard_insert_le</a>", [{"full_name": "Set.encard_insert_le", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [237, 9], "def_end_pos": [237, 25]}]], "state_before": "case inl\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 encard (insert a s) \u2264 encard s + 1", "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_singleton", "start": [1187, 1], "end": [1188, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "hasSum_two_pi_I_cauchyPowerSeries_integral", "start": [582, 1], "end": [606, 88], "traced_tactics": [{"tactic": "have hR : 0 < R := (Complex.abs.nonneg w).trans_lt hw", "annotated_tactic": ["have hR : 0 < R := (Complex.abs.nonneg w).<a>trans_lt</a> hw", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)\n    (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z)", "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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)\n    (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z)"}, {"tactic": "have hwR : abs w / R \u2208 Ico (0 : \u211d) 1 :=\n  \u27e8div_nonneg (Complex.abs.nonneg w) hR.le, (div_lt_one hR).2 hw\u27e9", "annotated_tactic": ["have hwR : <a>abs</a> w / R \u2208 <a>Ico</a> (0 : \u211d) 1 :=\n    \u27e8<a>div_nonneg</a> (Complex.abs.nonneg w) hR.le, (<a>div_lt_one</a> hR).2 hw\u27e9", [{"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"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": "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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)\n    (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z)", "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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)\n    (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z)"}, {"tactic": "refine' intervalIntegral.hasSum_integral_of_dominated_convergence\n    (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (abs w / R) ^ n) (fun n => _) (fun n => _) _ _ _", "annotated_tactic": ["refine' <a>intervalIntegral.hasSum_integral_of_dominated_convergence</a>\n      (fun n \u03b8 => \u2016f (<a>circleMap</a> c R \u03b8)\u2016 * (<a>abs</a> w / R) ^ n) (fun n => _) (fun n => _) _ _ _", [{"full_name": "intervalIntegral.hasSum_integral_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1025, 16], "def_end_pos": [1025, 56]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)\n    (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z)", "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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 AEStronglyMeasurable\n    (fun \u03b8 => deriv (circleMap c R) \u03b8 \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))\n    (Measure.restrict volume (\u0399 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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 \u2200\u1d50 (t : \u211d),\n    t \u2208 \u0399 0 (2 * \u03c0) \u2192\n      \u2016deriv (circleMap c R) t \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R t)\u2016 \u2264\n        (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t\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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 \u2200\u1d50 (t : \u211d), t \u2208 \u0399 0 (2 * \u03c0) \u2192 Summable fun n => (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t\n\ncase refine'_4\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 IntervalIntegrable (fun t => \u2211' (n : \u2115), (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t) volume 0\n    (2 * \u03c0)\n\ncase refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 \u2200\u1d50 (t : \u211d),\n    t \u2208 \u0399 0 (2 * \u03c0) \u2192\n      HasSum (fun n => deriv (circleMap c R) t \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R t))\n        (deriv (circleMap c R) t \u2022 (fun z => (z - (c + w))\u207b\u00b9 \u2022 f z) (circleMap c R t))"}, {"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 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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 AEStronglyMeasurable\n    (fun \u03b8 => deriv (circleMap c R) \u03b8 \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))\n    (Measure.restrict volume (\u0399 0 (2 * \u03c0)))", "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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 AEStronglyMeasurable\n    (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (w / (circleMap c R \u03b8 - c)) ^ n \u2022 (circleMap c R \u03b8 - c)\u207b\u00b9 \u2022 f (circleMap c R \u03b8))\n    (Measure.restrict volume (\u0399 0 (2 * \u03c0)))"}, {"tactic": "apply_rules [AEStronglyMeasurable.smul, hf.def.1] <;> apply Measurable.aestronglyMeasurable", "annotated_tactic": ["apply_rules [<a>AEStronglyMeasurable.smul</a>, hf.def.1] <;> apply <a>Measurable.aestronglyMeasurable</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.smul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1331, 19], "def_end_pos": [1331, 23]}, {"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1285, 9], "def_end_pos": [1285, 47]}]], "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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 AEStronglyMeasurable\n    (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (w / (circleMap c R \u03b8 - c)) ^ n \u2022 (circleMap c R \u03b8 - c)\u207b\u00b9 \u2022 f (circleMap c R \u03b8))\n    (Measure.restrict volume (\u0399 0 (2 * \u03c0)))", "state_after": "case refine'_1.hf.hf\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 Measurable fun x => circleMap 0 R x * I\n\ncase refine'_1.hg.hf.hf\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 Measurable fun x => (w / (circleMap c R x - c)) ^ n\n\ncase refine'_1.hg.hg.hf.hf\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 Measurable fun x => (circleMap c R x - c)\u207b\u00b9"}, {"tactic": "exact (measurable_circleMap 0 R).mul_const I", "annotated_tactic": ["exact (<a>measurable_circleMap</a> 0 R).<a>mul_const</a> <a>I</a>", [{"full_name": "measurable_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [190, 9], "def_end_pos": [190, 29]}, {"full_name": "Measurable.mul_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [120, 9], "def_end_pos": [120, 29]}, {"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [293, 5], "def_end_pos": [293, 6]}]], "state_before": "case refine'_1.hf.hf\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 Measurable fun x => circleMap 0 R x * I", "state_after": "no goals"}, {"tactic": "exact (((measurable_circleMap c R).sub measurable_const).const_div w).pow measurable_const", "annotated_tactic": ["exact (((<a>measurable_circleMap</a> c R).<a>sub</a> <a>measurable_const</a>).<a>const_div</a> w).<a>pow</a> <a>measurable_const</a>", [{"full_name": "measurable_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [190, 9], "def_end_pos": [190, 29]}, {"full_name": "Measurable.sub", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [333, 3], "def_end_pos": [333, 14]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Measurable.const_div", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [299, 9], "def_end_pos": [299, 29]}, {"full_name": "Measurable.pow", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [217, 9], "def_end_pos": [217, 23]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case refine'_1.hg.hf.hf\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 Measurable fun x => (w / (circleMap c R x - c)) ^ n", "state_after": "no goals"}, {"tactic": "exact ((measurable_circleMap c R).sub measurable_const).inv", "annotated_tactic": ["exact ((<a>measurable_circleMap</a> c R).<a>sub</a> <a>measurable_const</a>).<a>inv</a>", [{"full_name": "measurable_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [190, 9], "def_end_pos": [190, 29]}, {"full_name": "Measurable.sub", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [333, 3], "def_end_pos": [333, 14]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Measurable.inv", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}]], "state_before": "case refine'_1.hg.hg.hf.hf\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 Measurable fun x => (circleMap c R x - c)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simp [norm_smul, abs_of_pos hR, mul_left_comm R, inv_mul_cancel_left\u2080 hR.ne', mul_comm \u2016_\u2016]", "annotated_tactic": ["simp [<a>norm_smul</a>, <a>abs_of_pos</a> hR, <a>mul_left_comm</a> R, <a>inv_mul_cancel_left\u2080</a> hR.ne', <a>mul_comm</a> \u2016_\u2016]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}, {"full_name": "inv_mul_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 29]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\nn : \u2115\n\u22a2 \u2200\u1d50 (t : \u211d),\n    t \u2208 \u0399 0 (2 * \u03c0) \u2192\n      \u2016deriv (circleMap c R) t \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R t)\u2016 \u2264\n        (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun _ _ => (summable_geometric_of_lt_1 hwR.1 hwR.2).mul_left _", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun _ _ => (<a>summable_geometric_of_lt_1</a> hwR.1 hwR.2).<a>mul_left</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": "summable_geometric_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 35]}, {"full_name": "Summable.mul_left", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [42, 9], "def_end_pos": [42, 26]}]], "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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 \u2200\u1d50 (t : \u211d), t \u2208 \u0399 0 (2 * \u03c0) \u2192 Summable fun n => (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t", "state_after": "no goals"}, {"tactic": "simpa only [tsum_mul_left, tsum_geometric_of_lt_1 hwR.1 hwR.2] using\n  hf.norm.mul_continuousOn continuousOn_const", "annotated_tactic": ["simpa only [<a>tsum_mul_left</a>, <a>tsum_geometric_of_lt_1</a> hwR.1 hwR.2] using\n      hf.norm.mul_continuousOn <a>continuousOn_const</a>", [{"full_name": "tsum_mul_left", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [113, 9], "def_end_pos": [113, 22]}, {"full_name": "tsum_geometric_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 31]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}]], "state_before": "case refine'_4\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 IntervalIntegrable (fun t => \u2211' (n : \u2115), (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t) volume 0\n    (2 * \u03c0)", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall fun \u03b8 _ => HasSum.const_smul _ _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun \u03b8 _ => <a>HasSum.const_smul</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": "HasSum.const_smul", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 26]}]], "state_before": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u22a2 \u2200\u1d50 (t : \u211d),\n    t \u2208 \u0399 0 (2 * \u03c0) \u2192\n      HasSum (fun n => deriv (circleMap c R) t \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R t))\n        (deriv (circleMap c R) t \u2022 (fun z => (z - (c + w))\u207b\u00b9 \u2022 f z) (circleMap c R t))", "state_after": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 HasSum (fun n => (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))\n    ((fun z => (z - (c + w))\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))"}, {"tactic": "simp only [smul_smul]", "annotated_tactic": ["simp only [<a>smul_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}]], "state_before": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 HasSum (fun n => (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))\n    ((fun z => (z - (c + w))\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))", "state_after": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 HasSum (fun n => ((w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) \u2022 f (circleMap c R \u03b8))\n    ((circleMap c R \u03b8 - (c + w))\u207b\u00b9 \u2022 f (circleMap c R \u03b8))"}, {"tactic": "refine' HasSum.smul_const _ _", "annotated_tactic": ["refine' <a>HasSum.smul_const</a> _ _", [{"full_name": "HasSum.smul_const", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Module.lean", "def_pos": [21, 9], "def_end_pos": [21, 26]}]], "state_before": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 HasSum (fun n => ((w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) \u2022 f (circleMap c R \u03b8))\n    ((circleMap c R \u03b8 - (c + w))\u207b\u00b9 \u2022 f (circleMap c R \u03b8))", "state_after": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 HasSum (fun n => (w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) (circleMap c R \u03b8 - (c + w))\u207b\u00b9"}, {"tactic": "have : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1 := by simpa [abs_of_pos hR] using hwR.2", "annotated_tactic": ["have : \u2016w / (<a>circleMap</a> c R \u03b8 - c)\u2016 < 1 := by simpa [<a>abs_of_pos</a> hR] using hwR.2", [{"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}]], "state_before": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 HasSum (fun n => (w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) (circleMap c R \u03b8 - (c + w))\u207b\u00b9", "state_after": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nthis : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1\n\u22a2 HasSum (fun n => (w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) (circleMap c R \u03b8 - (c + w))\u207b\u00b9"}, {"tactic": "convert (hasSum_geometric_of_norm_lt_1 this).mul_right _ using 1", "annotated_tactic": ["convert (<a>hasSum_geometric_of_norm_lt_1</a> this).<a>mul_right</a> _ using 1", [{"full_name": "hasSum_geometric_of_norm_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "def_pos": [282, 9], "def_end_pos": [282, 38]}, {"full_name": "HasSum.mul_right", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [38, 9], "def_end_pos": [38, 25]}]], "state_before": "case refine'_5\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nthis : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1\n\u22a2 HasSum (fun n => (w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) (circleMap c R \u03b8 - (c + w))\u207b\u00b9", "state_after": "case h.e'_6\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nthis : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1\n\u22a2 (circleMap c R \u03b8 - (c + w))\u207b\u00b9 = (1 - w / (circleMap c R \u03b8 - c))\u207b\u00b9 * (circleMap c R \u03b8 - c)\u207b\u00b9"}, {"tactic": "simp [\u2190 sub_sub, \u2190 mul_inv, sub_mul, div_mul_cancel _ (circleMap_ne_center hR.ne')]", "annotated_tactic": ["simp [\u2190 <a>sub_sub</a>, \u2190 <a>mul_inv</a>, <a>sub_mul</a>, <a>div_mul_cancel</a> _ (<a>circleMap_ne_center</a> hR.ne')]", [{"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [526, 3], "def_end_pos": [526, 14]}, {"full_name": "mul_inv", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [482, 9], "def_end_pos": [482, 16]}, {"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [372, 7], "def_end_pos": [372, 14]}, {"full_name": "div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}, {"full_name": "circleMap_ne_center", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [168, 9], "def_end_pos": [168, 28]}]], "state_before": "case h.e'_6\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\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nthis : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1\n\u22a2 (circleMap c R \u03b8 - (c + w))\u207b\u00b9 = (1 - w / (circleMap c R \u03b8 - c))\u207b\u00b9 * (circleMap c R \u03b8 - c)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simpa [abs_of_pos hR] using hwR.2", "annotated_tactic": ["simpa [<a>abs_of_pos</a> hR] using hwR.2", [{"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 : \u211d\nw : \u2102\nhf : CircleIntegrable f c R\nhw : \u2191Complex.abs w < R\nhR : 0 < R\nhwR : \u2191Complex.abs w / R \u2208 Ico 0 1\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.toMeasure_map_apply", "start": [100, 1], "end": [104, 39], "traced_tactics": [{"tactic": "rw [toMeasure_apply_eq_toOuterMeasure_apply _ s hs,\n  toMeasure_apply_eq_toOuterMeasure_apply _ (f \u207b\u00b9' s) (measurableSet_preimage hf hs)]", "annotated_tactic": ["rw [<a>toMeasure_apply_eq_toOuterMeasure_apply</a> _ s hs,\n    <a>toMeasure_apply_eq_toOuterMeasure_apply</a> _ (f \u207b\u00b9' s) (<a>measurableSet_preimage</a> hf hs)]", [{"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.toMeasure_apply_eq_toOuterMeasure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 48]}, {"full_name": "measurableSet_preimage", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [325, 9], "def_end_pos": [325, 31]}]], "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\ns : Set \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nhf : Measurable f\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure (map f p)) s = \u2191\u2191(toMeasure p) (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : PMF \u03b1\nb : \u03b2\ns : Set \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nhf : Measurable f\nhs : MeasurableSet s\n\u22a2 \u2191(toOuterMeasure (map f p)) s = \u2191(toOuterMeasure p) (f \u207b\u00b9' s)"}, {"tactic": "exact toOuterMeasure_map_apply f p s", "annotated_tactic": ["exact <a>toOuterMeasure_map_apply</a> f p s", [{"full_name": "PMF.toOuterMeasure_map_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [95, 9], "def_end_pos": [95, 33]}]], "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\ns : Set \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nhf : Measurable f\nhs : MeasurableSet s\n\u22a2 \u2191(toOuterMeasure (map f p)) s = \u2191(toOuterMeasure p) (f \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.mem_union", "start": [569, 1], "end": [570, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup_himp_right", "start": [633, 1], "end": [635, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max_mem_insert_bot_image_coe", "start": [1624, 1], "end": [1626, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "full_name": "BoundedContinuousFunction.toLp_denseRange", "start": [368, 1], "end": [374, 85], "traced_tactics": [{"tactic": "haveI : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E", "annotated_tactic": ["haveI : <a>NormedSpace</a> \u211d E := <a>RestrictScalars.normedSpace</a> \u211d \ud835\udd5c E", [{"full_name": "NormedSpace", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [43, 7], "def_end_pos": [43, 18]}, {"full_name": "RestrictScalars.normedSpace", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [529, 10], "def_end_pos": [529, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis : NormedSpace \u211d E\n\u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c)"}, {"tactic": "rw [denseRange_iff_closure_range]", "annotated_tactic": ["rw [<a>denseRange_iff_closure_range</a>]", [{"full_name": "denseRange_iff_closure_range", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis : NormedSpace \u211d E\n\u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis : NormedSpace \u211d E\n\u22a2 closure (range \u2191(toLp p \u03bc \ud835\udd5c)) = univ"}, {"tactic": "suffices (LinearMap.range (toLp p \u03bc \ud835\udd5c : _ \u2192L[\ud835\udd5c] Lp E p \u03bc)).toAddSubgroup.topologicalClosure = \u22a4\n  by exact congr_arg ((\u2191) : AddSubgroup (Lp E p \u03bc) \u2192 Set (Lp E p \u03bc)) this", "annotated_tactic": ["suffices (<a>LinearMap.range</a> (<a>toLp</a> p \u03bc \ud835\udd5c : _ \u2192L[\ud835\udd5c] <a>Lp</a> E p \u03bc)).toAddSubgroup.topologicalClosure = \u22a4\n    by exact <a>congr_arg</a> ((\u2191) : <a>AddSubgroup</a> (<a>Lp</a> E p \u03bc) \u2192 <a>Set</a> (<a>Lp</a> E p \u03bc)) this", [{"full_name": "LinearMap.range", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1080, 5], "def_end_pos": [1080, 10]}, {"full_name": "BoundedContinuousFunction.toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1778, 5], "def_end_pos": [1778, 9]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "AddSubgroup", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [370, 11], "def_end_pos": [370, 22]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"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\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis : NormedSpace \u211d E\n\u22a2 closure (range \u2191(toLp p \u03bc \ud835\udd5c)) = univ", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis : NormedSpace \u211d E\n\u22a2 AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u22a4"}, {"tactic": "simpa [range_toLp p \u03bc] using MeasureTheory.Lp.boundedContinuousFunction_dense E hp", "annotated_tactic": ["simpa [<a>range_toLp</a> p \u03bc] using <a>MeasureTheory.Lp.boundedContinuousFunction_dense</a> E hp", [{"full_name": "BoundedContinuousFunction.range_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1792, 9], "def_end_pos": [1792, 19]}, {"full_name": "MeasureTheory.Lp.boundedContinuousFunction_dense", "def_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "def_pos": [331, 9], "def_end_pos": [331, 40]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis : NormedSpace \u211d E\n\u22a2 AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u22a4", "state_after": "no goals"}, {"tactic": "exact congr_arg ((\u2191) : AddSubgroup (Lp E p \u03bc) \u2192 Set (Lp E p \u03bc)) this", "annotated_tactic": ["exact <a>congr_arg</a> ((\u2191) : <a>AddSubgroup</a> (<a>Lp</a> E p \u03bc) \u2192 <a>Set</a> (<a>Lp</a> E p \u03bc)) this", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "AddSubgroup", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [370, 11], "def_end_pos": [370, 22]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"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\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b1\ninst\u271d\u2078 : T4Space \u03b1\ninst\u271d\u2077 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nthis\u271d : NormedSpace \u211d E\nthis : AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u22a4\n\u22a2 closure (range \u2191(toLp p \u03bc \ud835\udd5c)) = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Char.lean", "full_name": "Char.ofNat_toNat", "start": [41, 1], "end": [43, 6], "traced_tactics": [{"tactic": "rw [Char.ofNat, dif_pos h]", "annotated_tactic": ["rw [<a>Char.ofNat</a>, <a>dif_pos</a> h]", [{"full_name": "Char.ofNat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2077, 5], "def_end_pos": [2077, 15]}, {"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": "c : Char\nh : isValidCharNat (toNat c)\n\u22a2 ofNat (toNat c) = c", "state_after": "c : Char\nh : isValidCharNat (toNat c)\n\u22a2 ofNatAux (toNat c) h = c"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "c : Char\nh : isValidCharNat (toNat c)\n\u22a2 ofNatAux (toNat c) h = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.rfindOpt", "start": [367, 1], "end": [387, 12], "traced_tactics": [{"tactic": "simp only [Nat.rfindOpt, exists_prop, tsub_eq_zero_iff_le, PFun.coe_val, Part.mem_bind_iff,\n  Part.mem_some_iff, Option.mem_def, Part.mem_coe]", "annotated_tactic": ["simp only [<a>Nat.rfindOpt</a>, <a>exists_prop</a>, <a>tsub_eq_zero_iff_le</a>, <a>PFun.coe_val</a>, <a>Part.mem_bind_iff</a>,\n        <a>Part.mem_some_iff</a>, <a>Option.mem_def</a>, <a>Part.mem_coe</a>]", [{"full_name": "Nat.rfindOpt", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [124, 5], "def_end_pos": [124, 13]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"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": "PFun.coe_val", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"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": "Part.mem_coe", "def_path": "Mathlib/Data/Part.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}]], "state_before": "n : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (b \u2208 Part.bind (Nat.rfind fun n_1 => Part.some (decide (1 - f (n_1 ::\u1d65 v) = 0))) fun a => \u2191pred (f (a ::\u1d65 v))) \u2194\n    b \u2208 Nat.rfindOpt fun a => ofNat (Option \u2115) (f (a ::\u1d65 v))", "state_after": "n : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (\u2203 a, (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) \u2227 b = pred (f (a ::\u1d65 v))) \u2194\n    \u2203 a,\n      (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227\n        ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "refine'\n  exists_congr fun a => (and_congr (iff_of_eq _) Iff.rfl).trans (and_congr_right fun h => _)", "annotated_tactic": ["refine'\n        <a>exists_congr</a> fun a => (<a>and_congr</a> (<a>iff_of_eq</a> _) <a>Iff.rfl</a>).<a>trans</a> (<a>and_congr_right</a> fun h => _)", [{"full_name": "exists_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [379, 9], "def_end_pos": [379, 21]}, {"full_name": "and_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "iff_of_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Iff.rfl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [663, 19], "def_end_pos": [663, 26]}, {"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_congr_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [161, 9], "def_end_pos": [161, 24]}]], "state_before": "n : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (\u2203 a, (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) \u2227 b = pred (f (a ::\u1d65 v))) \u2194\n    \u2203 a,\n      (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227\n        ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_1\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) =\n    (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))))\n\ncase refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case refine'_1\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) =\n    (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))))", "state_after": "case refine'_1.e_a.e_p\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 =>\n    \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case refine'_1.e_a.e_p\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 =>\n    \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))", "state_after": "case refine'_1.e_a.e_p.h\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\nb a n : \u2115\n\u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v)))))"}, {"tactic": "cases f (n ::\u1d65 v) <;> simp [Nat.succ_le_succ]", "annotated_tactic": ["cases f (n ::\u1d65 v) <;> simp [<a>Nat.succ_le_succ</a>]", [{"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]}]], "state_before": "case refine'_1.e_a.e_p.h\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\nb a n : \u2115\n\u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v)))))", "state_after": "case refine'_1.e_a.e_p.h.succ\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\u00b9\nb a n n\u271d : \u2115\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_1.e_a.e_p.h.succ\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\u00b9\nb a n n\u271d : \u2115\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d))", "state_after": "no goals"}, {"tactic": "have := Nat.rfind_spec h", "annotated_tactic": ["have := <a>Nat.rfind_spec</a> h", [{"full_name": "Nat.rfind_spec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [79, 9], "def_end_pos": [79, 19]}]], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "simp only [Part.coe_some, Part.mem_some_iff] at this", "annotated_tactic": ["simp only [<a>Part.coe_some</a>, <a>Part.mem_some_iff</a>] at this", [{"full_name": "Part.coe_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [361, 9], "def_end_pos": [361, 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 refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192\n    (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b)"}, {"tactic": "cases' f (a ::\u1d65 v) with c <;> intro this", "annotated_tactic": ["cases' f (a ::\u1d65 v) with c <;> intro this", []], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192\n    (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b)", "state_after": "case refine'_2.zero\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) zero)\n\u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b\n\ncase refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b"}, {"tactic": "rw [\u2190 Option.some_inj, eq_comm]", "annotated_tactic": ["rw [\u2190 <a>Option.some_inj</a>, <a>eq_comm</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": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b", "state_after": "case refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b", "state_after": "no goals"}, {"tactic": "cases this", "annotated_tactic": ["cases this", []], "state_before": "case refine'_2.zero\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) zero)\n\u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.snorm_exponent_top_lim_le_liminf_snorm_exponent_top", "start": [1341, 1], "end": [1347, 69], "traced_tactics": [{"tactic": "rw [snorm_exponent_top_lim_eq_essSup_liminf h_lim]", "annotated_tactic": ["rw [<a>snorm_exponent_top_lim_eq_essSup_liminf</a> h_lim]", [{"full_name": "MeasureTheory.Lp.snorm_exponent_top_lim_eq_essSup_liminf", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 48]}]], "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\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 F\nf_lim : \u03b1 \u2192 F\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 \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop", "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\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 F\nf_lim : \u03b1 \u2192 F\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 essSup (fun x => liminf (fun m => \u2191\u2016f m x\u2016\u208a) atTop) \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop"}, {"tactic": "simp_rw [snorm_exponent_top, snormEssSup]", "annotated_tactic": ["simp_rw [<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": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 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\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 F\nf_lim : \u03b1 \u2192 F\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 essSup (fun x => liminf (fun m => \u2191\u2016f m x\u2016\u208a) atTop) \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop", "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\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 F\nf_lim : \u03b1 \u2192 F\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 essSup (fun x => liminf (fun m => \u2191\u2016f m x\u2016\u208a) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => \u2191\u2016f n x\u2016\u208a) \u03bc) atTop"}, {"tactic": "exact ENNReal.essSup_liminf_le fun n => fun x => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)", "annotated_tactic": ["exact <a>ENNReal.essSup_liminf_le</a> fun n => fun x => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)", [{"full_name": "ENNReal.essSup_liminf_le", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [325, 9], "def_end_pos": [325, 25]}]], "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\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 F\nf_lim : \u03b1 \u2192 F\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 essSup (fun x => liminf (fun m => \u2191\u2016f m x\u2016\u208a) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => \u2191\u2016f n x\u2016\u208a) \u03bc) atTop", "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_mutuallySingular", "start": [846, 1], "end": [861, 40], "traced_tactics": [{"tactic": "by_cases hl : s.HaveLebesgueDecomposition \u03bc", "annotated_tactic": ["by_cases hl : s.HaveLebesgueDecomposition \u03bc", []], "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 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos\n\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\nhl : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc\n\ncase neg\n\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\nhl : \u00acHaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "haveI := hl", "annotated_tactic": ["haveI := hl", []], "state_before": "case pos\n\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\nhl : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "obtain \u27e8i, hi, hpos, hneg\u27e9 := s.toJordanDecomposition.mutuallySingular", "annotated_tactic": ["obtain \u27e8i, hi, hpos, hneg\u27e9 := s.toJordanDecomposition.mutuallySingular", []], "state_before": "case pos\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos : \u2191\u2191(toJordanDecomposition s).posPart i = 0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart i\u1d9c = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc] at hpos", "annotated_tactic": ["rw [s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc] at hpos", []], "state_before": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos : \u2191\u2191(toJordanDecomposition s).posPart i = 0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart i\u1d9c = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart i\u1d9c = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc] at hneg", "annotated_tactic": ["rw [s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc] at hneg", []], "state_before": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart i\u1d9c = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc))\n      i\u1d9c =\n    0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [add_apply, add_eq_zero_iff] at hpos hneg", "annotated_tactic": ["rw [<a>add_apply</a>, <a>add_eq_zero_iff</a>] at hpos hneg", [{"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}, {"full_name": "add_eq_zero_iff", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}]], "state_before": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc))\n      i\u1d9c =\n    0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc) i = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc)) i = 0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc) i\u1d9c = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc)) i\u1d9c = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "exact \u27e8i, hi, hpos.1, hneg.1\u27e9", "annotated_tactic": ["exact \u27e8i, hi, hpos.1, hneg.1\u27e9", []], "state_before": "case pos.intro.intro.intro\n\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\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc) i = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc)) i = 0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc) i\u1d9c = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc)) i\u1d9c = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "no goals"}, {"tactic": "rw [not_haveLebesgueDecomposition_iff] at hl", "annotated_tactic": ["rw [<a>not_haveLebesgueDecomposition_iff</a>] at hl", [{"full_name": "MeasureTheory.SignedMeasure.not_haveLebesgueDecomposition_iff", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [786, 9], "def_end_pos": [786, 42]}]], "state_before": "case neg\n\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\nhl : \u00acHaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg\n\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\nhl :\n  \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u2228\n    \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "cases' hl with hp hn", "annotated_tactic": ["cases' hl with hp hn", []], "state_before": "case neg\n\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\nhl :\n  \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u2228\n    \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg.inl\n\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\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc\n\ncase neg.inr\n\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\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [Measure.singularPart, dif_neg hp]", "annotated_tactic": ["rw [<a>Measure.singularPart</a>, <a>dif_neg</a> hp]", [{"full_name": "MeasureTheory.Measure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [88, 17], "def_end_pos": [88, 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": "case neg.inl\n\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\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg.inl\n\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\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 0 \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "exact MutuallySingular.zero_left", "annotated_tactic": ["exact <a>MutuallySingular.zero_left</a>", [{"full_name": "MeasureTheory.Measure.MutuallySingular.zero_left", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}]], "state_before": "case neg.inl\n\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\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 0 \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "no goals"}, {"tactic": "rw [Measure.singularPart, Measure.singularPart, dif_neg hn]", "annotated_tactic": ["rw [<a>Measure.singularPart</a>, <a>Measure.singularPart</a>, <a>dif_neg</a> hn]", [{"full_name": "MeasureTheory.Measure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [88, 17], "def_end_pos": [88, 29]}, {"full_name": "MeasureTheory.Measure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [88, 17], "def_end_pos": [88, 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": "case neg.inr\n\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\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg.inr\n\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\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 (if h : Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc then\n      (Classical.choose\n          (_ : \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition s).posPart = p.1 + withDensity \u03bc p.2)).1\n    else 0) \u27c2\u2098\n    0"}, {"tactic": "exact MutuallySingular.zero_right", "annotated_tactic": ["exact <a>MutuallySingular.zero_right</a>", [{"full_name": "MeasureTheory.Measure.MutuallySingular.zero_right", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [57, 9], "def_end_pos": [57, 19]}]], "state_before": "case neg.inr\n\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\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 (if h : Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc then\n      (Classical.choose\n          (_ : \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition s).posPart = p.1 + withDensity \u03bc p.2)).1\n    else 0) \u27c2\u2098\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_const", "start": [972, 1], "end": [973, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "AlternatingMap.measure_parallelepiped", "start": [588, 1], "end": [593, 32], "traced_tactics": [{"tactic": "conv_rhs => rw [\u03c9.eq_smul_basis_det (finBasisOfFinrankEq \u211d G _i.out)]", "annotated_tactic": ["conv_rhs => rw [\u03c9.eq_smul_basis_det (<a>finBasisOfFinrankEq</a> \u211d G _i.out)]", [{"full_name": "FiniteDimensional.finBasisOfFinrankEq", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [253, 19], "def_end_pos": [253, 38]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n\u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) = ENNReal.ofReal |\u2191\u03c9 v|", "state_after": "E : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n\u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) =\n    ENNReal.ofReal\n      |\u2191(\u2191\u03c9 \u2191(finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)) \u2022\n              Basis.det (finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)))\n          v|"}, {"tactic": "simp only [addHaar_parallelepiped, AlternatingMap.measure, coe_nnreal_smul_apply,\n  AlternatingMap.smul_apply, Algebra.id.smul_eq_mul, abs_mul, ENNReal.ofReal_mul (abs_nonneg _),\n  Real.ennnorm_eq_ofReal_abs]", "annotated_tactic": ["simp only [<a>addHaar_parallelepiped</a>, <a>AlternatingMap.measure</a>, <a>coe_nnreal_smul_apply</a>,\n    <a>AlternatingMap.smul_apply</a>, <a>Algebra.id.smul_eq_mul</a>, <a>abs_mul</a>, <a>ENNReal.ofReal_mul</a> (<a>abs_nonneg</a> _),\n    <a>Real.ennnorm_eq_ofReal_abs</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [566, 9], "def_end_pos": [566, 31]}, {"full_name": "AlternatingMap.measure", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [583, 31], "def_end_pos": [583, 60]}, {"full_name": "MeasureTheory.Measure.coe_nnreal_smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [905, 9], "def_end_pos": [905, 30]}, {"full_name": "AlternatingMap.smul_apply", "def_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 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": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"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": "Real.ennnorm_eq_ofReal_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1812, 9], "def_end_pos": [1812, 30]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n\u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) =\n    ENNReal.ofReal\n      |\u2191(\u2191\u03c9 \u2191(finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)) \u2022\n              Basis.det (finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)))\n          v|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.image_core_gc", "start": [255, 1], "end": [256, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_sup_inf'", "start": [1163, 1], "end": [1165, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.nontrivial_iff_ne_singleton", "start": [823, 1], "end": [824, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_zero", "start": [180, 1], "end": [181, 60], "traced_tactics": [{"tactic": "simp only [variance, evariance_zero, ENNReal.zero_toReal]", "annotated_tactic": ["simp only [<a>variance</a>, <a>evariance_zero</a>, <a>ENNReal.zero_toReal</a>]", [{"full_name": "ProbabilityTheory.variance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "ProbabilityTheory.evariance_zero", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [147, 9], "def_end_pos": [147, 23]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc\u271d \u03bc : Measure \u03a9\n\u22a2 variance 0 \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.hasFiniteIntegral_of_fintype", "start": [187, 1], "end": [194, 44], "traced_tactics": [{"tactic": "apply ae_of_all \u03bc", "annotated_tactic": ["apply <a>ae_of_all</a> \u03bc", [{"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\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\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f a\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\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\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (a : \u03b1), \u2016f a\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a)"}, {"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\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (a : \u03b1), \u2016f a\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\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\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2016f x\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a)"}, {"tactic": "rw [\u2190 coe_nnnorm (f x)]", "annotated_tactic": ["rw [\u2190 <a>coe_nnnorm</a> (f x)]", [{"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 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\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2016f x\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\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\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2191\u2016f x\u2016\u208a \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a)"}, {"tactic": "apply NNReal.toReal_le_toReal", "annotated_tactic": ["apply <a>NNReal.toReal_le_toReal</a>", [{"full_name": "NNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [384, 9], "def_end_pos": [384, 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\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2191\u2016f x\u2016\u208a \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a)", "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\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2016f x\u2016\u208a \u2264 Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a"}, {"tactic": "apply Finset.le_sup (Finset.mem_univ x)", "annotated_tactic": ["apply <a>Finset.le_sup</a> (<a>Finset.mem_univ</a> x)", [{"full_name": "Finset.le_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [114, 9], "def_end_pos": [114, 15]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2016f x\u2016\u208a \u2264 Finset.sup Finset.univ fun a => \u2016f a\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.mapIdx_induction", "start": [193, 1], "end": [201, 18], "traced_tactics": [{"tactic": "have := SatisfiesM_mapIdxM (m := Id) (as := as) (f := f) motive h0", "annotated_tactic": ["have := <a>SatisfiesM_mapIdxM</a> (m := <a>Id</a>) (as := as) (f := f) motive h0", [{"full_name": "Array.SatisfiesM_mapIdxM", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [171, 9], "def_end_pos": [171, 27]}, {"full_name": "Id", "def_path": "lake-packages/lean4/src/lean/Init/Control/Id.lean", "def_pos": [13, 5], "def_end_pos": [13, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nf : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1)\n\u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i]", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nf : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1)\nthis :\n  \u2200 (p : Fin (size as) \u2192 \u03b2 \u2192 Prop),\n    (\u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f i as[i])) \u2192\n      SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])\n        (mapIdxM as f)\n\u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i]"}, {"tactic": "simp [SatisfiesM_Id_eq] at this", "annotated_tactic": ["simp [<a>SatisfiesM_Id_eq</a>] at this", [{"full_name": "SatisfiesM_Id_eq", "def_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "def_pos": [184, 17], "def_end_pos": [184, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nf : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1)\nthis :\n  \u2200 (p : Fin (size as) \u2192 \u03b2 \u2192 Prop),\n    (\u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f i as[i])) \u2192\n      SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])\n        (mapIdxM as f)\n\u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i]", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nf : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1)\nthis :\n  \u2200 (p : Fin (size as) \u2192 \u03b2 \u2192 Prop),\n    (\u2200 (i : Fin (size as)), motive i.val \u2192 p i (f i as[i.val]) \u2227 motive (i.val + 1)) \u2192\n      motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdxM as f)[i]\n\u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i]"}, {"tactic": "exact this _ hs", "annotated_tactic": ["exact this _ hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nf : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1)\nthis :\n  \u2200 (p : Fin (size as) \u2192 \u03b2 \u2192 Prop),\n    (\u2200 (i : Fin (size as)), motive i.val \u2192 p i (f i as[i.val]) \u2227 motive (i.val + 1)) \u2192\n      motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdxM as f)[i]\n\u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i]", "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": [173, 1], "end": [179, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_double_succ", "start": [857, 1], "end": [858, 34], "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": [856, 1], "end": [872, 46], "traced_tactics": [{"tactic": "let f := @zpowGroupHom G _ n", "annotated_tactic": ["let f := @<a>zpowGroupHom</a> G _ n", [{"full_name": "zpowGroupHom", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [407, 5], "def_end_pos": [407, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc", "state_after": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc"}, {"tactic": "have hf : Continuous f := continuous_zpow n", "annotated_tactic": ["have hf : <a>Continuous</a> f := <a>continuous_zpow</a> n", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "continuous_zpow", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc", "state_after": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc"}, {"tactic": "haveI : (\u03bc.map f).IsHaarMeasure :=\n  isHaarMeasure_map \u03bc f hf (RootableBy.surjective_pow G \u2124 hn) (by simp)", "annotated_tactic": ["haveI : (\u03bc.map f).<a>IsHaarMeasure</a> :=\n        <a>isHaarMeasure_map</a> \u03bc f hf (<a>RootableBy.surjective_pow</a> G \u2124 hn) (by simp)", [{"full_name": "MeasureTheory.Measure.IsHaarMeasure", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [733, 7], "def_end_pos": [733, 20]}, {"full_name": "MeasureTheory.Measure.isHaarMeasure_map", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [793, 9], "def_end_pos": [793, 26]}, {"full_name": "RootableBy.surjective_pow", "def_path": "Mathlib/GroupTheory/Divisible.lean", "def_pos": [276, 9], "def_end_pos": [276, 34]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc", "state_after": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc"}, {"tactic": "obtain \u27e8C, -, -, hC\u27e9 := isHaarMeasure_eq_smul_isHaarMeasure (\u03bc.map f) \u03bc", "annotated_tactic": ["obtain \u27e8C, -, -, hC\u27e9 := <a>isHaarMeasure_eq_smul_isHaarMeasure</a> (\u03bc.map f) \u03bc", [{"full_name": "MeasureTheory.Measure.isHaarMeasure_eq_smul_isHaarMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [725, 9], "def_end_pos": [725, 44]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc"}, {"tactic": "suffices C = 1 by rwa [this, one_smul] at hC", "annotated_tactic": ["suffices C = 1 by rwa [this, <a>one_smul</a>] at hC", [{"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 C = 1"}, {"tactic": "have h_univ : (\u03bc.map f) univ = \u03bc univ := by\n  rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ,\n    preimage_univ]", "annotated_tactic": ["have h_univ : (\u03bc.map f) <a>univ</a> = \u03bc <a>univ</a> := by\n        rw [<a>map_apply_of_aemeasurable</a> hf.measurable.aemeasurable <a>MeasurableSet.univ</a>,\n          <a>preimage_univ</a>]", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"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]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 C = 1", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\n\u22a2 C = 1"}, {"tactic": "have h\u03bc\u2080 : \u03bc univ \u2260 0 := IsOpenPosMeasure.open_pos univ isOpen_univ univ_nonempty", "annotated_tactic": ["have h\u03bc\u2080 : \u03bc <a>univ</a> \u2260 0 := <a>IsOpenPosMeasure.open_pos</a> <a>univ</a> <a>isOpen_univ</a> <a>univ_nonempty</a>", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.IsOpenPosMeasure.open_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "isOpen_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [123, 17], "def_end_pos": [123, 28]}, {"full_name": "Set.univ_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [535, 9], "def_end_pos": [535, 22]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\n\u22a2 C = 1", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\nh\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0\n\u22a2 C = 1"}, {"tactic": "have h\u03bc\u2081 : \u03bc univ \u2260 \u221e := CompactSpace.isFiniteMeasure.measure_univ_lt_top.ne", "annotated_tactic": ["have h\u03bc\u2081 : \u03bc <a>univ</a> \u2260 \u221e := CompactSpace.isFiniteMeasure.measure_univ_lt_top.ne", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\nh\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0\n\u22a2 C = 1", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\nh\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc univ \u2260 \u22a4\n\u22a2 C = 1"}, {"tactic": "rwa [hC, smul_apply, Algebra.id.smul_eq_mul, mul_comm, \u2190 ENNReal.eq_div_iff h\u03bc\u2080 h\u03bc\u2081,\n  ENNReal.div_self h\u03bc\u2080 h\u03bc\u2081] at h_univ", "annotated_tactic": ["rwa [hC, <a>smul_apply</a>, <a>Algebra.id.smul_eq_mul</a>, <a>mul_comm</a>, \u2190 <a>ENNReal.eq_div_iff</a> h\u03bc\u2080 h\u03bc\u2081,\n        <a>ENNReal.div_self</a> h\u03bc\u2080 h\u03bc\u2081] at h_univ", [{"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 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]}, {"full_name": "ENNReal.eq_div_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1754, 9], "def_end_pos": [1754, 19]}, {"full_name": "ENNReal.div_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1746, 19], "def_end_pos": [1746, 27]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\nh\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc univ \u2260 \u22a4\n\u22a2 C = 1", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\n\u22a2 Filter.Tendsto (\u2191f) (Filter.cocompact G) (Filter.cocompact G)", "state_after": "no goals"}, {"tactic": "rwa [this, one_smul] at hC", "annotated_tactic": ["rwa [this, <a>one_smul</a>] at hC", [{"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis\u271d : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nthis : C = 1\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc", "state_after": "no goals"}, {"tactic": "rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ,\n  preimage_univ]", "annotated_tactic": ["rw [<a>map_apply_of_aemeasurable</a> hf.measurable.aemeasurable <a>MeasurableSet.univ</a>,\n          <a>preimage_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]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "ContinuousOn.integral_sub_linear_isLittleO_ae", "start": [1068, 1], "end": [1076, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.integral_mul_eq_integral", "start": [288, 1], "end": [289, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "full_name": "MeasureTheory.withDensity\u1d65_sub'", "start": [107, 1], "end": [109, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "full_name": "Std.Range.forIn'_eq_forIn_range'", "start": [43, 1], "end": [86, 64], "traced_tactics": [{"tactic": "let \u27e8start, stop, step\u27e9 := r", "annotated_tactic": ["let \u27e8start, stop, step\u27e9 := r", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : (a : Nat) \u2192 a \u2208 r \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn' r init f =\n    forIn\n      (List.pmap Subtype.mk (List.range' r.start (numElems r) r.step)\n        (_ : \u2200 (x : Nat), x \u2208 List.range' r.start (numElems r) r.step \u2192 x \u2208 r))\n      init fun x =>\n      match (motive := { x // x \u2208 r } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn' { start := start, stop := stop, step := step } init f =\n    forIn\n      (List.pmap Subtype.mk\n        (List.range' { start := start, stop := stop, step := step }.start\n          (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)\n        (_ :\n          \u2200 (x : Nat),\n            x \u2208\n                List.range' { start := start, stop := stop, step := step }.start\n                  (numElems { start := start, stop := stop, step := step })\n                  { start := start, stop := stop, step := step }.step \u2192\n              x \u2208 { start := start, stop := stop, step := step }))\n      init fun x =>\n      match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h"}, {"tactic": "let L := List.range' start (numElems \u27e8start, stop, step\u27e9) step", "annotated_tactic": ["let L := <a>List.range'</a> start (<a>numElems</a> \u27e8start, stop, step\u27e9) step", [{"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "Std.Range.numElems", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [7, 5], "def_end_pos": [7, 13]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn' { start := start, stop := stop, step := step } init f =\n    forIn\n      (List.pmap Subtype.mk\n        (List.range' { start := start, stop := stop, step := step }.start\n          (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)\n        (_ :\n          \u2200 (x : Nat),\n            x \u2208\n                List.range' { start := start, stop := stop, step := step }.start\n                  (numElems { start := start, stop := stop, step := step })\n                  { start := start, stop := stop, step := step }.step \u2192\n              x \u2208 { start := start, stop := stop, step := step }))\n      init fun x =>\n      match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\n\u22a2 forIn' { start := start, stop := stop, step := step } init f =\n    forIn\n      (List.pmap Subtype.mk\n        (List.range' { start := start, stop := stop, step := step }.start\n          (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)\n        (_ :\n          \u2200 (x : Nat),\n            x \u2208\n                List.range' { start := start, stop := stop, step := step }.start\n                  (numElems { start := start, stop := stop, step := step })\n                  { start := start, stop := stop, step := step }.step \u2192\n              x \u2208 { start := start, stop := stop, step := step }))\n      init fun x =>\n      match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h"}, {"tactic": "let f' : { a // start \u2264 a \u2227 a < stop } \u2192 _ := fun \u27e8a, h\u27e9 => f a h", "annotated_tactic": ["let f' : { a // start \u2264 a \u2227 a < stop } \u2192 _ := fun \u27e8a, h\u27e9 => f a h", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\n\u22a2 forIn' { start := start, stop := stop, step := step } init f =\n    forIn\n      (List.pmap Subtype.mk\n        (List.range' { start := start, stop := stop, step := step }.start\n          (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)\n        (_ :\n          \u2200 (x : Nat),\n            x \u2208\n                List.range' { start := start, stop := stop, step := step }.start\n                  (numElems { start := start, stop := stop, step := step })\n                  { start := start, stop := stop, step := step }.step \u2192\n              x \u2208 { start := start, stop := stop, step := step }))\n      init fun x =>\n      match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\n\u22a2 forIn' { start := start, stop := stop, step := step } init f =\n    forIn\n      (List.pmap Subtype.mk\n        (List.range' { start := start, stop := stop, step := step }.start\n          (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)\n        (_ :\n          \u2200 (x : Nat),\n            x \u2208\n                List.range' { start := start, stop := stop, step := step }.start\n                  (numElems { start := start, stop := stop, step := step })\n                  { start := start, stop := stop, step := step }.step \u2192\n              x \u2208 { start := start, stop := stop, step := step }))\n      init fun x =>\n      match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h"}, {"tactic": "suffices \u2200 H, forIn' [start:stop:step] init f = forIn (L.pmap Subtype.mk H) init f' from this _", "annotated_tactic": ["suffices \u2200 H, <a>forIn'</a> [start:stop:step] init f = <a>forIn</a> (L.pmap <a>Subtype.mk</a> H) init f' from this _", [{"full_name": "ForIn'.forIn'", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [264, 3], "def_end_pos": [264, 9]}, {"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": "Subtype.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 19], "def_end_pos": [560, 46]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\n\u22a2 forIn' { start := start, stop := stop, step := step } init f =\n    forIn\n      (List.pmap Subtype.mk\n        (List.range' { start := start, stop := stop, step := step }.start\n          (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)\n        (_ :\n          \u2200 (x : Nat),\n            x \u2208\n                List.range' { start := start, stop := stop, step := step }.start\n                  (numElems { start := start, stop := stop, step := step })\n                  { start := start, stop := stop, step := step }.step \u2192\n              x \u2208 { start := start, stop := stop, step := step }))\n      init fun x =>\n      match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a h", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop),\n    forIn' { start := start, stop := stop, step := step } init f = forIn (List.pmap Subtype.mk L H) init f'"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop),\n    forIn' { start := start, stop := stop, step := step } init f = forIn (List.pmap Subtype.mk L H) init f'", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 forIn' { start := start, stop := stop, step := step } init f = forIn (List.pmap Subtype.mk L H) init f'"}, {"tactic": "dsimp only [forIn', Range.forIn']", "annotated_tactic": ["dsimp only [<a>forIn'</a>, <a>Range.forIn'</a>]", [{"full_name": "ForIn'.forIn'", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [264, 3], "def_end_pos": [264, 9]}, {"full_name": "Std.Range.forIn'", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [38, 25], "def_end_pos": [38, 31]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 forIn' { start := start, stop := stop, step := step } init f = forIn (List.pmap Subtype.mk L H) init f'", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn (List.pmap Subtype.mk (List.range' start (numElems { start := start, stop := stop, step := step }) step) H)\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)"}, {"tactic": "simp [numElems, Nat.not_le.2 h]", "annotated_tactic": ["simp [<a>numElems</a>, <a>Nat.not_le</a>.2 h]", [{"full_name": "Std.Range.numElems", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [7, 5], "def_end_pos": [7, 13]}, {"full_name": "Nat.not_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [147, 27], "def_end_pos": [147, 33]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn (List.pmap Subtype.mk (List.range' start (numElems { start := start, stop := stop, step := step }) step) H)\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start (if step = 0 then stop else (stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start (if step = 0 then stop else (stop - start + step - 1) / step) step \u2192\n              (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start (if step = 0 then stop else (stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start (if step = 0 then stop else (stop - start + step - 1) / step) step \u2192\n              (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nh\u271d : step = 0\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start stop step)\n        (_ : \u2200 (a : Nat), a \u2208 List.range' start stop step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)\n\ncase inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nh\u271d : \u00acstep = 0\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)"}, {"tactic": "subst step", "annotated_tactic": ["subst step", []], "state_before": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nh\u271d : step = 0\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start stop step)\n        (_ : \u2200 (a : Nat), a \u2208 List.range' start stop step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 forIn'.loop start stop 0 f stop start\n      (_ : { start := start, stop := stop, step := 0 }.start \u2264 { start := start, stop := stop, step := 0 }.start) init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start stop 0)\n        (_ : \u2200 (a : Nat), a \u2208 List.range' start stop 0 \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)"}, {"tactic": "suffices \u2200 n H init,\n    forIn'.loop start stop 0 f n start (Nat.le_refl _) init =\n    forIn ((List.range' start n 0).pmap Subtype.mk H) init f' from this _ ..", "annotated_tactic": ["suffices \u2200 n H init,\n          <a>forIn'.loop</a> start stop 0 f n start (<a>Nat.le_refl</a> _) init =\n          <a>forIn</a> ((<a>List.range'</a> start n 0).<a>pmap</a> <a>Subtype.mk</a> H) init f' from this _ ..", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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": "ForIn.forIn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [248, 3], "def_end_pos": [248, 8]}, {"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "List.pmap", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [754, 13], "def_end_pos": [754, 17]}, {"full_name": "Subtype.mk", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [560, 19], "def_end_pos": [560, 46]}]], "state_before": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 forIn'.loop start stop 0 f stop start\n      (_ : { start := start, stop := stop, step := 0 }.start \u2264 { start := start, stop := stop, step := 0 }.start) init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start stop 0)\n        (_ : \u2200 (a : Nat), a \u2208 List.range' start stop 0 \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 \u2200 (n : Nat) (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 \u2200 (n : Nat) (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'", "state_after": "case inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nn : Nat\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'"}, {"tactic": "induction n with (intro H init; unfold forIn'.loop; simp [*])\n| succ n ih => simp [ih (List.forall_mem_cons.1 H).2]; rfl", "annotated_tactic": ["induction n with (intro H init; unfold <a>forIn'.loop</a>; simp [*])\n      | <a>succ</a> n ih => simp [ih (<a>List.forall_mem_cons</a>.1 H).2]; rfl", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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": "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 inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nn : Nat\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'", "state_after": "no goals"}, {"tactic": "intro H init", "annotated_tactic": ["intro H init", []], "state_before": "case inl.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f Nat.zero start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f'", "state_after": "case inl.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nH : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 forIn'.loop start stop 0 f Nat.zero start (_ : start \u2264 start) init =\n    forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f'"}, {"tactic": "unfold forIn'.loop", "annotated_tactic": ["unfold <a>forIn'.loop</a>", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}]], "state_before": "case inl.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nH : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 forIn'.loop start stop 0 f Nat.zero start (_ : start \u2264 start) init =\n    forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f'", "state_after": "case inl.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nH : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 (if hu : start < stop then\n      match Nat.zero with\n      | 0 => pure init\n      | Nat.succ fuel => do\n        let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init\n        match __do_lift with\n          | ForInStep.done b => pure b\n          | ForInStep.yield b => forIn'.loop start stop 0 f fuel (start + 0) (_ : start \u2264 start + 0) b\n    else pure init) =\n    forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f'"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case inl.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nH : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 (if hu : start < stop then\n      match Nat.zero with\n      | 0 => pure init\n      | Nat.succ fuel => do\n        let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init\n        match __do_lift with\n          | ForInStep.done b => pure b\n          | ForInStep.yield b => forIn'.loop start stop 0 f fuel (start + 0) (_ : start \u2264 start + 0) b\n    else pure init) =\n    forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f'", "state_after": "no goals"}, {"tactic": "simp [ih (List.forall_mem_cons.1 H).2]", "annotated_tactic": ["simp [ih (<a>List.forall_mem_cons</a>.1 H).2]", [{"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 inl.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nn : Nat\nih :\n  \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'\nH : \u2200 (a : Nat), a \u2208 List.range' start (Nat.succ n) 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 (do\n      let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn'.loop start stop 0 f n start (_ : start \u2264 start) b) =\n    do\n    let x \u2190\n      f start\n          (_ :\n            start \u2264 { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val \u2227\n              { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val < stop)\n          init\n    match x with\n      | ForInStep.done b => pure b\n      | ForInStep.yield b =>\n        forIn\n          (List.pmap Subtype.mk (List.range' start n 0)\n            (_ : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 (fun a => (fun a => start \u2264 a \u2227 a < stop) a) a))\n          b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "case inl.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nn : Nat\nih :\n  \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'\nH : \u2200 (a : Nat), a \u2208 List.range' start (Nat.succ n) 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 (do\n      let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b =>\n          forIn\n            (List.pmap Subtype.mk (List.range' start n 0)\n              (_ : \u2200 (x : Nat), x \u2208 List.range' (start + 0) n 0 \u2192 start \u2264 x \u2227 x < stop))\n            b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)) =\n    do\n    let x \u2190\n      f start\n          (_ :\n            start \u2264 { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val \u2227\n              { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val < stop)\n          init\n    match x with\n      | ForInStep.done b => pure b\n      | ForInStep.yield b =>\n        forIn\n          (List.pmap Subtype.mk (List.range' start n 0)\n            (_ : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 (fun a => (fun a => start \u2264 a \u2227 a < stop) a) a))\n          b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case inl.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop : Nat\nh : start < stop\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nn : Nat\nih :\n  \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),\n    forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =\n      forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f'\nH : \u2200 (a : Nat), a \u2208 List.range' start (Nat.succ n) 0 \u2192 start \u2264 a \u2227 a < stop\ninit : \u03b2\n\u22a2 (do\n      let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b =>\n          forIn\n            (List.pmap Subtype.mk (List.range' start n 0)\n              (_ : \u2200 (x : Nat), x \u2208 List.range' (start + 0) n 0 \u2192 start \u2264 x \u2227 x < stop))\n            b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)) =\n    do\n    let x \u2190\n      f start\n          (_ :\n            start \u2264 { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val \u2227\n              { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val < stop)\n          init\n    match x with\n      | ForInStep.done b => pure b\n      | ForInStep.yield b =>\n        forIn\n          (List.pmap Subtype.mk (List.range' start n 0)\n            (_ : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 (fun a => (fun a => start \u2264 a \u2227 a < stop) a) a))\n          b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "no goals"}, {"tactic": "next step0 =>\nhave hstep := Nat.pos_of_ne_zero step0\nsuffices \u2200 fuel l i hle H, l \u2264 fuel \u2192\n    (\u2200 j, stop \u2264 i + step * j \u2194 l \u2264 j) \u2192 \u2200 init,\n    forIn'.loop start stop step f fuel i hle init =\n    List.forIn ((List.range' i l step).pmap Subtype.mk H) init f' by\n  refine this _ _ _ _ _\n    ((numElems_le_iff hstep).2 (Nat.le_trans ?_ (Nat.le_add_left ..)))\n    (fun _ => (numElems_le_iff hstep).symm) _\n  conv => lhs; rw [\u2190 Nat.one_mul stop]\n  exact Nat.mul_le_mul_right stop hstep\nintro fuel; induction fuel with intro l i hle H h1 h2 init\n| zero => simp [forIn'.loop, Nat.le_zero.1 h1]; split <;> simp\n| succ fuel ih =>\n  cases l with\n  | zero => rw [forIn'.loop]; simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]\n  | succ l =>\n    have ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n      (List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\n        rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]\n    have := h2 0; simp at this\n    rw [forIn'.loop]; simp [List.forIn, this, ih]; rfl", "annotated_tactic": ["next step0 =>\n      have hstep := <a>Nat.pos_of_ne_zero</a> step0\n      suffices \u2200 fuel l i hle H, l \u2264 fuel \u2192\n          (\u2200 j, stop \u2264 i + step * j \u2194 l \u2264 j) \u2192 \u2200 init,\n          <a>forIn'.loop</a> start stop step f fuel i hle init =\n          <a>List.forIn</a> ((<a>List.range'</a> i l step).<a>pmap</a> <a>Subtype.mk</a> H) init f' by\n        refine this _ _ _ _ _\n          ((<a>numElems_le_iff</a> hstep).2 (<a>Nat.le_trans</a> ?_ (<a>Nat.le_add_left</a> ..)))\n          (fun _ => (<a>numElems_le_iff</a> hstep).<a>symm</a>) _\n        conv => lhs; rw [\u2190 <a>Nat.one_mul</a> stop]\n        exact <a>Nat.mul_le_mul_right</a> stop hstep\n      intro fuel; induction fuel with intro l i hle H h1 h2 init\n      | <a>zero</a> => simp [<a>forIn'.loop</a>, <a>Nat.le_zero</a>.1 h1]; split <;> simp\n      | <a>succ</a> fuel ih =>\n        cases l with\n        | <a>zero</a> => rw [<a>forIn'.loop</a>]; simp [<a>Nat.not_lt</a>.2 <| by simpa using (h2 0).2 (<a>Nat.le_refl</a> _)]\n        | <a>succ</a> l =>\n          have ih := ih _ _ (<a>Nat.le_trans</a> hle (<a>Nat.le_add_right</a> ..))\n            (<a>List.forall_mem_cons</a>.1 H).2 (<a>Nat.le_of_succ_le_succ</a> h1) fun i => by\n              rw [<a>Nat.add_right_comm</a>, <a>Nat.add_assoc</a>, \u2190 <a>Nat.mul_succ</a>, h2, <a>Nat.succ_le_succ_iff</a>]\n          have := h2 0; simp at this\n          rw [<a>forIn'.loop</a>]; simp [<a>List.forIn</a>, this, ih]; rfl", [{"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": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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]}, {"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "List.pmap", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [754, 13], "def_end_pos": [754, 17]}, {"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": "_private.\u00ablake-packages\u00bb.std.Std.Data.Range.Lemmas.0.Std.Range.numElems_le_iff", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 32]}, {"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.Range.Lemmas.0.Std.Range.numElems_le_iff", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 32]}, {"full_name": "Iff.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [671, 9], "def_end_pos": [671, 17]}, {"full_name": "Nat.one_mul", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [181, 27], "def_end_pos": [181, 34]}, {"full_name": "Nat.mul_le_mul_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}, {"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"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.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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": "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": "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": "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.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.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.succ_le_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 25]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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": "case inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nh\u271d : \u00acstep = 0\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "no goals"}, {"tactic": "have hstep := Nat.pos_of_ne_zero step0", "annotated_tactic": ["have hstep := <a>Nat.pos_of_ne_zero</a> step0", [{"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": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)"}, {"tactic": "suffices \u2200 fuel l i hle H, l \u2264 fuel \u2192\n    (\u2200 j, stop \u2264 i + step * j \u2194 l \u2264 j) \u2192 \u2200 init,\n    forIn'.loop start stop step f fuel i hle init =\n    List.forIn ((List.range' i l step).pmap Subtype.mk H) init f' by\n  refine this _ _ _ _ _\n    ((numElems_le_iff hstep).2 (Nat.le_trans ?_ (Nat.le_add_left ..)))\n    (fun _ => (numElems_le_iff hstep).symm) _\n  conv => lhs; rw [\u2190 Nat.one_mul stop]\n  exact Nat.mul_le_mul_right stop hstep", "annotated_tactic": ["suffices \u2200 fuel l i hle H, l \u2264 fuel \u2192\n          (\u2200 j, stop \u2264 i + step * j \u2194 l \u2264 j) \u2192 \u2200 init,\n          <a>forIn'.loop</a> start stop step f fuel i hle init =\n          <a>List.forIn</a> ((<a>List.range'</a> i l step).<a>pmap</a> <a>Subtype.mk</a> H) init f' by\n        refine this _ _ _ _ _\n          ((<a>numElems_le_iff</a> hstep).2 (<a>Nat.le_trans</a> ?_ (<a>Nat.le_add_left</a> ..)))\n          (fun _ => (<a>numElems_le_iff</a> hstep).<a>symm</a>) _\n        conv => lhs; rw [\u2190 <a>Nat.one_mul</a> stop]\n        exact <a>Nat.mul_le_mul_right</a> stop hstep", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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]}, {"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "List.pmap", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [754, 13], "def_end_pos": [754, 17]}, {"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": "_private.\u00ablake-packages\u00bb.std.Std.Data.Range.Lemmas.0.Std.Range.numElems_le_iff", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 32]}, {"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.Range.Lemmas.0.Std.Range.numElems_le_iff", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 32]}, {"full_name": "Iff.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [671, 9], "def_end_pos": [671, 17]}, {"full_name": "Nat.one_mul", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [181, 27], "def_end_pos": [181, 34]}, {"full_name": "Nat.mul_le_mul_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\n\u22a2 \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'"}, {"tactic": "intro fuel", "annotated_tactic": ["intro fuel", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\n\u22a2 \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\n\u22a2 \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'"}, {"tactic": "induction fuel with intro l i hle H h1 h2 init\n| zero => simp [forIn'.loop, Nat.le_zero.1 h1]; split <;> simp\n| succ fuel ih =>\ncases l with\n| zero => rw [forIn'.loop]; simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]\n| succ l =>\nhave ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n(List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\nrw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]\nhave := h2 0; simp at this\nrw [forIn'.loop]; simp [List.forIn, this, ih]; rfl", "annotated_tactic": ["induction fuel with intro l i hle H h1 h2 init\n      | <a>zero</a> => simp [<a>forIn'.loop</a>, <a>Nat.le_zero</a>.1 h1]; split <;> simp\n      | <a>succ</a> fuel ih =>\n        cases l with\n        | <a>zero</a> => rw [<a>forIn'.loop</a>]; simp [<a>Nat.not_lt</a>.2 <| by simpa using (h2 0).2 (<a>Nat.le_refl</a> _)]\n        | <a>succ</a> l =>\n          have ih := ih _ _ (<a>Nat.le_trans</a> hle (<a>Nat.le_add_right</a> ..))\n            (<a>List.forall_mem_cons</a>.1 H).2 (<a>Nat.le_of_succ_le_succ</a> h1) fun i => by\n              rw [<a>Nat.add_right_comm</a>, <a>Nat.add_assoc</a>, \u2190 <a>Nat.mul_succ</a>, h2, <a>Nat.succ_le_succ_iff</a>]\n          have := h2 0; simp at this\n          rw [<a>forIn'.loop</a>]; simp [<a>List.forIn</a>, this, ih]; rfl", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"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.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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": "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": "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": "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.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.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.succ_le_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 25]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\n\u22a2 \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'", "state_after": "no goals"}, {"tactic": "refine this _ _ _ _ _\n  ((numElems_le_iff hstep).2 (Nat.le_trans ?_ (Nat.le_add_left ..)))\n  (fun _ => (numElems_le_iff hstep).symm) _", "annotated_tactic": ["refine this _ _ _ _ _\n          ((<a>numElems_le_iff</a> hstep).2 (<a>Nat.le_trans</a> ?_ (<a>Nat.le_add_left</a> ..)))\n          (fun _ => (<a>numElems_le_iff</a> hstep).<a>symm</a>) _", [{"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.Range.Lemmas.0.Std.Range.numElems_le_iff", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 32]}, {"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.Range.Lemmas.0.Std.Range.numElems_le_iff", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [23, 17], "def_end_pos": [23, 32]}, {"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": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nthis :\n  \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn\n      (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)\n        (_ :\n          \u2200 (a : Nat),\n            a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nthis :\n  \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\n\u22a2 stop \u2264 step * stop"}, {"tactic": "conv => lhs; rw [\u2190 Nat.one_mul stop]", "annotated_tactic": ["conv => lhs; rw [\u2190 <a>Nat.one_mul</a> stop]", [{"full_name": "Nat.one_mul", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [181, 27], "def_end_pos": [181, 34]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nthis :\n  \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\n\u22a2 stop \u2264 step * stop", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nthis :\n  \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\n\u22a2 1 * stop \u2264 step * stop"}, {"tactic": "exact Nat.mul_le_mul_right stop hstep", "annotated_tactic": ["exact <a>Nat.mul_le_mul_right</a> stop hstep", [{"full_name": "Nat.mul_le_mul_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nthis :\n  \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\n\u22a2 1 * stop \u2264 step * stop", "state_after": "no goals"}, {"tactic": "simp [forIn'.loop, Nat.le_zero.1 h1]", "annotated_tactic": ["simp [<a>forIn'.loop</a>, <a>Nat.le_zero</a>.1 h1]", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"full_name": "Nat.le_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 16]}]], "state_before": "case zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nl i : Nat\nhle : start \u2264 i\nH : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop\nh1 : l \u2264 Nat.zero\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j\ninit : \u03b2\n\u22a2 forIn'.loop start stop step f Nat.zero i hle init = List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'", "state_after": "case zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nl i : Nat\nhle : start \u2264 i\nH : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop\nh1 : l \u2264 Nat.zero\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j\ninit : \u03b2\n\u22a2 (if hu : i < stop then pure init else pure init) = pure init"}, {"tactic": "split <;> simp", "annotated_tactic": ["split <;> simp", []], "state_before": "case zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nl i : Nat\nhle : start \u2264 i\nH : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop\nh1 : l \u2264 Nat.zero\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j\ninit : \u03b2\n\u22a2 (if hu : i < stop then pure init else pure init) = pure init", "state_after": "no goals"}, {"tactic": "cases l with\n| zero => rw [forIn'.loop]; simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]\n| succ l =>\n  have ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n    (List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\n      rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]\n  have := h2 0; simp at this\n  rw [forIn'.loop]; simp [List.forIn, this, ih]; rfl", "annotated_tactic": ["cases l with\n        | <a>zero</a> => rw [<a>forIn'.loop</a>]; simp [<a>Nat.not_lt</a>.2 <| by simpa using (h2 0).2 (<a>Nat.le_refl</a> _)]\n        | <a>succ</a> l =>\n          have ih := ih _ _ (<a>Nat.le_trans</a> hle (<a>Nat.le_add_right</a> ..))\n            (<a>List.forall_mem_cons</a>.1 H).2 (<a>Nat.le_of_succ_le_succ</a> h1) fun i => by\n              rw [<a>Nat.add_right_comm</a>, <a>Nat.add_assoc</a>, \u2190 <a>Nat.mul_succ</a>, h2, <a>Nat.succ_le_succ_iff</a>]\n          have := h2 0; simp at this\n          rw [<a>forIn'.loop</a>]; simp [<a>List.forIn</a>, this, ih]; rfl", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"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.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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": "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": "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": "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.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.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.succ_le_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 25]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"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": "case succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\nl i : Nat\nhle : start \u2264 i\nH : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop\nh1 : l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j\ninit : \u03b2\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'", "state_after": "no goals"}, {"tactic": "rw [forIn'.loop]", "annotated_tactic": ["rw [<a>forIn'.loop</a>]", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}]], "state_before": "case succ.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nH : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.zero \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i Nat.zero step) H) init f'", "state_after": "case succ.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nH : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.zero \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j\n\u22a2 (if hu : i < stop then\n      match Nat.succ fuel with\n      | 0 => pure init\n      | Nat.succ fuel => do\n        let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init\n        match __do_lift with\n          | ForInStep.done b => pure b\n          | ForInStep.yield b => forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) b\n    else pure init) =\n    List.forIn (List.pmap Subtype.mk (List.range' i Nat.zero step) H) init f'"}, {"tactic": "simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]", "annotated_tactic": ["simp [<a>Nat.not_lt</a>.2 <| by simpa using (h2 0).2 (<a>Nat.le_refl</a> _)]", [{"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_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}]], "state_before": "case succ.zero\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nH : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.zero \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j\n\u22a2 (if hu : i < stop then\n      match Nat.succ fuel with\n      | 0 => pure init\n      | Nat.succ fuel => do\n        let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init\n        match __do_lift with\n          | ForInStep.done b => pure b\n          | ForInStep.yield b => forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) b\n    else pure init) =\n    List.forIn (List.pmap Subtype.mk (List.range' i Nat.zero step) H) init f'", "state_after": "no goals"}, {"tactic": "simpa using (h2 0).2 (Nat.le_refl _)", "annotated_tactic": ["simpa using (h2 0).2 (<a>Nat.le_refl</a> _)", [{"full_name": "Nat.le_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nH : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.zero \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j\n\u22a2 ?m.22038 \u2264 ?m.22037", "state_after": "no goals"}, {"tactic": "have ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n  (List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\n    rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]", "annotated_tactic": ["have ih := ih _ _ (<a>Nat.le_trans</a> hle (<a>Nat.le_add_right</a> ..))\n            (<a>List.forall_mem_cons</a>.1 H).2 (<a>Nat.le_of_succ_le_succ</a> h1) fun i => by\n              rw [<a>Nat.add_right_comm</a>, <a>Nat.add_assoc</a>, \u2190 <a>Nat.mul_succ</a>, h2, <a>Nat.succ_le_succ_iff</a>]", [{"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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": "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": "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": "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.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.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.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 succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'", "state_after": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'"}, {"tactic": "have := h2 0", "annotated_tactic": ["have := h2 0", []], "state_before": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'", "state_after": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : stop \u2264 i + step * 0 \u2194 Nat.succ l \u2264 0\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'"}, {"tactic": "simp at this", "annotated_tactic": ["simp at this", []], "state_before": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : stop \u2264 i + step * 0 \u2194 Nat.succ l \u2264 0\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'", "state_after": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : i < stop\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'"}, {"tactic": "rw [forIn'.loop]", "annotated_tactic": ["rw [<a>forIn'.loop</a>]", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}]], "state_before": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : i < stop\n\u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'", "state_after": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : i < stop\n\u22a2 (if hu : i < stop then\n      match Nat.succ fuel with\n      | 0 => pure init\n      | Nat.succ fuel => do\n        let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init\n        match __do_lift with\n          | ForInStep.done b => pure b\n          | ForInStep.yield b => forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) b\n    else pure init) =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'"}, {"tactic": "simp [List.forIn, this, ih]", "annotated_tactic": ["simp [<a>List.forIn</a>, this, ih]", [{"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": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : i < stop\n\u22a2 (if hu : i < stop then\n      match Nat.succ fuel with\n      | 0 => pure init\n      | Nat.succ fuel => do\n        let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init\n        match __do_lift with\n          | ForInStep.done b => pure b\n          | ForInStep.yield b => forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) b\n    else pure init) =\n    List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f'", "state_after": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : i < stop\n\u22a2 (do\n      let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b =>\n          List.forIn.loop (fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop))\n            (List.pmap Subtype.mk (List.range' (i + step) l step)\n              (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n            b) =\n    List.forIn.loop (fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop))\n      ({ val := i, property := (_ : start \u2264 i \u2227 i < stop) } ::\n        List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n      init"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.succ\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih\u271d :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni : Nat\nhle : start \u2264 i\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j\nih :\n  \u2200 (init : \u03b2),\n    forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =\n      List.forIn\n        (List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n        init f'\nthis : i < stop\n\u22a2 (do\n      let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b =>\n          List.forIn.loop (fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop))\n            (List.pmap Subtype.mk (List.range' (i + step) l step)\n              (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n            b) =\n    List.forIn.loop (fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop))\n      ({ val := i, property := (_ : start \u2264 i \u2227 i < stop) } ::\n        List.pmap Subtype.mk (List.range' (i + step) l step)\n          (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))\n      init", "state_after": "no goals"}, {"tactic": "rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]", "annotated_tactic": ["rw [<a>Nat.add_right_comm</a>, <a>Nat.add_assoc</a>, \u2190 <a>Nat.mul_succ</a>, h2, <a>Nat.succ_le_succ_iff</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]}, {"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.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.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": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit\u271d : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : start < stop\nstep0 : \u00acstep = 0\nhstep : 0 < step\nfuel : Nat\nih :\n  \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),\n    l \u2264 fuel \u2192\n      (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192\n        \u2200 (init : \u03b2),\n          forIn'.loop start stop step f fuel i hle init =\n            List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f'\ni\u271d : Nat\nhle : start \u2264 i\u271d\ninit : \u03b2\nl : Nat\nH : \u2200 (a : Nat), a \u2208 List.range' i\u271d (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop\nh1 : Nat.succ l \u2264 Nat.succ fuel\nh2 : \u2200 (j : Nat), stop \u2264 i\u271d + step * j \u2194 Nat.succ l \u2264 j\ni : Nat\n\u22a2 stop \u2264 i\u271d + step + step * i \u2194 l \u2264 i", "state_after": "no goals"}, {"tactic": "simp [List.range', h, numElems_stop_le_start \u27e8start, stop, step\u27e9 (Nat.not_lt.1 h)]", "annotated_tactic": ["simp [<a>List.range'</a>, h, <a>numElems_stop_le_start</a> \u27e8start, stop, step\u27e9 (<a>Nat.not_lt</a>.1 h)]", [{"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "Std.Range.numElems_stop_le_start", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [14, 9], "def_end_pos": [14, 31]}, {"full_name": "Nat.not_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [150, 27], "def_end_pos": [150, 33]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : \u00acstart < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    forIn (List.pmap Subtype.mk (List.range' start (numElems { start := start, stop := stop, step := step }) step) H)\n      init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)", "state_after": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : \u00acstart < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    pure init"}, {"tactic": "cases stop <;> unfold forIn'.loop <;> simp [List.forIn', h]", "annotated_tactic": ["cases stop <;> unfold <a>forIn'.loop</a> <;> simp [<a>List.forIn'</a>, h]", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"full_name": "List.forIn'", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Control.lean", "def_pos": [160, 25], "def_end_pos": [160, 31]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nstart stop step : Nat\nf : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step\nf' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=\n  fun x =>\n    match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n    | { val := a, property := h } => f a h\nH : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop\nh : \u00acstart < stop\n\u22a2 forIn'.loop start stop step f stop start\n      (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)\n      init =\n    pure init", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/OneDim.lean", "full_name": "Real.tendsto_Icc_vitaliFamily_right", "start": [33, 1], "end": [41, 44], "traced_tactics": [{"tactic": "refine' (VitaliFamily.tendsto_filterAt_iff _).2 \u27e8_, _\u27e9", "annotated_tactic": ["refine' (<a>VitaliFamily.tendsto_filterAt_iff</a> _).2 \u27e8_, _\u27e9", [{"full_name": "VitaliFamily.tendsto_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [261, 9], "def_end_pos": [261, 29]}]], "state_before": "x : \u211d\n\u22a2 Tendsto (fun y => Icc x y) (\ud835\udcdd[Ioi x] x) (VitaliFamily.filterAt (vitaliFamily volume 1) x)", "state_after": "case refine'_1\nx : \u211d\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) x\n\ncase refine'_2\nx : \u211d\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with y hy using <a>Icc_mem_vitaliFamily_at_right</a> hy", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Real.Icc_mem_vitaliFamily_at_right", "def_path": "Mathlib/MeasureTheory/Covering/OneDim.lean", "def_pos": [26, 9], "def_end_pos": [26, 38]}]], "state_before": "case refine'_1\nx : \u211d\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) x", "state_after": "no goals"}, {"tactic": "intro \u03b5 \u03b5pos", "annotated_tactic": ["intro \u03b5 \u03b5pos", []], "state_before": "case refine'_2\nx : \u211d\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5", "state_after": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5"}, {"tactic": "have : x \u2208 Ico x (x + \u03b5) := \u27e8le_refl _, by linarith\u27e9", "annotated_tactic": ["have : x \u2208 <a>Ico</a> x (x + \u03b5) := \u27e8<a>le_refl</a> _, by linarith\u27e9", [{"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5", "state_after": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5"}, {"tactic": "filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy", "annotated_tactic": ["filter_upwards [<a>Icc_mem_nhdsWithin_Ioi</a> this] with y hy", [{"full_name": "Icc_mem_nhdsWithin_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 31]}]], "state_before": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5", "state_after": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\ny : \u211d\nhy : y \u2208 Icc x (x + \u03b5)\n\u22a2 Icc x y \u2286 Metric.closedBall x \u03b5"}, {"tactic": "rw [closedBall_eq_Icc]", "annotated_tactic": ["rw [<a>closedBall_eq_Icc</a>]", [{"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}]], "state_before": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\ny : \u211d\nhy : y \u2208 Icc x (x + \u03b5)\n\u22a2 Icc x y \u2286 Metric.closedBall x \u03b5", "state_after": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\ny : \u211d\nhy : y \u2208 Icc x (x + \u03b5)\n\u22a2 Icc x y \u2286 Icc (x - \u03b5) (x + \u03b5)"}, {"tactic": "exact Icc_subset_Icc (by linarith) hy.2", "annotated_tactic": ["exact <a>Icc_subset_Icc</a> (by linarith) hy.2", [{"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}]], "state_before": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\ny : \u211d\nhy : y \u2208 Icc x (x + \u03b5)\n\u22a2 Icc x y \u2286 Icc (x - \u03b5) (x + \u03b5)", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "x \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 x < x + \u03b5", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "x \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ico x (x + \u03b5)\ny : \u211d\nhy : y \u2208 Icc x (x + \u03b5)\n\u22a2 x - \u03b5 \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Functor.lean", "full_name": "Finset.map_traverse", "start": [213, 1], "end": [217, 53], "traced_tactics": [{"tactic": "unfold traverse", "annotated_tactic": ["unfold <a>traverse</a>", [{"full_name": "Finset.traverse", "def_path": "Mathlib/Data/Finset/Functor.lean", "def_pos": [195, 5], "def_end_pos": [195, 13]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 Functor.map h <$> traverse g s = traverse (Functor.map h \u2218 g) s", "state_after": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =\n    Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val"}, {"tactic": "simp only [map_comp_coe, functor_norm]", "annotated_tactic": ["simp only [<a>map_comp_coe</a>, functor_norm]", [{"full_name": "Finset.map_comp_coe", "def_path": "Mathlib/Data/Finset/Functor.lean", "def_pos": [208, 9], "def_end_pos": [208, 21]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =\n    Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val", "state_after": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 (Multiset.toFinset \u2218 Functor.map h) <$> Multiset.traverse g s.val =\n    Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val"}, {"tactic": "rw [LawfulFunctor.comp_map, Multiset.map_traverse]", "annotated_tactic": ["rw [<a>LawfulFunctor.comp_map</a>, <a>Multiset.map_traverse</a>]", [{"full_name": "LawfulFunctor.comp_map", "def_path": "lake-packages/lean4/src/lean/Init/Control/Lawful.lean", "def_pos": [19, 3], "def_end_pos": [19, 11]}, {"full_name": "Multiset.map_traverse", "def_path": "Mathlib/Data/Multiset/Functor.lean", "def_pos": [117, 9], "def_end_pos": [117, 21]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 (Multiset.toFinset \u2218 Functor.map h) <$> Multiset.traverse g s.val =\n    Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val", "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.neg_lt_sub_right_of_lt_add", "start": [1116, 11], "end": [1117, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.emod_lt_of_pos", "start": [385, 1], "end": [388, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.data_join", "start": [493, 9], "end": [494, 15], "traced_tactics": [{"tactic": "rw [join_eq]", "annotated_tactic": ["rw [<a>join_eq</a>]", [{"full_name": "String.join_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [488, 9], "def_end_pos": [488, 16]}]], "state_before": "ss : List String\n\u22a2 (join ss).data = List.join (List.map data ss)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.zero_testAgainstNN_apply", "start": [354, 1], "end": [355, 91], "traced_tactics": [{"tactic": "simp only [testAgainstNN, toMeasure_zero, lintegral_zero_measure, ENNReal.zero_toNNReal]", "annotated_tactic": ["simp only [<a>testAgainstNN</a>, <a>toMeasure_zero</a>, <a>lintegral_zero_measure</a>, <a>ENNReal.zero_toNNReal</a>]", [{"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.toMeasure_zero", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [223, 9], "def_end_pos": [223, 23]}, {"full_name": "MeasureTheory.lintegral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [632, 9], "def_end_pos": [632, 31]}, {"full_name": "ENNReal.zero_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [239, 17], "def_end_pos": [239, 30]}]], "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 : TopologicalSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN 0 f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.maps_range_to", "start": [543, 1], "end": [544, 88], "traced_tactics": [{"tactic": "rw [\u2190 image_univ, maps_image_to]", "annotated_tactic": ["rw [\u2190 <a>image_univ</a>, <a>maps_image_to</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.maps_image_to", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [524, 9], "def_end_pos": [524, 22]}]], "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\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf\u271d f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng\u271d g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b1\ns : Set \u03b2\n\u22a2 MapsTo f (range g) s \u2194 MapsTo (f \u2218 g) univ s", "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_covering_ae", "start": [203, 1], "end": [395, 32], "traced_tactics": [{"tactic": "choose R hR0 hR1 hR\u03bc using this", "annotated_tactic": ["choose R hR0 hR1 hR\u03bc using this", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nthis : \u2200 (x : \u03b1), \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0"}, {"tactic": "let t' := { a \u2208 t | r a \u2264 R (c a) }", "annotated_tactic": ["let t' := { a \u2208 t | r a \u2264 R (c a) }", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0"}, {"tactic": "obtain \u27e8u, ut', u_disj, hu\u27e9 : \u2203 (u : _) (_ : u \u2286 t'),\n    u.PairwiseDisjoint B \u2227 \u2200 a \u2208 t', \u2203 b \u2208 u, (B a \u2229 B b).Nonempty \u2227 r a \u2264 2 * r b := by\n  have A : \u2200 a \u2208 t', r a \u2264 1 := by\n    intro a ha\n    apply ha.2.trans (hR1 (c a))\n  have A' : \u2200 a \u2208 t', (B a).Nonempty :=\n    fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1)\n  refine' exists_disjoint_subfamily_covering_enlargment B t' r 2 one_lt_two (fun a ha => _) 1 A A'\n  exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))", "annotated_tactic": ["obtain \u27e8u, ut', u_disj, hu\u27e9 : \u2203 (u : _) (_ : u \u2286 t'),\n      u.PairwiseDisjoint B \u2227 \u2200 a \u2208 t', \u2203 b \u2208 u, (B a \u2229 B b).<a>Nonempty</a> \u2227 r a \u2264 2 * r b := by\n    have A : \u2200 a \u2208 t', r a \u2264 1 := by\n      intro a ha\n      apply ha.2.<a>trans</a> (hR1 (c a))\n    have A' : \u2200 a \u2208 t', (B a).<a>Nonempty</a> :=\n      fun a hat' => <a>Set.Nonempty.mono</a> <a>interior_subset</a> (ht a hat'.1)\n    refine' <a>exists_disjoint_subfamily_covering_enlargment</a> B t' r 2 <a>one_lt_two</a> (fun a ha => _) 1 A A'\n    exact <a>nonempty_closedBall</a>.1 ((A' a ha).<a>mono</a> (hB a ha.1))", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.Nonempty.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}, {"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 24]}, {"full_name": "Vitali.exists_disjoint_subfamily_covering_enlargment", "def_path": "Mathlib/MeasureTheory/Covering/Vitali.lean", "def_pos": [59, 9], "def_end_pos": [59, 54]}, {"full_name": "one_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [106, 7], "def_end_pos": [106, 17]}, {"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.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0"}, {"tactic": "have ut : u \u2286 t := fun a hau => (ut' hau).1", "annotated_tactic": ["have ut : u \u2286 t := fun a hau => (ut' hau).1", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0"}, {"tactic": "have u_count : u.Countable := u_disj.countable_of_nonempty_interior fun a ha => ht a (ut ha)", "annotated_tactic": ["have u_count : u.Countable := u_disj.countable_of_nonempty_interior fun a ha => ht a (ut ha)", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0"}, {"tactic": "refine' \u27e8u, fun a hat' => (ut' hat').1, u_count, u_disj, _\u27e9", "annotated_tactic": ["refine' \u27e8u, fun a hat' => (ut' hat').1, u_count, u_disj, _\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\n\u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0"}, {"tactic": "refine' null_of_locally_null _ fun x _ => _", "annotated_tactic": ["refine' <a>null_of_locally_null</a> _ fun x _ => _", [{"full_name": "MeasureTheory.null_of_locally_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3939, 9], "def_end_pos": [3939, 29]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\n\u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0"}, {"tactic": "let v := { a \u2208 u | (B a \u2229 ball x (R x)).Nonempty }", "annotated_tactic": ["let v := { a \u2208 u | (B a \u2229 <a>ball</a> x (R x)).<a>Nonempty</a> }", [{"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]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\n\u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\n\u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0"}, {"tactic": "have vu : v \u2286 u := fun a ha => ha.1", "annotated_tactic": ["have vu : v \u2286 u := fun a ha => ha.1", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\n\u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\n\u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0"}, {"tactic": "refine' \u27e8_ \u2229 ball x (R x), inter_mem_nhdsWithin _ (ball_mem_nhds _ (hR0 _)),\n  nonpos_iff_eq_zero.mp (le_of_forall_le_of_dense fun \u03b5 \u03b5pos => _)\u27e9", "annotated_tactic": ["refine' \u27e8_ \u2229 <a>ball</a> x (R x), <a>inter_mem_nhdsWithin</a> _ (<a>ball_mem_nhds</a> _ (hR0 _)),\n    nonpos_iff_eq_zero.mp (<a>le_of_forall_le_of_dense</a> fun \u03b5 \u03b5pos => _)\u27e9", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "inter_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"full_name": "le_of_forall_le_of_dense", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1329, 9], "def_end_pos": [1329, 33]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5"}, {"tactic": "have I : (\u2211' a : v, \u03bc (B a)) < \u221e := by\n  calc\n    (\u2211' a : v, \u03bc (B a)) = \u03bc (\u22c3 a \u2208 v, B a) := by\n      rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]\n      exact u_disj.subset vu\n    _ \u2264 \u03bc (closedBall x K) := (measure_mono (iUnion\u2082_subset fun a ha => hK a (vu ha) ha.2))\n    _ < \u221e := \u03bcK", "annotated_tactic": ["have I : (\u2211' a : v, \u03bc (B a)) < \u221e := by\n    calc\n      (\u2211' a : v, \u03bc (B a)) = \u03bc (\u22c3 a \u2208 v, B a) := by\n        rw [<a>measure_biUnion</a> (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).<a>measurableSet</a>]\n        exact u_disj.subset vu\n      _ \u2264 \u03bc (<a>closedBall</a> x K) := (<a>measure_mono</a> (<a>iUnion\u2082_subset</a> fun a ha => hK a (vu ha) ha.2))\n      _ < \u221e := \u03bcK", [{"full_name": "MeasureTheory.measure_biUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [172, 9], "def_end_pos": [172, 24]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}, {"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.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.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5"}, {"tactic": "obtain \u27e8w, hw\u27e9 : \u2203 w : Finset v, (\u2211' a : { a // a \u2209 w }, \u03bc (B a)) < \u03b5 / C :=\n  haveI : 0 < \u03b5 / C := by\n    simp only [ENNReal.div_pos_iff, \u03b5pos.ne', ENNReal.coe_ne_top, Ne.def, not_false_iff,\n      and_self_iff]\n  ((tendsto_order.1 (ENNReal.tendsto_tsum_compl_atTop_zero I.ne)).2 _ this).exists", "annotated_tactic": ["obtain \u27e8w, hw\u27e9 : \u2203 w : <a>Finset</a> v, (\u2211' a : { a // a \u2209 w }, \u03bc (B a)) < \u03b5 / C :=\n    haveI : 0 < \u03b5 / C := by\n      simp only [<a>ENNReal.div_pos_iff</a>, \u03b5pos.ne', <a>ENNReal.coe_ne_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>,\n        <a>and_self_iff</a>]\n    ((<a>tendsto_order</a>.1 (<a>ENNReal.tendsto_tsum_compl_atTop_zero</a> I.ne)).2 _ 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": "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]}, {"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": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "ENNReal.tendsto_tsum_compl_atTop_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [962, 9], "def_end_pos": [962, 38]}, {"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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5"}, {"tactic": "have M : (s \\ \u22c3 a \u2208 u, B a) \u2229\n    ball x (R x) \u2286 \u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a) := by\n  intro z hz\n  set k := \u22c3 (a : v) (_ : a \u2208 w), B a\n  have k_closed : IsClosed k := isClosed_biUnion_finset fun i _ => h't _ (ut (vu i.2))\n  have z_notmem_k : z \u2209 k := by\n    simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,\n      SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]\n    intro b hbv _ h'z\n    have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n      mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)\n    simpa only [diff_inter_self]\n  have : ball x (R x) \\ k \u2208 \ud835\udcdd z := by\n    apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _\n    exact (mem_diff _).2 \u27e8mem_of_mem_inter_right hz, z_notmem_k\u27e9\n  obtain \u27e8d, dpos, hd\u27e9 : \u2203 d, 0 < d \u2227 closedBall z d \u2286 ball x (R x) \\ k :=\n    nhds_basis_closedBall.mem_iff.1 this\n  obtain \u27e8a, hat, ad, rfl\u27e9 : \u2203 a \u2208 t, r a \u2264 min d (R z) \u2227 c a = z\n  exact hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))\n  have ax : B a \u2286 ball x (R x) := by\n    refine' (hB a hat).trans _\n    refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))\n    exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))\n  obtain \u27e8b, bu, ab, bdiam\u27e9 : \u2203 b \u2208 u, (B a \u2229 B b).Nonempty \u2227 r a \u2264 2 * r b\n  exact hu a \u27e8hat, ad.trans (min_le_right _ _)\u27e9\n  have bv : b \u2208 v := by\n    refine' \u27e8bu, ab.mono _\u27e9\n    rw [inter_comm]\n    exact inter_subset_inter_right _ ax\n  let b' : v := \u27e8b, bv\u27e9\n  have b'_notmem_w : b' \u2209 w := by\n    intro b'w\n    have b'k : B b' \u2286 k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w\n    have : (ball x (R x) \\ k \u2229 k).Nonempty := by\n      apply ab.mono (inter_subset_inter _ b'k)\n      refine' ((hB _ hat).trans _).trans hd\n      exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))\n    simpa only [diff_inter_self, Set.not_nonempty_empty]\n  let b'' : { a // a \u2209 w } := \u27e8b', b'_notmem_w\u27e9\n  have zb : c a \u2208 closedBall (c b) (3 * r b) := by\n    rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9\n    have A : dist (c a) e \u2264 r a := mem_closedBall'.1 (hB a hat ea)\n    have B : dist e (c b) \u2264 r b := mem_closedBall.1 (hB b (ut bu) eb)\n    simp only [mem_closedBall]\n    linarith only [dist_triangle (c a) e (c b), A, B, bdiam]\n  suffices H : closedBall (c b'') (3 * r b'') \u2286 \u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a)\n  exact H zb\n  exact subset_iUnion (fun a : { a // a \u2209 w } => closedBall (c a) (3 * r a)) b''", "annotated_tactic": ["have M : (s \\ \u22c3 a \u2208 u, B a) \u2229\n      <a>ball</a> x (R x) \u2286 \u22c3 a : { a // a \u2209 w }, <a>closedBall</a> (c a) (3 * r a) := by\n    intro z hz\n    set k := \u22c3 (a : v) (_ : a \u2208 w), B a\n    have k_closed : <a>IsClosed</a> k := <a>isClosed_biUnion_finset</a> fun i _ => h't _ (ut (vu i.2))\n    have z_notmem_k : z \u2209 k := by\n      simp only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_sep_iff</a>, <a>forall_exists_index</a>,\n        <a>SetCoe.exists</a>, <a>not_and</a>, <a>exists_and_right</a>, <a>Subtype.coe_mk</a>]\n      intro b hbv _ h'z\n      have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n        <a>mem_inter</a> (<a>mem_of_mem_inter_left</a> hz) (<a>mem_biUnion</a> (vu hbv) h'z)\n      simpa only [<a>diff_inter_self</a>]\n    -- since the elements of `w` are closed and finitely many, one can find a small ball around `z`\n    -- not intersecting them\n    have : <a>ball</a> x (R x) \\ k \u2208 \ud835\udcdd z := by\n      apply <a>IsOpen.mem_nhds</a> (isOpen_ball.sdiff k_closed) _\n      exact (<a>mem_diff</a> _).2 \u27e8<a>mem_of_mem_inter_right</a> hz, z_notmem_k\u27e9\n    obtain \u27e8d, dpos, hd\u27e9 : \u2203 d, 0 < d \u2227 <a>closedBall</a> z d \u2286 <a>ball</a> x (R x) \\ k :=\n      nhds_basis_closedBall.mem_iff.1 this\n    -- choose an element `a` of the family `t` contained in this small ball\n    obtain \u27e8a, hat, ad, rfl\u27e9 : \u2203 a \u2208 t, r a \u2264 <a>min</a> d (R z) \u2227 c a = z\n    exact hf z ((<a>mem_diff</a> _).1 (<a>mem_of_mem_inter_left</a> hz)).1 (<a>min</a> d (R z)) (<a>lt_min</a> dpos (hR0 z))\n    have ax : B a \u2286 <a>ball</a> x (R x) := by\n      refine' (hB a hat).<a>trans</a> _\n      refine' <a>Subset.trans</a> _ (hd.trans (<a>diff_subset</a> (<a>ball</a> x (R x)) k))\n      exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))\n    -- it intersects an element `b` of `u` with comparable diameter, by definition of `u`\n    obtain \u27e8b, bu, ab, bdiam\u27e9 : \u2203 b \u2208 u, (B a \u2229 B b).<a>Nonempty</a> \u2227 r a \u2264 2 * r b\n    exact hu a \u27e8hat, ad.trans (<a>min_le_right</a> _ _)\u27e9\n    have bv : b \u2208 v := by\n      refine' \u27e8bu, ab.mono _\u27e9\n      rw [<a>inter_comm</a>]\n      exact <a>inter_subset_inter_right</a> _ ax\n    let b' : v := \u27e8b, bv\u27e9\n    -- `b` cannot belong to `w`, as the elements of `w` do not intersect `closedBall z d`,\n    -- contrary to `b`\n    have b'_notmem_w : b' \u2209 w := by\n      intro b'w\n      have b'k : B b' \u2286 k := @<a>Finset.subset_set_biUnion_of_mem</a> _ _ _ (fun y : v => B y) _ b'w\n      have : (<a>ball</a> x (R x) \\ k \u2229 k).<a>Nonempty</a> := by\n        apply ab.mono (<a>inter_subset_inter</a> _ b'k)\n        refine' ((hB _ hat).<a>trans</a> _).<a>trans</a> hd\n        exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))\n      simpa only [<a>diff_inter_self</a>, <a>Set.not_nonempty_empty</a>]\n    let b'' : { a // a \u2209 w } := \u27e8b', b'_notmem_w\u27e9\n    -- since `a` and `b` have comparable diameters, it follows that `z` belongs to the\n    -- enlargement of `b`\n    have zb : c a \u2208 <a>closedBall</a> (c b) (3 * r b) := by\n      rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9\n      have A : <a>dist</a> (c a) e \u2264 r a := <a>mem_closedBall'</a>.1 (hB a hat ea)\n      have B : <a>dist</a> e (c b) \u2264 r b := <a>mem_closedBall</a>.1 (hB b (ut bu) eb)\n      simp only [<a>mem_closedBall</a>]\n      linarith only [<a>dist_triangle</a> (c a) e (c b), A, B, bdiam]\n    suffices H : <a>closedBall</a> (c b'') (3 * r b'') \u2286 \u22c3 a : { a // a \u2209 w }, <a>closedBall</a> (c a) (3 * r a)\n    exact H zb\n    exact <a>subset_iUnion</a> (fun a : { a // a \u2209 w } => <a>closedBall</a> (c a) (3 * r a)) b''", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"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": "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": "Set.mem_sep_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1433, 9], "def_end_pos": [1433, 20]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"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": "Set.mem_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 18]}, {"full_name": "Set.mem_of_mem_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [917, 9], "def_end_pos": [917, 30]}, {"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}, {"full_name": "Set.diff_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2053, 9], "def_end_pos": [2053, 24]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [928, 9], "def_end_pos": [928, 24]}, {"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_of_mem_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [921, 9], "def_end_pos": [921, 31]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"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_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1819, 9], "def_end_pos": [1819, 17]}, {"full_name": "Set.mem_of_mem_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [917, 9], "def_end_pos": [917, 30]}, {"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": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"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_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "Finset.subset_set_biUnion_of_mem", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2119, 9], "def_end_pos": [2119, 34]}, {"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": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "Set.diff_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2053, 9], "def_end_pos": [2053, 24]}, {"full_name": "Set.not_nonempty_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 27]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 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": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}, {"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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nM : (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5"}, {"tactic": "haveI : Encodable v := (u_count.mono vu).toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> v := (u_count.mono vu).<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": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nM : (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nM : (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\nthis : Encodable \u2191v\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5"}, {"tactic": "calc\n  \u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03bc (\u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a)) :=\n    measure_mono M\n  _ \u2264 \u2211' a : { a // a \u2209 w }, \u03bc (closedBall (c a) (3 * r a)) := (measure_iUnion_le _)\n  _ \u2264 \u2211' a : { a // a \u2209 w }, C * \u03bc (B a) := (ENNReal.tsum_le_tsum fun a => \u03bcB a (ut (vu a.1.2)))\n  _ = C * \u2211' a : { a // a \u2209 w }, \u03bc (B a) := ENNReal.tsum_mul_left\n  _ \u2264 C * (\u03b5 / C) := (mul_le_mul_left' hw.le _)\n  _ \u2264 \u03b5 := ENNReal.mul_div_le", "annotated_tactic": ["calc\n    \u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 <a>ball</a> x (R x)) \u2264 \u03bc (\u22c3 a : { a // a \u2209 w }, <a>closedBall</a> (c a) (3 * r a)) :=\n      <a>measure_mono</a> M\n    _ \u2264 \u2211' a : { a // a \u2209 w }, \u03bc (<a>closedBall</a> (c a) (3 * r a)) := (<a>measure_iUnion_le</a> _)\n    _ \u2264 \u2211' a : { a // a \u2209 w }, C * \u03bc (B a) := (<a>ENNReal.tsum_le_tsum</a> fun a => \u03bcB a (ut (vu a.1.2)))\n    _ = C * \u2211' a : { a // a \u2209 w }, \u03bc (B a) := <a>ENNReal.tsum_mul_left</a>\n    _ \u2264 C * (\u03b5 / C) := (<a>mul_le_mul_left'</a> hw.le _)\n    _ \u2264 \u03b5 := <a>ENNReal.mul_div_le</a>", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"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": "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": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"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]}, {"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 intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nM : (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\nthis : Encodable \u2191v\n\u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\n\u22a2 \u2200 (x : \u03b1), \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\n\u22a2 \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4"}, {"tactic": "obtain \u27e8R, Rpos, \u03bcR\u27e9 : \u2203 R, 0 < R \u2227 \u03bc (closedBall x R) < \u221e :=\n  (\u03bc.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall", "annotated_tactic": ["obtain \u27e8R, Rpos, \u03bcR\u27e9 : \u2203 R, 0 < R \u2227 \u03bc (<a>closedBall</a> x R) < \u221e :=\n      (\u03bc.finiteAt_nhds x).<a>exists_mem_basis</a> <a>nhds_basis_closedBall</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "MeasureTheory.Measure.FiniteAtFilter.exists_mem_basis", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3257, 9], "def_end_pos": [3257, 40]}, {"full_name": "Metric.nhds_basis_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [989, 9], "def_end_pos": [989, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\n\u22a2 \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4"}, {"tactic": "refine' \u27e8min 1 (R / 20), _, min_le_left _ _, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>min</a> 1 (R / 20), _, <a>min_le_left</a> _ _, _\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": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 0 < min 1 (R / 20)\n\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 \u2191\u2191\u03bc (closedBall x (20 * min 1 (R / 20))) < \u22a4"}, {"tactic": "simp only [true_and_iff, lt_min_iff, zero_lt_one]", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>lt_min_iff</a>, <a>zero_lt_one</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "lt_min_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [53, 9], "def_end_pos": [53, 19]}, {"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.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 0 < min 1 (R / 20)", "state_after": "case intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 0 < R / 20"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 0 < R / 20", "state_after": "no goals"}, {"tactic": "apply lt_of_le_of_lt (measure_mono _) \u03bcR", "annotated_tactic": ["apply <a>lt_of_le_of_lt</a> (<a>measure_mono</a> _) \u03bcR", [{"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]}]], "state_before": "case intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 \u2191\u2191\u03bc (closedBall x (20 * min 1 (R / 20))) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 closedBall x (20 * min 1 (R / 20)) \u2286 closedBall x 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": [609, 9], "def_end_pos": [609, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 closedBall x (20 * min 1 (R / 20)) \u2286 closedBall x R", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 20 * min 1 (R / 20) \u2264 R"}, {"tactic": "calc\n  20 * min 1 (R / 20) \u2264 20 * (R / 20) :=\n    mul_le_mul_of_nonneg_left (min_le_right _ _) (by norm_num)\n  _ = R := by ring", "annotated_tactic": ["calc\n        20 * <a>min</a> 1 (R / 20) \u2264 20 * (R / 20) :=\n          <a>mul_le_mul_of_nonneg_left</a> (<a>min_le_right</a> _ _) (by norm_num)\n        _ = R := by ring", [{"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": "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": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 20 * min 1 (R / 20) \u2264 R", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 0 \u2264 20", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nx : \u03b1\nR : \u211d\nRpos : 0 < R\n\u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4\n\u22a2 20 * (R / 20) = R", "state_after": "no goals"}, {"tactic": "have A : \u2200 a \u2208 t', r a \u2264 1 := by\n  intro a ha\n  apply ha.2.trans (hR1 (c a))", "annotated_tactic": ["have A : \u2200 a \u2208 t', r a \u2264 1 := by\n      intro a ha\n      apply ha.2.<a>trans</a> (hR1 (c a))", [{"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\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c 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 r a \u2264 2 * r b", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nA : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1\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 r a \u2264 2 * r b"}, {"tactic": "have A' : \u2200 a \u2208 t', (B a).Nonempty :=\n  fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1)", "annotated_tactic": ["have A' : \u2200 a \u2208 t', (B a).<a>Nonempty</a> :=\n      fun a hat' => <a>Set.Nonempty.mono</a> <a>interior_subset</a> (ht a hat'.1)", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.Nonempty.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}, {"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\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nA : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1\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 r a \u2264 2 * r b", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nA : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1\nA' : \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 r a \u2264 2 * r b"}, {"tactic": "refine' exists_disjoint_subfamily_covering_enlargment B t' r 2 one_lt_two (fun a ha => _) 1 A A'", "annotated_tactic": ["refine' <a>exists_disjoint_subfamily_covering_enlargment</a> B t' r 2 <a>one_lt_two</a> (fun a ha => _) 1 A A'", [{"full_name": "Vitali.exists_disjoint_subfamily_covering_enlargment", "def_path": "Mathlib/MeasureTheory/Covering/Vitali.lean", "def_pos": [59, 9], "def_end_pos": [59, 54]}, {"full_name": "one_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [106, 7], "def_end_pos": [106, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nA : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1\nA' : \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 r a \u2264 2 * r b", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nA : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1\nA' : \u2200 (a : \u03b9), a \u2208 t' \u2192 Set.Nonempty (B a)\na : \u03b9\nha : a \u2208 t'\n\u22a2 0 \u2264 r a"}, {"tactic": "exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))", "annotated_tactic": ["exact <a>nonempty_closedBall</a>.1 ((A' a ha).<a>mono</a> (hB a ha.1))", [{"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.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nA : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1\nA' : \u2200 (a : \u03b9), a \u2208 t' \u2192 Set.Nonempty (B a)\na : \u03b9\nha : a \u2208 t'\n\u22a2 0 \u2264 r a", "state_after": "no goals"}, {"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\n\u22a2 \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\na : \u03b9\nha : a \u2208 t'\n\u22a2 r a \u2264 1"}, {"tactic": "apply ha.2.trans (hR1 (c a))", "annotated_tactic": ["apply ha.2.<a>trans</a> (hR1 (c a))", [{"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\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\na : \u03b9\nha : a \u2208 t'\n\u22a2 r a \u2264 1", "state_after": "no goals"}, {"tactic": "have Idist_v : \u2200 a \u2208 v, dist (c a) x \u2264 r a + R x := by\n  intro a hav\n  apply dist_le_add_of_nonempty_closedBall_inter_closedBall\n  refine' hav.2.mono _\n  apply inter_subset_inter _ ball_subset_closedBall\n  exact hB a (ut (vu hav))", "annotated_tactic": ["have Idist_v : \u2200 a \u2208 v, <a>dist</a> (c a) x \u2264 r a + R x := by\n      intro a hav\n      apply <a>dist_le_add_of_nonempty_closedBall_inter_closedBall</a>\n      refine' hav.2.<a>mono</a> _\n      apply <a>inter_subset_inter</a> _ <a>ball_subset_closedBall</a>\n      exact hB a (ut (vu hav))", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [633, 9], "def_end_pos": [633, 60]}, {"full_name": "Set.Nonempty.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}, {"full_name": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "set R0 := sSup (r '' v) with R0_def", "annotated_tactic": ["set R0 := <a>sSup</a> (r '' v) with R0_def", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "have R0_bdd : BddAbove (r '' v) := by\n  refine' \u27e81, fun r' hr' => _\u27e9\n  rcases (mem_image _ _ _).1 hr' with \u27e8b, hb, rfl\u27e9\n  exact le_trans (ut' (vu hb)).2 (hR1 (c b))", "annotated_tactic": ["have R0_bdd : <a>BddAbove</a> (r '' v) := by\n      refine' \u27e81, fun r' hr' => _\u27e9\n      rcases (<a>mem_image</a> _ _ _).1 hr' with \u27e8b, hb, rfl\u27e9\n      exact <a>le_trans</a> (ut' (vu hb)).2 (hR1 (c b))", [{"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]}, {"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\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "rcases le_total R0 (R x) with (H | H)", "annotated_tactic": ["rcases <a>le_total</a> R0 (R x) with (H | H)", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\ncase inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "intro a hav", "annotated_tactic": ["intro a hav", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\n\u22a2 \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 dist (c a) x \u2264 r a + R x"}, {"tactic": "apply dist_le_add_of_nonempty_closedBall_inter_closedBall", "annotated_tactic": ["apply <a>dist_le_add_of_nonempty_closedBall_inter_closedBall</a>", [{"full_name": "Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [633, 9], "def_end_pos": [633, 60]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 dist (c a) x \u2264 r a + R x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 Set.Nonempty (closedBall (c a) (r a) \u2229 closedBall x (R x))"}, {"tactic": "refine' hav.2.mono _", "annotated_tactic": ["refine' hav.2.<a>mono</a> _", [{"full_name": "Set.Nonempty.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 Set.Nonempty (closedBall (c a) (r a) \u2229 closedBall x (R x))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 B a \u2229 ball x (R x) \u2286 closedBall (c a) (r a) \u2229 closedBall x (R x)"}, {"tactic": "apply inter_subset_inter _ ball_subset_closedBall", "annotated_tactic": ["apply <a>inter_subset_inter</a> _ <a>ball_subset_closedBall</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": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 B a \u2229 ball x (R x) \u2286 closedBall (c a) (r a) \u2229 closedBall x (R x)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 B a \u2286 closedBall (c a) (r a)"}, {"tactic": "exact hB a (ut (vu hav))", "annotated_tactic": ["exact hB a (ut (vu hav))", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\na : \u03b9\nhav : a \u2208 v\n\u22a2 B a \u2286 closedBall (c a) (r a)", "state_after": "no goals"}, {"tactic": "refine' \u27e81, fun r' hr' => _\u27e9", "annotated_tactic": ["refine' \u27e81, fun r' hr' => _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\n\u22a2 BddAbove (r '' v)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nr' : \u211d\nhr' : r' \u2208 r '' v\n\u22a2 r' \u2264 1"}, {"tactic": "rcases (mem_image _ _ _).1 hr' with \u27e8b, hb, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hr' with \u27e8b, hb, 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\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nr' : \u211d\nhr' : r' \u2208 r '' v\n\u22a2 r' \u2264 1", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nb : \u03b9\nhb : b \u2208 v\nhr' : r b \u2208 r '' v\n\u22a2 r b \u2264 1"}, {"tactic": "exact le_trans (ut' (vu hb)).2 (hR1 (c b))", "annotated_tactic": ["exact <a>le_trans</a> (ut' (vu hb)).2 (hR1 (c b))", [{"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\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nb : \u03b9\nhb : b \u2208 v\nhr' : r b \u2208 r '' v\n\u22a2 r b \u2264 1", "state_after": "no goals"}, {"tactic": "refine' \u27e820 * R x, hR\u03bc x, fun a au hax => _\u27e9", "annotated_tactic": ["refine' \u27e820 * R x, hR\u03bc x, fun a au hax => _\u27e9", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\n\u22a2 B a \u2286 closedBall x (20 * R x)"}, {"tactic": "refine' (hB a (ut au)).trans _", "annotated_tactic": ["refine' (hB a (ut au)).<a>trans</a> _", [{"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\n\u22a2 B a \u2286 closedBall x (20 * R x)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\n\u22a2 closedBall (c a) (r a) \u2286 closedBall x (20 * R x)"}, {"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 inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\n\u22a2 closedBall (c a) (r a) \u2286 closedBall x (20 * R x)", "state_after": "case inl.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\n\u22a2 r a + dist (c a) x \u2264 20 * R x"}, {"tactic": "have : r a \u2264 R0 := le_csSup R0_bdd (mem_image_of_mem _ \u27e8au, hax\u27e9)", "annotated_tactic": ["have : r a \u2264 R0 := <a>le_csSup</a> R0_bdd (<a>mem_image_of_mem</a> _ \u27e8au, hax\u27e9)", [{"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": "case inl.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\n\u22a2 r a + dist (c a) x \u2264 20 * R x", "state_after": "case inl.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\nthis : r a \u2264 R0\n\u22a2 r a + dist (c a) x \u2264 20 * R x"}, {"tactic": "linarith [Idist_v a \u27e8au, hax\u27e9, hR0 x]", "annotated_tactic": ["linarith [Idist_v a \u27e8au, hax\u27e9, hR0 x]", []], "state_before": "case inl.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R0 \u2264 R x\na : \u03b9\nau : a \u2208 u\nhax : Set.Nonempty (B a \u2229 ball x (R x))\nthis : r a \u2264 R0\n\u22a2 r a + dist (c a) x \u2264 20 * R x", "state_after": "no goals"}, {"tactic": "have R0pos : 0 < R0 := (hR0 x).trans_le H", "annotated_tactic": ["have R0pos : 0 < R0 := (hR0 x).<a>trans_le</a> H", [{"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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "have vnonempty : v.Nonempty := by\n  by_contra h\n  rw [nonempty_iff_ne_empty, Classical.not_not] at h\n  rw [h, image_empty, Real.sSup_empty] at R0_def\n  exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def))", "annotated_tactic": ["have vnonempty : v.Nonempty := by\n        by_contra h\n        rw [<a>nonempty_iff_ne_empty</a>, <a>Classical.not_not</a>] at h\n        rw [h, <a>image_empty</a>, <a>Real.sSup_empty</a>] at R0_def\n        exact <a>lt_irrefl</a> _ (R0pos.trans_le (<a>le_of_eq</a> R0_def))", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}, {"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.image_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [337, 9], "def_end_pos": [337, 20]}, {"full_name": "Real.sSup_empty", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [797, 9], "def_end_pos": [797, 19]}, {"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]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "obtain \u27e8a, hav, R0a\u27e9 : \u2203 a \u2208 v, R0 / 2 < r a := by\n  obtain \u27e8r', r'mem, hr'\u27e9 : \u2203 r' \u2208 r '' v, R0 / 2 < r' :=\n    exists_lt_of_lt_csSup (nonempty_image_iff.2 vnonempty) (half_lt_self R0pos)\n  rcases (mem_image _ _ _).1 r'mem with \u27e8a, hav, rfl\u27e9\n  exact \u27e8a, hav, hr'\u27e9", "annotated_tactic": ["obtain \u27e8a, hav, R0a\u27e9 : \u2203 a \u2208 v, R0 / 2 < r a := by\n        obtain \u27e8r', r'mem, hr'\u27e9 : \u2203 r' \u2208 r '' v, R0 / 2 < r' :=\n          <a>exists_lt_of_lt_csSup</a> (<a>nonempty_image_iff</a>.2 vnonempty) (<a>half_lt_self</a> R0pos)\n        rcases (<a>mem_image</a> _ _ _).1 r'mem with \u27e8a, hav, rfl\u27e9\n        exact \u27e8a, hav, hr'\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]}, {"full_name": "half_lt_self", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [525, 11], "def_end_pos": [525, 23]}, {"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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K"}, {"tactic": "refine' \u27e88 * R0, _, _\u27e9", "annotated_tactic": ["refine' \u27e88 * R0, _, _\u27e9", []], "state_before": "case inr.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K", "state_after": "case inr.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 \u2191\u2191\u03bc (closedBall x (8 * R0)) < \u22a4\n\ncase inr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x (8 * R0)"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\n\u22a2 Set.Nonempty v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nh : \u00acSet.Nonempty v\n\u22a2 False"}, {"tactic": "rw [nonempty_iff_ne_empty, Classical.not_not] at h", "annotated_tactic": ["rw [<a>nonempty_iff_ne_empty</a>, <a>Classical.not_not</a>] at h", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}, {"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\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nh : \u00acSet.Nonempty v\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nh : v = \u2205\n\u22a2 False"}, {"tactic": "rw [h, image_empty, Real.sSup_empty] at R0_def", "annotated_tactic": ["rw [h, <a>image_empty</a>, <a>Real.sSup_empty</a>] at R0_def", [{"full_name": "Set.image_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [337, 9], "def_end_pos": [337, 20]}, {"full_name": "Real.sSup_empty", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [797, 9], "def_end_pos": [797, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nh : v = \u2205\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = 0\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nh : v = \u2205\n\u22a2 False"}, {"tactic": "exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def))", "annotated_tactic": ["exact <a>lt_irrefl</a> _ (R0pos.trans_le (<a>le_of_eq</a> R0_def))", [{"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]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = 0\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nh : v = \u2205\n\u22a2 False", "state_after": "no goals"}, {"tactic": "obtain \u27e8r', r'mem, hr'\u27e9 : \u2203 r' \u2208 r '' v, R0 / 2 < r' :=\n  exists_lt_of_lt_csSup (nonempty_image_iff.2 vnonempty) (half_lt_self R0pos)", "annotated_tactic": ["obtain \u27e8r', r'mem, hr'\u27e9 : \u2203 r' \u2208 r '' v, R0 / 2 < r' :=\n          <a>exists_lt_of_lt_csSup</a> (<a>nonempty_image_iff</a>.2 vnonempty) (<a>half_lt_self</a> R0pos)", [{"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]}, {"full_name": "half_lt_self", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [525, 11], "def_end_pos": [525, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\n\u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\nr' : \u211d\nr'mem : r' \u2208 r '' v\nhr' : R0 / 2 < r'\n\u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a"}, {"tactic": "rcases (mem_image _ _ _).1 r'mem with \u27e8a, hav, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 r'mem with \u27e8a, hav, 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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\nr' : \u211d\nr'mem : r' \u2208 r '' v\nhr' : R0 / 2 < r'\n\u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nr'mem : r a \u2208 r '' v\nhr' : R0 / 2 < r a\n\u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a"}, {"tactic": "exact \u27e8a, hav, hr'\u27e9", "annotated_tactic": ["exact \u27e8a, hav, hr'\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nr'mem : r a \u2208 r '' v\nhr' : R0 / 2 < r a\n\u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a", "state_after": "no goals"}, {"tactic": "apply lt_of_le_of_lt (measure_mono _) (hR\u03bc (c a))", "annotated_tactic": ["apply <a>lt_of_le_of_lt</a> (<a>measure_mono</a> _) (hR\u03bc (c 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_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "case inr.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 \u2191\u2191\u03bc (closedBall x (8 * R0)) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 closedBall x (8 * R0) \u2286 closedBall (c a) (20 * R (c a))"}, {"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\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 closedBall x (8 * R0) \u2286 closedBall (c a) (20 * R (c a))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 8 * R0 + dist x (c a) \u2264 20 * R (c a)"}, {"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 h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 8 * R0 + dist x (c a) \u2264 20 * R (c a)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 8 * R0 + dist (c a) x \u2264 20 * R (c a)"}, {"tactic": "linarith [Idist_v a hav, (ut' (vu hav)).2]", "annotated_tactic": ["linarith [Idist_v a hav, (ut' (vu hav)).2]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 8 * R0 + dist (c a) x \u2264 20 * R (c a)", "state_after": "no goals"}, {"tactic": "intro b bu hbx", "annotated_tactic": ["intro b bu hbx", []], "state_before": "case inr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\n\u22a2 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x (8 * R0)", "state_after": "case inr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\n\u22a2 B b \u2286 closedBall x (8 * R0)"}, {"tactic": "refine' (hB b (ut bu)).trans _", "annotated_tactic": ["refine' (hB b (ut bu)).<a>trans</a> _", [{"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}]], "state_before": "case inr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\n\u22a2 B b \u2286 closedBall x (8 * R0)", "state_after": "case inr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\n\u22a2 closedBall (c b) (r b) \u2286 closedBall x (8 * R0)"}, {"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 inr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\n\u22a2 closedBall (c b) (r b) \u2286 closedBall x (8 * R0)", "state_after": "case inr.intro.intro.refine'_2.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\n\u22a2 r b + dist (c b) x \u2264 8 * R0"}, {"tactic": "have : r b \u2264 R0 := le_csSup R0_bdd (mem_image_of_mem _ \u27e8bu, hbx\u27e9)", "annotated_tactic": ["have : r b \u2264 R0 := <a>le_csSup</a> R0_bdd (<a>mem_image_of_mem</a> _ \u27e8bu, hbx\u27e9)", [{"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": "case inr.intro.intro.refine'_2.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\n\u22a2 r b + dist (c b) x \u2264 8 * R0", "state_after": "case inr.intro.intro.refine'_2.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\nthis : r b \u2264 R0\n\u22a2 r b + dist (c b) x \u2264 8 * R0"}, {"tactic": "linarith [Idist_v b \u27e8bu, hbx\u27e9]", "annotated_tactic": ["linarith [Idist_v b \u27e8bu, hbx\u27e9]", []], "state_before": "case inr.intro.intro.refine'_2.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nIdist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x\nR0 : \u211d := sSup (r '' v)\nR0_def : R0 = sSup (r '' v)\nR0_bdd : BddAbove (r '' v)\nH : R x \u2264 R0\nR0pos : 0 < R0\nvnonempty : Set.Nonempty v\na : \u03b9\nhav : a \u2208 v\nR0a : R0 / 2 < r a\nb : \u03b9\nbu : b \u2208 u\nhbx : Set.Nonempty (B b \u2229 ball x (R x))\nthis : r b \u2264 R0\n\u22a2 r b + dist (c b) x \u2264 8 * R0", "state_after": "no goals"}, {"tactic": "calc\n  (\u2211' a : v, \u03bc (B a)) = \u03bc (\u22c3 a \u2208 v, B a) := by\n    rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]\n    exact u_disj.subset vu\n  _ \u2264 \u03bc (closedBall x K) := (measure_mono (iUnion\u2082_subset fun a ha => hK a (vu ha) ha.2))\n  _ < \u221e := \u03bcK", "annotated_tactic": ["calc\n      (\u2211' a : v, \u03bc (B a)) = \u03bc (\u22c3 a \u2208 v, B a) := by\n        rw [<a>measure_biUnion</a> (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).<a>measurableSet</a>]\n        exact u_disj.subset vu\n      _ \u2264 \u03bc (<a>closedBall</a> x K) := (<a>measure_mono</a> (<a>iUnion\u2082_subset</a> fun a ha => hK a (vu ha) ha.2))\n      _ < \u221e := \u03bcK", [{"full_name": "MeasureTheory.measure_biUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [172, 9], "def_end_pos": [172, 24]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}, {"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.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4", "state_after": "no goals"}, {"tactic": "rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]", "annotated_tactic": ["rw [<a>measure_biUnion</a> (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).<a>measurableSet</a>]", [{"full_name": "MeasureTheory.measure_biUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [172, 9], "def_end_pos": [172, 24]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) = \u2191\u2191\u03bc (\u22c3 a \u2208 v, B a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 PairwiseDisjoint v fun a => B a"}, {"tactic": "exact u_disj.subset vu", "annotated_tactic": ["exact u_disj.subset vu", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 PairwiseDisjoint v fun a => B a", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.div_pos_iff, \u03b5pos.ne', ENNReal.coe_ne_top, Ne.def, not_false_iff,\n  and_self_iff]", "annotated_tactic": ["simp only [<a>ENNReal.div_pos_iff</a>, \u03b5pos.ne', <a>ENNReal.coe_ne_top</a>, <a>Ne.def</a>, <a>not_false_iff</a>,\n        <a>and_self_iff</a>]", [{"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]}, {"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": "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\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\n\u22a2 0 < \u03b5 / \u2191C", "state_after": "no goals"}, {"tactic": "intro z hz", "annotated_tactic": ["intro z hz", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\n\u22a2 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "set k := \u22c3 (a : v) (_ : a \u2208 w), B a", "annotated_tactic": ["set k := \u22c3 (a : v) (_ : a \u2208 w), B a", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have k_closed : IsClosed k := isClosed_biUnion_finset fun i _ => h't _ (ut (vu i.2))", "annotated_tactic": ["have k_closed : <a>IsClosed</a> k := <a>isClosed_biUnion_finset</a> fun i _ => h't _ (ut (vu i.2))", [{"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]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have z_notmem_k : z \u2209 k := by\n  simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,\n    SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]\n  intro b hbv _ h'z\n  have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n    mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)\n  simpa only [diff_inter_self]", "annotated_tactic": ["have z_notmem_k : z \u2209 k := by\n      simp only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_sep_iff</a>, <a>forall_exists_index</a>,\n        <a>SetCoe.exists</a>, <a>not_and</a>, <a>exists_and_right</a>, <a>Subtype.coe_mk</a>]\n      intro b hbv _ h'z\n      have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n        <a>mem_inter</a> (<a>mem_of_mem_inter_left</a> hz) (<a>mem_biUnion</a> (vu hbv) h'z)\n      simpa only [<a>diff_inter_self</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": "Set.mem_sep_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1433, 9], "def_end_pos": [1433, 20]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"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": "Set.mem_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 18]}, {"full_name": "Set.mem_of_mem_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [917, 9], "def_end_pos": [917, 30]}, {"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}, {"full_name": "Set.diff_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2053, 9], "def_end_pos": [2053, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have : ball x (R x) \\ k \u2208 \ud835\udcdd z := by\n  apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _\n  exact (mem_diff _).2 \u27e8mem_of_mem_inter_right hz, z_notmem_k\u27e9", "annotated_tactic": ["have : <a>ball</a> x (R x) \\ k \u2208 \ud835\udcdd z := by\n      apply <a>IsOpen.mem_nhds</a> (isOpen_ball.sdiff k_closed) _\n      exact (<a>mem_diff</a> _).2 \u27e8<a>mem_of_mem_inter_right</a> hz, z_notmem_k\u27e9", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [928, 9], "def_end_pos": [928, 24]}, {"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_of_mem_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [921, 9], "def_end_pos": [921, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd z\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "obtain \u27e8d, dpos, hd\u27e9 : \u2203 d, 0 < d \u2227 closedBall z d \u2286 ball x (R x) \\ k :=\n  nhds_basis_closedBall.mem_iff.1 this", "annotated_tactic": ["obtain \u27e8d, dpos, hd\u27e9 : \u2203 d, 0 < d \u2227 <a>closedBall</a> z d \u2286 <a>ball</a> x (R x) \\ k :=\n      nhds_basis_closedBall.mem_iff.1 this", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd z\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd z\nd : \u211d\ndpos : 0 < d\nhd : closedBall z d \u2286 ball x (R x) \\ k\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "obtain \u27e8a, hat, ad, rfl\u27e9 : \u2203 a \u2208 t, r a \u2264 min d (R z) \u2227 c a = z", "annotated_tactic": ["obtain \u27e8a, hat, ad, rfl\u27e9 : \u2203 a \u2208 t, r a \u2264 <a>min</a> d (R z) \u2227 c a = z", [{"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 intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd z\nd : \u211d\ndpos : 0 < d\nhd : closedBall z d \u2286 ball x (R x) \\ k\n\u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd z\nd : \u211d\ndpos : 0 < d\nhd : closedBall z d \u2286 ball x (R x) \\ k\n\u22a2 \u2203 a, a \u2208 t \u2227 r a \u2264 min d (R z) \u2227 c a = z\n\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "exact hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))", "annotated_tactic": ["exact hf z ((<a>mem_diff</a> _).1 (<a>mem_of_mem_inter_left</a> hz)).1 (<a>min</a> d (R z)) (<a>lt_min</a> dpos (hR0 z))", [{"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_of_mem_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [917, 9], "def_end_pos": [917, 30]}, {"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]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd z\nd : \u211d\ndpos : 0 < d\nhd : closedBall z d \u2286 ball x (R x) \\ k\n\u22a2 \u2203 a, a \u2208 t \u2227 r a \u2264 min d (R z) \u2227 c a = z\n\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have ax : B a \u2286 ball x (R x) := by\n  refine' (hB a hat).trans _\n  refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))\n  exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))", "annotated_tactic": ["have ax : B a \u2286 <a>ball</a> x (R x) := by\n      refine' (hB a hat).<a>trans</a> _\n      refine' <a>Subset.trans</a> _ (hd.trans (<a>diff_subset</a> (<a>ball</a> x (R x)) k))\n      exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "obtain \u27e8b, bu, ab, bdiam\u27e9 : \u2203 b \u2208 u, (B a \u2229 B b).Nonempty \u2227 r a \u2264 2 * r b", "annotated_tactic": ["obtain \u27e8b, bu, ab, bdiam\u27e9 : \u2203 b \u2208 u, (B a \u2229 B b).<a>Nonempty</a> \u2227 r a \u2264 2 * r b", [{"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 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\n\u22a2 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\n\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "exact hu a \u27e8hat, ad.trans (min_le_right _ _)\u27e9", "annotated_tactic": ["exact hu a \u27e8hat, ad.trans (<a>min_le_right</a> _ _)\u27e9", [{"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\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\n\u22a2 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\n\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have bv : b \u2208 v := by\n  refine' \u27e8bu, ab.mono _\u27e9\n  rw [inter_comm]\n  exact inter_subset_inter_right _ ax", "annotated_tactic": ["have bv : b \u2208 v := by\n      refine' \u27e8bu, ab.mono _\u27e9\n      rw [<a>inter_comm</a>]\n      exact <a>inter_subset_inter_right</a> _ ax", [{"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_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "let b' : v := \u27e8b, bv\u27e9", "annotated_tactic": ["let b' : v := \u27e8b, bv\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have b'_notmem_w : b' \u2209 w := by\n  intro b'w\n  have b'k : B b' \u2286 k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w\n  have : (ball x (R x) \\ k \u2229 k).Nonempty := by\n    apply ab.mono (inter_subset_inter _ b'k)\n    refine' ((hB _ hat).trans _).trans hd\n    exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))\n  simpa only [diff_inter_self, Set.not_nonempty_empty]", "annotated_tactic": ["have b'_notmem_w : b' \u2209 w := by\n      intro b'w\n      have b'k : B b' \u2286 k := @<a>Finset.subset_set_biUnion_of_mem</a> _ _ _ (fun y : v => B y) _ b'w\n      have : (<a>ball</a> x (R x) \\ k \u2229 k).<a>Nonempty</a> := by\n        apply ab.mono (<a>inter_subset_inter</a> _ b'k)\n        refine' ((hB _ hat).<a>trans</a> _).<a>trans</a> hd\n        exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))\n      simpa only [<a>diff_inter_self</a>, <a>Set.not_nonempty_empty</a>]", [{"full_name": "Finset.subset_set_biUnion_of_mem", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2119, 9], "def_end_pos": [2119, 34]}, {"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": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "Set.diff_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2053, 9], "def_end_pos": [2053, 24]}, {"full_name": "Set.not_nonempty_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 27]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "let b'' : { a // a \u2209 w } := \u27e8b', b'_notmem_w\u27e9", "annotated_tactic": ["let b'' : { a // a \u2209 w } := \u27e8b', b'_notmem_w\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "have zb : c a \u2208 closedBall (c b) (3 * r b) := by\n  rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9\n  have A : dist (c a) e \u2264 r a := mem_closedBall'.1 (hB a hat ea)\n  have B : dist e (c b) \u2264 r b := mem_closedBall.1 (hB b (ut bu) eb)\n  simp only [mem_closedBall]\n  linarith only [dist_triangle (c a) e (c b), A, B, bdiam]", "annotated_tactic": ["have zb : c a \u2208 <a>closedBall</a> (c b) (3 * r b) := by\n      rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9\n      have A : <a>dist</a> (c a) e \u2264 r a := <a>mem_closedBall'</a>.1 (hB a hat ea)\n      have B : <a>dist</a> e (c b) \u2264 r b := <a>mem_closedBall</a>.1 (hB b (ut bu) eb)\n      simp only [<a>mem_closedBall</a>]\n      linarith only [<a>dist_triangle</a> (c a) e (c b), A, B, bdiam]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closedBall'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "suffices H : closedBall (c b'') (3 * r b'') \u2286 \u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a)", "annotated_tactic": ["suffices H : <a>closedBall</a> (c b'') (3 * r b'') \u2286 \u22c3 a : { a // a \u2209 w }, <a>closedBall</a> (c a) (3 * r 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]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\nH : closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\n\ncase H\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\n\u22a2 closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "exact H zb", "annotated_tactic": ["exact H zb", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\nH : closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\n\u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)\n\ncase H\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\n\u22a2 closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "case H\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\n\u22a2 closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)"}, {"tactic": "exact subset_iUnion (fun a : { a // a \u2209 w } => closedBall (c a) (3 * r a)) b''", "annotated_tactic": ["exact <a>subset_iUnion</a> (fun a : { a // a \u2209 w } => <a>closedBall</a> (c a) (3 * r a)) b''", [{"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}]], "state_before": "case H\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\nzb : c a \u2208 closedBall (c b) (3 * r b)\n\u22a2 closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)", "state_after": "no goals"}, {"tactic": "simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,\n  SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>mem_sep_iff</a>, <a>forall_exists_index</a>,\n        <a>SetCoe.exists</a>, <a>not_and</a>, <a>exists_and_right</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": "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_sep_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1433, 9], "def_end_pos": [1433, 20]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"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": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\n\u22a2 \u00acz \u2208 k", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\n\u22a2 \u2200 (x_1 : \u03b9) (x_2 : x_1 \u2208 u \u2227 Set.Nonempty (B x_1 \u2229 ball x (R x))),\n    { val := x_1, property := (_ : x_1 \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w \u2192 \u00acz \u2208 B x_1"}, {"tactic": "intro b hbv _ h'z", "annotated_tactic": ["intro b hbv _ h'z", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\n\u22a2 \u2200 (x_1 : \u03b9) (x_2 : x_1 \u2208 u \u2227 Set.Nonempty (B x_1 \u2229 ball x (R x))),\n    { val := x_1, property := (_ : x_1 \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w \u2192 \u00acz \u2208 B x_1", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nb : \u03b9\nhbv : b \u2208 u \u2227 Set.Nonempty (B b \u2229 ball x (R x))\nh\u271d : { val := b, property := (_ : b \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w\nh'z : z \u2208 B b\n\u22a2 False"}, {"tactic": "have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n  mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)", "annotated_tactic": ["have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n        <a>mem_inter</a> (<a>mem_of_mem_inter_left</a> hz) (<a>mem_biUnion</a> (vu hbv) h'z)", [{"full_name": "Set.mem_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 18]}, {"full_name": "Set.mem_of_mem_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [917, 9], "def_end_pos": [917, 30]}, {"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nb : \u03b9\nhbv : b \u2208 u \u2227 Set.Nonempty (B b \u2229 ball x (R x))\nh\u271d : { val := b, property := (_ : b \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w\nh'z : z \u2208 B b\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nb : \u03b9\nhbv : b \u2208 u \u2227 Set.Nonempty (B b \u2229 ball x (R x))\nh\u271d : { val := b, property := (_ : b \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w\nh'z : z \u2208 B b\nthis : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a\n\u22a2 False"}, {"tactic": "simpa only [diff_inter_self]", "annotated_tactic": ["simpa only [<a>diff_inter_self</a>]", [{"full_name": "Set.diff_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2053, 9], "def_end_pos": [2053, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nb : \u03b9\nhbv : b \u2208 u \u2227 Set.Nonempty (B b \u2229 ball x (R x))\nh\u271d : { val := b, property := (_ : b \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w\nh'z : z \u2208 B b\nthis : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _", "annotated_tactic": ["apply <a>IsOpen.mem_nhds</a> (isOpen_ball.sdiff k_closed) _", [{"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\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\n\u22a2 ball x (R x) \\ k \u2208 \ud835\udcdd z", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\n\u22a2 z \u2208 ball x (R x) \\ k"}, {"tactic": "exact (mem_diff _).2 \u27e8mem_of_mem_inter_right hz, z_notmem_k\u27e9", "annotated_tactic": ["exact (<a>mem_diff</a> _).2 \u27e8<a>mem_of_mem_inter_right</a> hz, z_notmem_k\u27e9", [{"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_of_mem_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [921, 9], "def_end_pos": [921, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nz : \u03b1\nhz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nz_notmem_k : \u00acz \u2208 k\n\u22a2 z \u2208 ball x (R x) \\ k", "state_after": "no goals"}, {"tactic": "refine' (hB a hat).trans _", "annotated_tactic": ["refine' (hB a hat).<a>trans</a> _", [{"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\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 B a \u2286 ball x (R x)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 closedBall (c a) (r a) \u2286 ball x (R x)"}, {"tactic": "refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))", "annotated_tactic": ["refine' <a>Subset.trans</a> _ (hd.trans (<a>diff_subset</a> (<a>ball</a> x (R x)) k))", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 closedBall (c a) (r a) \u2286 ball x (R x)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 closedBall (c a) (r a) \u2286 closedBall (c a) d"}, {"tactic": "exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))", "annotated_tactic": ["exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))", [{"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"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\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\n\u22a2 closedBall (c a) (r a) \u2286 closedBall (c a) d", "state_after": "no goals"}, {"tactic": "refine' \u27e8bu, ab.mono _\u27e9", "annotated_tactic": ["refine' \u27e8bu, ab.mono _\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 b \u2208 v", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 B a \u2229 B b \u2286 B b \u2229 ball x (R x)"}, {"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\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 B a \u2229 B b \u2286 B b \u2229 ball x (R x)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 B b \u2229 B a \u2286 B b \u2229 ball x (R x)"}, {"tactic": "exact inter_subset_inter_right _ ax", "annotated_tactic": ["exact <a>inter_subset_inter_right</a> _ ax", [{"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": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\n\u22a2 B b \u2229 B a \u2286 B b \u2229 ball x (R x)", "state_after": "no goals"}, {"tactic": "intro b'w", "annotated_tactic": ["intro b'w", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\n\u22a2 \u00acb' \u2208 w", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\n\u22a2 False"}, {"tactic": "have b'k : B b' \u2286 k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w", "annotated_tactic": ["have b'k : B b' \u2286 k := @<a>Finset.subset_set_biUnion_of_mem</a> _ _ _ (fun y : v => B y) _ b'w", [{"full_name": "Finset.subset_set_biUnion_of_mem", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2119, 9], "def_end_pos": [2119, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 False"}, {"tactic": "have : (ball x (R x) \\ k \u2229 k).Nonempty := by\n  apply ab.mono (inter_subset_inter _ b'k)\n  refine' ((hB _ hat).trans _).trans hd\n  exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))", "annotated_tactic": ["have : (<a>ball</a> x (R x) \\ k \u2229 k).<a>Nonempty</a> := by\n        apply ab.mono (<a>inter_subset_inter</a> _ b'k)\n        refine' ((hB _ hat).<a>trans</a> _).<a>trans</a> hd\n        exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))", [{"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": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"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\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis\u271d : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\nthis : Set.Nonempty (ball x (R x) \\ k \u2229 k)\n\u22a2 False"}, {"tactic": "simpa only [diff_inter_self, Set.not_nonempty_empty]", "annotated_tactic": ["simpa only [<a>diff_inter_self</a>, <a>Set.not_nonempty_empty</a>]", [{"full_name": "Set.diff_inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2053, 9], "def_end_pos": [2053, 24]}, {"full_name": "Set.not_nonempty_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis\u271d : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\nthis : Set.Nonempty (ball x (R x) \\ k \u2229 k)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply ab.mono (inter_subset_inter _ b'k)", "annotated_tactic": ["apply ab.mono (<a>inter_subset_inter</a> _ b'k)", [{"full_name": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 Set.Nonempty (ball x (R x) \\ k \u2229 k)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 B a \u2286 ball x (R x) \\ k"}, {"tactic": "refine' ((hB _ hat).trans _).trans hd", "annotated_tactic": ["refine' ((hB _ hat).<a>trans</a> _).<a>trans</a> hd", [{"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"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\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 B a \u2286 ball x (R x) \\ k", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 closedBall (c a) (r a) \u2286 closedBall (c a) d"}, {"tactic": "exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))", "annotated_tactic": ["exact <a>closedBall_subset_closedBall</a> (ad.trans (<a>min_le_left</a> _ _))", [{"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"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\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'w : b' \u2208 w\nb'k : B \u2191b' \u2286 k\n\u22a2 closedBall (c a) (r a) \u2286 closedBall (c a) d", "state_after": "no goals"}, {"tactic": "rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9", "annotated_tactic": ["rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nab : Set.Nonempty (B a \u2229 B b)\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B a\neb : e \u2208 B b\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)"}, {"tactic": "have A : dist (c a) e \u2264 r a := mem_closedBall'.1 (hB a hat ea)", "annotated_tactic": ["have A : <a>dist</a> (c a) e \u2264 r a := <a>mem_closedBall'</a>.1 (hB a hat ea)", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"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\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B a\neb : e \u2208 B b\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B a\neb : e \u2208 B b\nA : dist (c a) e \u2264 r a\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)"}, {"tactic": "have B : dist e (c b) \u2264 r b := mem_closedBall.1 (hB b (ut bu) eb)", "annotated_tactic": ["have B : <a>dist</a> e (c b) \u2264 r b := <a>mem_closedBall</a>.1 (hB b (ut bu) eb)", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B a\neb : e \u2208 B b\nA : dist (c a) e \u2264 r a\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB\u271d : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B\u271d a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B\u271d a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B\u271d a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B\u271d a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\u271d\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 B\u271d b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B\u271d a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B\u271d a \u2229 ball x (R x)) \u2192 B\u271d a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B\u271d \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B\u271d \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B\u271d \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B\u271d a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B\u271d a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B\u271d a\neb : e \u2208 B\u271d b\nA : dist (c a) e \u2264 r a\nB : dist e (c b) \u2264 r b\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)"}, {"tactic": "simp only [mem_closedBall]", "annotated_tactic": ["simp only [<a>mem_closedBall</a>]", [{"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB\u271d : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B\u271d a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B\u271d a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B\u271d a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B\u271d a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\u271d\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 B\u271d b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B\u271d a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B\u271d a \u2229 ball x (R x)) \u2192 B\u271d a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B\u271d \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B\u271d \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B\u271d \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B\u271d a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B\u271d a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B\u271d a\neb : e \u2208 B\u271d b\nA : dist (c a) e \u2264 r a\nB : dist e (c b) \u2264 r b\n\u22a2 c a \u2208 closedBall (c b) (3 * r b)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB\u271d : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B\u271d a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B\u271d a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B\u271d a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B\u271d a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\u271d\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 B\u271d b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B\u271d a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B\u271d a \u2229 ball x (R x)) \u2192 B\u271d a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B\u271d \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B\u271d \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B\u271d \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B\u271d a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B\u271d a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B\u271d a\neb : e \u2208 B\u271d b\nA : dist (c a) e \u2264 r a\nB : dist e (c b) \u2264 r b\n\u22a2 dist (c a) (c b) \u2264 3 * r b"}, {"tactic": "linarith only [dist_triangle (c a) e (c b), A, B, bdiam]", "annotated_tactic": ["linarith only [<a>dist_triangle</a> (c a) e (c b), A, B, bdiam]", [{"full_name": "dist_triangle", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MetricSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b1\nt : Set \u03b9\nC : \u211d\u22650\nr : \u03b9 \u2192 \u211d\nc : \u03b9 \u2192 \u03b1\nB\u271d : \u03b9 \u2192 Set \u03b1\nhB : \u2200 (a : \u03b9), a \u2208 t \u2192 B\u271d a \u2286 closedBall (c a) (r a)\n\u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B\u271d a)\nht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B\u271d a))\nh't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B\u271d a)\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x\nR : \u03b1 \u2192 \u211d\nhR0 : \u2200 (x : \u03b1), 0 < R x\nhR1 : \u2200 (x : \u03b1), R x \u2264 1\nhR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4\nt' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)}\nu : Set \u03b9\nut' : u \u2286 t'\nu_disj : PairwiseDisjoint u B\u271d\nhu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 B\u271d b) \u2227 r a \u2264 2 * r b\nut : u \u2286 t\nu_count : Set.Countable u\nx : \u03b1\nx\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B\u271d a\nv : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 ball x (R x))}\nvu : v \u2286 u\nK : \u211d\n\u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4\nhK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B\u271d a \u2229 ball x (R x)) \u2192 B\u271d a \u2286 closedBall x K\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nI : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B\u271d \u2191a) < \u22a4\nw : Finset \u2191v\nhw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B\u271d \u2191\u2191a) < \u03b5 / \u2191C\nk : Set \u03b1 := \u22c3 a \u2208 w, B\u271d \u2191a\nk_closed : IsClosed k\nd : \u211d\ndpos : 0 < d\na : \u03b9\nhat : a \u2208 t\nhz : c a \u2208 (s \\ \u22c3 a \u2208 u, B\u271d a) \u2229 ball x (R x)\nz_notmem_k : \u00acc a \u2208 k\nthis : ball x (R x) \\ k \u2208 \ud835\udcdd (c a)\nhd : closedBall (c a) d \u2286 ball x (R x) \\ k\nad : r a \u2264 min d (R (c a))\nax : B\u271d a \u2286 ball x (R x)\nb : \u03b9\nbu : b \u2208 u\nbdiam : r a \u2264 2 * r b\nbv : b \u2208 v\nb' : \u2191v := { val := b, property := bv }\nb'_notmem_w : \u00acb' \u2208 w\nb'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w }\ne : \u03b1\nea : e \u2208 B\u271d a\neb : e \u2208 B\u271d b\nA : dist (c a) e \u2264 r a\nB : dist e (c b) \u2264 r b\n\u22a2 dist (c a) (c b) \u2264 3 * r 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_iff", "start": [749, 1], "end": [751, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.lt_to_int", "start": [1380, 1], "end": [1385, 49], "traced_tactics": [{"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nh : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt", "state_after": "\u03b1 : Type u_1\nm n : ZNum\nh : \u2191m < \u2191n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nh : \u2191m < \u2191n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt", "state_after": "no goals"}, {"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nh : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt", "state_after": "\u03b1 : Type u_1\nm n : ZNum\nh : m = n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt"}, {"tactic": "simp [h, lt_irrefl]", "annotated_tactic": ["simp [h, <a>lt_irrefl</a>]", [{"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\nm n : ZNum\nh : m = n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt", "state_after": "no goals"}, {"tactic": "simp [not_lt_of_gt h]", "annotated_tactic": ["simp [<a>not_lt_of_gt</a> h]", [{"full_name": "not_lt_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [307, 9], "def_end_pos": [307, 21]}]], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nh : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.gt = Ordering.lt", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.product_singleton", "start": [227, 1], "end": [229, 32], "traced_tactics": [{"tactic": "ext \u27e8x, y\u27e9", "annotated_tactic": ["ext \u27e8x, y\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s' : Finset \u03b1\nt t' : Finset \u03b2\na : \u03b1\nb\u271d b : \u03b2\n\u22a2 s \u00d7\u02e2 {b} = map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s", "state_after": "case a.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s' : Finset \u03b1\nt t' : Finset \u03b2\na : \u03b1\nb\u271d b : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 {b} \u2194 (x, y) \u2208 map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s"}, {"tactic": "simp [and_left_comm, eq_comm]", "annotated_tactic": ["simp [<a>and_left_comm</a>, <a>eq_comm</a>]", [{"full_name": "and_left_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}, {"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.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s' : Finset \u03b1\nt t' : Finset \u03b2\na : \u03b1\nb\u271d b : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 {b} \u2194 (x, y) \u2208 map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.setLaverage_congr", "start": [122, 1], "end": [123, 69], "traced_tactics": [{"tactic": "simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h]", "annotated_tactic": ["simp only [<a>setLaverage_eq</a>, <a>set_lintegral_congr</a> h, <a>measure_congr</a> h]", [{"full_name": "MeasureTheory.setLaverage_eq", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [107, 9], "def_end_pos": [107, 23]}, {"full_name": "MeasureTheory.set_lintegral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [312, 9], "def_end_pos": [312, 28]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 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\nh : s =\u1da0[ae \u03bc] t\n\u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc", "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": [529, 1], "end": [545, 38], "traced_tactics": [{"tactic": "rw [Cofix.corec', Cofix.dest_corec]", "annotated_tactic": ["rw [<a>Cofix.corec'</a>, <a>Cofix.dest_corec</a>]", [{"full_name": "MvQPF.Cofix.corec'", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [171, 5], "def_end_pos": [171, 17]}, {"full_name": "MvQPF.Cofix.dest_corec", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [184, 9], "def_end_pos": [184, 25]}]], "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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 dest (corec' g x) = (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$>\n      Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g (Sum.inr x) =\n    (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$>\n      Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g (Sum.inr x) =\n    (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$> g x =\n    (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x"}, {"tactic": "congr!", "annotated_tactic": ["congr!", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$> g x =\n    (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x", "state_after": "case h.e'_6.h.e'_7\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) = Sum.elim _root_.id (corec' g)"}, {"tactic": "ext (i | i) <;> erw [corec_roll] <;> dsimp [Cofix.corec']", "annotated_tactic": ["ext (i | i) <;> erw [<a>corec_roll</a>] <;> dsimp [<a>Cofix.corec'</a>]", [{"full_name": "MvQPF.corec_roll", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [520, 9], "def_end_pos": [520, 19]}, {"full_name": "MvQPF.Cofix.corec'", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [171, 5], "def_end_pos": [171, 17]}]], "state_before": "case h.e'_6.h.e'_7\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) = Sum.elim _root_.id (corec' g)", "state_after": "case h.e'_6.h.e'_7.h.inl\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\ni : Cofix F \u03b1\n\u22a2 corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl i) =\n    i\n\ncase h.e'_6.h.e'_7.h.inr\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\n\u22a2 corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inr i) =\n    corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) (Sum.inr i)"}, {"tactic": "mv_bisim i with R a b x Ha Hb", "annotated_tactic": ["mv_bisim i with R a b x Ha Hb", []], "state_before": "case h.e'_6.h.e'_7.h.inl\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\ni : Cofix F \u03b1\n\u22a2 corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl i) =\n    i", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest a) (dest b)"}, {"tactic": "rw [Ha, Hb, Cofix.dest_corec]", "annotated_tactic": ["rw [Ha, Hb, <a>Cofix.dest_corec</a>]", [{"full_name": "MvQPF.Cofix.dest_corec", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [184, 9], "def_end_pos": [184, 25]}]], "state_before": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest a) (dest b)", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR' (RelLast' \u03b1 R)\n    ((TypeVec.id :::\n        corec\n          (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n            Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)) <$$>\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n          Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n        (Sum.inl x))\n    (dest x)"}, {"tactic": "dsimp [Function.comp]", "annotated_tactic": ["dsimp [<a>Function.comp</a>]", [{"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR' (RelLast' \u03b1 R)\n    ((TypeVec.id :::\n        corec\n          (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n            Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)) <$$>\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n          Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n        (Sum.inl x))\n    (dest x)", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n    (RelLast' \u03b1 fun a b =>\n      \u2203 x,\n        a =\n            corec\n              (fun x =>\n                (TypeVec.id ::: fun x => x) <$$>\n                  Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n              (Sum.inl x) \u2227\n          b = x)\n    ((TypeVec.id :::\n        corec fun x =>\n          (TypeVec.id ::: fun x => x) <$$>\n            Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) <$$>\n      (TypeVec.id ::: fun x => x) <$$> (TypeVec.id ::: Sum.inl) <$$> dest x)\n    (dest x)"}, {"tactic": "repeat rw [MvFunctor.map_map, \u2190 appendFun_comp_id]", "annotated_tactic": ["repeat rw [<a>MvFunctor.map_map</a>, \u2190 <a>appendFun_comp_id</a>]", [{"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}]], "state_before": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n    (RelLast' \u03b1 fun a b =>\n      \u2203 x,\n        a =\n            corec\n              (fun x =>\n                (TypeVec.id ::: fun x => x) <$$>\n                  Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n              (Sum.inl x) \u2227\n          b = x)\n    ((TypeVec.id :::\n        corec fun x =>\n          (TypeVec.id ::: fun x => x) <$$>\n            Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) <$$>\n      (TypeVec.id ::: fun x => x) <$$> (TypeVec.id ::: Sum.inl) <$$> dest x)\n    (dest x)", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n    (RelLast' \u03b1 fun a b =>\n      \u2203 x,\n        a =\n            corec\n              (fun x =>\n                (TypeVec.id ::: fun x => x) <$$>\n                  Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n              (Sum.inl x) \u2227\n          b = x)\n    ((TypeVec.id :::\n        ((corec fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n            fun x => x) \u2218\n          Sum.inl) <$$>\n      dest x)\n    (dest x)"}, {"tactic": "apply liftR_map_last'", "annotated_tactic": ["apply <a>liftR_map_last'</a>", [{"full_name": "MvQPF.liftR_map_last'", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [422, 9], "def_end_pos": [422, 24]}]], "state_before": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n    (RelLast' \u03b1 fun a b =>\n      \u2203 x,\n        a =\n            corec\n              (fun x =>\n                (TypeVec.id ::: fun x => x) <$$>\n                  Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n              (Sum.inl x) \u2227\n          b = x)\n    ((TypeVec.id :::\n        ((corec fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n            fun x => x) \u2218\n          Sum.inl) <$$>\n      dest x)\n    (dest x)", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 \u2200 (x : Cofix F \u03b1),\n    \u2203 x_1,\n      (((corec fun x =>\n                  (TypeVec.id ::: fun x => x) <$$>\n                    Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n                fun x => x) \u2218\n              Sum.inl)\n            x =\n          corec\n            (fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n            (Sum.inl x_1) \u2227\n        x = x_1"}, {"tactic": "dsimp [Function.comp]", "annotated_tactic": ["dsimp [<a>Function.comp</a>]", [{"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.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 \u2200 (x : Cofix F \u03b1),\n    \u2203 x_1,\n      (((corec fun x =>\n                  (TypeVec.id ::: fun x => x) <$$>\n                    Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n                fun x => x) \u2218\n              Sum.inl)\n            x =\n          corec\n            (fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n            (Sum.inl x_1) \u2227\n        x = x_1", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 \u2200 (x : Cofix F \u03b1),\n    \u2203 x_1,\n      corec\n            (fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n            (Sum.inl x) =\n          corec\n            (fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n            (Sum.inl x_1) \u2227\n        x = x_1"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 \u2200 (x : Cofix F \u03b1),\n    \u2203 x_1,\n      corec\n            (fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n            (Sum.inl x) =\n          corec\n            (fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n            (Sum.inl x_1) \u2227\n        x = x_1", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d\u00b9 : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\nx\u271d : Cofix F \u03b1\n\u22a2 \u2203 x,\n    corec\n          (fun x =>\n            (TypeVec.id ::: fun x => x) <$$>\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n          (Sum.inl x\u271d) =\n        corec\n          (fun x =>\n            (TypeVec.id ::: fun x => x) <$$>\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n          (Sum.inl x) \u2227\n      x\u271d = x"}, {"tactic": "exact \u27e8_, rfl, rfl\u27e9", "annotated_tactic": ["exact \u27e8_, <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": "case h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d\u00b9 : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\nx\u271d : Cofix F \u03b1\n\u22a2 \u2203 x,\n    corec\n          (fun x =>\n            (TypeVec.id ::: fun x => x) <$$>\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n          (Sum.inl x\u271d) =\n        corec\n          (fun x =>\n            (TypeVec.id ::: fun x => x) <$$>\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n          (Sum.inl x) \u2227\n      x\u271d = x", "state_after": "no goals"}, {"tactic": "rw [MvFunctor.map_map, \u2190 appendFun_comp_id]", "annotated_tactic": ["rw [<a>MvFunctor.map_map</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": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}]], "state_before": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n    (RelLast' \u03b1 fun a b =>\n      \u2203 x,\n        a =\n            corec\n              (fun x =>\n                (TypeVec.id ::: fun x => x) <$$>\n                  Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n              (Sum.inl x) \u2227\n          b = x)\n    ((TypeVec.id :::\n        (corec fun x =>\n            (TypeVec.id ::: fun x => x) <$$>\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n          fun x => x) <$$>\n      (TypeVec.id ::: Sum.inl) <$$> dest x)\n    (dest x)", "state_after": "case h.e'_6.h.e'_7.h.inl.intro.intro\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n  fun a b =>\n    \u2203 x,\n      a =\n          corec\n            (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n              Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n            (Sum.inl x) \u2227\n        b = x\na b x : Cofix F \u03b1\nHa :\n  a =\n    corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n    (RelLast' \u03b1 fun a b =>\n      \u2203 x,\n        a =\n            corec\n              (fun x =>\n                (TypeVec.id ::: fun x => x) <$$>\n                  Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n              (Sum.inl x) \u2227\n          b = x)\n    ((TypeVec.id :::\n        ((corec fun x =>\n              (TypeVec.id ::: fun x => x) <$$>\n                Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n            fun x => x) \u2218\n          Sum.inl) <$$>\n      dest x)\n    (dest x)"}, {"tactic": "congr with y", "annotated_tactic": ["congr with y", []], "state_before": "case h.e'_6.h.e'_7.h.inr\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\n\u22a2 corec\n      (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      (Sum.inr i) =\n    corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) (Sum.inr i)", "state_after": "case h.e'_6.h.e'_7.h.inr.e_g.h\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\ny : Cofix F \u03b1 \u2295 \u03b2\n\u22a2 (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      y =\n    Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y"}, {"tactic": "erw [appendFun_id_id]", "annotated_tactic": ["erw [<a>appendFun_id_id</a>]", [{"full_name": "TypeVec.appendFun_id_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [292, 9], "def_end_pos": [292, 24]}]], "state_before": "case h.e'_6.h.e'_7.h.inr.e_g.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\ny : Cofix F \u03b1 \u2295 \u03b2\n\u22a2 (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n        Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n      y =\n    Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y", "state_after": "case h.e'_6.h.e'_7.h.inr.e_g.h\nn : \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 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\ny : Cofix F \u03b1 \u2295 \u03b2\n\u22a2 (MvFunctor.map TypeVec.id \u2218 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val) y =\n    Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y"}, {"tactic": "simp [MvFunctor.id_map, Sum.elim]", "annotated_tactic": ["simp [<a>MvFunctor.id_map</a>, <a>Sum.elim</a>]", [{"full_name": "MvFunctor.id_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [106, 9], "def_end_pos": [106, 15]}, {"full_name": "Sum.elim", "def_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "def_pos": [94, 15], "def_end_pos": [94, 19]}]], "state_before": "case h.e'_6.h.e'_7.h.inr.e_g.h\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\ny : Cofix F \u03b1 \u2295 \u03b2\n\u22a2 (MvFunctor.map TypeVec.id \u2218 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val) y =\n    Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.withDensity_le_mul", "start": [542, 1], "end": [605, 101], "traced_tactics": [{"tactic": "have t_ne_zero' : t \u2260 0 := (zero_lt_one.trans ht).ne'", "annotated_tactic": ["have t_ne_zero' : t \u2260 0 := (zero_lt_one.trans ht).<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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "have t_ne_zero : (t : \u211d\u22650\u221e) \u2260 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'", "annotated_tactic": ["have t_ne_zero : (t : \u211d\u22650\u221e) \u2260 0 := by simpa only [<a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>] using t_ne_zero'", [{"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]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "let \u03bd := \u03bc.withDensity (v.limRatioMeas h\u03c1)", "annotated_tactic": ["let \u03bd := \u03bc.withDensity (v.limRatioMeas h\u03c1)", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "let f := v.limRatioMeas h\u03c1", "annotated_tactic": ["let f := v.limRatioMeas h\u03c1", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "have f_meas : Measurable f := v.limRatioMeas_measurable h\u03c1", "annotated_tactic": ["have f_meas : <a>Measurable</a> f := v.limRatioMeas_measurable h\u03c1", [{"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "have A : \u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) := by\n  apply le_trans _ (zero_le _)\n  have M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) := hs.inter (f_meas (measurableSet_singleton _))\n  simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]\n  apply (ae_restrict_iff' M).2\n  exact eventually_of_forall fun x hx => hx.2", "annotated_tactic": ["have A : \u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) := by\n    apply <a>le_trans</a> _ (<a>zero_le</a> _)\n    have M : <a>MeasurableSet</a> (s \u2229 f \u207b\u00b9' {0}) := hs.inter (f_meas (<a>measurableSet_singleton</a> _))\n    simp only [<a>nonpos_iff_eq_zero</a>, M, <a>withDensity_apply</a>, <a>lintegral_eq_zero_iff</a> f_meas]\n    apply (<a>ae_restrict_iff'</a> M).2\n    exact <a>eventually_of_forall</a> fun x hx => hx.2", [{"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]}, {"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": "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.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]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "have B : \u03bd (s \u2229 f \u207b\u00b9' {\u221e}) \u2264 ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u221e}) := by\n  apply le_trans (le_of_eq _) (zero_le _)\n  apply withDensity_absolutelyContinuous \u03bc _\n  rw [\u2190 nonpos_iff_eq_zero]\n  exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top h\u03c1).le", "annotated_tactic": ["have B : \u03bd (s \u2229 f \u207b\u00b9' {\u221e}) \u2264 ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u221e}) := by\n    apply <a>le_trans</a> (<a>le_of_eq</a> _) (<a>zero_le</a> _)\n    apply <a>withDensity_absolutelyContinuous</a> \u03bc _\n    rw [\u2190 <a>nonpos_iff_eq_zero</a>]\n    exact (<a>measure_mono</a> (<a>inter_subset_right</a> _ _)).<a>trans</a> (v.measure_limRatioMeas_top h\u03c1).<a>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_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 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.withDensity_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [117, 9], "def_end_pos": [117, 41]}, {"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": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "have C :\n  \u2200 n : \u2124,\n    \u03bd (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) \u2264\n      ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n  intro n\n  let I := Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))\n  have M : MeasurableSet (s \u2229 f \u207b\u00b9' I) := hs.inter (f_meas measurableSet_Ico)\n  simp only [M, withDensity_apply, coe_nnreal_smul_apply]\n  calc\n    (\u222b\u207b x in s \u2229 f \u207b\u00b9' I, f x \u2202\u03bc) \u2264 \u222b\u207b x in s \u2229 f \u207b\u00b9' I, (t : \u211d\u22650\u221e) ^ (n + 1) \u2202\u03bc :=\n      lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))\n    _ = (t : \u211d\u22650\u221e) ^ (n + 1) * \u03bc (s \u2229 f \u207b\u00b9' I) := by\n      simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n    _ = (t : \u211d\u22650\u221e) ^ (2 : \u2124) * ((t : \u211d\u22650\u221e) ^ (n - 1) * \u03bc (s \u2229 f \u207b\u00b9' I)) := by\n      rw [\u2190 mul_assoc, \u2190 ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]\n      congr 2\n      abel\n    _ \u2264 (t : \u211d\u22650\u221e) ^ 2 * \u03c1 (s \u2229 f \u207b\u00b9' I) := by\n      refine' mul_le_mul_left' _ _\n      rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne']\n      apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1\n      intro x hx\n      apply lt_of_lt_of_le _ hx.2.1\n      rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n        zpow_add\u2080 t_ne_zero']\n      conv_rhs => rw [\u2190 mul_one (t ^ n)]\n      refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)\n      rw [zpow_neg_one]\n      exact inv_lt_one ht", "annotated_tactic": ["have C :\n    \u2200 n : \u2124,\n      \u03bd (s \u2229 f \u207b\u00b9' <a>Ico</a> ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) \u2264\n        ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' <a>Ico</a> ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n    intro n\n    let I := <a>Ico</a> ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))\n    have M : <a>MeasurableSet</a> (s \u2229 f \u207b\u00b9' I) := hs.inter (f_meas <a>measurableSet_Ico</a>)\n    simp only [M, <a>withDensity_apply</a>, <a>coe_nnreal_smul_apply</a>]\n    calc\n      (\u222b\u207b x in s \u2229 f \u207b\u00b9' I, f x \u2202\u03bc) \u2264 \u222b\u207b x in s \u2229 f \u207b\u00b9' I, (t : \u211d\u22650\u221e) ^ (n + 1) \u2202\u03bc :=\n        <a>lintegral_mono_ae</a> ((<a>ae_restrict_iff'</a> M).2 (<a>eventually_of_forall</a> fun x hx => hx.2.2.<a>le</a>))\n      _ = (t : \u211d\u22650\u221e) ^ (n + 1) * \u03bc (s \u2229 f \u207b\u00b9' I) := by\n        simp only [<a>lintegral_const</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>, <a>univ_inter</a>]\n      _ = (t : \u211d\u22650\u221e) ^ (2 : \u2124) * ((t : \u211d\u22650\u221e) ^ (n - 1) * \u03bc (s \u2229 f \u207b\u00b9' I)) := by\n        rw [\u2190 <a>mul_assoc</a>, \u2190 <a>ENNReal.zpow_add</a> t_ne_zero <a>ENNReal.coe_ne_top</a>]\n        congr 2\n        abel\n      _ \u2264 (t : \u211d\u22650\u221e) ^ 2 * \u03c1 (s \u2229 f \u207b\u00b9' I) := by\n        refine' <a>mul_le_mul_left'</a> _ _\n        rw [\u2190 <a>ENNReal.coe_zpow</a> (zero_lt_one.trans ht).<a>ne'</a>]\n        apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1\n        intro x hx\n        apply <a>lt_of_lt_of_le</a> _ hx.2.1\n        rw [\u2190 <a>ENNReal.coe_zpow</a> (zero_lt_one.trans ht).<a>ne'</a>, <a>ENNReal.coe_lt_coe</a>, <a>sub_eq_add_neg</a>,\n          <a>zpow_add\u2080</a> t_ne_zero']\n        conv_rhs => rw [\u2190 <a>mul_one</a> (t ^ n)]\n        refine' <a>mul_lt_mul'</a> <a>le_rfl</a> _ (<a>zero_le</a> _) (<a>NNReal.zpow_pos</a> t_ne_zero' _)\n        rw [<a>zpow_neg_one</a>]\n        exact <a>inv_lt_one</a> ht", [{"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_Ico", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 26]}, {"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.coe_nnreal_smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [905, 9], "def_end_pos": [905, 30]}, {"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"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": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.zpow_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1986, 19], "def_end_pos": [1986, 27]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"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.coe_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"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_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "zpow_add\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Power.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"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'", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [525, 9], "def_end_pos": [525, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "NNReal.zpow_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "zpow_neg_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}, {"full_name": "inv_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 19]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nC : \u2200 (n : \u2124), \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "calc\n  \u03bd s =\n    \u03bd (s \u2229 f \u207b\u00b9' {0}) + \u03bd (s \u2229 f \u207b\u00b9' {\u221e}) +\n      \u2211' n : \u2124, \u03bd (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) :=\n    measure_eq_measure_preimage_add_measure_tsum_Ico_zpow \u03bd f_meas hs ht\n  _ \u2264\n      ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) + ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u221e}) +\n        \u2211' n : \u2124, ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) :=\n    (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C))\n  _ = ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) s :=\n    (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) f_meas hs ht).symm", "annotated_tactic": ["calc\n    \u03bd s =\n      \u03bd (s \u2229 f \u207b\u00b9' {0}) + \u03bd (s \u2229 f \u207b\u00b9' {\u221e}) +\n        \u2211' n : \u2124, \u03bd (s \u2229 f \u207b\u00b9' <a>Ico</a> ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) :=\n      <a>measure_eq_measure_preimage_add_measure_tsum_Ico_zpow</a> \u03bd f_meas hs ht\n    _ \u2264\n        ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) + ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u221e}) +\n          \u2211' n : \u2124, ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' <a>Ico</a> ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) :=\n      (<a>add_le_add</a> (<a>add_le_add</a> A B) (<a>ENNReal.tsum_le_tsum</a> C))\n    _ = ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) s :=\n      (<a>measure_eq_measure_preimage_add_measure_tsum_Ico_zpow</a> ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) f_meas hs ht).<a>symm</a>", [{"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"full_name": "measure_eq_measure_preimage_add_measure_tsum_Ico_zpow", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1615, 9], "def_end_pos": [1615, 62]}, {"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}, {"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": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "measure_eq_measure_preimage_add_measure_tsum_Ico_zpow", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1615, 9], "def_end_pos": [1615, 62]}, {"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nC : \u2200 (n : \u2124), \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s", "state_after": "no goals"}, {"tactic": "simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'", "annotated_tactic": ["simpa only [<a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>] using t_ne_zero'", [{"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]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\n\u22a2 \u2191t \u2260 0", "state_after": "no goals"}, {"tactic": "apply le_trans _ (zero_le _)", "annotated_tactic": ["apply <a>le_trans</a> _ (<a>zero_le</a> _)", [{"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 0"}, {"tactic": "have M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) := hs.inter (f_meas (measurableSet_singleton _))", "annotated_tactic": ["have M : <a>MeasurableSet</a> (s \u2229 f \u207b\u00b9' {0}) := hs.inter (f_meas (<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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nM : MeasurableSet (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 0"}, {"tactic": "simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]", "annotated_tactic": ["simp only [<a>nonpos_iff_eq_zero</a>, M, <a>withDensity_apply</a>, <a>lintegral_eq_zero_iff</a> f_meas]", [{"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.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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nM : MeasurableSet (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nM : MeasurableSet (s \u2229 f \u207b\u00b9' {0})\n\u22a2 limRatioMeas v h\u03c1 =\u1da0[ae (Measure.restrict \u03bc (s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' {0}))] 0"}, {"tactic": "apply (ae_restrict_iff' M).2", "annotated_tactic": ["apply (<a>ae_restrict_iff'</a> M).2", [{"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nM : MeasurableSet (s \u2229 f \u207b\u00b9' {0})\n\u22a2 limRatioMeas v h\u03c1 =\u1da0[ae (Measure.restrict \u03bc (s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' {0}))] 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nM : MeasurableSet (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2229 f \u207b\u00b9' {0} \u2192 limRatioMeas v h\u03c1 x = OfNat.ofNat 0 x"}, {"tactic": "exact eventually_of_forall fun x hx => hx.2", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x hx => hx.2", [{"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nM : MeasurableSet (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2229 f \u207b\u00b9' {0} \u2192 limRatioMeas v h\u03c1 x = OfNat.ofNat 0 x", "state_after": "no goals"}, {"tactic": "apply le_trans (le_of_eq _) (zero_le _)", "annotated_tactic": ["apply <a>le_trans</a> (<a>le_of_eq</a> _) (<a>zero_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_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) = 0"}, {"tactic": "apply withDensity_absolutelyContinuous \u03bc _", "annotated_tactic": ["apply <a>withDensity_absolutelyContinuous</a> \u03bc _", [{"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) = 0", "state_after": "case 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) = 0"}, {"tactic": "rw [\u2190 nonpos_iff_eq_zero]", "annotated_tactic": ["rw [\u2190 <a>nonpos_iff_eq_zero</a>]", [{"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 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) = 0", "state_after": "case 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 0"}, {"tactic": "exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top h\u03c1).le", "annotated_tactic": ["exact (<a>measure_mono</a> (<a>inter_subset_right</a> _ _)).<a>trans</a> (v.measure_limRatioMeas_top h\u03c1).<a>le</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": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "state_before": "case 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\n\u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 0", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\n\u22a2 \u2200 (n : \u2124), \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))"}, {"tactic": "let I := Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))", "annotated_tactic": ["let I := <a>Ico</a> ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))", [{"full_name": "Set.Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 8]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))"}, {"tactic": "have M : MeasurableSet (s \u2229 f \u207b\u00b9' I) := hs.inter (f_meas measurableSet_Ico)", "annotated_tactic": ["have M : <a>MeasurableSet</a> (s \u2229 f \u207b\u00b9' I) := hs.inter (f_meas <a>measurableSet_Ico</a>)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))"}, {"tactic": "simp only [M, withDensity_apply, coe_nnreal_smul_apply]", "annotated_tactic": ["simp only [M, <a>withDensity_apply</a>, <a>coe_nnreal_smul_apply</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.Measure.coe_nnreal_smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [905, 9], "def_end_pos": [905, 30]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u222b\u207b (a : \u03b1) in s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)), limRatioMeas v h\u03c1 a \u2202\u03bc \u2264\n    \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))"}, {"tactic": "calc\n  (\u222b\u207b x in s \u2229 f \u207b\u00b9' I, f x \u2202\u03bc) \u2264 \u222b\u207b x in s \u2229 f \u207b\u00b9' I, (t : \u211d\u22650\u221e) ^ (n + 1) \u2202\u03bc :=\n    lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))\n  _ = (t : \u211d\u22650\u221e) ^ (n + 1) * \u03bc (s \u2229 f \u207b\u00b9' I) := by\n    simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n  _ = (t : \u211d\u22650\u221e) ^ (2 : \u2124) * ((t : \u211d\u22650\u221e) ^ (n - 1) * \u03bc (s \u2229 f \u207b\u00b9' I)) := by\n    rw [\u2190 mul_assoc, \u2190 ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]\n    congr 2\n    abel\n  _ \u2264 (t : \u211d\u22650\u221e) ^ 2 * \u03c1 (s \u2229 f \u207b\u00b9' I) := by\n    refine' mul_le_mul_left' _ _\n    rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne']\n    apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1\n    intro x hx\n    apply lt_of_lt_of_le _ hx.2.1\n    rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n      zpow_add\u2080 t_ne_zero']\n    conv_rhs => rw [\u2190 mul_one (t ^ n)]\n    refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)\n    rw [zpow_neg_one]\n    exact inv_lt_one ht", "annotated_tactic": ["calc\n      (\u222b\u207b x in s \u2229 f \u207b\u00b9' I, f x \u2202\u03bc) \u2264 \u222b\u207b x in s \u2229 f \u207b\u00b9' I, (t : \u211d\u22650\u221e) ^ (n + 1) \u2202\u03bc :=\n        <a>lintegral_mono_ae</a> ((<a>ae_restrict_iff'</a> M).2 (<a>eventually_of_forall</a> fun x hx => hx.2.2.<a>le</a>))\n      _ = (t : \u211d\u22650\u221e) ^ (n + 1) * \u03bc (s \u2229 f \u207b\u00b9' I) := by\n        simp only [<a>lintegral_const</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>, <a>univ_inter</a>]\n      _ = (t : \u211d\u22650\u221e) ^ (2 : \u2124) * ((t : \u211d\u22650\u221e) ^ (n - 1) * \u03bc (s \u2229 f \u207b\u00b9' I)) := by\n        rw [\u2190 <a>mul_assoc</a>, \u2190 <a>ENNReal.zpow_add</a> t_ne_zero <a>ENNReal.coe_ne_top</a>]\n        congr 2\n        abel\n      _ \u2264 (t : \u211d\u22650\u221e) ^ 2 * \u03c1 (s \u2229 f \u207b\u00b9' I) := by\n        refine' <a>mul_le_mul_left'</a> _ _\n        rw [\u2190 <a>ENNReal.coe_zpow</a> (zero_lt_one.trans ht).<a>ne'</a>]\n        apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1\n        intro x hx\n        apply <a>lt_of_lt_of_le</a> _ hx.2.1\n        rw [\u2190 <a>ENNReal.coe_zpow</a> (zero_lt_one.trans ht).<a>ne'</a>, <a>ENNReal.coe_lt_coe</a>, <a>sub_eq_add_neg</a>,\n          <a>zpow_add\u2080</a> t_ne_zero']\n        conv_rhs => rw [\u2190 <a>mul_one</a> (t ^ n)]\n        refine' <a>mul_lt_mul'</a> <a>le_rfl</a> _ (<a>zero_le</a> _) (<a>NNReal.zpow_pos</a> t_ne_zero' _)\n        rw [<a>zpow_neg_one</a>]\n        exact <a>inv_lt_one</a> ht", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [265, 9], "def_end_pos": [265, 26]}, {"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"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": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.zpow_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1986, 19], "def_end_pos": [1986, 27]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"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.coe_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"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_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "zpow_add\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Power.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"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'", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [525, 9], "def_end_pos": [525, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "NNReal.zpow_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [918, 9], "def_end_pos": [918, 17]}, {"full_name": "zpow_neg_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}, {"full_name": "inv_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 19]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u222b\u207b (a : \u03b1) in s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)), limRatioMeas v h\u03c1 a \u2202\u03bc \u2264\n    \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)))", "state_after": "no goals"}, {"tactic": "simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</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": "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]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u222b\u207b (x : \u03b1) in s \u2229 f \u207b\u00b9' I, \u2191t ^ (n + 1) \u2202\u03bc = \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I)", "state_after": "no goals"}, {"tactic": "rw [\u2190 mul_assoc, \u2190 ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>, \u2190 <a>ENNReal.zpow_add</a> t_ne_zero <a>ENNReal.coe_ne_top</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.zpow_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1986, 19], "def_end_pos": [1986, 27]}, {"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) = \u2191t ^ 2 * (\u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) = \u2191t ^ (2 + (n - 1)) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I)"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) = \u2191t ^ (2 + (n - 1)) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I)", "state_after": "case e_a.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 n + 1 = 2 + (n - 1)"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case e_a.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 n + 1 = 2 + (n - 1)", "state_after": "no goals"}, {"tactic": "refine' mul_le_mul_left' _ _", "annotated_tactic": ["refine' <a>mul_le_mul_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]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191t ^ 2 * (\u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I)) \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I)", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) \u2264 \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I)"}, {"tactic": "rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne']", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_zpow</a> (zero_lt_one.trans ht).<a>ne'</a>]", [{"full_name": "ENNReal.coe_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 17]}, {"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) \u2264 \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I)", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191(t ^ (n - 1)) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) \u2264 \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I)"}, {"tactic": "apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1", "annotated_tactic": ["apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 \u2191(t ^ (n - 1)) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) \u2264 \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I)", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 s \u2229 f \u207b\u00b9' I \u2286 {x | \u2191(t ^ (n - 1)) < limRatioMeas v h\u03c1 x}"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\n\u22a2 s \u2229 f \u207b\u00b9' I \u2286 {x | \u2191(t ^ (n - 1)) < limRatioMeas v h\u03c1 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 x \u2208 {x | \u2191(t ^ (n - 1)) < limRatioMeas v h\u03c1 x}"}, {"tactic": "apply lt_of_lt_of_le _ hx.2.1", "annotated_tactic": ["apply <a>lt_of_lt_of_le</a> _ hx.2.1", [{"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": "\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 x \u2208 {x | \u2191(t ^ (n - 1)) < limRatioMeas v h\u03c1 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 \u2191(t ^ (n - 1)) < \u2191t ^ n"}, {"tactic": "rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n  zpow_add\u2080 t_ne_zero']", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_zpow</a> (zero_lt_one.trans ht).<a>ne'</a>, <a>ENNReal.coe_lt_coe</a>, <a>sub_eq_add_neg</a>,\n          <a>zpow_add\u2080</a> t_ne_zero']", [{"full_name": "ENNReal.coe_zpow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1907, 9], "def_end_pos": [1907, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "zpow_add\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Power.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 \u2191(t ^ (n - 1)) < \u2191t ^ n", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t ^ n * t ^ (-1) < t ^ n"}, {"tactic": "conv_rhs => rw [\u2190 mul_one (t ^ n)]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>mul_one</a> (t ^ n)]", [{"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t ^ n * t ^ (-1) < t ^ n", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t ^ n * t ^ (-1) < t ^ n * 1"}, {"tactic": "refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)", "annotated_tactic": ["refine' <a>mul_lt_mul'</a> <a>le_rfl</a> _ (<a>zero_le</a> _) (<a>NNReal.zpow_pos</a> t_ne_zero' _)", [{"full_name": "mul_lt_mul'", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [525, 9], "def_end_pos": [525, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "NNReal.zpow_pos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [918, 9], "def_end_pos": [918, 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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t ^ n * t ^ (-1) < t ^ n * 1", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t ^ (-1) < 1"}, {"tactic": "rw [zpow_neg_one]", "annotated_tactic": ["rw [<a>zpow_neg_one</a>]", [{"full_name": "zpow_neg_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t ^ (-1) < 1", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t\u207b\u00b9 < 1"}, {"tactic": "exact inv_lt_one ht", "annotated_tactic": ["exact <a>inv_lt_one</a> ht", [{"full_name": "inv_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 19]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : \u211d\u22650\nht : 1 < t\nt_ne_zero' : t \u2260 0\nt_ne_zero : \u2191t \u2260 0\n\u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1)\nf : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1\nf_meas : Measurable f\nA : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0})\nB : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4})\nn : \u2124\nI : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1))\nM : MeasurableSet (s \u2229 f \u207b\u00b9' I)\nx : \u03b1\nhx : x \u2208 s \u2229 f \u207b\u00b9' I\n\u22a2 t\u207b\u00b9 < 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_congr_norm_ae", "start": [486, 1], "end": [488, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.ppred_to_nat", "start": [844, 1], "end": [849, 8], "traced_tactics": [{"tactic": "rw [ppred, Option.map_some, Nat.ppred_eq_some.2]", "annotated_tactic": ["rw [<a>ppred</a>, <a>Option.map_some</a>, <a>Nat.ppred_eq_some</a>.2]", [{"full_name": "Num.ppred", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [445, 5], "def_end_pos": [445, 10]}, {"full_name": "Option.map_some", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [125, 9], "def_end_pos": [125, 17]}, {"full_name": "Nat.ppred_eq_some", "def_path": "Mathlib/Data/Nat/PSub.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}]], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 castNum <$> ppred (pos p) = Nat.ppred \u2191(pos p)", "state_after": "\u03b1 : Type u_1\np : PosNum\n\u22a2 Nat.succ \u2191(pred' p) = \u2191(pos p)"}, {"tactic": "rw [PosNum.pred'_to_nat, Nat.succ_pred_eq_of_pos (PosNum.to_nat_pos _)]", "annotated_tactic": ["rw [<a>PosNum.pred'_to_nat</a>, <a>Nat.succ_pred_eq_of_pos</a> (<a>PosNum.to_nat_pos</a> _)]", [{"full_name": "PosNum.pred'_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [533, 9], "def_end_pos": [533, 21]}, {"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": "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\np : PosNum\n\u22a2 Nat.succ \u2191(pred' p) = \u2191(pos p)", "state_after": "\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191p = \u2191(pos p)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191p = \u2191(pos p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "List.foldr_sup_eq_sup_toFinset", "start": [213, 1], "end": [217, 6], "traced_tactics": [{"tactic": "rw [\u2190 coe_fold_r, \u2190 Multiset.fold_dedup_idem, sup_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>sup_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.sup_def", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [44, 9], "def_end_pos": [44, 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 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \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 \u2294 x_1) \u22a5 l = sup (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 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \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 \u2294 x_1) \u22a5 (dedup \u2191l) = Multiset.sup (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 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \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 \u2294 x_1) \u22a5 (dedup \u2191l) = Multiset.sup (dedup \u2191l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "map_finset_sup'", "start": [895, 1], "end": [897, 56], "traced_tactics": [{"tactic": "refine' hs.cons_induction _ _ <;> intros <;> simp [*]", "annotated_tactic": ["refine' hs.cons_induction _ _ <;> intros <;> simp [*]", []], "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 : SemilatticeSup \u03b1\ns\u271d : Finset \u03b2\nH : Finset.Nonempty s\u271d\nf\u271d : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : SupHomClass F \u03b1 \u03b2\nf : F\ns : Finset \u03b9\nhs : Finset.Nonempty s\ng : \u03b9 \u2192 \u03b1\n\u22a2 \u2191f (sup' s hs g) = sup' s hs (\u2191f \u2218 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_of_exists_lt", "start": [2529, 1], "end": [2532, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.totalDegree_finset_prod", "start": [725, 1], "end": [729, 6], "traced_tactics": [{"tactic": "refine' le_trans (totalDegree_multiset_prod _) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>totalDegree_multiset_prod</a> _) _", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MvPolynomial.totalDegree_multiset_prod", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [718, 9], "def_end_pos": [718, 34]}]], "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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 totalDegree (Finset.prod s f) \u2264 \u2211 i in s, totalDegree (f 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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 Multiset.sum (Multiset.map totalDegree (Multiset.map f s.val)) \u2264 \u2211 i in s, totalDegree (f i)"}, {"tactic": "rw [Multiset.map_map]", "annotated_tactic": ["rw [<a>Multiset.map_map</a>]", [{"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 Multiset.sum (Multiset.map totalDegree (Multiset.map f s.val)) \u2264 \u2211 i in s, totalDegree (f 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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 Multiset.sum (Multiset.map (totalDegree \u2218 f) s.val) \u2264 \u2211 i in s, totalDegree (f i)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 Multiset.sum (Multiset.map (totalDegree \u2218 f) s.val) \u2264 \u2211 i in s, totalDegree (f 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.neg_le_sub_left_of_le_add", "start": [1020, 11], "end": [1022, 26], "traced_tactics": [{"tactic": "have h := Int.le_neg_add_of_add_le (Int.sub_left_le_of_le_add h)", "annotated_tactic": ["have h := <a>Int.le_neg_add_of_add_le</a> (<a>Int.sub_left_le_of_le_add</a> h)", [{"full_name": "Int.le_neg_add_of_add_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [957, 19], "def_end_pos": [957, 39]}, {"full_name": "Int.sub_left_le_of_le_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [989, 19], "def_end_pos": [989, 40]}]], "state_before": "a b c : Int\nh : c \u2264 a + b\n\u22a2 -a \u2264 b - c", "state_after": "a b c : Int\nh\u271d : c \u2264 a + b\nh : -a \u2264 -c + b\n\u22a2 -a \u2264 b - c"}, {"tactic": "rwa [Int.add_comm] at h", "annotated_tactic": ["rwa [<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\u271d : c \u2264 a + b\nh : -a \u2264 -c + b\n\u22a2 -a \u2264 b - c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.isPartition_classes", "start": [216, 1], "end": [217, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass", "start": [505, 1], "end": [518, 24], "traced_tactics": [{"tactic": "apply tendsto_iff_dist_tendsto_zero.mpr", "annotated_tactic": ["apply tendsto_iff_dist_tendsto_zero.mpr", []], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)"}, {"tactic": "have obs := fun i => (\u03bcs i).testAgainstNN_lipschitz_estimate f 0", "annotated_tactic": ["have obs := fun i => (\u03bcs i).<a>testAgainstNN_lipschitz_estimate</a> f 0", [{"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [390, 9], "def_end_pos": [390, 41]}]], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 testAgainstNN (\u03bcs i) 0 + nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)"}, {"tactic": "simp_rw [testAgainstNN_zero, zero_add] at obs", "annotated_tactic": ["simp_rw [<a>testAgainstNN_zero</a>, <a>zero_add</a>] at obs", [{"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_zero", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [343, 9], "def_end_pos": [343, 27]}, {"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\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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 testAgainstNN (\u03bcs i) 0 + nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)"}, {"tactic": "simp_rw [show \u2200 i, dist ((\u03bcs i).testAgainstNN f) 0 = (\u03bcs i).testAgainstNN f by\n    simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]]", "annotated_tactic": ["simp_rw [show \u2200 i, <a>dist</a> ((\u03bcs i).<a>testAgainstNN</a> f) 0 = (\u03bcs i).<a>testAgainstNN</a> f by\n      simp only [<a>dist_nndist</a>, <a>NNReal.nndist_zero_eq_val'</a>, <a>eq_self_iff_true</a>, <a>imp_true_iff</a>]]", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"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.testAgainstNN", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [321, 5], "def_end_pos": [321, 18]}, {"full_name": "dist_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 20]}, {"full_name": "NNReal.nndist_zero_eq_val'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1711, 9], "def_end_pos": [1711, 35]}, {"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": "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\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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => \u2191(testAgainstNN (\u03bcs b) f)) F (\ud835\udcdd 0)"}, {"tactic": "refine' squeeze_zero (fun i => NNReal.coe_nonneg _) obs _", "annotated_tactic": ["refine' <a>squeeze_zero</a> (fun i => <a>NNReal.coe_nonneg</a> _) obs _", [{"full_name": "squeeze_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1460, 9], "def_end_pos": [1460, 21]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 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\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => \u2191(testAgainstNN (\u03bcs b) f)) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)"}, {"tactic": "have lim_pair : Tendsto (fun i => (\u27e8nndist f 0, (\u03bcs i).mass\u27e9 : \u211d \u00d7 \u211d)) F (\ud835\udcdd \u27e8nndist f 0, 0\u27e9) := by\n  refine' (Prod.tendsto_iff _ _).mpr \u27e8tendsto_const_nhds, _\u27e9\n  exact (NNReal.continuous_coe.tendsto 0).comp mass_lim", "annotated_tactic": ["have lim_pair : <a>Tendsto</a> (fun i => (\u27e8<a>nndist</a> f 0, (\u03bcs i).<a>mass</a>\u27e9 : \u211d \u00d7 \u211d)) F (\ud835\udcdd \u27e8<a>nndist</a> f 0, 0\u27e9) := by\n    refine' (<a>Prod.tendsto_iff</a> _ _).<a>mpr</a> \u27e8<a>tendsto_const_nhds</a>, _\u27e9\n    exact (NNReal.continuous_coe.tendsto 0).<a>comp</a> mass_lim", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 9]}, {"full_name": "MeasureTheory.FiniteMeasure.mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [163, 5], "def_end_pos": [163, 9]}, {"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 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]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\nlim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)"}, {"tactic": "have key := tendsto_mul.comp lim_pair", "annotated_tactic": ["have key := tendsto_mul.comp lim_pair", []], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\nlim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\nlim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\nkey : Tendsto ((fun p => p.1 * p.2) \u2218 fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0) * 0))\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)"}, {"tactic": "rwa [mul_zero] at key", "annotated_tactic": ["rwa [<a>mul_zero</a>] at key", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\nlim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\nkey : Tendsto ((fun p => p.1 * p.2) \u2218 fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0) * 0))\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]", "annotated_tactic": ["simp only [<a>dist_nndist</a>, <a>NNReal.nndist_zero_eq_val'</a>, <a>eq_self_iff_true</a>, <a>imp_true_iff</a>]", [{"full_name": "dist_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 20]}, {"full_name": "NNReal.nndist_zero_eq_val'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1711, 9], "def_end_pos": [1711, 35]}, {"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": "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\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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 \u2200 (i : \u03b3), dist (testAgainstNN (\u03bcs i) f) 0 = \u2191(testAgainstNN (\u03bcs i) f)", "state_after": "no goals"}, {"tactic": "refine' (Prod.tendsto_iff _ _).mpr \u27e8tendsto_const_nhds, _\u27e9", "annotated_tactic": ["refine' (<a>Prod.tendsto_iff</a> _ _).<a>mpr</a> \u27e8<a>tendsto_const_nhds</a>, _\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]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))", "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun n => (\u2191(nndist f 0), \u2191(mass (\u03bcs n))).2) F (\ud835\udcdd (\u2191(nndist f 0), 0).2)"}, {"tactic": "exact (NNReal.continuous_coe.tendsto 0).comp mass_lim", "annotated_tactic": ["exact (NNReal.continuous_coe.tendsto 0).<a>comp</a> mass_lim", [{"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "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\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun n => (\u2191(nndist f 0), \u2191(mass (\u03bcs n))).2) F (\ud835\udcdd (\u2191(nndist f 0), 0).2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Infinite.exists_superset_ncard_eq", "start": [972, 1], "end": [977, 27], "traced_tactics": [{"tactic": "obtain \u27e8s\u2081, hs\u2081, hs\u2081fin, hs\u2081card\u27e9 := (ht.diff hs).exists_subset_ncard_eq (k - s.ncard)", "annotated_tactic": ["obtain \u27e8s\u2081, hs\u2081, hs\u2081fin, hs\u2081card\u27e9 := (ht.diff hs).<a>exists_subset_ncard_eq</a> (k - s.ncard)", [{"full_name": "Set.Infinite.exists_subset_ncard_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [963, 9], "def_end_pos": [963, 40]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nht : Set.Infinite t\nhst : s \u2286 t\nhs : Set.Finite s\nk : \u2115\nhsk : Set.ncard s \u2264 k\n\u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 Set.ncard s' = k", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nht : Set.Infinite t\nhst : s \u2286 t\nhs : Set.Finite s\nk : \u2115\nhsk : Set.ncard s \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : Set.Finite s\u2081\nhs\u2081card : Set.ncard s\u2081 = k - Set.ncard s\n\u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 Set.ncard s' = k"}, {"tactic": "refine' \u27e8s \u222a s\u2081, subset_union_left _ _, union_subset hst (hs\u2081.trans (diff_subset _ _)), _\u27e9", "annotated_tactic": ["refine' \u27e8s \u222a s\u2081, <a>subset_union_left</a> _ _, <a>union_subset</a> hst (hs\u2081.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 intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nht : Set.Infinite t\nhst : s \u2286 t\nhs : Set.Finite s\nk : \u2115\nhsk : Set.ncard s \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : Set.Finite s\u2081\nhs\u2081card : Set.ncard s\u2081 = k - Set.ncard s\n\u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 Set.ncard s' = k", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nht : Set.Infinite t\nhst : s \u2286 t\nhs : Set.Finite s\nk : \u2115\nhsk : Set.ncard s \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : Set.Finite s\u2081\nhs\u2081card : Set.ncard s\u2081 = k - Set.ncard s\n\u22a2 Set.ncard (s \u222a s\u2081) = k"}, {"tactic": "rwa [ncard_union_eq (disjoint_of_subset_right hs\u2081 disjoint_sdiff_right) hs hs\u2081fin, hs\u2081card,\n  add_tsub_cancel_of_le]", "annotated_tactic": ["rwa [<a>ncard_union_eq</a> (<a>disjoint_of_subset_right</a> hs\u2081 <a>disjoint_sdiff_right</a>) hs hs\u2081fin, hs\u2081card,\n    <a>add_tsub_cancel_of_le</a>]", [{"full_name": "Set.ncard_union_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}, {"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]}, {"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 intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Set \u03b1\nht : Set.Infinite t\nhst : s \u2286 t\nhs : Set.Finite s\nk : \u2115\nhsk : Set.ncard s \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : Set.Finite s\u2081\nhs\u2081card : Set.ncard s\u2081 = k - Set.ncard s\n\u22a2 Set.ncard (s \u222a s\u2081) = k", "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.zero_dvd", "start": [601, 11], "end": [602, 76], "traced_tactics": [{"tactic": "rw [e, Int.zero_mul]", "annotated_tactic": ["rw [e, <a>Int.zero_mul</a>]", [{"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": "n : Int\nx\u271d : 0 \u2223 n\nk : Int\ne : n = 0 * k\n\u22a2 n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.some_orElse'", "start": [282, 1], "end": [283, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.powerset_insert", "start": [98, 1], "end": [112, 46], "traced_tactics": [{"tactic": "ext t", "annotated_tactic": ["ext t", []], "state_before": "\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 powerset (insert a s) = powerset s \u222a image (insert a) (powerset s)", "state_after": "case a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\n\u22a2 t \u2208 powerset (insert a s) \u2194 t \u2208 powerset s \u222a image (insert a) (powerset s)"}, {"tactic": "simp only [exists_prop, mem_powerset, mem_image, mem_union, subset_insert_iff]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_powerset</a>, <a>mem_image</a>, <a>mem_union</a>, <a>subset_insert_iff</a>]", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Finset.mem_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}, {"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_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"full_name": "Finset.subset_insert_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2024, 9], "def_end_pos": [2024, 26]}]], "state_before": "case a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\n\u22a2 t \u2208 powerset (insert a s) \u2194 t \u2208 powerset s \u222a image (insert a) (powerset s)", "state_after": "case a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t"}, {"tactic": "by_cases h : a \u2208 t", "annotated_tactic": ["by_cases h : a \u2208 t", []], "state_before": "case a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t", "state_after": "case pos\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\ncase neg\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case pos\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t", "state_after": "case pos.mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 erase t a \u2286 s \u2192 t \u2286 s \u2228 \u2203 a_2, a_2 \u2286 s \u2227 insert a a_2 = t\n\ncase pos.mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 (t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t) \u2192 erase t a \u2286 s"}, {"tactic": "exact fun H => Or.inr \u27e8_, H, insert_erase h\u27e9", "annotated_tactic": ["exact fun H => <a>Or.inr</a> \u27e8_, H, <a>insert_erase</a> h\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.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}]], "state_before": "case pos.mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 erase t a \u2286 s \u2192 t \u2286 s \u2228 \u2203 a_2, a_2 \u2286 s \u2227 insert a a_2 = t", "state_after": "no goals"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case pos.mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 (t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t) \u2192 erase t a \u2286 s", "state_after": "case pos.mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\u22a2 erase t a \u2286 s"}, {"tactic": "cases' H with H H", "annotated_tactic": ["cases' H with H H", []], "state_before": "case pos.mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\u22a2 erase t a \u2286 s", "state_after": "case pos.mpr.inl\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : t \u2286 s\n\u22a2 erase t a \u2286 s\n\ncase pos.mpr.inr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\u22a2 erase t a \u2286 s"}, {"tactic": "exact Subset.trans (erase_subset a t) H", "annotated_tactic": ["exact <a>Subset.trans</a> (<a>erase_subset</a> a t) H", [{"full_name": "Finset.Subset.trans", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 21]}, {"full_name": "Finset.erase_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1976, 9], "def_end_pos": [1976, 21]}]], "state_before": "case pos.mpr.inl\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : t \u2286 s\n\u22a2 erase t a \u2286 s", "state_after": "no goals"}, {"tactic": "rcases H with \u27e8u, hu\u27e9", "annotated_tactic": ["rcases H with \u27e8u, hu\u27e9", []], "state_before": "case pos.mpr.inr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\u22a2 erase t a \u2286 s", "state_after": "case pos.mpr.inr.intro\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nu : Finset \u03b1\nhu : u \u2286 s \u2227 insert a u = t\n\u22a2 erase t a \u2286 s"}, {"tactic": "rw [\u2190 hu.2]", "annotated_tactic": ["rw [\u2190 hu.2]", []], "state_before": "case pos.mpr.inr.intro\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nu : Finset \u03b1\nhu : u \u2286 s \u2227 insert a u = t\n\u22a2 erase t a \u2286 s", "state_after": "case pos.mpr.inr.intro\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nu : Finset \u03b1\nhu : u \u2286 s \u2227 insert a u = t\n\u22a2 erase (insert a u) a \u2286 s"}, {"tactic": "exact Subset.trans (erase_insert_subset a u) hu.1", "annotated_tactic": ["exact <a>Subset.trans</a> (<a>erase_insert_subset</a> a u) hu.1", [{"full_name": "Finset.Subset.trans", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 21]}, {"full_name": "Finset.erase_insert_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2029, 9], "def_end_pos": [2029, 28]}]], "state_before": "case pos.mpr.inr.intro\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nu : Finset \u03b1\nhu : u \u2286 s \u2227 insert a u = t\n\u22a2 erase (insert a u) a \u2286 s", "state_after": "no goals"}, {"tactic": "have : \u00ac\u2203 u : Finset \u03b1, u \u2286 s \u2227 insert a u = t := by simp [Ne.symm (ne_insert_of_not_mem _ _ h)]", "annotated_tactic": ["have : \u00ac\u2203 u : <a>Finset</a> \u03b1, u \u2286 s \u2227 <a>insert</a> a u = t := by simp [<a>Ne.symm</a> (<a>ne_insert_of_not_mem</a> _ _ h)]", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"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": "Finset.ne_insert_of_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1182, 9], "def_end_pos": [1182, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t", "state_after": "case neg\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\nthis : \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t"}, {"tactic": "simp [Finset.erase_eq_of_not_mem h, this]", "annotated_tactic": ["simp [<a>Finset.erase_eq_of_not_mem</a> h, this]", [{"full_name": "Finset.erase_eq_of_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 28]}]], "state_before": "case neg\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\nthis : \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t", "state_after": "no goals"}, {"tactic": "simp [Ne.symm (ne_insert_of_not_mem _ _ h)]", "annotated_tactic": ["simp [<a>Ne.symm</a> (<a>ne_insert_of_not_mem</a> _ _ h)]", [{"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": "Finset.ne_insert_of_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1182, 9], "def_end_pos": [1182, 29]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\n\u22a2 \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Constructions.lean", "full_name": "WType.leftInverse_nat", "start": [66, 1], "end": [74, 8], "traced_tactics": [{"tactic": "rw [toNat, ofNat]", "annotated_tactic": ["rw [<a>toNat</a>, <a>ofNat</a>]", [{"full_name": "WType.toNat", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [61, 5], "def_end_pos": [61, 10]}, {"full_name": "WType.ofNat", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [54, 5], "def_end_pos": [54, 10]}]], "state_before": "f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 ofNat (toNat (mk Nat\u03b1.zero f)) = mk Nat\u03b1.zero f", "state_after": "f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 mk Nat\u03b1.zero Empty.elim = mk Nat\u03b1.zero f"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 mk Nat\u03b1.zero Empty.elim = mk Nat\u03b1.zero f", "state_after": "case e_f\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 Empty.elim = f"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_f\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 Empty.elim = f", "state_after": "case e_f.h\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\nx : Empty\n\u22a2 Empty.elim x = f x"}, {"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "case e_f.h\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\nx : Empty\n\u22a2 Empty.elim x = f x", "state_after": "no goals"}, {"tactic": "simp only [toNat, ofNat, leftInverse_nat (f ()), mk.injEq, heq_eq_eq, true_and]", "annotated_tactic": ["simp only [<a>toNat</a>, <a>ofNat</a>, leftInverse_nat (f ()), mk.injEq, <a>heq_eq_eq</a>, <a>true_and</a>]", [{"full_name": "WType.toNat", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [61, 5], "def_end_pos": [61, 10]}, {"full_name": "WType.ofNat", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [54, 5], "def_end_pos": [54, 10]}, {"full_name": "heq_eq_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [143, 17], "def_end_pos": [143, 26]}, {"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": "f : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2\n\u22a2 ofNat (toNat (mk Nat\u03b1.succ f)) = mk Nat\u03b1.succ f", "state_after": "f : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2\n\u22a2 (fun x => f ()) = f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "f : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2\n\u22a2 (fun x => f ()) = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Comap.lean", "full_name": "MvPolynomial.comap_rename", "start": [90, 1], "end": [92, 40], "traced_tactics": [{"tactic": "funext", "annotated_tactic": ["funext", []], "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 : \u03c3 \u2192 \u03c4\nx : \u03c4 \u2192 R\n\u22a2 comap (rename f) x = x \u2218 f", "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 : \u03c3 \u2192 \u03c4\nx : \u03c4 \u2192 R\nx\u271d : \u03c3\n\u22a2 comap (rename f) x x\u271d = (x \u2218 f) x\u271d"}, {"tactic": "simp [rename_X, comap_apply, aeval_X]", "annotated_tactic": ["simp [<a>rename_X</a>, <a>comap_apply</a>, <a>aeval_X</a>]", [{"full_name": "MvPolynomial.rename_X", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}, {"full_name": "MvPolynomial.comap_apply", "def_path": "Mathlib/Data/MvPolynomial/Comap.lean", "def_pos": [42, 9], "def_end_pos": [42, 20]}, {"full_name": "MvPolynomial.aeval_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1474, 9], "def_end_pos": [1474, 16]}]], "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 : \u03c3 \u2192 \u03c4\nx : \u03c4 \u2192 R\nx\u271d : \u03c3\n\u22a2 comap (rename f) x x\u271d = (x \u2218 f) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint\u2080", "start": [201, 1], "end": [209, 84], "traced_tactics": [{"tactic": "rcases show Summable fun i => \u03bc (As i) from ENNReal.summable with \u27e8S, hS\u27e9", "annotated_tactic": ["rcases show <a>Summable</a> fun i => \u03bc (As i) from <a>ENNReal.summable</a> with \u27e8S, hS\u27e9", [{"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\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 i, As 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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 i, As i)"}, {"tactic": "rw [hS.tsum_eq]", "annotated_tactic": ["rw [hS.tsum_eq]", []], "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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 i, As 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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 S \u2264 \u2191\u2191\u03bc (\u22c3 i, As i)"}, {"tactic": "refine' tendsto_le_of_eventuallyLE hS tendsto_const_nhds (eventually_of_forall _)", "annotated_tactic": ["refine' <a>tendsto_le_of_eventuallyLE</a> hS <a>tendsto_const_nhds</a> (<a>eventually_of_forall</a> _)", [{"full_name": "tendsto_le_of_eventuallyLE", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [229, 7], "def_end_pos": [229, 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 intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 S \u2264 \u2191\u2191\u03bc (\u22c3 i, As 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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 \u2200 (x : Finset \u03b9), (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) x \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 i, As i)) x"}, {"tactic": "intro s", "annotated_tactic": ["intro s", []], "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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 \u2200 (x : Finset \u03b9), (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) x \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 i, As i)) x", "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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) s \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 i, As i)) s"}, {"tactic": "simp only [\u2190 measure_biUnion_finset\u2080 (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i]", "annotated_tactic": ["simp only [\u2190 <a>measure_biUnion_finset\u2080</a> (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i]", [{"full_name": "MeasureTheory.measure_biUnion_finset\u2080", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [187, 9], "def_end_pos": [187, 32]}]], "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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) s \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 i, As i)) s", "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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\ns : Finset \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 s, As b) \u2264 \u2191\u2191\u03bc (\u22c3 i, As i)"}, {"tactic": "exact measure_mono (iUnion\u2082_subset_iUnion (fun i : \u03b9 => i \u2208 s) fun i : \u03b9 => As i)", "annotated_tactic": ["exact <a>measure_mono</a> (<a>iUnion\u2082_subset_iUnion</a> (fun i : \u03b9 => i \u2208 s) fun i : \u03b9 => As i)", [{"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_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [536, 9], "def_end_pos": [536, 30]}]], "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\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i)\nAs_disj : Pairwise (AEDisjoint \u03bc on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\ns : Finset \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 s, As b) \u2264 \u2191\u2191\u03bc (\u22c3 i, As i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.map_rightInverse", "start": [1345, 1], "end": [1347, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/Basic.lean", "full_name": "Function.LeftInverse.Prod_map", "start": [334, 1], "end": [336, 73], "traced_tactics": [{"tactic": "rw [Prod.map_map, hf.comp_eq_id, hg.comp_eq_id, map_id, id]", "annotated_tactic": ["rw [<a>Prod.map_map</a>, hf.comp_eq_id, hg.comp_eq_id, <a>map_id</a>, <a>id</a>]", [{"full_name": "Prod.map_map", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [95, 9], "def_end_pos": [95, 16]}, {"full_name": "Prod.map_id", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [142, 9], "def_end_pos": [142, 15]}, {"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\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nhf : LeftInverse f\u2081 f\u2082\nhg : LeftInverse g\u2081 g\u2082\na : \u03b2 \u00d7 \u03b4\n\u22a2 map f\u2081 g\u2081 (map f\u2082 g\u2082 a) = 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.extractP_eq_find?_eraseP", "start": [1137, 9], "end": [1145, 21], "traced_tactics": [{"tactic": "exact go #[] _ rfl", "annotated_tactic": ["exact go #[] _ <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\np : \u03b1 \u2192 Bool\nl : List \u03b1\n\u22a2 extractP p l = (find? p l, eraseP p l)", "state_after": "no goals"}, {"tactic": "simp [extractP.go, find?, eraseP, h]", "annotated_tactic": ["simp [<a>extractP.go</a>, <a>find?</a>, <a>eraseP</a>, h]", [{"full_name": "List.extractP.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1010, 3], "def_end_pos": [1010, 5]}, {"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.eraseP", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [980, 5], "def_end_pos": [980, 11]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nh : l = acc.data ++ []\n\u22a2 extractP.go p l [] acc = (find? p [], acc.data ++ eraseP p [])", "state_after": "no goals"}, {"tactic": "simp [extractP.go, find?, eraseP]", "annotated_tactic": ["simp [<a>extractP.go</a>, <a>find?</a>, <a>eraseP</a>]", [{"full_name": "List.extractP.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1010, 3], "def_end_pos": [1010, 5]}, {"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.eraseP", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [980, 5], "def_end_pos": [980, 11]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 l = acc.data ++ x :: xs \u2192 extractP.go p l (x :: xs) acc = (find? p (x :: xs), acc.data ++ eraseP p (x :: xs))", "state_after": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 l = acc.data ++ x :: xs \u2192\n    (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (Array.push acc x)) =\n      (match p x with\n        | true => some x\n        | false => find? p xs,\n        acc.data ++ bif p x then xs else x :: eraseP p xs)"}, {"tactic": "cases p x <;> simp", "annotated_tactic": ["cases p x <;> simp", []], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 l = acc.data ++ x :: xs \u2192\n    (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (Array.push acc x)) =\n      (match p x with\n        | true => some x\n        | false => find? p xs,\n        acc.data ++ bif p x then xs else x :: eraseP p xs)", "state_after": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 l = acc.data ++ x :: xs \u2192 extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 l = acc.data ++ x :: xs \u2192 extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)", "state_after": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)"}, {"tactic": "rw [go _ xs]", "annotated_tactic": ["rw [go _ xs]", []], "state_before": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)", "state_after": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 (find? p xs, (Array.push acc x).data ++ eraseP p xs) = (find? p xs, acc.data ++ x :: eraseP p xs)\n\ncase false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs"}, {"tactic": "{simp}", "annotated_tactic": ["{simp}", []], "state_before": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 (find? p xs, (Array.push acc x).data ++ eraseP p xs) = (find? p xs, acc.data ++ x :: eraseP p xs)\n\ncase false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs", "state_after": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case false\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.piecewise_mem_Icc_of_mem_of_mem", "start": [2625, 1], "end": [2628, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.one_def", "start": [92, 1], "end": [93, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_pow'", "start": [396, 1], "end": [397, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_sub_zpow_of_ne", "start": [499, 1], "end": [514, 64], "traced_tactics": [{"tactic": "rcases em (w \u2208 sphere c |R| \u2227 n < -1) with (\u27e8hw, hn\u27e9 | H)", "annotated_tactic": ["rcases <a>em</a> (w \u2208 <a>sphere</a> c |R| \u2227 n < -1) with (\u27e8hw, hn\u27e9 | H)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 9]}, {"full_name": "Metric.sphere", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [485, 5], "def_end_pos": [485, 11]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0", "state_after": "case inl.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn\u271d : n \u2260 -1\nc w : \u2102\nR : \u211d\nhw : w \u2208 sphere c |R|\nhn : n < -1\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0\n\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : \u00ac(w \u2208 sphere c |R| \u2227 n < -1)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0"}, {"tactic": "push_neg at H", "annotated_tactic": ["push_neg at H", []], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : \u00ac(w \u2208 sphere c |R| \u2227 n < -1)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0"}, {"tactic": "refine' integral_eq_zero_of_hasDerivWithinAt' fun z hz => (hd z _).hasDerivWithinAt", "annotated_tactic": ["refine' <a>integral_eq_zero_of_hasDerivWithinAt'</a> fun z hz => (hd z _).<a>hasDerivWithinAt</a>", [{"full_name": "circleIntegral.integral_eq_zero_of_hasDerivWithinAt'", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [469, 9], "def_end_pos": [469, 46]}, {"full_name": "HasDerivAt.hasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 36]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nhd : \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nhd : \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z\nz : \u2102\nhz : z \u2208 sphere c |R|\n\u22a2 z \u2260 w \u2228 -1 \u2264 n"}, {"tactic": "exact (ne_or_eq z w).imp_right fun (h : z = w) => H <| h \u25b8 hz", "annotated_tactic": ["exact (<a>ne_or_eq</a> z w).<a>imp_right</a> fun (h : z = w) => H <| h \u25b8 hz", [{"full_name": "ne_or_eq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}, {"full_name": "Or.imp_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [253, 9], "def_end_pos": [253, 21]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nhd : \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z\nz : \u2102\nhz : z \u2208 sphere c |R|\n\u22a2 z \u2260 w \u2228 -1 \u2264 n", "state_after": "no goals"}, {"tactic": "exact integral_sub_zpow_of_undef (hn.trans (by decide)) hw", "annotated_tactic": ["exact <a>integral_sub_zpow_of_undef</a> (hn.trans (by decide)) hw", [{"full_name": "circleIntegral.integral_sub_zpow_of_undef", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [489, 9], "def_end_pos": [489, 35]}]], "state_before": "case inl.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn\u271d : n \u2260 -1\nc w : \u2102\nR : \u211d\nhw : w \u2208 sphere c |R|\nhn : n < -1\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn\u271d : n \u2260 -1\nc w : \u2102\nR : \u211d\nhw : w \u2208 sphere c |R|\nhn : n < -1\n\u22a2 -1 < 0", "state_after": "no goals"}, {"tactic": "intro z hne", "annotated_tactic": ["intro z hne", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\n\u22a2 \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z"}, {"tactic": "convert ((hasDerivAt_zpow (n + 1) _ (hne.imp _ _)).comp z\n  ((hasDerivAt_id z).sub_const w)).div_const _ using 1", "annotated_tactic": ["convert ((<a>hasDerivAt_zpow</a> (n + 1) _ (hne.imp _ _)).<a>comp</a> z\n      ((<a>hasDerivAt_id</a> z).<a>sub_const</a> w)).<a>div_const</a> _ using 1", [{"full_name": "hasDerivAt_zpow", "def_path": "Mathlib/Analysis/Calculus/Deriv/ZPow.lean", "def_pos": [64, 9], "def_end_pos": [64, 24]}, {"full_name": "HasDerivAt.comp", "def_path": "Mathlib/Analysis/Calculus/Deriv/Comp.lean", "def_pos": [174, 16], "def_end_pos": [174, 31]}, {"full_name": "hasDerivAt_id", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [663, 9], "def_end_pos": [663, 22]}, {"full_name": "HasDerivAt.sub_const", "def_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "def_pos": [333, 16], "def_end_pos": [333, 36]}, {"full_name": "HasDerivAt.div_const", "def_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "def_pos": [290, 9], "def_end_pos": [290, 29]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z", "state_after": "case h.e'_7\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 (z - w) ^ n = \u2191(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (\u2191n + 1)\n\ncase convert_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 z \u2260 w \u2192 id z - w \u2260 0\n\ncase convert_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 -1 \u2264 n \u2192 0 \u2264 n + 1"}, {"tactic": "exacts [sub_ne_zero.2, neg_le_iff_add_nonneg.1]", "annotated_tactic": ["exacts [<a>sub_ne_zero</a>.2, <a>neg_le_iff_add_nonneg</a>.1]", [{"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}, {"full_name": "neg_le_iff_add_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [229, 15], "def_end_pos": [229, 36]}]], "state_before": "case convert_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 z \u2260 w \u2192 id z - w \u2260 0\n\ncase convert_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 -1 \u2264 n \u2192 0 \u2264 n + 1", "state_after": "no goals"}, {"tactic": "have hn' : (n + 1 : \u2102) \u2260 0 := by\n  rwa [Ne, \u2190 eq_neg_iff_add_eq_zero, \u2190 Int.cast_one, \u2190 Int.cast_neg, Int.cast_inj]", "annotated_tactic": ["have hn' : (n + 1 : \u2102) \u2260 0 := by\n        rwa [<a>Ne</a>, \u2190 <a>eq_neg_iff_add_eq_zero</a>, \u2190 <a>Int.cast_one</a>, \u2190 <a>Int.cast_neg</a>, <a>Int.cast_inj</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": "eq_neg_iff_add_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [666, 3], "def_end_pos": [666, 14]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [77, 9], "def_end_pos": [77, 17]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}, {"full_name": "Int.cast_inj", "def_path": "Mathlib/Data/Int/CharZero.lean", "def_pos": [34, 9], "def_end_pos": [34, 17]}]], "state_before": "case h.e'_7\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 (z - w) ^ n = \u2191(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (\u2191n + 1)", "state_after": "case h.e'_7\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\nhn' : \u2191n + 1 \u2260 0\n\u22a2 (z - w) ^ n = \u2191(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (\u2191n + 1)"}, {"tactic": "simp [mul_assoc, mul_div_cancel_left _ hn']", "annotated_tactic": ["simp [<a>mul_assoc</a>, <a>mul_div_cancel_left</a> _ hn']", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_div_cancel_left", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [165, 9], "def_end_pos": [165, 28]}]], "state_before": "case h.e'_7\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\nhn' : \u2191n + 1 \u2260 0\n\u22a2 (z - w) ^ n = \u2191(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (\u2191n + 1)", "state_after": "no goals"}, {"tactic": "rwa [Ne, \u2190 eq_neg_iff_add_eq_zero, \u2190 Int.cast_one, \u2190 Int.cast_neg, Int.cast_inj]", "annotated_tactic": ["rwa [<a>Ne</a>, \u2190 <a>eq_neg_iff_add_eq_zero</a>, \u2190 <a>Int.cast_one</a>, \u2190 <a>Int.cast_neg</a>, <a>Int.cast_inj</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": "eq_neg_iff_add_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [666, 3], "def_end_pos": [666, 14]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [77, 9], "def_end_pos": [77, 17]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}, {"full_name": "Int.cast_inj", "def_path": "Mathlib/Data/Int/CharZero.lean", "def_pos": [34, 9], "def_end_pos": [34, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nn : \u2124\nhn : n \u2260 -1\nc w : \u2102\nR : \u211d\nH : w \u2208 sphere c |R| \u2192 -1 \u2264 n\nz : \u2102\nhne : z \u2260 w \u2228 -1 \u2264 n\n\u22a2 \u2191n + 1 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.hasFiniteIntegral_smul_iff", "start": [415, 1], "end": [421, 33], "traced_tactics": [{"tactic": "obtain \u27e8c, rfl\u27e9 := hc", "annotated_tactic": ["obtain \u27e8c, rfl\u27e9 := hc", []], "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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nc : \ud835\udd5c\nhc : IsUnit c\nf : \u03b1 \u2192 \u03b2\n\u22a2 HasFiniteIntegral (c \u2022 f) \u2194 HasFiniteIntegral f", "state_after": "case 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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\n\u22a2 HasFiniteIntegral (\u2191c \u2022 f) \u2194 HasFiniteIntegral f"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case 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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\n\u22a2 HasFiniteIntegral (\u2191c \u2022 f) \u2194 HasFiniteIntegral f", "state_after": "case intro.mp\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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\n\u22a2 HasFiniteIntegral (\u2191c \u2022 f) \u2192 HasFiniteIntegral f\n\ncase intro.mpr\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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\n\u22a2 HasFiniteIntegral f \u2192 HasFiniteIntegral (\u2191c \u2022 f)"}, {"tactic": "exact HasFiniteIntegral.smul _", "annotated_tactic": ["exact <a>HasFiniteIntegral.smul</a> _", [{"full_name": "MeasureTheory.HasFiniteIntegral.smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [402, 9], "def_end_pos": [402, 31]}]], "state_before": "case intro.mpr\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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\n\u22a2 HasFiniteIntegral f \u2192 HasFiniteIntegral (\u2191c \u2022 f)", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case intro.mp\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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\n\u22a2 HasFiniteIntegral (\u2191c \u2022 f) \u2192 HasFiniteIntegral f", "state_after": "case intro.mp\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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\nh : HasFiniteIntegral (\u2191c \u2022 f)\n\u22a2 HasFiniteIntegral f"}, {"tactic": "simpa only [smul_smul, Units.inv_mul, one_smul] using h.smul ((c\u207b\u00b9 : \ud835\udd5c\u02e3) : \ud835\udd5c)", "annotated_tactic": ["simpa only [<a>smul_smul</a>, <a>Units.inv_mul</a>, <a>one_smul</a>] using h.smul ((c\u207b\u00b9 : \ud835\udd5c\u02e3) : \ud835\udd5c)", [{"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "Units.inv_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [262, 9], "def_end_pos": [262, 16]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case intro.mp\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\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 : MulActionWithZero \ud835\udd5c \u03b2\ninst\u271d : BoundedSMul \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\u02e3\nh : HasFiniteIntegral (\u2191c \u2022 f)\n\u22a2 HasFiniteIntegral f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector3.lean", "full_name": "exists_vector_zero", "start": [234, 1], "end": [235, 68], "traced_tactics": [{"tactic": "rw [\u2190 eq_nil v]", "annotated_tactic": ["rw [\u2190 <a>eq_nil</a> v]", [{"full_name": "Vector3.eq_nil", "def_path": "Mathlib/Data/Vector3.lean", "def_pos": [104, 9], "def_end_pos": [104, 15]}]], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 0 \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 0\nfv : f v\n\u22a2 f []", "state_after": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 0 \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 0\nfv : f v\n\u22a2 f v"}, {"tactic": "exact fv", "annotated_tactic": ["exact fv", []], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 0 \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 0\nfv : f v\n\u22a2 f v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top", "start": [178, 1], "end": [213, 43], "traced_tactics": [{"tactic": "have hp0_lt : 0 < p := lt_of_lt_of_le zero_lt_one hp1", "annotated_tactic": ["have hp0_lt : 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4"}, {"tactic": "have hp0 : 0 \u2264 p := le_of_lt hp0_lt", "annotated_tactic": ["have hp0 : 0 \u2264 p := <a>le_of_lt</a> hp0_lt", [{"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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4"}, {"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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p"}, {"tactic": "have h_zero_lt_half_rpow : (0 : \u211d\u22650\u221e) < (1 / 2 : \u211d\u22650\u221e) ^ p := by\n  rw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt]\n  exact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt", "annotated_tactic": ["have h_zero_lt_half_rpow : (0 : \u211d\u22650\u221e) < (1 / 2 : \u211d\u22650\u221e) ^ p := by\n        rw [\u2190 <a>ENNReal.zero_rpow_of_pos</a> hp0_lt]\n        exact <a>ENNReal.rpow_lt_rpow</a> (by simp [<a>zero_lt_one</a>]) hp0_lt", [{"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.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [646, 9], "def_end_pos": [646, 21]}, {"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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p"}, {"tactic": "have h_rw : (1 / 2 : \u211d\u22650\u221e) ^ p * (2 : \u211d\u22650\u221e) ^ (p - 1) = 1 / 2 := by\n  rw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n    ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div,\n    ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top, ENNReal.one_rpow, one_mul,\n    ENNReal.rpow_neg_one]", "annotated_tactic": ["have h_rw : (1 / 2 : \u211d\u22650\u221e) ^ p * (2 : \u211d\u22650\u221e) ^ (p - 1) = 1 / 2 := by\n        rw [<a>sub_eq_add_neg</a>, <a>ENNReal.rpow_add</a> _ _ <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>, \u2190 <a>mul_assoc</a>, \u2190\n          <a>ENNReal.mul_rpow_of_nonneg</a> _ _ hp0, <a>one_div</a>,\n          <a>ENNReal.inv_mul_cancel</a> <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>, <a>ENNReal.one_rpow</a>, <a>one_mul</a>,\n          <a>ENNReal.rpow_neg_one</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": "ENNReal.rpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [507, 9], "def_end_pos": [507, 17]}, {"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": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"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": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 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.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.one_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [459, 9], "def_end_pos": [459, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "ENNReal.rpow_neg_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [529, 9], "def_end_pos": [529, 21]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p"}, {"tactic": "rw [\u2190 ENNReal.mul_le_mul_left (ne_of_lt h_zero_lt_half_rpow).symm _]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.mul_le_mul_left</a> (<a>ne_of_lt</a> h_zero_lt_half_rpow).<a>symm</a> _]", [{"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": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p * (f a + g a) ^ p \u2264 (1 / 2) ^ p * (2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p)\n\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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p \u2260 \u22a4"}, {"tactic": "rw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.zero_rpow_of_pos</a> hp0_lt]", [{"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": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 < (1 / 2) ^ 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 ^ p < (1 / 2) ^ p"}, {"tactic": "exact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt", "annotated_tactic": ["exact <a>ENNReal.rpow_lt_rpow</a> (by simp [<a>zero_lt_one</a>]) hp0_lt", [{"full_name": "ENNReal.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [646, 9], "def_end_pos": [646, 21]}, {"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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 ^ p < (1 / 2) ^ p", "state_after": "no goals"}, {"tactic": "simp [zero_lt_one]", "annotated_tactic": ["simp [<a>zero_lt_one</a>]", [{"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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 < 1 / 2", "state_after": "no goals"}, {"tactic": "rw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n  ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div,\n  ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top, ENNReal.one_rpow, one_mul,\n  ENNReal.rpow_neg_one]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>ENNReal.rpow_add</a> _ _ <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>, \u2190 <a>mul_assoc</a>, \u2190\n          <a>ENNReal.mul_rpow_of_nonneg</a> _ _ hp0, <a>one_div</a>,\n          <a>ENNReal.inv_mul_cancel</a> <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>, <a>ENNReal.one_rpow</a>, <a>one_mul</a>,\n          <a>ENNReal.rpow_neg_one</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": "ENNReal.rpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [507, 9], "def_end_pos": [507, 17]}, {"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": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"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": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 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.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.one_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [459, 9], "def_end_pos": [459, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "ENNReal.rpow_neg_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [529, 9], "def_end_pos": [529, 21]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\n\u22a2 (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2", "state_after": "no goals"}, {"tactic": "rw [mul_add, \u2190 mul_assoc, \u2190 mul_assoc, h_rw, \u2190 ENNReal.mul_rpow_of_nonneg _ _ hp0, mul_add]", "annotated_tactic": ["rw [<a>mul_add</a>, \u2190 <a>mul_assoc</a>, \u2190 <a>mul_assoc</a>, h_rw, \u2190 <a>ENNReal.mul_rpow_of_nonneg</a> _ _ hp0, <a>mul_add</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"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": "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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p * (f a + g a) ^ p \u2264 (1 / 2) ^ p * (2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2 * f a + 1 / 2 * g a) ^ p \u2264 1 / 2 * f a ^ p + 1 / 2 * g a ^ p"}, {"tactic": "refine'\n  ENNReal.rpow_arith_mean_le_arith_mean2_rpow (1 / 2 : \u211d\u22650\u221e) (1 / 2 : \u211d\u22650\u221e) (f a) (g a) _\n    hp1", "annotated_tactic": ["refine'\n          <a>ENNReal.rpow_arith_mean_le_arith_mean2_rpow</a> (1 / 2 : \u211d\u22650\u221e) (1 / 2 : \u211d\u22650\u221e) (f a) (g a) _\n            hp1", [{"full_name": "ENNReal.rpow_arith_mean_le_arith_mean2_rpow", "def_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "def_pos": [284, 9], "def_end_pos": [284, 44]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2 * f a + 1 / 2 * g a) ^ p \u2264 1 / 2 * f a ^ p + 1 / 2 * g a ^ 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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 1 / 2 + 1 / 2 = 1"}, {"tactic": "rw [ENNReal.div_add_div_same, one_add_one_eq_two,\n  ENNReal.div_self two_ne_zero ENNReal.coe_ne_top]", "annotated_tactic": ["rw [<a>ENNReal.div_add_div_same</a>, <a>one_add_one_eq_two</a>,\n          <a>ENNReal.div_self</a> <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>]", [{"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": "one_add_one_eq_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [218, 9], "def_end_pos": [218, 27]}, {"full_name": "ENNReal.div_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1746, 19], "def_end_pos": [1746, 27]}, {"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]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 1 / 2 + 1 / 2 = 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 lt_top_iff_ne_top]", "annotated_tactic": ["rw [\u2190 <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\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p < \u22a4"}, {"tactic": "refine' ENNReal.rpow_lt_top_of_nonneg hp0 _", "annotated_tactic": ["refine' <a>ENNReal.rpow_lt_top_of_nonneg</a> hp0 _", [{"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]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p < \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 1 / 2 \u2260 \u22a4"}, {"tactic": "rw [one_div, ENNReal.inv_ne_top]", "annotated_tactic": ["rw [<a>one_div</a>, <a>ENNReal.inv_ne_top</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.inv_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 19]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 1 / 2 \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 2 \u2260 0"}, {"tactic": "exact two_ne_zero", "annotated_tactic": ["exact <a>two_ne_zero</a>", [{"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "have h_two : (2 : \u211d\u22650\u221e) ^ (p - 1) \u2260 \u22a4 :=\n  ENNReal.rpow_ne_top_of_nonneg (by simp [hp1]) ENNReal.coe_ne_top", "annotated_tactic": ["have h_two : (2 : \u211d\u22650\u221e) ^ (p - 1) \u2260 \u22a4 :=\n        <a>ENNReal.rpow_ne_top_of_nonneg</a> (by simp [hp1]) <a>ENNReal.coe_ne_top</a>", [{"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.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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \u22a4"}, {"tactic": "rw [lintegral_add_left', lintegral_const_mul'' _ (hf.pow_const p),\n  lintegral_const_mul' _ _ h_two, ENNReal.add_lt_top]", "annotated_tactic": ["rw [<a>lintegral_add_left'</a>, <a>lintegral_const_mul''</a> _ (hf.pow_const p),\n        <a>lintegral_const_mul'</a> _ _ h_two, <a>ENNReal.add_lt_top</a>]", [{"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_const_mul''", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [691, 9], "def_end_pos": [691, 30]}, {"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.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 27]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 \u2227 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\n\ncase hf\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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 AEMeasurable fun a => 2 ^ (p - 1) * f a ^ p"}, {"tactic": "simp [hp1]", "annotated_tactic": ["simp [hp1]", []], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 0 \u2264 p - 1", "state_after": "no goals"}, {"tactic": "exact \u27e8ENNReal.mul_lt_top h_two hf_top.ne, ENNReal.mul_lt_top h_two hg_top.ne\u27e9", "annotated_tactic": ["exact \u27e8<a>ENNReal.mul_lt_top</a> h_two hf_top.ne, <a>ENNReal.mul_lt_top</a> h_two hg_top.ne\u27e9", [{"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.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}]], "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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 \u2227 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "exact (hf.pow_const p).const_mul _", "annotated_tactic": ["exact (hf.pow_const p).<a>const_mul</a> _", [{"full_name": "AEMeasurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [113, 9], "def_end_pos": [113, 31]}]], "state_before": "case hf\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\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 AEMeasurable fun a => 2 ^ (p - 1) * f a ^ p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.monotone_comap", "start": [147, 1], "end": [147, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.iInf_apply'", "start": [1192, 1], "end": [1195, 25], "traced_tactics": [{"tactic": "rw [iInf, sInf_apply' hs]", "annotated_tactic": ["rw [<a>iInf</a>, <a>sInf_apply'</a> hs]", [{"full_name": "iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}, {"full_name": "MeasureTheory.OuterMeasure.sInf_apply'", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1174, 9], "def_end_pos": [1174, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2a05 i, m i) s = \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (t n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 \u03bc \u2208 range fun i => m i, \u2191\u03bc (t n) =\n    \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (t n)"}, {"tactic": "simp only [iInf_range]", "annotated_tactic": ["simp only [<a>iInf_range</a>]", [{"full_name": "iInf_range", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1448, 9], "def_end_pos": [1448, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 \u03bc \u2208 range fun i => m i, \u2191\u03bc (t n) =\n    \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (t n)", "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_Ioo_of_lt", "start": [187, 1], "end": [188, 52], "traced_tactics": [{"tactic": "rw [card_fintype_Ioo, sub_sub, toNat_sub_of_le h]", "annotated_tactic": ["rw [<a>card_fintype_Ioo</a>, <a>sub_sub</a>, <a>toNat_sub_of_le</a> h]", [{"full_name": "Int.card_fintype_Ioo", "def_path": "Mathlib/Data/Int/Interval.lean", "def_pos": [167, 9], "def_end_pos": [167, 25]}, {"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [526, 3], "def_end_pos": [526, 14]}, {"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 < b\n\u22a2 \u2191(Fintype.card \u2191(Set.Ioo a b)) = b - a - 1", "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.getLast_eq_get", "start": [712, 1], "end": [716, 99], "traced_tactics": [{"tactic": "rw [getLast_singleton, get_singleton]", "annotated_tactic": ["rw [<a>getLast_singleton</a>, <a>get_singleton</a>]", [{"full_name": "List.getLast_singleton", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [547, 9], "def_end_pos": [547, 26]}, {"full_name": "List.get_singleton", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [671, 17], "def_end_pos": [671, 30]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nh : [a] \u2260 []\n\u22a2 getLast [a] h = get [a] { val := length [a] - 1, isLt := (_ : length [a] - 1 < length [a]) }", "state_after": "no goals"}, {"tactic": "rw [getLast_cons', getLast_eq_get (b :: l)]", "annotated_tactic": ["rw [<a>getLast_cons'</a>, getLast_eq_get (b :: l)]", [{"full_name": "List.getLast_cons'", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [345, 9], "def_end_pos": [345, 22]}]], "state_before": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 getLast (a :: b :: l) h =\n    get (a :: b :: l) { val := length (a :: b :: l) - 1, isLt := (_ : length (a :: b :: l) - 1 < length (a :: b :: l)) }", "state_after": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 get (b :: l) { val := length (b :: l) - 1, isLt := (_ : length (b :: l) - 1 < length (b :: l)) } =\n    get (a :: b :: l) { val := length (a :: b :: l) - 1, isLt := (_ : length (a :: b :: l) - 1 < length (a :: b :: l)) }\n\n\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 b :: l \u2260 []"}, {"tactic": "{rfl}", "annotated_tactic": ["{rfl}", []], "state_before": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 get (b :: l) { val := length (b :: l) - 1, isLt := (_ : length (b :: l) - 1 < length (b :: l)) } =\n    get (a :: b :: l) { val := length (a :: b :: l) - 1, isLt := (_ : length (a :: b :: l) - 1 < length (a :: b :: l)) }\n\n\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 b :: l \u2260 []", "state_after": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 b :: l \u2260 []"}, {"tactic": "exact cons_ne_nil b l", "annotated_tactic": ["exact <a>cons_ne_nil</a> b l", [{"full_name": "List.cons_ne_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [20, 9], "def_end_pos": [20, 20]}]], "state_before": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\nh : a :: b :: l \u2260 []\n\u22a2 b :: l \u2260 []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.rnDeriv_sub", "start": [1177, 1], "end": [1182, 99], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg] at hst", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>] at hst", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t\u271d s t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc\ninst\u271d : HaveLebesgueDecomposition t \u03bc\nhst : HaveLebesgueDecomposition (s - t) \u03bc\n\u22a2 rnDeriv (s - t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc - rnDeriv t \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t\u271d s t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc\ninst\u271d : HaveLebesgueDecomposition t \u03bc\nhst : HaveLebesgueDecomposition (s + -t) \u03bc\n\u22a2 rnDeriv (s - t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc - rnDeriv t \u03bc"}, {"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <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]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t\u271d s t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc\ninst\u271d : HaveLebesgueDecomposition t \u03bc\nhst : HaveLebesgueDecomposition (s + -t) \u03bc\n\u22a2 rnDeriv (s - t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc - rnDeriv t \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t\u271d s t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc\ninst\u271d : HaveLebesgueDecomposition t \u03bc\nhst : HaveLebesgueDecomposition (s + -t) \u03bc\n\u22a2 rnDeriv (s + -t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc + -rnDeriv t \u03bc"}, {"tactic": "exact ae_eq_trans (rnDeriv_add _ _ _) (Filter.EventuallyEq.add (ae_eq_refl _) (rnDeriv_neg _ _))", "annotated_tactic": ["exact <a>ae_eq_trans</a> (<a>rnDeriv_add</a> _ _ _) (<a>Filter.EventuallyEq.add</a> (<a>ae_eq_refl</a> _) (<a>rnDeriv_neg</a> _ _))", [{"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.SignedMeasure.rnDeriv_add", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 20]}, {"full_name": "Filter.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}, {"full_name": "MeasureTheory.ae_eq_refl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}, {"full_name": "MeasureTheory.SignedMeasure.rnDeriv_neg", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [1143, 9], "def_end_pos": [1143, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t\u271d s t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc\ninst\u271d : HaveLebesgueDecomposition t \u03bc\nhst : HaveLebesgueDecomposition (s + -t) \u03bc\n\u22a2 rnDeriv (s + -t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc + -rnDeriv t \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean", "full_name": "LinearMap.exists_map_addHaar_eq_smul_addHaar'", "start": [42, 1], "end": [96, 45], "traced_tactics": [{"tactic": "have : ProperSpace E := properSpace_of_locallyCompactSpace \ud835\udd5c", "annotated_tactic": ["have : <a>ProperSpace</a> E := <a>properSpace_of_locallyCompactSpace</a> \ud835\udd5c", [{"full_name": "ProperSpace", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2198, 7], "def_end_pos": [2198, 18]}, {"full_name": "properSpace_of_locallyCompactSpace", "def_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "def_pos": [497, 7], "def_end_pos": [497, 41]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis : ProperSpace E\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have : FiniteDimensional \ud835\udd5c E := finiteDimensional_of_locallyCompactSpace \ud835\udd5c", "annotated_tactic": ["have : <a>FiniteDimensional</a> \ud835\udd5c E := <a>finiteDimensional_of_locallyCompactSpace</a> \ud835\udd5c", [{"full_name": "FiniteDimensional", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [87, 5], "def_end_pos": [87, 22]}, {"full_name": "finiteDimensional_of_locallyCompactSpace", "def_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "def_pos": [466, 9], "def_end_pos": [466, 49]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis : ProperSpace E\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d : ProperSpace E\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "let S : Submodule \ud835\udd5c E := LinearMap.ker L", "annotated_tactic": ["let S : <a>Submodule</a> \ud835\udd5c E := <a>LinearMap.ker</a> L", [{"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [41, 11], "def_end_pos": [41, 20]}, {"full_name": "LinearMap.ker", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1225, 5], "def_end_pos": [1225, 8]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "obtain \u27e8T, hT\u27e9 : \u2203 T : Submodule \ud835\udd5c E, IsCompl S T := Submodule.exists_isCompl S", "annotated_tactic": ["obtain \u27e8T, hT\u27e9 : \u2203 T : <a>Submodule</a> \ud835\udd5c E, <a>IsCompl</a> S T := <a>Submodule.exists_isCompl</a> S", [{"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [41, 11], "def_end_pos": [41, 20]}, {"full_name": "IsCompl", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [450, 11], "def_end_pos": [450, 18]}, {"full_name": "Submodule.exists_isCompl", "def_path": "Mathlib/LinearAlgebra/Basis/VectorSpace.lean", "def_pos": [230, 9], "def_end_pos": [230, 33]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "let M : (S \u00d7 T) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT", "annotated_tactic": ["let M : (S \u00d7 T) \u2243\u2097[\ud835\udd5c] E := <a>Submodule.prodEquivOfIsCompl</a> S T hT", [{"full_name": "Submodule.prodEquivOfIsCompl", "def_path": "Mathlib/LinearAlgebra/Projection.lean", "def_pos": [99, 5], "def_end_pos": [99, 23]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have M_cont : Continuous M.symm := LinearMap.continuous_of_finiteDimensional _", "annotated_tactic": ["have M_cont : <a>Continuous</a> M.symm := <a>LinearMap.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\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "let P : S \u00d7 T \u2192\u2097[\ud835\udd5c] T := LinearMap.snd \ud835\udd5c S T", "annotated_tactic": ["let P : S \u00d7 T \u2192\u2097[\ud835\udd5c] T := <a>LinearMap.snd</a> \ud835\udd5c S T", [{"full_name": "LinearMap.snd", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [66, 5], "def_end_pos": [66, 8]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have P_cont : Continuous P := LinearMap.continuous_of_finiteDimensional _", "annotated_tactic": ["have P_cont : <a>Continuous</a> P := <a>LinearMap.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\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have I : Function.Bijective (LinearMap.domRestrict L T) :=\n  \u27e8LinearMap.injective_domRestrict_iff.2 (IsCompl.inf_eq_bot hT.symm),\n  (LinearMap.surjective_domRestrict_iff h).2 hT.symm.sup_eq_top\u27e9", "annotated_tactic": ["have I : <a>Function.Bijective</a> (<a>LinearMap.domRestrict</a> L T) :=\n    \u27e8<a>LinearMap.injective_domRestrict_iff</a>.2 (<a>IsCompl.inf_eq_bot</a> hT.symm),\n    (<a>LinearMap.surjective_domRestrict_iff</a> h).2 hT.symm.sup_eq_top\u27e9", [{"full_name": "Function.Bijective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [131, 5], "def_end_pos": [131, 14]}, {"full_name": "LinearMap.domRestrict", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [176, 5], "def_end_pos": [176, 16]}, {"full_name": "LinearMap.injective_domRestrict_iff", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1446, 15], "def_end_pos": [1446, 40]}, {"full_name": "IsCompl.inf_eq_bot", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [495, 9], "def_end_pos": [495, 19]}, {"full_name": "LinearMap.surjective_domRestrict_iff", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [868, 15], "def_end_pos": [868, 58]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "let L' : T \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (LinearMap.domRestrict L T) I", "annotated_tactic": ["let L' : T \u2243\u2097[\ud835\udd5c] F := <a>LinearEquiv.ofBijective</a> (<a>LinearMap.domRestrict</a> L T) I", [{"full_name": "LinearEquiv.ofBijective", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [2112, 19], "def_end_pos": [2112, 30]}, {"full_name": "LinearMap.domRestrict", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [176, 5], "def_end_pos": [176, 16]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have L'_cont : Continuous L' := LinearMap.continuous_of_finiteDimensional _", "annotated_tactic": ["have L'_cont : <a>Continuous</a> L' := <a>LinearMap.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\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have A : L = (L' : T \u2192\u2097[\ud835\udd5c] F).comp (P.comp (M.symm : E \u2192\u2097[\ud835\udd5c] (S \u00d7 T))) := by\n  ext x\n  obtain \u27e8y, z, hyz\u27e9 : \u2203 (y : S) (z : T), M.symm x = (y, z) := \u27e8_, _, rfl\u27e9\n  have : x = M (y, z) := by\n    rw [\u2190 hyz]; simp only [LinearEquiv.apply_symm_apply]\n  simp [this]", "annotated_tactic": ["have A : L = (L' : T \u2192\u2097[\ud835\udd5c] F).<a>comp</a> (P.comp (M.symm : E \u2192\u2097[\ud835\udd5c] (S \u00d7 T))) := by\n    ext x\n    obtain \u27e8y, z, hyz\u27e9 : \u2203 (y : S) (z : T), M.symm x = (y, z) := \u27e8_, _, <a>rfl</a>\u27e9\n    have : x = M (y, z) := by\n      rw [\u2190 hyz]; simp only [<a>LinearEquiv.apply_symm_apply</a>]\n    simp [this]", [{"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]}, {"full_name": "LinearEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [374, 9], "def_end_pos": [374, 25]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "let \u03bcS : Measure S := addHaar", "annotated_tactic": ["let \u03bcS : <a>Measure</a> S := <a>addHaar</a>", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure.addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [673, 3], "def_end_pos": [673, 14]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "let \u03bcT : Measure T := addHaar", "annotated_tactic": ["let \u03bcT : <a>Measure</a> T := <a>addHaar</a>", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}, {"full_name": "MeasureTheory.Measure.addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [673, 3], "def_end_pos": [673, 14]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "obtain \u27e8c\u2080, c\u2080_pos, c\u2080_fin, h\u2080\u27e9 :\n    \u2203 c\u2080 : \u211d\u22650\u221e, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u221e \u2227 \u03bc.map M.symm = c\u2080 \u2022 \u03bcS.prod \u03bcT := by\n  have : IsAddHaarMeasure (\u03bc.map M.symm) :=\n    M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map \u03bc\n  exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _", "annotated_tactic": ["obtain \u27e8c\u2080, c\u2080_pos, c\u2080_fin, h\u2080\u27e9 :\n      \u2203 c\u2080 : \u211d\u22650\u221e, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u221e \u2227 \u03bc.map M.symm = c\u2080 \u2022 \u03bcS.prod \u03bcT := by\n    have : <a>IsAddHaarMeasure</a> (\u03bc.map M.symm) :=\n      M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map \u03bc\n    exact <a>isAddHaarMeasure_eq_smul_isAddHaarMeasure</a> _ _", [{"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.isAddHaarMeasure_eq_smul_isAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [724, 15], "def_end_pos": [724, 56]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "have J : (\u03bcS.prod \u03bcT).map P = (\u03bcS univ) \u2022 \u03bcT := map_snd_prod", "annotated_tactic": ["have J : (\u03bcS.prod \u03bcT).<a>map</a> P = (\u03bcS <a>univ</a>) \u2022 \u03bcT := <a>map_snd_prod</a>", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.map_snd_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [351, 15], "def_end_pos": [351, 27]}]], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "obtain \u27e8c\u2081, c\u2081_pos, c\u2081_fin, h\u2081\u27e9 : \u2203 c\u2081 : \u211d\u22650\u221e, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u221e \u2227 \u03bcT.map L' = c\u2081 \u2022 \u03bd := by\n  have : IsAddHaarMeasure (\u03bcT.map L') :=\n    L'.toContinuousLinearEquiv.isAddHaarMeasure_map \u03bcT\n  exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _", "annotated_tactic": ["obtain \u27e8c\u2081, c\u2081_pos, c\u2081_fin, h\u2081\u27e9 : \u2203 c\u2081 : \u211d\u22650\u221e, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u221e \u2227 \u03bcT.map L' = c\u2081 \u2022 \u03bd := by\n    have : <a>IsAddHaarMeasure</a> (\u03bcT.map L') :=\n      L'.toContinuousLinearEquiv.isAddHaarMeasure_map \u03bcT\n    exact <a>isAddHaarMeasure_eq_smul_isAddHaarMeasure</a> _ _", [{"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.isAddHaarMeasure_eq_smul_isAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [724, 15], "def_end_pos": [724, 56]}]], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "refine \u27e8c\u2080 * c\u2081, by simp [pos_iff_ne_zero, c\u2080_pos, c\u2081_pos], ENNReal.mul_lt_top c\u2080_fin c\u2081_fin, ?_\u27e9", "annotated_tactic": ["refine \u27e8c\u2080 * c\u2081, by simp [<a>pos_iff_ne_zero</a>, c\u2080_pos, c\u2081_pos], <a>ENNReal.mul_lt_top</a> c\u2080_fin c\u2081_fin, ?_\u27e9", [{"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": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 map (\u2191L) \u03bc = (c\u2080 * c\u2081 * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "simp only [I, h\u2080, Measure.map_smul, J, smul_smul, h\u2081]", "annotated_tactic": ["simp only [I, h\u2080, <a>Measure.map_smul</a>, J, <a>smul_smul</a>, h\u2081]", [{"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]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 map (\u2191L) \u03bc = (c\u2080 * c\u2081 * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 (c\u2080 * \u2191\u2191addHaar univ * c\u2081) \u2022 \u03bd = (c\u2080 * c\u2081 * \u2191\u2191addHaar univ) \u2022 \u03bd"}, {"tactic": "rw [mul_assoc, mul_comm _ c\u2081, \u2190 mul_assoc]", "annotated_tactic": ["rw [<a>mul_assoc</a>, <a>mul_comm</a> _ c\u2081, \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]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"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.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 (c\u2080 * \u2191\u2191addHaar univ * c\u2081) \u2022 \u03bd = (c\u2080 * c\u2081 * \u2191\u2191addHaar univ) \u2022 \u03bd", "state_after": "no goals"}, {"tactic": "rcases subsingleton_or_nontrivial E with hE|hE", "annotated_tactic": ["rcases <a>subsingleton_or_nontrivial</a> E with hE|hE", [{"full_name": "subsingleton_or_nontrivial", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [94, 9], "def_end_pos": [94, 35]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d : ProperSpace E\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 ProperSpace F", "state_after": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d : ProperSpace E\nthis : FiniteDimensional \ud835\udd5c E\nhE : Subsingleton E\n\u22a2 ProperSpace F\n\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d : ProperSpace E\nthis : FiniteDimensional \ud835\udd5c E\nhE : Nontrivial E\n\u22a2 ProperSpace F"}, {"tactic": "have : Subsingleton F := Function.Surjective.subsingleton h", "annotated_tactic": ["have : <a>Subsingleton</a> F := <a>Function.Surjective.subsingleton</a> 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": "Function.Surjective.subsingleton", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [232, 19], "def_end_pos": [232, 42]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d : ProperSpace E\nthis : FiniteDimensional \ud835\udd5c E\nhE : Subsingleton E\n\u22a2 ProperSpace F", "state_after": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nhE : Subsingleton E\nthis : Subsingleton F\n\u22a2 ProperSpace F"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nhE : Subsingleton E\nthis : Subsingleton F\n\u22a2 ProperSpace F", "state_after": "no goals"}, {"tactic": "have : ProperSpace \ud835\udd5c := properSpace_of_locallyCompact_module \ud835\udd5c E", "annotated_tactic": ["have : <a>ProperSpace</a> \ud835\udd5c := <a>properSpace_of_locallyCompact_module</a> \ud835\udd5c E", [{"full_name": "ProperSpace", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2198, 7], "def_end_pos": [2198, 18]}, {"full_name": "properSpace_of_locallyCompact_module", "def_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "def_pos": [522, 7], "def_end_pos": [522, 43]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d : ProperSpace E\nthis : FiniteDimensional \ud835\udd5c E\nhE : Nontrivial E\n\u22a2 ProperSpace F", "state_after": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nhE : Nontrivial E\nthis : ProperSpace \ud835\udd5c\n\u22a2 ProperSpace F"}, {"tactic": "have : FiniteDimensional \ud835\udd5c F := Module.Finite.of_surjective L h", "annotated_tactic": ["have : <a>FiniteDimensional</a> \ud835\udd5c F := <a>Module.Finite.of_surjective</a> L h", [{"full_name": "FiniteDimensional", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [87, 5], "def_end_pos": [87, 22]}, {"full_name": "Module.Finite.of_surjective", "def_path": "Mathlib/RingTheory/Finiteness.lean", "def_pos": [564, 9], "def_end_pos": [564, 22]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nhE : Nontrivial E\nthis : ProperSpace \ud835\udd5c\n\u22a2 ProperSpace F", "state_after": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nhE : Nontrivial E\nthis\u271d : ProperSpace \ud835\udd5c\nthis : FiniteDimensional \ud835\udd5c F\n\u22a2 ProperSpace F"}, {"tactic": "exact FiniteDimensional.proper \ud835\udd5c F", "annotated_tactic": ["exact <a>FiniteDimensional.proper</a> \ud835\udd5c F", [{"full_name": "FiniteDimensional.proper", "def_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "def_pos": [591, 9], "def_end_pos": [591, 33]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nhE : Nontrivial E\nthis\u271d : ProperSpace \ud835\udd5c\nthis : FiniteDimensional \ud835\udd5c F\n\u22a2 ProperSpace F", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\n\u22a2 L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\n\u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x"}, {"tactic": "obtain \u27e8y, z, hyz\u27e9 : \u2203 (y : S) (z : T), M.symm x = (y, z) := \u27e8_, _, rfl\u27e9", "annotated_tactic": ["obtain \u27e8y, z, hyz\u27e9 : \u2203 (y : S) (z : T), M.symm x = (y, z) := \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 h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\n\u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x", "state_after": "case h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\n\u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x"}, {"tactic": "have : x = M (y, z) := by\n  rw [\u2190 hyz]; simp only [LinearEquiv.apply_symm_apply]", "annotated_tactic": ["have : x = M (y, z) := by\n      rw [\u2190 hyz]; simp only [<a>LinearEquiv.apply_symm_apply</a>]", [{"full_name": "LinearEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [374, 9], "def_end_pos": [374, 25]}]], "state_before": "case h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\n\u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x", "state_after": "case h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nthis\u271d : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\nthis : x = \u2191M (y, z)\n\u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nthis\u271d : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\nthis : x = \u2191M (y, z)\n\u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x", "state_after": "no goals"}, {"tactic": "rw [\u2190 hyz]", "annotated_tactic": ["rw [\u2190 hyz]", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\n\u22a2 x = \u2191M (y, z)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\n\u22a2 x = \u2191M (\u2191(LinearEquiv.symm M) x)"}, {"tactic": "simp only [LinearEquiv.apply_symm_apply]", "annotated_tactic": ["simp only [<a>LinearEquiv.apply_symm_apply</a>]", [{"full_name": "LinearEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [374, 9], "def_end_pos": [374, 25]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nx : E\ny : { x // x \u2208 S }\nz : { x // x \u2208 T }\nhyz : \u2191(LinearEquiv.symm M) x = (y, z)\n\u22a2 x = \u2191M (\u2191(LinearEquiv.symm M) x)", "state_after": "no goals"}, {"tactic": "rw [Measure.map_map, Measure.map_map, A]", "annotated_tactic": ["rw [<a>Measure.map_map</a>, <a>Measure.map_map</a>, A]", [{"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_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 map (\u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)))) \u03bc = map ((\u2191L' \u2218 \u2191P) \u2218 \u2191(LinearEquiv.symm M)) \u03bc\n\ncase hg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable (\u2191L' \u2218 \u2191P)\n\ncase hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable \u2191(LinearEquiv.symm M)\n\ncase hg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable \u2191L'\n\ncase hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable \u2191P"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 map (\u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)))) \u03bc = map ((\u2191L' \u2218 \u2191P) \u2218 \u2191(LinearEquiv.symm M)) \u03bc", "state_after": "no goals"}, {"tactic": "exact L'_cont.measurable.comp P_cont.measurable", "annotated_tactic": ["exact L'_cont.measurable.comp P_cont.measurable", []], "state_before": "case hg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable (\u2191L' \u2218 \u2191P)", "state_after": "no goals"}, {"tactic": "exact M_cont.measurable", "annotated_tactic": ["exact M_cont.measurable", []], "state_before": "case hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable \u2191(LinearEquiv.symm M)", "state_after": "no goals"}, {"tactic": "exact L'_cont.measurable", "annotated_tactic": ["exact L'_cont.measurable", []], "state_before": "case hg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable \u2191L'", "state_after": "no goals"}, {"tactic": "exact P_cont.measurable", "annotated_tactic": ["exact P_cont.measurable", []], "state_before": "case hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\n\u22a2 Measurable \u2191P", "state_after": "no goals"}, {"tactic": "have : IsAddHaarMeasure (\u03bc.map M.symm) :=\n  M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map \u03bc", "annotated_tactic": ["have : <a>IsAddHaarMeasure</a> (\u03bc.map M.symm) :=\n      M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map \u03bc", [{"full_name": "MeasureTheory.Measure.IsAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [725, 7], "def_end_pos": [725, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\n\u22a2 \u2203 c\u2080, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u22a4 \u2227 map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nthis\u271d : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nthis : IsAddHaarMeasure (map (\u2191(LinearEquiv.symm M)) \u03bc)\n\u22a2 \u2203 c\u2080, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u22a4 \u2227 map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT"}, {"tactic": "exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _", "annotated_tactic": ["exact <a>isAddHaarMeasure_eq_smul_isAddHaarMeasure</a> _ _", [{"full_name": "MeasureTheory.Measure.isAddHaarMeasure_eq_smul_isAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [724, 15], "def_end_pos": [724, 56]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nthis\u271d : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nthis : IsAddHaarMeasure (map (\u2191(LinearEquiv.symm M)) \u03bc)\n\u22a2 \u2203 c\u2080, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u22a4 \u2227 map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT", "state_after": "no goals"}, {"tactic": "have : IsAddHaarMeasure (\u03bcT.map L') :=\n  L'.toContinuousLinearEquiv.isAddHaarMeasure_map \u03bcT", "annotated_tactic": ["have : <a>IsAddHaarMeasure</a> (\u03bcT.map L') :=\n      L'.toContinuousLinearEquiv.isAddHaarMeasure_map \u03bcT", [{"full_name": "MeasureTheory.Measure.IsAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [725, 7], "def_end_pos": [725, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\n\u22a2 \u2203 c\u2081, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u22a4 \u2227 map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nthis\u271d : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nthis : IsAddHaarMeasure (map (\u2191L') \u03bcT)\n\u22a2 \u2203 c\u2081, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u22a4 \u2227 map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd"}, {"tactic": "exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _", "annotated_tactic": ["exact <a>isAddHaarMeasure_eq_smul_isAddHaarMeasure</a> _ _", [{"full_name": "MeasureTheory.Measure.isAddHaarMeasure_eq_smul_isAddHaarMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [724, 15], "def_end_pos": [724, 56]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b2 : ProperSpace E\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nthis\u271d : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nthis : IsAddHaarMeasure (map (\u2191L') \u03bcT)\n\u22a2 \u2203 c\u2081, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u22a4 \u2227 map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd", "state_after": "no goals"}, {"tactic": "simp [pos_iff_ne_zero, c\u2080_pos, c\u2081_pos]", "annotated_tactic": ["simp [<a>pos_iff_ne_zero</a>, c\u2080_pos, c\u2081_pos]", [{"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": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : MeasurableSpace E\ninst\u271d\u2078 : BorelSpace E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : MeasurableSpace F\ninst\u271d\u2074 : BorelSpace F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192\u2097[\ud835\udd5c] F\n\u03bc : Measure E\n\u03bd : Measure F\ninst\u271d\u00b2 : IsAddHaarMeasure \u03bc\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bd\ninst\u271d : LocallyCompactSpace E\nh : Function.Surjective \u2191L\nthis\u271d\u00b9 : ProperSpace E\nthis\u271d : FiniteDimensional \ud835\udd5c E\nthis : ProperSpace F\nS : Submodule \ud835\udd5c E := ker L\nT : Submodule \ud835\udd5c E\nhT : IsCompl S T\nM : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT\nM_cont : Continuous \u2191(LinearEquiv.symm M)\nP : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T }\nP_cont : Continuous \u2191P\nI\u271d : Function.Bijective \u2191(domRestrict L T)\nL' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d\nL'_cont : Continuous \u2191L'\nA : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))\nI : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc))\n\u03bcS : Measure { x // x \u2208 S } := addHaar\n\u03bcT : Measure { x // x \u2208 T } := addHaar\nc\u2080 : \u211d\u22650\u221e\nc\u2080_pos : c\u2080 \u2260 0\nc\u2080_fin : c\u2080 \u2260 \u22a4\nh\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT\nJ : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT\nc\u2081 : \u211d\u22650\u221e\nc\u2081_pos : c\u2081 \u2260 0\nc\u2081_fin : c\u2081 \u2260 \u22a4\nh\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd\n\u22a2 0 < c\u2080 * c\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono", "start": [181, 1], "end": [185, 80], "traced_tactics": [{"tactic": "refine' strictMono_nat_of_lt_succ fun n => _", "annotated_tactic": ["refine' <a>strictMono_nat_of_lt_succ</a> fun n => _", [{"full_name": "strictMono_nat_of_lt_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1033, 9], "def_end_pos": [1033, 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\n\u22a2 StrictMono (seqTendstoAeSeq hfg)", "state_after": "\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 < seqTendstoAeSeq hfg (n + 1)"}, {"tactic": "rw [seqTendstoAeSeq_succ]", "annotated_tactic": ["rw [<a>seqTendstoAeSeq_succ</a>]", [{"full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [166, 9], "def_end_pos": [166, 29]}]], "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 < seqTendstoAeSeq hfg (n + 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 : 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 < max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1)"}, {"tactic": "exact lt_of_lt_of_le (lt_add_one <| seqTendstoAeSeq hfg n) (le_max_right _ _)", "annotated_tactic": ["exact <a>lt_of_lt_of_le</a> (<a>lt_add_one</a> <| <a>seqTendstoAeSeq</a> hfg n) (<a>le_max_right</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": "lt_add_one", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [20, 7], "def_end_pos": [20, 17]}, {"full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [161, 19], "def_end_pos": [161, 34]}, {"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\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 < 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/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.snd_map_prod_mk", "start": [999, 1], "end": [1001, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_op_distrib", "start": [79, 1], "end": [81, 33], "traced_tactics": [{"tactic": "simp only [fold, fold_distrib]", "annotated_tactic": ["simp only [<a>fold</a>, <a>fold_distrib</a>]", [{"full_name": "Finset.fold", "def_path": "Mathlib/Data/Finset/Fold.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}, {"full_name": "Multiset.fold_distrib", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [95, 9], "def_end_pos": [95, 21]}]], "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\u271d : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\nb\u2081 b\u2082 : \u03b2\n\u22a2 fold op (op b\u2081 b\u2082) (fun x => op (f x) (g x)) s = op (fold op b\u2081 f s) (fold op b\u2082 g s)", "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_add", "start": [402, 1], "end": [404, 77], "traced_tactics": [{"tactic": "simp_rw [condexpIndSMul]", "annotated_tactic": ["simp_rw [<a>condexpIndSMul</a>]", [{"full_name": "MeasureTheory.condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [384, 19], "def_end_pos": [384, 33]}]], "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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 condexpIndSMul hm hs h\u03bcs (x + y) = condexpIndSMul hm hs h\u03bcs x + condexpIndSMul hm hs h\u03bcs y", "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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (x + y))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\n    \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n      \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d y)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))"}, {"tactic": "rw [toSpanSingleton_add, add_compLpL, add_apply]", "annotated_tactic": ["rw [<a>toSpanSingleton_add</a>, <a>add_compLpL</a>, <a>add_apply</a>]", [{"full_name": "ContinuousLinearMap.toSpanSingleton_add", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1252, 9], "def_end_pos": [1252, 28]}, {"full_name": "ContinuousLinearMap.add_compLpL", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1187, 9], "def_end_pos": [1187, 20]}, {"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\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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (x + y))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\n    \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n      \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d y)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_reverse", "start": [1028, 1], "end": [1029, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.iUnion_null_iff'", "start": [129, 1], "end": [132, 101], "traced_tactics": [{"tactic": "by_cases i : \u03b9 <;> simp [i]", "annotated_tactic": ["by_cases i : \u03b9 <;> simp [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\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (\u22c3 (i : \u03b9), s i) = 0", "state_after": "case pos\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\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (s (_ : \u03b9)) = 0"}, {"tactic": "exact (fun h => h (Iff.mpr (Iff.of_eq (eq_true i)) trivial))", "annotated_tactic": ["exact (fun h => h (<a>Iff.mpr</a> (<a>Iff.of_eq</a> (<a>eq_true</a> i)) <a>trivial</a>))", [{"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": "Iff.of_eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [677, 9], "def_end_pos": [677, 18]}, {"full_name": "eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [14, 9], "def_end_pos": [14, 16]}, {"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 pos\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\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (s (_ : \u03b9)) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ManyOneEquiv.of_equiv", "start": [235, 1], "end": [237, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le", "start": [1210, 1], "end": [1221, 84], "traced_tactics": [{"tactic": "have : SigmaFinite (\u03bc.trim h\u03c4.measurableSpace_le) :=\n  haveI h_le : (h\u03c4.min h\u03c3).measurableSpace \u2264 h\u03c4.measurableSpace := by\n    rw [IsStoppingTime.measurableSpace_min]\n    exact inf_le_left; simp_all only\n  sigmaFiniteTrim_mono _ h_le", "annotated_tactic": ["have : <a>SigmaFinite</a> (\u03bc.trim h\u03c4.measurableSpace_le) :=\n    haveI h_le : (h\u03c4.min h\u03c3).<a>measurableSpace</a> \u2264 h\u03c4.measurableSpace := by\n      rw [<a>IsStoppingTime.measurableSpace_min</a>]\n      exact <a>inf_le_left</a>; simp_all only\n    <a>sigmaFiniteTrim_mono</a> _ h_le", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}, {"full_name": "MeasureTheory.IsStoppingTime.measurableSpace", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [318, 15], "def_end_pos": [318, 30]}, {"full_name": "MeasureTheory.IsStoppingTime.measurableSpace_min", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [591, 9], "def_end_pos": [591, 28]}, {"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [388, 9], "def_end_pos": [388, 20]}, {"full_name": "MeasureTheory.sigmaFiniteTrim_mono", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [107, 9], "def_end_pos": [107, 29]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 \u03bc[f|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[f|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\n\u22a2 \u03bc[f|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[f|IsStoppingTime.measurableSpace h\u03c4]"}, {"tactic": "refine' (condexp_ae_eq_restrict_of_measurableSpace_eq_on h\u03c4.measurableSpace_le\n  (h\u03c4.min h\u03c3).measurableSpace_le (h\u03c4.measurableSet_le_stopping_time h\u03c3) fun t => _).symm", "annotated_tactic": ["refine' (<a>condexp_ae_eq_restrict_of_measurableSpace_eq_on</a> h\u03c4.measurableSpace_le\n    (h\u03c4.min h\u03c3).<a>measurableSpace_le</a> (h\u03c4.measurableSet_le_stopping_time h\u03c3) fun t => _).<a>symm</a>", [{"full_name": "MeasureTheory.condexp_ae_eq_restrict_of_measurableSpace_eq_on", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean", "def_pos": [145, 9], "def_end_pos": [145, 56]}, {"full_name": "MeasureTheory.IsStoppingTime.measurableSpace_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [379, 9], "def_end_pos": [379, 27]}, {"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\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\n\u22a2 \u03bc[f|IsStoppingTime.measurableSpace\n        (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]\n    \u03bc[f|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nt : Set \u03a9\n\u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u2194 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)"}, {"tactic": "rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_le_iff]", "annotated_tactic": ["rw [<a>Set.inter_comm</a> _ t, <a>IsStoppingTime.measurableSet_inter_le_iff</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.IsStoppingTime.measurableSet_inter_le_iff", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [655, 9], "def_end_pos": [655, 35]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nt : Set \u03a9\n\u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u2194 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t)", "state_after": "case h\u03c0\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nt : Set \u03a9\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9"}, {"tactic": "simp_all only", "annotated_tactic": ["simp_all only", []], "state_before": "case h\u03c0\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\nthis : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nt : Set \u03a9\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9", "state_after": "no goals"}, {"tactic": "rw [IsStoppingTime.measurableSpace_min]", "annotated_tactic": ["rw [<a>IsStoppingTime.measurableSpace_min</a>]", [{"full_name": "MeasureTheory.IsStoppingTime.measurableSpace_min", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [591, 9], "def_end_pos": [591, 28]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 IsStoppingTime.measurableSpace h\u03c4", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime.measurableSpace ?h\u03c4 \u2293 IsStoppingTime.measurableSpace ?h\u03c0 \u2264 IsStoppingTime.measurableSpace h\u03c4\n\ncase h\u03c4\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c4 \u03c9\n\ncase h\u03c0\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9"}, {"tactic": "exact inf_le_left", "annotated_tactic": ["exact <a>inf_le_left</a>", [{"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [388, 9], "def_end_pos": [388, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime.measurableSpace ?h\u03c4 \u2293 IsStoppingTime.measurableSpace ?h\u03c0 \u2264 IsStoppingTime.measurableSpace h\u03c4\n\ncase h\u03c4\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c4 \u03c9\n\ncase h\u03c0\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9", "state_after": "case h\u03c0\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9"}, {"tactic": "simp_all only", "annotated_tactic": ["simp_all only", []], "state_before": "case h\u03c0\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9\u2070 : LinearOrder \u03b9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u03b9 m\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : CompleteSpace E\nf : \u03a9 \u2192 E\ninst\u271d\u2076 : IsCountablyGenerated atTop\ninst\u271d\u2075 : TopologicalSpace \u03b9\ninst\u271d\u2074 : OrderTopology \u03b9\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : SecondCountableTopology \u03b9\ninst\u271d\u00b9 : BorelSpace \u03b9\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c3 : IsStoppingTime \u2131 \u03c3\ninst\u271d :\n  SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m))\n\u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_lt_top_iff", "start": [133, 9], "end": [134, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean", "full_name": "Orientation.measure_eq_volume", "start": [50, 1], "end": [56, 36], "traced_tactics": [{"tactic": "have A : o.volumeForm.measure (stdOrthonormalBasis \u211d F).toBasis.parallelepiped = 1 :=\n  Orientation.measure_orthonormalBasis o (stdOrthonormalBasis \u211d F)", "annotated_tactic": ["have A : o.volumeForm.measure (<a>stdOrthonormalBasis</a> \u211d F).toBasis.parallelepiped = 1 :=\n    <a>Orientation.measure_orthonormalBasis</a> o (<a>stdOrthonormalBasis</a> \u211d F)", [{"full_name": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}, {"full_name": "Orientation.measure_orthonormalBasis", "def_path": "Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean", "def_pos": [36, 9], "def_end_pos": [36, 45]}, {"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\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)\n\u22a2 AlternatingMap.measure (volumeForm o) = volume", "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)\nA :\n  \u2191\u2191(AlternatingMap.measure (volumeForm o))\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =\n    1\n\u22a2 AlternatingMap.measure (volumeForm o) = volume"}, {"tactic": "rw [addHaarMeasure_unique o.volumeForm.measure\n  (stdOrthonormalBasis \u211d F).toBasis.parallelepiped, A, one_smul]", "annotated_tactic": ["rw [<a>addHaarMeasure_unique</a> o.volumeForm.measure\n    (<a>stdOrthonormalBasis</a> \u211d F).toBasis.parallelepiped, A, <a>one_smul</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": "stdOrthonormalBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [833, 17], "def_end_pos": [833, 36]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "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)\nA :\n  \u2191\u2191(AlternatingMap.measure (volumeForm o))\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =\n    1\n\u22a2 AlternatingMap.measure (volumeForm o) = volume", "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)\nA :\n  \u2191\u2191(AlternatingMap.measure (volumeForm o))\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =\n    1\n\u22a2 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) = volume"}, {"tactic": "simp only [volume, Basis.addHaar]", "annotated_tactic": ["simp only [<a>volume</a>, <a>Basis.addHaar</a>]", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Basis.addHaar", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [220, 1], "def_end_pos": [223, 42]}]], "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)\nA :\n  \u2191\u2191(AlternatingMap.measure (volumeForm o))\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =\n    1\n\u22a2 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) = volume", "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'.trim_pre", "start": [311, 1], "end": [317, 21], "traced_tactics": [{"tactic": "refine' le_antisymm (le_pre.2 fun s hs => _) (le_trim _)", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>le_pre</a>.2 fun s hs => _) (<a>le_trim</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.le_pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [272, 9], "def_end_pos": [272, 15]}, {"full_name": "MeasureTheory.OuterMeasure.le_trim", "def_path": 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OuterRegular (x \u2022 \u03bc)"}, {"tactic": "rw [zero_smul]", "annotated_tactic": ["rw [<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 inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nhx : 0 \u2260 \u22a4\n\u22a2 OuterRegular (0 \u2022 \u03bc)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nhx : 0 \u2260 \u22a4\n\u22a2 OuterRegular 0"}, {"tactic": "exact OuterRegular.zero", "annotated_tactic": ["exact <a>OuterRegular.zero</a>", [{"full_name": "MeasureTheory.Measure.OuterRegular.zero", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", 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OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\nA : Set \u03b1\nx\u271d : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191(x \u2022 \u03bc) A\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r"}, {"tactic": "rw [smul_apply, A.measure_eq_iInf_isOpen, smul_eq_mul] at hr", "annotated_tactic": ["rw [<a>smul_apply</a>, A.measure_eq_iInf_isOpen, <a>smul_eq_mul</a>] at hr", [{"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]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\nA : Set \u03b1\nx\u271d : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191(x \u2022 \u03bc) A\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\nA : Set \u03b1\nx\u271d : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > x * \u2a05 U, \u2a05 (_ : A \u2286 U), \u2a05 (_ : IsOpen U), \u2191\u2191\u03bc U\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r"}, {"tactic": "simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr", "annotated_tactic": ["simpa only [<a>ENNReal.mul_iInf_of_ne</a> h0 hx, <a>gt_iff_lt</a>, <a>iInf_lt_iff</a>, <a>exists_prop</a>] using hr", [{"full_name": "ENNReal.mul_iInf_of_ne", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2527, 9], "def_end_pos": [2527, 23]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"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": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\nA : Set \u03b1\nx\u271d : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > x * \u2a05 U, \u2a05 (_ : A \u2286 U), \u2a05 (_ : IsOpen U), \u2191\u2191\u03bc U\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r", "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_Lp_mul_Lq", "start": [159, 1], "end": [175, 101], "traced_tactics": [{"tactic": "by_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0", "annotated_tactic": ["by_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "by_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0", "annotated_tactic": ["by_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"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": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "by_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4", "annotated_tactic": ["by_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "exact ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top hpq hf hf_top hg_top hf_zero hg_zero", "annotated_tactic": ["exact <a>ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top</a> hpq hf hf_top hg_top hf_zero hg_zero", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [105, 9], "def_end_pos": [105, 56]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}, {"tactic": "refine' Eq.trans_le _ (zero_le _)", "annotated_tactic": ["refine' <a>Eq.trans_le</a> _ (<a>zero_le</a> _)", [{"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0"}, {"tactic": "exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.nonneg hf hf_zero", "annotated_tactic": ["exact <a>lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero</a> hpq.nonneg hf hf_zero", [{"full_name": "ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [137, 9], "def_end_pos": [137, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\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": "no goals"}, {"tactic": "refine' Eq.trans_le _ (zero_le _)", "annotated_tactic": ["refine' <a>Eq.trans_le</a> _ (<a>zero_le</a> _)", [{"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0"}, {"tactic": "rw [mul_comm]", "annotated_tactic": ["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 pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0"}, {"tactic": "exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.symm.nonneg hg hg_zero", "annotated_tactic": ["exact <a>lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero</a> hpq.symm.nonneg hg hg_zero", [{"full_name": "ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [137, 9], "def_end_pos": [137, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.pos hpq.symm.nonneg hf_top hg_zero", "annotated_tactic": ["exact <a>lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top</a> hpq.pos hpq.symm.nonneg hf_top hg_zero", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [147, 9], "def_end_pos": [147, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}, {"tactic": "rw [mul_comm, mul_comm ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>mul_comm</a> ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.symm.pos hpq.nonneg hg_top hf_zero", "annotated_tactic": ["exact <a>lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top</a> hpq.symm.pos hpq.nonneg hg_top hf_zero", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [147, 9], "def_end_pos": [147, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.integrableOn_finset_iUnion", "start": [209, 1], "end": [211, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.comap_compl", "start": [1831, 1], "end": [1837, 89], "traced_tactics": [{"tactic": "rw [\u2190Function.comp_def, \u2190MeasurableSpace.comap_comp]", "annotated_tactic": ["rw [\u2190<a>Function.comp_def</a>, \u2190<a>MeasurableSpace.comap_comp</a>]", [{"full_name": "Function.comp_def", "def_path": "Mathlib/Init/Function.lean", "def_pos": [26, 7], "def_end_pos": [26, 15]}, {"full_name": "MeasurableSpace.comap_comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 19]}]], "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' : MeasurableSpace \u03b2\ninst\u271d : BooleanAlgebra \u03b2\nh : Measurable compl\nf : \u03b1 \u2192 \u03b2\n\u22a2 MeasurableSpace.comap (fun a => (f a)\u1d9c) inferInstance = MeasurableSpace.comap f inferInstance", "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 t u : Set \u03b1\nm' : MeasurableSpace \u03b2\ninst\u271d : BooleanAlgebra \u03b2\nh : Measurable compl\nf : \u03b1 \u2192 \u03b2\n\u22a2 MeasurableSpace.comap (fun a => f a) (MeasurableSpace.comap compl inferInstance) =\n    MeasurableSpace.comap f inferInstance"}, {"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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm' : MeasurableSpace \u03b2\ninst\u271d : BooleanAlgebra \u03b2\nh : Measurable compl\nf : \u03b1 \u2192 \u03b2\n\u22a2 MeasurableSpace.comap (fun a => f a) (MeasurableSpace.comap compl inferInstance) =\n    MeasurableSpace.comap f inferInstance", "state_after": "case e_m\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' : MeasurableSpace \u03b2\ninst\u271d : BooleanAlgebra \u03b2\nh : Measurable compl\nf : \u03b1 \u2192 \u03b2\n\u22a2 MeasurableSpace.comap compl inferInstance = inferInstance"}, {"tactic": "exact (MeasurableEquiv.ofInvolutive _ compl_involutive h).measurableEmbedding.comap_eq", "annotated_tactic": ["exact (<a>MeasurableEquiv.ofInvolutive</a> _ <a>compl_involutive</a> h).measurableEmbedding.comap_eq", [{"full_name": "MeasurableEquiv.ofInvolutive", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1706, 5], "def_end_pos": [1706, 17]}, {"full_name": "compl_involutive", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [643, 9], "def_end_pos": [643, 25]}]], "state_before": "case e_m\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' : MeasurableSpace \u03b2\ninst\u271d : BooleanAlgebra \u03b2\nh : Measurable compl\nf : \u03b1 \u2192 \u03b2\n\u22a2 MeasurableSpace.comap compl inferInstance = inferInstance", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sup_eq_union", "start": [1343, 1], "end": [1344, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1.stmts_supportsStmt", "start": [1374, 1], "end": [1378, 60], "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\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q"}, {"tactic": "exact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls)", "annotated_tactic": ["exact fun l ls h \u21a6 <a>stmts\u2081_supportsStmt_mono</a> h (ss.2 _ ls)", [{"full_name": "Turing.TM1.stmts\u2081_supportsStmt_mono", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1341, 9], "def_end_pos": [1341, 33]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.casesCons_append1", "start": [320, 11], "end": [323, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.TendstoInMeasure.congr", "start": [84, 11], "end": [86, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.zero_emod", "start": [252, 9], "end": [252, 76], "traced_tactics": [{"tactic": "simp [mod_def', emod]", "annotated_tactic": ["simp [<a>mod_def'</a>, <a>emod</a>]", [{"full_name": "Int.mod_def'", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [233, 9], "def_end_pos": [233, 17]}, {"full_name": "Int.emod", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 9]}]], "state_before": "b : Int\n\u22a2 0 % b = 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.ediv_mul_cancel", "start": [704, 11], "end": [705, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup_sdiff_left", "start": [607, 1], "end": [611, 42], "traced_tactics": [{"tactic": "refine' Finset.cons_induction_on s _ fun b t _ h => _", "annotated_tactic": ["refine' <a>Finset.cons_induction_on</a> s _ fun b t _ h => _", [{"full_name": "Finset.cons_induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1234, 9], "def_end_pos": [1234, 26]}]], "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\u271d s : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (sup s fun b => a \\ f b) = a \\ inf s f", "state_after": "case refine'_1\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\u271d s : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (sup \u2205 fun b => a \\ f b) = a \\ inf \u2205 f\n\ncase refine'_2\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\u271d s : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nb : \u03b9\nt : Finset \u03b9\nx\u271d : \u00acb \u2208 t\nh : (sup t fun b => a \\ f b) = a \\ inf t f\n\u22a2 (sup (cons b t x\u271d) fun b => a \\ f b) = a \\ inf (cons b t x\u271d) f"}, {"tactic": "rw [sup_empty, inf_empty, sdiff_top]", "annotated_tactic": ["rw [<a>sup_empty</a>, <a>inf_empty</a>, <a>sdiff_top</a>]", [{"full_name": "Finset.sup_empty", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Finset.inf_empty", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [318, 9], "def_end_pos": [318, 18]}, {"full_name": "sdiff_top", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [1000, 9], "def_end_pos": [1000, 18]}]], "state_before": "case refine'_1\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\u271d s : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (sup \u2205 fun b => a \\ f b) = a \\ inf \u2205 f", "state_after": "no goals"}, {"tactic": "rw [sup_cons, inf_cons, h, sdiff_inf]", "annotated_tactic": ["rw [<a>sup_cons</a>, <a>inf_cons</a>, h, <a>sdiff_inf</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.inf_cons", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [323, 9], "def_end_pos": [323, 17]}, {"full_name": "sdiff_inf", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [675, 9], "def_end_pos": [675, 18]}]], "state_before": "case refine'_2\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\u271d s : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nb : \u03b9\nt : Finset \u03b9\nx\u271d : \u00acb \u2208 t\nh : (sup t fun b => a \\ f b) = a \\ inf t f\n\u22a2 (sup (cons b t x\u271d) fun b => a \\ f b) = a \\ inf (cons b t x\u271d) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "full_name": "MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''", "start": [335, 1], "end": [346, 19], "traced_tactics": [{"tactic": "change\n  StronglyMeasurable\n    ((fun y => \u222b x, (fun u : \u03b3 \u00d7 \u03b1 \u00d7 \u03b2 => f (u.1, u.2.2)) (x, y) \u2202\u03b7 y) \u2218 fun x => (a, x))", "annotated_tactic": ["change\n    <a>StronglyMeasurable</a>\n      ((fun y => \u222b x, (fun u : \u03b3 \u00d7 \u03b1 \u00d7 \u03b2 => f (u.1, u.2.2)) (x, y) \u2202\u03b7 y) \u2218 fun x => (a, x))", [{"full_name": "MeasureTheory.StronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [78, 5], "def_end_pos": [78, 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 : \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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, y)", "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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable ((fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y) \u2218 fun x => (a, x))"}, {"tactic": "refine' StronglyMeasurable.comp_measurable _ measurable_prod_mk_left", "annotated_tactic": ["refine' <a>StronglyMeasurable.comp_measurable</a> _ <a>measurable_prod_mk_left</a>", [{"full_name": "MeasureTheory.StronglyMeasurable.comp_measurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [387, 9], "def_end_pos": [387, 24]}, {"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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable ((fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y) \u2218 fun x => (a, x))", "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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y\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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 MeasurableSpace \u03b1"}, {"tactic": "have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' (\u03ba := \u03b7)\n  (hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd))", "annotated_tactic": ["have := <a>MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'</a> (\u03ba := \u03b7)\n    (hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd))", [{"full_name": "MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [330, 9], "def_end_pos": [330, 54]}]], "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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y\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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 MeasurableSpace \u03b1", "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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\nthis : StronglyMeasurable fun y => \u222b (x : \u03b3), (f \u2218 fun a => (a.1, a.2.2)) (x, y) \u2202\u2191\u03b7 y\n\u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y\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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 MeasurableSpace \u03b1"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\nthis : StronglyMeasurable fun y => \u222b (x : \u03b3), (f \u2218 fun a => (a.1, a.2.2)) (x, y) \u2202\u2191\u03b7 y\n\u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y\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\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 : \u03b3 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 MeasurableSpace \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.injective_codRestrict", "start": [164, 1], "end": [166, 64], "traced_tactics": [{"tactic": "simp only [Injective, Subtype.ext_iff, val_codRestrict_apply]", "annotated_tactic": ["simp only [<a>Injective</a>, <a>Subtype.ext_iff</a>, <a>val_codRestrict_apply</a>]", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Set.val_codRestrict_apply", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [152, 9], "def_end_pos": [152, 30]}]], "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\nf : \u03b9 \u2192 \u03b1\ns : Set \u03b1\nh : \u2200 (x : \u03b9), f x \u2208 s\n\u22a2 Injective (codRestrict f s h) \u2194 Injective f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upcrossingsBefore_pos_eq", "start": [718, 1], "end": [720, 55], "traced_tactics": [{"tactic": "simp_rw [upcrossingsBefore, (crossing_pos_eq hab).1]", "annotated_tactic": ["simp_rw [<a>upcrossingsBefore</a>, (<a>crossing_pos_eq</a> hab).1]", [{"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}, {"full_name": "MeasureTheory.crossing_pos_eq", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [670, 9], "def_end_pos": [670, 24]}]], "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 upcrossingsBefore 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N \u03c9 = upcrossingsBefore a b f N \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "generateFrom_measurableSet_of_generatePiSystem", "start": [277, 1], "end": [280, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepSets.bUnion", "start": [294, 1], "end": [297, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.smul_map_diagonal_volume_pi", "start": [353, 1], "end": [375, 16], "traced_tactics": [{"tactic": "refine' (Measure.pi_eq fun s hs => _).symm", "annotated_tactic": ["refine' (<a>Measure.pi_eq</a> fun s hs => _).<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\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\n\u22a2 ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = volume", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191volume (s i)"}, {"tactic": "simp only [det_diagonal, Measure.coe_smul, Algebra.id.smul_eq_mul, Pi.smul_apply]", "annotated_tactic": ["simp only [<a>det_diagonal</a>, <a>Measure.coe_smul</a>, <a>Algebra.id.smul_eq_mul</a>, <a>Pi.smul_apply</a>]", [{"full_name": "Matrix.det_diagonal", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [79, 9], "def_end_pos": [79, 21]}, {"full_name": "MeasureTheory.Measure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [836, 9], "def_end_pos": [836, 17]}, {"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.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191volume (s i)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191(Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)"}, {"tactic": "rw [Measure.map_apply _ (MeasurableSet.univ_pi hs)]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> _ (<a>MeasurableSet.univ_pi</a> hs)]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasurableSet.univ_pi", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [961, 19], "def_end_pos": [961, 40]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191(Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D))"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D))", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D))\n\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)"}, {"tactic": "rw [this, volume_pi_pi, Finset.abs_prod,\n  ENNReal.ofReal_prod_of_nonneg fun i _ => abs_nonneg (D i), \u2190 Finset.prod_mul_distrib]", "annotated_tactic": ["rw [this, <a>volume_pi_pi</a>, <a>Finset.abs_prod</a>,\n    <a>ENNReal.ofReal_prod_of_nonneg</a> fun i _ => <a>abs_nonneg</a> (D i), \u2190 <a>Finset.prod_mul_distrib</a>]", [{"full_name": "MeasureTheory.volume_pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [676, 9], "def_end_pos": [676, 21]}, {"full_name": "Finset.abs_prod", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [273, 9], "def_end_pos": [273, 17]}, {"full_name": "ENNReal.ofReal_prod_of_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2389, 9], "def_end_pos": [2389, 30]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [658, 9], "def_end_pos": [658, 25]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\nB : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\nB : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n\u22a2 \u220f x : \u03b9, ofReal |D x| * \u2191\u2191volume ((fun x_1 => D x * x_1) \u207b\u00b9' s x) = \u220f x : \u03b9, \u2191\u2191volume (s x)"}, {"tactic": "simp only [B]", "annotated_tactic": ["simp only [B]", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\nB : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n\u22a2 \u220f x : \u03b9, ofReal |D x| * \u2191\u2191volume ((fun x_1 => D x * x_1) \u207b\u00b9' s x) = \u220f x : \u03b9, \u2191\u2191volume (s x)", "state_after": "no goals"}, {"tactic": "exact Continuous.measurable (LinearMap.continuous_on_pi _)", "annotated_tactic": ["exact <a>Continuous.measurable</a> (<a>LinearMap.continuous_on_pi</a> _)", [{"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}, {"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": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D))", "state_after": "no goals"}, {"tactic": "ext f", "annotated_tactic": ["ext f", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i", "state_after": "case h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nf : \u03b9 \u2192 \u211d\n\u22a2 (f \u2208 \u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) \u2194\n    f \u2208 Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i"}, {"tactic": "simp only [LinearMap.coe_proj, Algebra.id.smul_eq_mul, LinearMap.smul_apply, mem_univ_pi,\n  mem_preimage, LinearMap.pi_apply, diagonal_toLin']", "annotated_tactic": ["simp only [<a>LinearMap.coe_proj</a>, <a>Algebra.id.smul_eq_mul</a>, <a>LinearMap.smul_apply</a>, <a>mem_univ_pi</a>,\n      <a>mem_preimage</a>, <a>LinearMap.pi_apply</a>, <a>diagonal_toLin'</a>]", [{"full_name": "LinearMap.coe_proj", "def_path": "Mathlib/LinearAlgebra/Pi.lean", "def_pos": [88, 9], "def_end_pos": [88, 17]}, {"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": "LinearMap.smul_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"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.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "LinearMap.pi_apply", "def_path": "Mathlib/LinearAlgebra/Pi.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Matrix.diagonal_toLin'", "def_path": "Mathlib/LinearAlgebra/Matrix/Diagonal.lean", "def_pos": [45, 9], "def_end_pos": [45, 24]}]], "state_before": "case h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nf : \u03b9 \u2192 \u211d\n\u22a2 (f \u2208 \u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) \u2194\n    f \u2208 Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\n\u22a2 \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\n\u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)"}, {"tactic": "have A : D i \u2260 0 := by\n  simp only [det_diagonal, Ne.def] at h\n  exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i)", "annotated_tactic": ["have A : D i \u2260 0 := by\n      simp only [<a>det_diagonal</a>, <a>Ne.def</a>] at h\n      exact <a>Finset.prod_ne_zero_iff</a>.1 h i (<a>Finset.mem_univ</a> i)", [{"full_name": "Matrix.det_diagonal", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [79, 9], "def_end_pos": [79, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Finset.prod_ne_zero_iff", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 25]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\n\u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\nA : D i \u2260 0\n\u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)"}, {"tactic": "rw [volume_preimage_mul_left A, \u2190 mul_assoc, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul,\n  mul_inv_cancel A, abs_one, ENNReal.ofReal_one, one_mul]", "annotated_tactic": ["rw [<a>volume_preimage_mul_left</a> A, \u2190 <a>mul_assoc</a>, \u2190 <a>ENNReal.ofReal_mul</a> (<a>abs_nonneg</a> _), \u2190 <a>abs_mul</a>,\n      <a>mul_inv_cancel</a> A, <a>abs_one</a>, <a>ENNReal.ofReal_one</a>, <a>one_mul</a>]", [{"full_name": "Real.volume_preimage_mul_left", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 33]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 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_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"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_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\nA : D i \u2260 0\n\u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)", "state_after": "no goals"}, {"tactic": "simp only [det_diagonal, Ne.def] at h", "annotated_tactic": ["simp only [<a>det_diagonal</a>, <a>Ne.def</a>] at h", [{"full_name": "Matrix.det_diagonal", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [79, 9], "def_end_pos": [79, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\n\u22a2 D i \u2260 0", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\nh : \u00ac\u220f i : \u03b9, D i = 0\n\u22a2 D i \u2260 0"}, {"tactic": "exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i)", "annotated_tactic": ["exact <a>Finset.prod_ne_zero_iff</a>.1 h i (<a>Finset.mem_univ</a> i)", [{"full_name": "Finset.prod_ne_zero_iff", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 25]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\nh : \u00ac\u220f i : \u03b9, D i = 0\n\u22a2 D i \u2260 0", "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_measure", "start": [1492, 1], "end": [1500, 75], "traced_tactics": [{"tactic": "induction s using Finset.cons_induction_on with\n| h\u2081 => simp\n| h\u2082 h ih =>\n  rw [Finset.forall_mem_cons] at hf\n  rw [Finset.sum_cons, Finset.sum_cons, \u2190 ih hf.2]\n  exact integral_add_measure hf.1 (integrable_finset_sum_measure.2 hf.2)", "annotated_tactic": ["induction s using <a>Finset.cons_induction_on</a> with\n  | h\u2081 => simp\n  | h\u2082 h ih =>\n    rw [<a>Finset.forall_mem_cons</a>] at hf\n    rw [<a>Finset.sum_cons</a>, <a>Finset.sum_cons</a>, \u2190 ih hf.2]\n    exact <a>integral_add_measure</a> hf.1 (<a>integrable_finset_sum_measure</a>.2 hf.2)", [{"full_name": "Finset.cons_induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1234, 9], "def_end_pos": [1234, 26]}, {"full_name": "Finset.forall_mem_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}, {"full_name": "Finset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [310, 3], "def_end_pos": [310, 14]}, {"full_name": "Finset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [310, 3], "def_end_pos": [310, 14]}, {"full_name": "MeasureTheory.integral_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1462, 9], "def_end_pos": [1462, 29]}, {"full_name": "MeasureTheory.integrable_finset_sum_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [572, 9], "def_end_pos": [572, 38]}]], "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\ns : Finset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i = \u2211 i in s, \u222b (a : \u03b1), f a \u2202\u03bc i", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\u2081\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 : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in \u2205, \u03bc i = \u2211 i in \u2205, \u222b (a : \u03b1), f a \u2202\u03bc i", "state_after": "no goals"}, {"tactic": "rw [Finset.forall_mem_cons] at hf", "annotated_tactic": ["rw [<a>Finset.forall_mem_cons</a>] at hf", [{"full_name": "Finset.forall_mem_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}]], "state_before": "case h\u2082\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\na\u271d : \u03b9\ns\u271d : Finset \u03b9\nh : \u00aca\u271d \u2208 s\u271d\nih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i\nhf : \u2200 (i : \u03b9), i \u2208 Finset.cons a\u271d s\u271d h \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in Finset.cons a\u271d s\u271d h, \u03bc i = \u2211 i in Finset.cons a\u271d s\u271d h, \u222b (a : \u03b1), f a \u2202\u03bc i", "state_after": "case h\u2082\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\na\u271d : \u03b9\ns\u271d : Finset \u03b9\nh : \u00aca\u271d \u2208 s\u271d\nih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i\nhf : Integrable f \u2227 \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in Finset.cons a\u271d s\u271d h, \u03bc i = \u2211 i in Finset.cons a\u271d s\u271d h, \u222b (a : \u03b1), f a \u2202\u03bc i"}, {"tactic": "rw [Finset.sum_cons, Finset.sum_cons, \u2190 ih hf.2]", "annotated_tactic": ["rw [<a>Finset.sum_cons</a>, <a>Finset.sum_cons</a>, \u2190 ih hf.2]", [{"full_name": "Finset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [310, 3], "def_end_pos": [310, 14]}, {"full_name": "Finset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [310, 3], "def_end_pos": [310, 14]}]], "state_before": "case h\u2082\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\na\u271d : \u03b9\ns\u271d : Finset \u03b9\nh : \u00aca\u271d \u2208 s\u271d\nih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i\nhf : Integrable f \u2227 \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in Finset.cons a\u271d s\u271d h, \u03bc i = \u2211 i in Finset.cons a\u271d s\u271d h, \u222b (a : \u03b1), f a \u2202\u03bc i", "state_after": "case h\u2082\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\na\u271d : \u03b9\ns\u271d : Finset \u03b9\nh : \u00aca\u271d \u2208 s\u271d\nih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i\nhf : Integrable f \u2227 \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202(\u03bc a\u271d + \u2211 x in s\u271d, \u03bc x) = \u222b (a : \u03b1), f a \u2202\u03bc a\u271d + \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i"}, {"tactic": "exact integral_add_measure hf.1 (integrable_finset_sum_measure.2 hf.2)", "annotated_tactic": ["exact <a>integral_add_measure</a> hf.1 (<a>integrable_finset_sum_measure</a>.2 hf.2)", [{"full_name": "MeasureTheory.integral_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1462, 9], "def_end_pos": [1462, 29]}, {"full_name": "MeasureTheory.integrable_finset_sum_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [572, 9], "def_end_pos": [572, 38]}]], "state_before": "case h\u2082\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\na\u271d : \u03b9\ns\u271d : Finset \u03b9\nh : \u00aca\u271d \u2208 s\u271d\nih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i\nhf : Integrable f \u2227 \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 Integrable f\n\u22a2 \u222b (a : \u03b1), f a \u2202(\u03bc a\u271d + \u2211 x in s\u271d, \u03bc x) = \u222b (a : \u03b1), f a \u2202\u03bc a\u271d + \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ENNReal.ofReal_cinfi", "start": [119, 1], "end": [130, 45], "traced_tactics": [{"tactic": "by_cases hf : BddBelow (range f)", "annotated_tactic": ["by_cases hf : <a>BddBelow</a> (<a>range</a> f)", [{"full_name": "BddBelow", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"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\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\n\u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : BddBelow (range f)\n\u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i)"}, {"tactic": "exact\n  Monotone.map_ciInf_of_continuousAt ENNReal.continuous_ofReal.continuousAt\n    (fun i j hij => ENNReal.ofReal_le_ofReal hij) hf", "annotated_tactic": ["exact\n      <a>Monotone.map_ciInf_of_continuousAt</a> ENNReal.continuous_ofReal.continuousAt\n        (fun i j hij => <a>ENNReal.ofReal_le_ofReal</a> hij) hf", [{"full_name": "Monotone.map_ciInf_of_continuousAt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2922, 9], "def_end_pos": [2922, 43]}, {"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\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : BddBelow (range f)\n\u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i)", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 \u2a05 i, ENNReal.ofReal (f i) = ENNReal.ofReal (\u2a05 i, f i)"}, {"tactic": "rw [Real.iInf_of_not_bddBelow hf, ENNReal.ofReal_zero, \u2190 ENNReal.bot_eq_zero, iInf_eq_bot]", "annotated_tactic": ["rw [<a>Real.iInf_of_not_bddBelow</a> hf, <a>ENNReal.ofReal_zero</a>, \u2190 <a>ENNReal.bot_eq_zero</a>, <a>iInf_eq_bot</a>]", [{"full_name": "Real.iInf_of_not_bddBelow", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [847, 9], "def_end_pos": [847, 29]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}, {"full_name": "ENNReal.bot_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [682, 9], "def_end_pos": [682, 20]}, {"full_name": "iInf_eq_bot", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 \u2a05 i, ENNReal.ofReal (f i) = ENNReal.ofReal (\u2a05 i, f i)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b"}, {"tactic": "obtain \u27e8y, hy_mem, hy_neg\u27e9 := not_bddBelow_iff.mp hf 0", "annotated_tactic": ["obtain \u27e8y, hy_mem, hy_neg\u27e9 := not_bddBelow_iff.mp hf 0", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ny : \u211d\nhy_mem : y \u2208 range f\nhy_neg : y < 0\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b"}, {"tactic": "obtain \u27e8i, rfl\u27e9 := mem_range.mpr hy_mem", "annotated_tactic": ["obtain \u27e8i, rfl\u27e9 := mem_range.mpr hy_mem", []], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ny : \u211d\nhy_mem : y \u2208 range f\nhy_neg : y < 0\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b", "state_after": "case neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ni : \u03b1\nhy_mem : (fun y => f y) i \u2208 range f\nhy_neg : (fun y => f y) i < 0\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b"}, {"tactic": "refine' fun x hx => \u27e8i, _\u27e9", "annotated_tactic": ["refine' fun x hx => \u27e8i, _\u27e9", []], "state_before": "case neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ni : \u03b1\nhy_mem : (fun y => f y) i \u2208 range f\nhy_neg : (fun y => f y) i < 0\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b", "state_after": "case neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ni : \u03b1\nhy_mem : (fun y => f y) i \u2208 range f\nhy_neg : (fun y => f y) i < 0\nx : \u211d\u22650\u221e\nhx : x > \u22a5\n\u22a2 ENNReal.ofReal (f i) < x"}, {"tactic": "rwa [ENNReal.ofReal_of_nonpos hy_neg.le]", "annotated_tactic": ["rwa [<a>ENNReal.ofReal_of_nonpos</a> hy_neg.le]", [{"full_name": "ENNReal.ofReal_of_nonpos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2178, 11], "def_end_pos": [2178, 27]}]], "state_before": "case neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ni : \u03b1\nhy_mem : (fun y => f y) i \u2208 range f\nhy_neg : (fun y => f y) i < 0\nx : \u211d\u22650\u221e\nhx : x > \u22a5\n\u22a2 ENNReal.ofReal (f i) < x", "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_sub'", "start": [414, 1], "end": [417, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "MeasureTheory.Measure.IicSnd_apply", "start": [183, 1], "end": [187, 56], "traced_tactics": [{"tactic": "rw [IicSnd, fst_apply hs,\n  restrict_apply' (MeasurableSet.univ.prod (measurableSet_Iic : MeasurableSet (Iic r))), \u2190\n  prod_univ, prod_inter_prod, inter_univ, univ_inter]", "annotated_tactic": ["rw [<a>IicSnd</a>, <a>fst_apply</a> hs,\n    <a>restrict_apply'</a> (MeasurableSet.univ.prod (<a>measurableSet_Iic</a> : <a>MeasurableSet</a> (<a>Iic</a> r))), \u2190\n    <a>prod_univ</a>, <a>prod_inter_prod</a>, <a>inter_univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.Measure.IicSnd", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [179, 19], "def_end_pos": [179, 25]}, {"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.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "Set.prod_univ", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [133, 9], "def_end_pos": [133, 18]}, {"full_name": "Set.prod_inter_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [174, 9], "def_end_pos": [174, 24]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(IicSnd \u03c1 r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pi.lean", "full_name": "Finset.mem_pi", "start": [49, 1], "end": [51, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.add_lt_add_iff_right", "start": [789, 11], "end": [790, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_const_lt_top", "start": [165, 1], "end": [166, 88], "traced_tactics": [{"tactic": "simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set \u03b1) hc", "annotated_tactic": ["simpa only [<a>Measure.restrict_univ</a>] using <a>set_lintegral_const_lt_top</a> (<a>univ</a> : <a>Set</a> \u03b1) hc", [{"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "MeasureTheory.set_lintegral_const_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [159, 9], "def_end_pos": [159, 35]}, {"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": "\u03b1 : Type u_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 : IsFiniteMeasure \u03bc\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), c \u2202\u03bc < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.martingale_martingalePart", "start": [93, 1], "end": [131, 6], "traced_tactics": [{"tactic": "refine' \u27e8adapted_martingalePart hf, fun i j hij => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>adapted_martingalePart</a> hf, fun i j hij => _\u27e9", [{"full_name": "MeasureTheory.adapted_martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [82, 9], "def_end_pos": [82, 31]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\n\u22a2 Martingale (martingalePart f \u2131 \u03bc) \u2131 \u03bc", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i"}, {"tactic": "refine' h_eq_sum.trans _", "annotated_tactic": ["refine' h_eq_sum.trans _", []], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i"}, {"tactic": "have h_ge : \u2200 k, i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] 0 := by\n  intro k hk\n  have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] :=\n    condexp_condexp_of_le (\u2131.mono hk) (\u2131.le k)\n  filter_upwards [this] with x hx\n  rw [Pi.sub_apply, Pi.zero_apply, hx, sub_self]", "annotated_tactic": ["have h_ge : \u2200 k, i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] 0 := by\n    intro k hk\n    have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] :=\n      <a>condexp_condexp_of_le</a> (\u2131.mono hk) (\u2131.le k)\n    filter_upwards [this] with x hx\n    rw [<a>Pi.sub_apply</a>, <a>Pi.zero_apply</a>, hx, <a>sub_self</a>]", [{"full_name": "MeasureTheory.condexp_condexp_of_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [330, 9], "def_end_pos": [330, 30]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"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\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i"}, {"tactic": "rw [martingalePart_eq_sum]", "annotated_tactic": ["rw [<a>martingalePart_eq_sum</a>]", [{"full_name": "MeasureTheory.martingalePart_eq_sum", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [75, 9], "def_end_pos": [75, 30]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc]\n    (fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])) i"}, {"tactic": "refine' EventuallyEq.add EventuallyEq.rfl _", "annotated_tactic": ["refine' <a>EventuallyEq.add</a> <a>EventuallyEq.rfl</a> _", [{"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.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc]\n    (fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])) i", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => Finset.sum (Finset.range i) (fun i => f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) x"}, {"tactic": "rw [\u2190 Finset.sum_range_add_sum_Ico _ hij, \u2190\n  add_zero (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2131 i]))]", "annotated_tactic": ["rw [\u2190 <a>Finset.sum_range_add_sum_Ico</a> _ hij, \u2190\n    <a>add_zero</a> (\u2211 i in <a>Finset.range</a> i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2131 i]))]", [{"full_name": "Finset.sum_range_add_sum_Ico", "def_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "def_pos": [88, 3], "def_end_pos": [88, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"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\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => Finset.sum (Finset.range i) (fun i => f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) x", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x =>\n      (\u2211 k in Finset.range i, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) +\n          \u2211 k in Finset.Ico i j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]))\n        x) =\u1d50[\u03bc]\n    fun x => (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) + 0) x"}, {"tactic": "refine' (eventuallyEq_sum fun k hk => h_lt k (Finset.mem_range.mp hk)).add _", "annotated_tactic": ["refine' (<a>eventuallyEq_sum</a> fun k hk => h_lt k (Finset.mem_range.mp hk)).<a>add</a> _", [{"full_name": "eventuallyEq_sum", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [774, 3], "def_end_pos": [774, 14]}, {"full_name": "Filter.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x =>\n      (\u2211 k in Finset.range i, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) +\n          \u2211 k in Finset.Ico i j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]))\n        x) =\u1d50[\u03bc]\n    fun x => (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) + 0) x", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x => Finset.sum (Finset.Ico i j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => OfNat.ofNat 0 x"}, {"tactic": "refine' (eventuallyEq_sum fun k hk => h_ge k (Finset.mem_Ico.mp hk).1).trans _", "annotated_tactic": ["refine' (<a>eventuallyEq_sum</a> fun k hk => h_ge k (Finset.mem_Ico.mp hk).1).<a>trans</a> _", [{"full_name": "eventuallyEq_sum", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [774, 3], "def_end_pos": [774, 14]}, {"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\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x => Finset.sum (Finset.Ico i j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => OfNat.ofNat 0 x", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 \u2211 i in Finset.Ico i j, 0 =\u1d50[\u03bc] fun x => OfNat.ofNat 0 x"}, {"tactic": "simp only [Finset.sum_const_zero, Pi.zero_apply]", "annotated_tactic": ["simp only [<a>Finset.sum_const_zero</a>, <a>Pi.zero_apply</a>]", [{"full_name": "Finset.sum_const_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [356, 3], "def_end_pos": [356, 14]}, {"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\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 \u2211 i in Finset.Ico i j, 0 =\u1d50[\u03bc] fun x => OfNat.ofNat 0 x", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 0 =\u1d50[\u03bc] fun x => 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 0 =\u1d50[\u03bc] fun x => 0", "state_after": "no goals"}, {"tactic": "rw [martingalePart_eq_sum]", "annotated_tactic": ["rw [<a>martingalePart_eq_sum</a>]", [{"full_name": "MeasureTheory.martingalePart_eq_sum", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [75, 9], "def_end_pos": [75, 30]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \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])) j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])"}, {"tactic": "refine' (condexp_add (hf_int 0) _).trans _", "annotated_tactic": ["refine' (<a>condexp_add</a> (hf_int 0) _).<a>trans</a> _", [{"full_name": "MeasureTheory.condexp_add", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [280, 9], "def_end_pos": [280, 20]}, {"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\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \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])) j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])", "state_after": "case refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 Integrable (\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]))\n\ncase refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[f 0|\u2191\u2131 i] + \u03bc[\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])"}, {"tactic": "refine' (EventuallyEq.add EventuallyEq.rfl (condexp_finset_sum fun i _ => _)).trans _", "annotated_tactic": ["refine' (<a>EventuallyEq.add</a> <a>EventuallyEq.rfl</a> (<a>condexp_finset_sum</a> fun i _ => _)).<a>trans</a> _", [{"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.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}, {"full_name": "MeasureTheory.condexp_finset_sum", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [293, 9], "def_end_pos": [293, 27]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[f 0|\u2191\u2131 i] + \u03bc[\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])", "state_after": "case refine'_2.refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni\u271d j : \u2115\nhij : i\u271d \u2264 j\ni : \u2115\nx\u271d : i \u2208 Finset.range j\n\u22a2 Integrable (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])\n\ncase refine'_2.refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x =>\n      (\u03bc[f 0|\u2191\u2131 i]) x +\n        Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])"}, {"tactic": "refine' EventuallyEq.add _ _", "annotated_tactic": ["refine' <a>EventuallyEq.add</a> _ _", [{"full_name": "Filter.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}]], "state_before": "case refine'_2.refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x =>\n      (\u03bc[f 0|\u2191\u2131 i]) x +\n        Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])", "state_after": "case refine'_2.refine'_2.refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x => (\u03bc[f 0|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => f 0 x\n\ncase refine'_2.refine'_2.refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x =>\n      Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x"}, {"tactic": "exact integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp", "annotated_tactic": ["exact <a>integrable_finset_sum'</a> _ fun i _ => ((hf_int _).<a>sub</a> (hf_int _)).<a>sub</a> <a>integrable_condexp</a>", [{"full_name": "MeasureTheory.integrable_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [682, 9], "def_end_pos": [682, 31]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}, {"full_name": "MeasureTheory.integrable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [210, 9], "def_end_pos": [210, 27]}]], "state_before": "case refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 Integrable (\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]))", "state_after": "no goals"}, {"tactic": "exact ((hf_int _).sub (hf_int _)).sub integrable_condexp", "annotated_tactic": ["exact ((hf_int _).<a>sub</a> (hf_int _)).<a>sub</a> <a>integrable_condexp</a>", [{"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}, {"full_name": "MeasureTheory.integrable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [210, 9], "def_end_pos": [210, 27]}]], "state_before": "case refine'_2.refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni\u271d j : \u2115\nhij : i\u271d \u2264 j\ni : \u2115\nx\u271d : i \u2208 Finset.range j\n\u22a2 Integrable (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])", "state_after": "no goals"}, {"tactic": "rw [condexp_of_stronglyMeasurable (\u2131.le _) _ (hf_int 0)]", "annotated_tactic": ["rw [<a>condexp_of_stronglyMeasurable</a> (\u2131.le _) _ (hf_int 0)]", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}]], "state_before": "case refine'_2.refine'_2.refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x => (\u03bc[f 0|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => f 0 x", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 StronglyMeasurable (f 0)"}, {"tactic": "exact (hf 0).mono (\u2131.mono (zero_le i))", "annotated_tactic": ["exact (hf 0).<a>mono</a> (\u2131.mono (<a>zero_le</a> i))", [{"full_name": "MeasureTheory.StronglyMeasurable.mono", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [369, 19], "def_end_pos": [369, 23]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 StronglyMeasurable (f 0)", "state_after": "no goals"}, {"tactic": "exact eventuallyEq_sum fun k _ => condexp_sub ((hf_int _).sub (hf_int _)) integrable_condexp", "annotated_tactic": ["exact <a>eventuallyEq_sum</a> fun k _ => <a>condexp_sub</a> ((hf_int _).<a>sub</a> (hf_int _)) <a>integrable_condexp</a>", [{"full_name": "eventuallyEq_sum", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [774, 3], "def_end_pos": [774, 14]}, {"full_name": "MeasureTheory.condexp_sub", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 20]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}, {"full_name": "MeasureTheory.integrable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [210, 9], "def_end_pos": [210, 27]}]], "state_before": "case refine'_2.refine'_2.refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x =>\n      Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x", "state_after": "no goals"}, {"tactic": "intro k hk", "annotated_tactic": ["intro k hk", []], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\n\u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0"}, {"tactic": "have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] :=\n  condexp_condexp_of_le (\u2131.mono hk) (\u2131.le k)", "annotated_tactic": ["have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] :=\n      <a>condexp_condexp_of_le</a> (\u2131.mono hk) (\u2131.le k)", [{"full_name": "MeasureTheory.condexp_condexp_of_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [330, 9], "def_end_pos": [330, 30]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\n\u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0", "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\nthis : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i]\n\u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0"}, {"tactic": "filter_upwards [this] with x hx", "annotated_tactic": ["filter_upwards [this] with x hx", []], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\nthis : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i]\n\u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\nthis : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i]\nx : \u03a9\nhx : (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x\n\u22a2 (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = OfNat.ofNat 0 x"}, {"tactic": "rw [Pi.sub_apply, Pi.zero_apply, hx, sub_self]", "annotated_tactic": ["rw [<a>Pi.sub_apply</a>, <a>Pi.zero_apply</a>, hx, <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": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 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\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\nthis : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i]\nx : \u03a9\nhx : (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x\n\u22a2 (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = OfNat.ofNat 0 x", "state_after": "no goals"}, {"tactic": "refine' fun k hk => EventuallyEq.sub _ _", "annotated_tactic": ["refine' fun k hk => <a>EventuallyEq.sub</a> _ _", [{"full_name": "Filter.EventuallyEq.sub", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1544, 3], "def_end_pos": [1544, 14]}]], "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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n\u22a2 \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]", "state_after": "case refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 (fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (f (k + 1) - f k) x\n\ncase refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 (fun x => (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 k]) x"}, {"tactic": "rw [condexp_of_stronglyMeasurable]", "annotated_tactic": ["rw [<a>condexp_of_stronglyMeasurable</a>]", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}]], "state_before": "case refine'_1\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 (fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (f (k + 1) - f k) x", "state_after": "case refine'_1.hf\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 StronglyMeasurable (f (k + 1) - f k)\n\ncase refine'_1.hfi\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 Integrable (f (k + 1) - f k)"}, {"tactic": "exact ((hf (k + 1)).mono (\u2131.mono (Nat.succ_le_of_lt hk))).sub ((hf k).mono (\u2131.mono hk.le))", "annotated_tactic": ["exact ((hf (k + 1)).<a>mono</a> (\u2131.mono (<a>Nat.succ_le_of_lt</a> hk))).<a>sub</a> ((hf k).<a>mono</a> (\u2131.mono hk.le))", [{"full_name": "MeasureTheory.StronglyMeasurable.mono", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [369, 19], "def_end_pos": [369, 23]}, {"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": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 14]}, {"full_name": "MeasureTheory.StronglyMeasurable.mono", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [369, 19], "def_end_pos": [369, 23]}]], "state_before": "case refine'_1.hf\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 StronglyMeasurable (f (k + 1) - f k)", "state_after": "no goals"}, {"tactic": "exact (hf_int _).sub (hf_int _)", "annotated_tactic": ["exact (hf_int _).<a>sub</a> (hf_int _)", [{"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "state_before": "case refine'_1.hfi\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 Integrable (f (k + 1) - f k)", "state_after": "no goals"}, {"tactic": "rw [condexp_of_stronglyMeasurable]", "annotated_tactic": ["rw [<a>condexp_of_stronglyMeasurable</a>]", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}]], "state_before": "case refine'_2\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 (fun x => (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 k]) x", "state_after": "case refine'_2.hf\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 StronglyMeasurable (\u03bc[f (k + 1) - f k|\u2191\u2131 k])\n\ncase refine'_2.hfi\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 Integrable (\u03bc[f (k + 1) - f k|\u2191\u2131 k])"}, {"tactic": "exact stronglyMeasurable_condexp.mono (\u2131.mono hk.le)", "annotated_tactic": ["exact stronglyMeasurable_condexp.mono (\u2131.mono hk.le)", []], "state_before": "case refine'_2.hf\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 StronglyMeasurable (\u03bc[f (k + 1) - f k|\u2191\u2131 k])", "state_after": "no goals"}, {"tactic": "exact integrable_condexp", "annotated_tactic": ["exact <a>integrable_condexp</a>", [{"full_name": "MeasureTheory.integrable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [210, 9], "def_end_pos": [210, 27]}]], "state_before": "case refine'_2.hfi\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 : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 Integrable (\u03bc[f (k + 1) - f k|\u2191\u2131 k])", "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_addChar_right_cancel", "start": [209, 1], "end": [210, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_ssubset_Icc_right", "start": [281, 1], "end": [284, 43], "traced_tactics": [{"tactic": "rw [\u2190 coe_ssubset, coe_Icc, coe_Icc]", "annotated_tactic": ["rw [\u2190 <a>coe_ssubset</a>, <a>coe_Icc</a>, <a>coe_Icc</a>]", [{"full_name": "Finset.coe_ssubset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [411, 9], "def_end_pos": [411, 20]}, {"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\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 < b\u2082\n\u22a2 Icc a\u2081 b\u2081 \u2282 Icc a\u2082 b\u2082", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 < b\u2082\n\u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082"}, {"tactic": "exact Set.Icc_ssubset_Icc_right hI ha hb", "annotated_tactic": ["exact <a>Set.Icc_ssubset_Icc_right</a> hI ha hb", [{"full_name": "Set.Icc_ssubset_Icc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [590, 9], "def_end_pos": [590, 30]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 < b\u2082\n\u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082", "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_le_integral_le", "start": [422, 1], "end": [448, 38], "traced_tactics": [{"tactic": "lift \u03b5 to \u211d\u22650 using \u03b5pos.le", "annotated_tactic": ["lift \u03b5 to \u211d\u22650 using \u03b5pos.le", []], "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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc", "state_after": "case 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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc"}, {"tactic": "have If : (\u222b\u207b x, f x \u2202\u03bc) < \u221e := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral", "annotated_tactic": ["have If : (\u222b\u207b x, f x \u2202\u03bc) < \u221e := <a>hasFiniteIntegral_iff_ofNNReal</a>.1 fint.hasFiniteIntegral", [{"full_name": "MeasureTheory.hasFiniteIntegral_iff_ofNNReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [132, 9], "def_end_pos": [132, 39]}]], "state_before": "case 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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc", "state_after": "case 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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc"}, {"tactic": "rcases exists_upperSemicontinuous_le_lintegral_le f If.ne \u03b5pos.ne' with \u27e8g, gf, gcont, gint\u27e9", "annotated_tactic": ["rcases <a>exists_upperSemicontinuous_le_lintegral_le</a> f If.ne \u03b5pos.ne' with \u27e8g, gf, gcont, gint\u27e9", [{"full_name": "MeasureTheory.exists_upperSemicontinuous_le_lintegral_le", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [387, 9], "def_end_pos": [387, 51]}]], "state_before": "case 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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc", "state_after": "case 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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc"}, {"tactic": "have Ig : (\u222b\u207b x, g x \u2202\u03bc) < \u221e := by\n  refine' lt_of_le_of_lt (lintegral_mono fun x => _) If\n  simpa using gf x", "annotated_tactic": ["have Ig : (\u222b\u207b x, g x \u2202\u03bc) < \u221e := by\n    refine' <a>lt_of_le_of_lt</a> (<a>lintegral_mono</a> fun x => _) If\n    simpa using gf x", [{"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", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "case 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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc", "state_after": "case 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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc"}, {"tactic": "refine' \u27e8g, gf, gcont, _, _\u27e9", "annotated_tactic": ["refine' \u27e8g, gf, gcont, _, _\u27e9", []], "state_before": "case 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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n      UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc", "state_after": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 Integrable fun x => \u2191(g x)\n\ncase intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc"}, {"tactic": "refine' lt_of_le_of_lt (lintegral_mono fun x => _) If", "annotated_tactic": ["refine' <a>lt_of_le_of_lt</a> (<a>lintegral_mono</a> fun x => _) If", [{"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", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4", "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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nx : \u03b1\n\u22a2 \u2191(g x) \u2264 \u2191(f x)"}, {"tactic": "simpa using gf x", "annotated_tactic": ["simpa using gf x", []], "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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nx : \u03b1\n\u22a2 \u2191(g x) \u2264 \u2191(f x)", "state_after": "no goals"}, {"tactic": "refine'\n  Integrable.mono fint gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable _", "annotated_tactic": ["refine'\n      <a>Integrable.mono</a> fint gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable _", [{"full_name": "MeasureTheory.Integrable.mono", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [471, 9], "def_end_pos": [471, 24]}]], "state_before": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 Integrable fun x => \u2191(g x)", "state_after": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016\u2191(g a)\u2016 \u2264 \u2016\u2191(f a)\u2016"}, {"tactic": "exact Filter.eventually_of_forall fun x => by simp [gf x]", "annotated_tactic": ["exact <a>Filter.eventually_of_forall</a> fun x => by simp [gf 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 intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016\u2191(g a)\u2016 \u2264 \u2016\u2191(f a)\u2016", "state_after": "no goals"}, {"tactic": "simp [gf x]", "annotated_tactic": ["simp [gf x]", []], "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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\nx : \u03b1\n\u22a2 \u2016\u2191(g x)\u2016 \u2264 \u2016\u2191(f x)\u2016", "state_after": "no goals"}, {"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": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc", "state_after": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) - \u2191\u03b5 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc)\n\ncase intro.intro.intro.intro.refine'_2.hf\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(g x)\n\ncase intro.intro.intro.intro.refine'_2.hfm\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => \u2191(g x)) \u03bc\n\ncase intro.intro.intro.intro.refine'_2.hf\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)\n\ncase intro.intro.intro.intro.refine'_2.hfm\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc"}, {"tactic": "rw [sub_le_iff_le_add]", "annotated_tactic": ["rw [<a>sub_le_iff_le_add</a>]", [{"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 intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) - \u2191\u03b5 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc)", "state_after": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5"}, {"tactic": "convert ENNReal.toReal_mono _ gint", "annotated_tactic": ["convert <a>ENNReal.toReal_mono</a> _ gint", [{"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}]], "state_before": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5", "state_after": "case h.e'_3.h.e'_1.h.e'_4.h\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\nx\u271d : \u03b1\n\u22a2 ENNReal.ofReal \u2191(f x\u271d) = \u2191(f x\u271d)\n\ncase 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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5)\n\ncase intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 \u2260 \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3.h.e'_1.h.e'_4.h\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\nx\u271d : \u03b1\n\u22a2 ENNReal.ofReal \u2191(f x\u271d) = \u2191(f x\u271d)", "state_after": "no goals"}, {"tactic": "rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]", "annotated_tactic": ["rw [<a>ENNReal.toReal_add</a> Ig.ne <a>ENNReal.coe_ne_top</a>]", [{"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 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.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 =\n    ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) + ENNReal.toReal \u2191\u03b5"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 =\n    ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) + ENNReal.toReal \u2191\u03b5", "state_after": "no goals"}, {"tactic": "simpa using Ig.ne", "annotated_tactic": ["simpa using Ig.ne", []], "state_before": "case intro.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 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "apply Filter.eventually_of_forall", "annotated_tactic": ["apply <a>Filter.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 intro.intro.intro.intro.refine'_2.hf\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(g x)", "state_after": "case intro.intro.intro.intro.refine'_2.hf.hp\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(g x)) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro.intro.intro.intro.refine'_2.hf.hp\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(g x)) x", "state_after": "no goals"}, {"tactic": "exact gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable", "annotated_tactic": ["exact gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable", []], "state_before": "case intro.intro.intro.intro.refine'_2.hfm\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => \u2191(g x)) \u03bc", "state_after": "no goals"}, {"tactic": "apply Filter.eventually_of_forall", "annotated_tactic": ["apply <a>Filter.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 intro.intro.intro.intro.refine'_2.hf\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)", "state_after": "case intro.intro.intro.intro.refine'_2.hf.hp\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro.intro.intro.intro.refine'_2.hf.hp\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x", "state_after": "no goals"}, {"tactic": "exact fint.aestronglyMeasurable", "annotated_tactic": ["exact fint.aestronglyMeasurable", []], "state_before": "case intro.intro.intro.intro.refine'_2.hfm\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\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.eval\u2082Hom_X", "start": [157, 1], "end": [168, 34], "traced_tactics": [{"tactic": "apply MvPolynomial.induction_on x\n  (fun n => by\n    rw [hom_C f, eval\u2082_C]\n    exact eq_intCast c n)\n  (fun p q hp hq => by\n    rw [eval\u2082_add, hp, hq]\n    exact (f.map_add _ _).symm)\n  (fun p n hp => by\n    rw [eval\u2082_mul, eval\u2082_X, hp]\n    exact (f.map_mul _ _).symm)", "annotated_tactic": ["apply <a>MvPolynomial.induction_on</a> x\n    (fun n => by\n      rw [<a>hom_C</a> f, <a>eval\u2082_C</a>]\n      exact <a>eq_intCast</a> c n)\n    (fun p q hp hq => by\n      rw [<a>eval\u2082_add</a>, hp, hq]\n      exact (f.map_add _ _).<a>symm</a>)\n    (fun p n hp => by\n      rw [<a>eval\u2082_mul</a>, <a>eval\u2082_X</a>, hp]\n      exact (f.map_mul _ _).<a>symm</a>)", [{"full_name": "MvPolynomial.induction_on", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [451, 9], "def_end_pos": [451, 21]}, {"full_name": "MvPolynomial.hom_C", "def_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "def_pos": [149, 9], "def_end_pos": [149, 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": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "MvPolynomial.eval\u2082_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 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": "MvPolynomial.eval\u2082_mul", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 18]}, {"full_name": "MvPolynomial.eval\u2082_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 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": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\n\u22a2 eval\u2082 c (\u2191f \u2218 X) x = \u2191f x", "state_after": "no goals"}, {"tactic": "rw [hom_C f, eval\u2082_C]", "annotated_tactic": ["rw [<a>hom_C</a> f, <a>eval\u2082_C</a>]", [{"full_name": "MvPolynomial.hom_C", "def_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "def_pos": [149, 9], "def_end_pos": [149, 14]}, {"full_name": "MvPolynomial.eval\u2082_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [988, 9], "def_end_pos": [988, 16]}]], "state_before": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\nn : \u2124\n\u22a2 eval\u2082 c (\u2191f \u2218 X) (\u2191C n) = \u2191f (\u2191C n)", "state_after": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\nn : \u2124\n\u22a2 \u2191c n = \u2191n"}, {"tactic": "exact eq_intCast c n", "annotated_tactic": ["exact <a>eq_intCast</a> c n", [{"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}]], "state_before": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\nn : \u2124\n\u22a2 \u2191c n = \u2191n", "state_after": "no goals"}, {"tactic": "rw [eval\u2082_add, hp, hq]", "annotated_tactic": ["rw [<a>eval\u2082_add</a>, hp, hq]", [{"full_name": "MvPolynomial.eval\u2082_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 18]}]], 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R \u2124\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\nhq : eval\u2082 c (\u2191f \u2218 X) q = \u2191f q\n\u22a2 \u2191f p + \u2191f q = \u2191f (p + q)"}, {"tactic": "exact (f.map_add _ _).symm", "annotated_tactic": ["exact (f.map_add _ _).<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": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q\u271d : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p q : MvPolynomial R \u2124\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\nhq : eval\u2082 c (\u2191f \u2218 X) q = \u2191f q\n\u22a2 \u2191f p + \u2191f q = \u2191f (p + q)", "state_after": "no goals"}, {"tactic": "rw [eval\u2082_mul, eval\u2082_X, hp]", "annotated_tactic": ["rw [<a>eval\u2082_mul</a>, <a>eval\u2082_X</a>, hp]", [{"full_name": "MvPolynomial.eval\u2082_mul", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 18]}, {"full_name": "MvPolynomial.eval\u2082_X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 16]}]], "state_before": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p : MvPolynomial R \u2124\nn : R\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\n\u22a2 eval\u2082 c (\u2191f \u2218 X) (p * X n) = \u2191f (p * X n)", "state_after": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p : MvPolynomial R \u2124\nn : R\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\n\u22a2 \u2191f p * (\u2191f \u2218 X) n = \u2191f (p * X n)"}, {"tactic": "exact (f.map_mul _ _).symm", "annotated_tactic": ["exact (f.map_mul _ _).<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": "R\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p : MvPolynomial R \u2124\nn : R\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\n\u22a2 \u2191f p * (\u2191f \u2218 X) n = \u2191f (p * X n)", "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.map", "start": [1135, 1], "end": [1141, 45], "traced_tactics": [{"tactic": "by_cases hf : Measurable f", "annotated_tactic": ["by_cases hf : <a>Measurable</a> f", [{"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\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\n\u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f", "state_after": "case pos\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f\n\ncase neg\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : \u00acMeasurable f\n\u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f"}, {"tactic": "refine' mk fun s hs hws => _", "annotated_tactic": ["refine' <a>mk</a> fun s hs hws => _", [{"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 pos\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f", "state_after": "case pos\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nhws : \u2191(VectorMeasure.map w f) s = 0\n\u22a2 \u2191(VectorMeasure.map v f) s = 0"}, {"tactic": "rw [map_apply _ hf hs] at hws \u22a2", "annotated_tactic": ["rw [<a>map_apply</a> _ hf hs] at hws \u22a2", [{"full_name": "MeasureTheory.VectorMeasure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [581, 9], "def_end_pos": [581, 18]}]], "state_before": "case pos\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nhws : \u2191(VectorMeasure.map w f) s = 0\n\u22a2 \u2191(VectorMeasure.map v f) s = 0", "state_after": "case pos\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nhws : \u2191w (f \u207b\u00b9' s) = 0\n\u22a2 \u2191v (f \u207b\u00b9' s) = 0"}, {"tactic": "exact h hws", "annotated_tactic": ["exact h hws", []], "state_before": "case pos\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nhws : \u2191w (f \u207b\u00b9' s) = 0\n\u22a2 \u2191v (f \u207b\u00b9' s) = 0", "state_after": "no goals"}, {"tactic": "intro s _", "annotated_tactic": ["intro s _", []], "state_before": "case neg\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : \u00acMeasurable f\n\u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f", "state_after": "case neg\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : \u00acMeasurable f\ns : Set \u03b2\na\u271d : \u2191(VectorMeasure.map w f) s = 0\n\u22a2 \u2191(VectorMeasure.map v f) s = 0"}, {"tactic": "rw [map_not_measurable v hf, zero_apply]", "annotated_tactic": ["rw [<a>map_not_measurable</a> v hf, <a>zero_apply</a>]", [{"full_name": "MeasureTheory.VectorMeasure.map_not_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [577, 9], "def_end_pos": [577, 27]}, {"full_name": "MeasureTheory.VectorMeasure.zero_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}]], "state_before": "case neg\n\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\u2076 : AddCommMonoid L\ninst\u271d\u2075 : TopologicalSpace L\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\ninst\u271d : MeasureSpace \u03b2\nh : v \u226a\u1d65 w\nf : \u03b1 \u2192 \u03b2\nhf : \u00acMeasurable f\ns : Set \u03b2\na\u271d : \u2191(VectorMeasure.map w f) s = 0\n\u22a2 \u2191(VectorMeasure.map v f) s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.mapAlgEquiv_trans", "start": [144, 1], "end": [148, 6], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "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\u2076 : CommSemiring R\nA\u2081 : Type u_2\nA\u2082 : Type u_3\nA\u2083 : Type u_4\ninst\u271d\u2075 : CommSemiring A\u2081\ninst\u271d\u2074 : CommSemiring A\u2082\ninst\u271d\u00b3 : CommSemiring A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081\ninst\u271d\u00b9 : Algebra R A\u2082\ninst\u271d : Algebra R A\u2083\ne : A\u2081 \u2243\u2090[R] A\u2082\nf : A\u2082 \u2243\u2090[R] A\u2083\n\u22a2 AlgEquiv.trans (mapAlgEquiv \u03c3 e) (mapAlgEquiv \u03c3 f) = mapAlgEquiv \u03c3 (AlgEquiv.trans e f)", "state_after": "case h.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\u271d : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u2076 : CommSemiring R\nA\u2081 : Type u_2\nA\u2082 : Type u_3\nA\u2083 : Type u_4\ninst\u271d\u2075 : CommSemiring A\u2081\ninst\u271d\u2074 : CommSemiring A\u2082\ninst\u271d\u00b3 : CommSemiring A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081\ninst\u271d\u00b9 : Algebra R A\u2082\ninst\u271d : Algebra R A\u2083\ne : A\u2081 \u2243\u2090[R] A\u2082\nf : A\u2082 \u2243\u2090[R] A\u2083\na\u271d : MvPolynomial \u03c3 A\u2081\nm\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m\u271d (\u2191(AlgEquiv.trans (mapAlgEquiv \u03c3 e) (mapAlgEquiv \u03c3 f)) a\u271d) =\n    coeff m\u271d (\u2191(mapAlgEquiv \u03c3 (AlgEquiv.trans e f)) a\u271d)"}, {"tactic": "simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map]", "annotated_tactic": ["simp only [<a>AlgEquiv.trans_apply</a>, <a>mapAlgEquiv_apply</a>, <a>map_map</a>]", [{"full_name": "AlgEquiv.trans_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [423, 9], "def_end_pos": [423, 20]}, {"full_name": "MvPolynomial.mapAlgEquiv_apply", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [128, 9], "def_end_pos": [128, 14]}, {"full_name": "MvPolynomial.map_map", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 16]}]], "state_before": "case h.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\u271d : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u2076 : CommSemiring R\nA\u2081 : Type u_2\nA\u2082 : Type u_3\nA\u2083 : Type u_4\ninst\u271d\u2075 : CommSemiring A\u2081\ninst\u271d\u2074 : CommSemiring A\u2082\ninst\u271d\u00b3 : CommSemiring A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081\ninst\u271d\u00b9 : Algebra R A\u2082\ninst\u271d : Algebra R A\u2083\ne : A\u2081 \u2243\u2090[R] A\u2082\nf : A\u2082 \u2243\u2090[R] A\u2083\na\u271d : MvPolynomial \u03c3 A\u2081\nm\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m\u271d (\u2191(AlgEquiv.trans (mapAlgEquiv \u03c3 e) (mapAlgEquiv \u03c3 f)) a\u271d) =\n    coeff m\u271d (\u2191(mapAlgEquiv \u03c3 (AlgEquiv.trans e f)) a\u271d)", "state_after": "case h.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\u271d : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u2076 : CommSemiring R\nA\u2081 : Type u_2\nA\u2082 : Type u_3\nA\u2083 : Type u_4\ninst\u271d\u2075 : CommSemiring A\u2081\ninst\u271d\u2074 : CommSemiring A\u2082\ninst\u271d\u00b3 : CommSemiring A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081\ninst\u271d\u00b9 : Algebra R A\u2082\ninst\u271d : Algebra R A\u2083\ne : A\u2081 \u2243\u2090[R] A\u2082\nf : A\u2082 \u2243\u2090[R] A\u2083\na\u271d : MvPolynomial \u03c3 A\u2081\nm\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m\u271d (\u2191(map (RingHom.comp \u2191f \u2191e)) a\u271d) = coeff m\u271d (\u2191(map \u2191(AlgEquiv.trans e f)) a\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.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\u271d : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u2076 : CommSemiring R\nA\u2081 : Type u_2\nA\u2082 : Type u_3\nA\u2083 : Type u_4\ninst\u271d\u2075 : CommSemiring A\u2081\ninst\u271d\u2074 : CommSemiring A\u2082\ninst\u271d\u00b3 : CommSemiring A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081\ninst\u271d\u00b9 : Algebra R A\u2082\ninst\u271d : Algebra R A\u2083\ne : A\u2081 \u2243\u2090[R] A\u2082\nf : A\u2082 \u2243\u2090[R] A\u2083\na\u271d : MvPolynomial \u03c3 A\u2081\nm\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m\u271d (\u2191(map (RingHom.comp \u2191f \u2191e)) a\u271d) = coeff m\u271d (\u2191(map \u2191(AlgEquiv.trans e f)) a\u271d)", "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": [569, 1], "end": [571, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.ModEq.mul_left'", "start": [128, 11], "end": [133, 56], "traced_tactics": [{"tactic": "obtain hc | rfl | hc := lt_trichotomy c 0", "annotated_tactic": ["obtain hc | rfl | hc := <a>lt_trichotomy</a> c 0", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 22]}]], "state_before": "m n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 c * a \u2261 c * b [ZMOD c * n]", "state_after": "case inl\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : c < 0\n\u22a2 c * a \u2261 c * b [ZMOD c * n]\n\ncase inr.inl\nm n a b d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 0 * a \u2261 0 * b [ZMOD 0 * n]\n\ncase inr.inr\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : 0 < c\n\u22a2 c * a \u2261 c * b [ZMOD c * n]"}, {"tactic": "rw [\u2190 neg_modEq_neg, \u2190 modEq_neg, \u2190 neg_mul, \u2190 neg_mul, \u2190 neg_mul]", "annotated_tactic": ["rw [\u2190 <a>neg_modEq_neg</a>, \u2190 <a>modEq_neg</a>, \u2190 <a>neg_mul</a>, \u2190 <a>neg_mul</a>, \u2190 <a>neg_mul</a>]", [{"full_name": "Int.neg_modEq_neg", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [113, 9], "def_end_pos": [113, 22]}, {"full_name": "Int.modEq_neg", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [119, 9], "def_end_pos": [119, 18]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}]], "state_before": "case inl\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : c < 0\n\u22a2 c * a \u2261 c * b [ZMOD c * n]", "state_after": "case inl\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : c < 0\n\u22a2 -c * a \u2261 -c * b [ZMOD -c * n]"}, {"tactic": "simp only [ModEq, mul_emod_mul_of_pos _ _ (neg_pos.2 hc), h.eq]", "annotated_tactic": ["simp only [<a>ModEq</a>, <a>mul_emod_mul_of_pos</a> _ _ (<a>neg_pos</a>.2 hc), h.eq]", [{"full_name": "Int.ModEq", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [31, 5], "def_end_pos": [31, 10]}, {"full_name": "Int.mul_emod_mul_of_pos", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [544, 17], "def_end_pos": [544, 36]}, {"full_name": "neg_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [677, 24], "def_end_pos": [677, 31]}]], "state_before": "case inl\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : c < 0\n\u22a2 -c * a \u2261 -c * b [ZMOD -c * n]", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inr.inl\nm n a b d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 0 * a \u2261 0 * b [ZMOD 0 * n]", "state_after": "no goals"}, {"tactic": "simp only [ModEq, mul_emod_mul_of_pos _ _ hc, h.eq]", "annotated_tactic": ["simp only [<a>ModEq</a>, <a>mul_emod_mul_of_pos</a> _ _ hc, h.eq]", [{"full_name": "Int.ModEq", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [31, 5], "def_end_pos": [31, 10]}, {"full_name": "Int.mul_emod_mul_of_pos", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [544, 17], "def_end_pos": [544, 36]}]], "state_before": "case inr.inr\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : 0 < c\n\u22a2 c * a \u2261 c * b [ZMOD c * 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_eq_succ", "start": [1094, 1], "end": [1098, 85], "traced_tactics": [{"tactic": "refine' \u27e8eq_insert_of_ncard_eq_succ, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>eq_insert_of_ncard_eq_succ</a>, _\u27e9", [{"full_name": "Set.eq_insert_of_ncard_eq_succ", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 35]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard s = n + 1 \u2194 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n) \u2192 ncard s = n + 1"}, {"tactic": "rintro \u27e8a, t, hat, h, rfl\u27e9", "annotated_tactic": ["rintro \u27e8a, t, hat, h, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nn : \u2115\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n) \u2192 ncard s = n + 1", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\na : \u03b1\nt : Set \u03b1\nhat : \u00aca \u2208 t\nh : insert a t = s\n\u22a2 ncard s = ncard t + 1"}, {"tactic": "rw [\u2190 h, ncard_insert_of_not_mem hat (hs.subset ((subset_insert a t).trans_eq h))]", "annotated_tactic": ["rw [\u2190 h, <a>ncard_insert_of_not_mem</a> hat (hs.subset ((<a>subset_insert</a> a t).<a>trans_eq</a> h))]", [{"full_name": "Set.ncard_insert_of_not_mem", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [568, 17], "def_end_pos": [568, 40]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 22]}, {"full_name": "LE.le.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [211, 7], "def_end_pos": [211, 21]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns t\u271d : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\na : \u03b1\nt : Set \u03b1\nhat : \u00aca \u2208 t\nh : insert a t = s\n\u22a2 ncard s = ncard t + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_hasDerivAt_of_tendsto_ae_left", "start": [781, 1], "end": [784, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.fdiv_eq_ediv_of_dvd", "start": [798, 1], "end": [800, 62], "traced_tactics": [{"tactic": "if bz : b = 0 then simp [bz] else\nrw [mul_fdiv_cancel_left _ bz, mul_ediv_cancel_left _ bz]", "annotated_tactic": ["if bz : b = 0 then simp [bz] else\n    rw [<a>mul_fdiv_cancel_left</a> _ bz, <a>mul_ediv_cancel_left</a> _ bz]", [{"full_name": "Int.mul_fdiv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [216, 17], "def_end_pos": [216, 37]}, {"full_name": "Int.mul_ediv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [219, 17], "def_end_pos": [219, 37]}]], "state_before": "b c : Int\n\u22a2 fdiv (b * c) b = b * c / b", "state_after": "no goals"}, {"tactic": "simp [bz]", "annotated_tactic": ["simp [bz]", []], "state_before": "b c : Int\nbz : b = 0\n\u22a2 fdiv (b * c) b = b * c / b", "state_after": "no goals"}, {"tactic": "rw [mul_fdiv_cancel_left _ bz, mul_ediv_cancel_left _ bz]", "annotated_tactic": ["rw [<a>mul_fdiv_cancel_left</a> _ bz, <a>mul_ediv_cancel_left</a> _ bz]", [{"full_name": "Int.mul_fdiv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [216, 17], "def_end_pos": [216, 37]}, {"full_name": "Int.mul_ediv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [219, 17], "def_end_pos": [219, 37]}]], "state_before": "b c : Int\nbz : \u00acb = 0\n\u22a2 fdiv (b * c) b = b * c / b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.singleton_eq_singleton_iff", "start": [1298, 1], "end": [1299, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.ofReal_set_integral_one", "start": [199, 1], "end": [201, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.piecewise_univ", "start": [247, 1], "end": [248, 59], "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\ninst\u271d : MeasurableSpace \u03b1\nf g : \u03b1 \u2192\u209b \u03b2\n\u22a2 \u2191(piecewise univ (_ : MeasurableSet univ) f g) = \u2191f", "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 g : \u03b1 \u2192\u209b \u03b2\n\u22a2 Set.piecewise univ \u2191f \u2191g = \u2191f"}, {"tactic": "convert Set.piecewise_univ f g", "annotated_tactic": ["convert <a>Set.piecewise_univ</a> f g", [{"full_name": "Set.piecewise_univ", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1392, 9], "def_end_pos": [1392, 23]}]], "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 g : \u03b1 \u2192\u209b \u03b2\n\u22a2 Set.piecewise univ \u2191f \u2191g = \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.lintegral_edist_lt_top", "start": [651, 1], "end": [656, 58], "traced_tactics": [{"tactic": "simp_rw [Pi.zero_apply, \u2190 hasFiniteIntegral_iff_edist]", "annotated_tactic": ["simp_rw [<a>Pi.zero_apply</a>, \u2190 <a>hasFiniteIntegral_iff_edist</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.hasFiniteIntegral_iff_edist", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [122, 9], "def_end_pos": [122, 36]}]], "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\n\u22a2 \u222b\u207b (a : \u03b1), edist (f a) (OfNat.ofNat 0 a) \u2202\u03bc < \u22a4 \u2227 \u222b\u207b (a : \u03b1), edist (g a) (OfNat.ofNat 0 a) \u2202\u03bc < \u22a4", "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 g : \u03b1 \u2192 \u03b2\nhf : Integrable f\nhg : Integrable g\n\u22a2 (HasFiniteIntegral fun a => f a) \u2227 HasFiniteIntegral fun a => g a"}, {"tactic": "exact \u27e8hf.hasFiniteIntegral, hg.hasFiniteIntegral\u27e9", "annotated_tactic": ["exact \u27e8hf.hasFiniteIntegral, hg.hasFiniteIntegral\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\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\n\u22a2 (HasFiniteIntegral fun a => f a) \u2227 HasFiniteIntegral fun a => g a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.const_mem_Lp", "start": [240, 1], "end": [242, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr\u2082_bisim", "start": [192, 1], "end": [203, 12], "traced_tactics": [{"tactic": "induction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082", "annotated_tactic": ["induction xs, ys using <a>Vector.revInductionOn\u2082</a> generalizing s\u2081 s\u2082", [{"full_name": "Vector.revInductionOn\u2082", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [95, 5], "def_end_pos": [95, 20]}]], "state_before": "\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\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) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\n\u22a2 R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2", "state_after": "case nil\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).1 (mapAccumr\u2082 f\u2082 nil nil s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 nil nil s\u2081).2 = (mapAccumr\u2082 f\u2082 nil nil s\u2082).2\n\ncase snoc\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).1 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).1 \u2227\n        (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).2 = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).2"}, {"tactic": "next => exact \u27e8h\u2080, rfl\u27e9", "annotated_tactic": ["next => exact \u27e8h\u2080, <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 nil\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).1 (mapAccumr\u2082 f\u2082 nil nil s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 nil nil s\u2081).2 = (mapAccumr\u2082 f\u2082 nil nil s\u2082).2\n\ncase snoc\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).1 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).1 \u2227\n        (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).2 = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).2", "state_after": "case snoc\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).1 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).1 \u2227\n        (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).2 = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).2"}, {"tactic": "next xs ys x y ih =>\n  rcases (hR x y h\u2080) with \u27e8hR, _\u27e9\n  simp only [mapAccumr\u2082_snoc, ih hR, true_and]\n  congr 1", "annotated_tactic": ["next xs ys x y ih =>\n    rcases (hR x y h\u2080) with \u27e8hR, _\u27e9\n    simp only [<a>mapAccumr\u2082_snoc</a>, ih hR, <a>true_and</a>]\n    congr 1", [{"full_name": "Vector.mapAccumr\u2082_snoc", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [157, 9], "def_end_pos": [157, 24]}, {"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 snoc\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).1 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).1 \u2227\n        (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).2 = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).2", "state_after": "no goals"}, {"tactic": "exact \u27e8h\u2080, rfl\u27e9", "annotated_tactic": ["exact \u27e8h\u2080, <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": "\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).1 (mapAccumr\u2082 f\u2082 nil nil s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 nil nil s\u2081).2 = (mapAccumr\u2082 f\u2082 nil nil s\u2082).2", "state_after": "no goals"}, {"tactic": "rcases (hR x y h\u2080) with \u27e8hR, _\u27e9", "annotated_tactic": ["rcases (hR x y h\u2080) with \u27e8hR, _\u27e9", []], "state_before": "\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).2", "state_after": "case intro\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x y s\u2081).1 (f\u2082 x y s\u2082).1\nright\u271d : (f\u2081 x y s\u2081).2 = (f\u2082 x y s\u2082).2\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).2"}, {"tactic": "simp only [mapAccumr\u2082_snoc, ih hR, true_and]", "annotated_tactic": ["simp only [<a>mapAccumr\u2082_snoc</a>, ih hR, <a>true_and</a>]", [{"full_name": "Vector.mapAccumr\u2082_snoc", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [157, 9], "def_end_pos": [157, 24]}, {"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\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x y s\u2081).1 (f\u2082 x y s\u2082).1\nright\u271d : (f\u2081 x y s\u2081).2 = (f\u2082 x y s\u2082).2\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).1 \u2227\n    (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).2", "state_after": "case intro\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x y s\u2081).1 (f\u2082 x y s\u2082).1\nright\u271d : (f\u2081 x y s\u2081).2 = (f\u2082 x y s\u2082).2\n\u22a2 snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).1).2 (f\u2081 x y s\u2081).2 = snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).1).2 (f\u2082 x y s\u2082).2"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case intro\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n  \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n    R s\u2081 s\u2082 \u2192\n      R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x y s\u2081).1 (f\u2082 x y s\u2082).1\nright\u271d : (f\u2081 x y s\u2081).2 = (f\u2082 x y s\u2082).2\n\u22a2 snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).1).2 (f\u2081 x y s\u2081).2 = snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).1).2 (f\u2082 x y s\u2082).2", "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", "start": [1582, 1], "end": [1584, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/EpsilonNFA.lean", "full_name": "NFA.to\u03b5NFA_correct", "start": [207, 1], "end": [209, 6], "traced_tactics": [{"tactic": "rw [\u03b5NFA.accepts, \u03b5NFA.eval, to\u03b5NFA_evalFrom_match]", "annotated_tactic": ["rw [<a>\u03b5NFA.accepts</a>, <a>\u03b5NFA.eval</a>, <a>to\u03b5NFA_evalFrom_match</a>]", [{"full_name": "\u03b5NFA.accepts", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [139, 5], "def_end_pos": [139, 12]}, {"full_name": "\u03b5NFA.eval", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [119, 5], "def_end_pos": [119, 9]}, {"full_name": "NFA.to\u03b5NFA_evalFrom_match", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [193, 9], "def_end_pos": [193, 30]}]], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\n\u22a2 \u03b5NFA.accepts (to\u03b5NFA M) = accepts M", "state_after": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\n\u22a2 {x | \u2203 S, S \u2208 (to\u03b5NFA M).accept \u2227 S \u2208 evalFrom M (to\u03b5NFA M).start x} = accepts M"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\n\u22a2 {x | \u2203 S, S \u2208 (to\u03b5NFA M).accept \u2227 S \u2208 evalFrom M (to\u03b5NFA M).start x} = accepts M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.vars_sub_of_disjoint", "start": [119, 1], "end": [122, 69], "traced_tactics": [{"tactic": "rw [\u2190 vars_neg q] at hpq", "annotated_tactic": ["rw [\u2190 <a>vars_neg</a> q] at hpq", [{"full_name": "MvPolynomial.vars_neg", "def_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "def_pos": [111, 9], "def_end_pos": [111, 17]}]], "state_before": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars q)\n\u22a2 vars (p - q) = vars p \u222a vars q", "state_after": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars (-q))\n\u22a2 vars (p - q) = vars p \u222a vars q"}, {"tactic": "convert vars_add_of_disjoint hpq using 2 <;> simp [sub_eq_add_neg]", "annotated_tactic": ["convert <a>vars_add_of_disjoint</a> hpq using 2 <;> simp [<a>sub_eq_add_neg</a>]", [{"full_name": "MvPolynomial.vars_add_of_disjoint", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [331, 9], "def_end_pos": [331, 29]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "R : Type u\nS : Type v\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 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars (-q))\n\u22a2 vars (p - q) = vars p \u222a vars q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.union_diff_cancel_right", "start": [1861, 1], "end": [1862, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_ssubset", "start": [2240, 1], "end": [2241, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.hausdorffMeasure_le_liminf_sum", "start": [610, 1], "end": [614, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.isEmpty_iff", "start": [178, 1], "end": [179, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_pi_le_diam_pow", "start": [281, 1], "end": [287, 76], "traced_tactics": [{"tactic": "simp only [ENNReal.coe_one, one_mul, Finset.prod_const, Fintype.card]", "annotated_tactic": ["simp only [<a>ENNReal.coe_one</a>, <a>one_mul</a>, <a>Finset.prod_const</a>, <a>Fintype.card</a>]", [{"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 19]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\n\u22a2 \u220f _i : \u03b9, \u21911 * EMetric.diam s = EMetric.diam s ^ Fintype.card \u03b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toMeasure_mono", "start": [292, 1], "end": [294, 95], "traced_tactics": [{"tactic": "simpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using toOuterMeasure_mono p h", "annotated_tactic": ["simpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using <a>toOuterMeasure_mono</a> p h", [{"full_name": "PMF.toOuterMeasure_mono", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 28]}]], "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 \u2286 t\n\u22a2 \u2191\u2191(toMeasure p) s \u2264 \u2191\u2191(toMeasure p) t", "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_zero_left'", "start": [1045, 1], "end": [1051, 36], "traced_tactics": [{"tactic": "suffices setToL1 hT = 0 by rw [this]; simp", "annotated_tactic": ["suffices <a>setToL1</a> hT = 0 by rw [this]; simp", [{"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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 hT) f = 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\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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = 0"}, {"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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = 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\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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp 0 (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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp 0 (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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp 0 (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "rw [setToL1SCLM_zero_left' hT h_zero f, ContinuousLinearMap.zero_comp,\n  ContinuousLinearMap.zero_apply]", "annotated_tactic": ["rw [<a>setToL1SCLM_zero_left'</a> hT h_zero f, <a>ContinuousLinearMap.zero_comp</a>,\n    <a>ContinuousLinearMap.zero_apply</a>]", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [895, 9], "def_end_pos": [895, 31]}, {"full_name": "ContinuousLinearMap.zero_comp", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 18]}, {"full_name": "ContinuousLinearMap.zero_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [644, 9], "def_end_pos": [644, 19]}]], "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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp 0 (coeToLp \u03b1 E \u211d)) 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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = 0\n\u22a2 \u2191(setToL1 hT) f = 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\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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = 0\n\u22a2 \u21910 f = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = 0\n\u22a2 \u21910 f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf_eq_zero_of_not_measurable", "start": [97, 1], "end": [99, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_eq_two", "start": [315, 1], "end": [322, 54], "traced_tactics": [{"tactic": "refine' \u27e8fun h \u21a6 _, fun \u27e8x, y, hne, hs\u27e9 \u21a6 by rw [hs, encard_pair hne]\u27e9", "annotated_tactic": ["refine' \u27e8fun h \u21a6 _, fun \u27e8x, y, hne, hs\u27e9 \u21a6 by rw [hs, <a>encard_pair</a> hne]\u27e9", [{"full_name": "Set.encard_pair", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 20]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 encard s = 2 \u2194 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "obtain \u27e8x, hx\u27e9 := nonempty_of_encard_ne_zero (s := s) (by rw [h]; simp)", "annotated_tactic": ["obtain \u27e8x, hx\u27e9 := <a>nonempty_of_encard_ne_zero</a> (s := s) (by rw [h]; simp)", [{"full_name": "Set.nonempty_of_encard_ne_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [100, 9], "def_end_pos": [100, 35]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "rw [\u2190insert_eq_of_mem hx, \u2190insert_diff_singleton, encard_insert_of_not_mem (fun h \u21a6 h.2 rfl),\n  \u2190one_add_one_eq_two, WithTop.add_right_cancel_iff (WithTop.one_ne_top), encard_eq_one] at h", "annotated_tactic": ["rw [\u2190<a>insert_eq_of_mem</a> hx, \u2190<a>insert_diff_singleton</a>, <a>encard_insert_of_not_mem</a> (fun h \u21a6 h.2 <a>rfl</a>),\n    \u2190<a>one_add_one_eq_two</a>, <a>WithTop.add_right_cancel_iff</a> (<a>WithTop.one_ne_top</a>), <a>encard_eq_one</a>] at h", [{"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.encard_insert_of_not_mem", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [118, 9], "def_end_pos": [118, 33]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "one_add_one_eq_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [218, 9], "def_end_pos": [218, 27]}, {"full_name": "WithTop.add_right_cancel_iff", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [183, 9], "def_end_pos": [183, 29]}, {"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_eq_one", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [290, 9], "def_end_pos": [290, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nh : \u2203 x_1, s \\ {x} = {x_1}\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "obtain \u27e8y, h\u27e9 := h", "annotated_tactic": ["obtain \u27e8y, h\u27e9 := h", []], "state_before": "case intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nh : \u2203 x_1, s \\ {x} = {x_1}\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "refine' \u27e8x, y, by rintro rfl; exact (h.symm.subset rfl).2 rfl, _\u27e9", "annotated_tactic": ["refine' \u27e8x, y, by rintro rfl; exact (h.symm.subset <a>rfl</a>).2 <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": "case intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 s = {x, y}"}, {"tactic": "rw [\u2190h, insert_diff_singleton, insert_eq_of_mem hx]", "annotated_tactic": ["rw [\u2190h, <a>insert_diff_singleton</a>, <a>insert_eq_of_mem</a> hx]", [{"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.insert_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 s = {x, y}", "state_after": "no goals"}, {"tactic": "rw [hs, encard_pair hne]", "annotated_tactic": ["rw [hs, <a>encard_pair</a> hne]", [{"full_name": "Set.encard_pair", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 20]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u2203 x y, x \u2260 y \u2227 s = {x, y}\nx y : \u03b1\nhne : x \u2260 y\nhs : s = {x, y}\n\u22a2 encard s = 2", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\n\u22a2 encard s \u2260 0", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\n\u22a2 2 \u2260 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 2\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 x \u2260 y", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : s \\ {x} = {x}\n\u22a2 False"}, {"tactic": "exact (h.symm.subset rfl).2 rfl", "annotated_tactic": ["exact (h.symm.subset <a>rfl</a>).2 <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": "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\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : s \\ {x} = {x}\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "full_name": "MeasureTheory.L2.inner_indicatorConstLp_one", "start": [274, 1], "end": [276, 80], "traced_tactics": [{"tactic": "rw [L2.inner_indicatorConstLp_eq_inner_set_integral \ud835\udd5c hs h\u03bcs (1 : \ud835\udd5c) f]", "annotated_tactic": ["rw [<a>L2.inner_indicatorConstLp_eq_inner_set_integral</a> \ud835\udd5c hs h\u03bcs (1 : \ud835\udd5c) f]", [{"full_name": "MeasureTheory.L2.inner_indicatorConstLp_eq_inner_set_integral", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [263, 9], "def_end_pos": [263, 53]}]], "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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\n\u22a2 inner (indicatorConstLp 2 hs h\u03bcs 1) f = \u222b (x : \u03b1) in s, \u2191\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\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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\n\u22a2 inner 1 (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\n\u22a2 inner 1 (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\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.univ_pi_Ioc_ae_eq_Icc", "start": [525, 1], "end": [527, 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 => Ioc (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 => Ioc (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i)"}, {"tactic": "exact pi_Ioc_ae_eq_pi_Icc", "annotated_tactic": ["exact <a>pi_Ioc_ae_eq_pi_Icc</a>", [{"full_name": "MeasureTheory.Measure.pi_Ioc_ae_eq_pi_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [520, 9], "def_end_pos": [520, 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 => Ioc (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/Data/Finset/Lattice.lean", "full_name": "Finset.not_mem_of_max_lt", "start": [1313, 1], "end": [1314, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_union", "start": [970, 1], "end": [975, 29], "traced_tactics": [{"tactic": "rw [Set.union_eq_iUnion]", "annotated_tactic": ["rw [<a>Set.union_eq_iUnion</a>]", [{"full_name": "Set.union_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1478, 9], "def_end_pos": [1478, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 restrict v (i \u222a j) \u2264 restrict w (i \u222a j)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 restrict v (\u22c3 b, bif b then i else j) \u2264 restrict w (\u22c3 b, bif b then i else j)"}, {"tactic": "refine' restrict_le_restrict_countable_iUnion v w _ _", "annotated_tactic": ["refine' <a>restrict_le_restrict_countable_iUnion</a> v w _ _", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_countable_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [957, 9], "def_end_pos": [957, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 restrict v (\u22c3 b, bif b then i else j) \u2264 restrict w (\u22c3 b, bif b then i else j)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 \u2200 (b : Bool), MeasurableSet (bif b then i else j)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 \u2200 (b : Bool), restrict v (bif b then i else j) \u2264 restrict w (bif b then i else j)"}, {"tactic": "measurability", "annotated_tactic": ["measurability", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 \u2200 (b : Bool), MeasurableSet (bif b then i else j)", "state_after": "no goals"}, {"tactic": "rintro (_ | _) <;> simpa", "annotated_tactic": ["rintro (_ | _) <;> simpa", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : OrderedAddCommMonoid M\ninst\u271d : OrderClosedTopology M\nv w : VectorMeasure \u03b1 M\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : restrict v i \u2264 restrict w i\nhj\u2081 : MeasurableSet j\nhj\u2082 : restrict v j \u2264 restrict w j\n\u22a2 \u2200 (b : Bool), restrict v (bif b then i else j) \u2264 restrict w (bif b then i else j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "Set.Infinite.meas_eq_top", "start": [3447, 1], "end": [3455, 23], "traced_tactics": [{"tactic": "simpa [Subtype.val_inj]", "annotated_tactic": ["simpa [<a>Subtype.val_inj</a>]", [{"full_name": "Subtype.val_inj", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [130, 9], "def_end_pos": [130, 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\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\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Set \u03b1\nhs : Set.Infinite s\nh' : \u2203 \u03b5, \u03b5 \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x}\n\u03b5 : \u211d\u22650\u221e\nhne\u271d : \u03b5 \u2260 0\nh\u03b5 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x}\nthis : Infinite \u2191s\nx y : \u2191s\nhne : x \u2260 y\n\u22a2 (Disjoint on fun x => {\u2191x}) x y", "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\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\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Set \u03b1\nhs : Set.Infinite s\nh' : \u2203 \u03b5, \u03b5 \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x}\n\u03b5 : \u211d\u22650\u221e\nhne : \u03b5 \u2260 0\nh\u03b5 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x}\nthis : Infinite \u2191s\n\u22a2 \u2191\u2191\u03bc (\u22c3 x, {\u2191x}) = \u2191\u2191\u03bc 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_Icc_eq_integral_Ioo", "start": [691, 1], "end": [692, 66], "traced_tactics": [{"tactic": "rw [integral_Icc_eq_integral_Ico, integral_Ico_eq_integral_Ioo]", "annotated_tactic": ["rw [<a>integral_Icc_eq_integral_Ico</a>, <a>integral_Ico_eq_integral_Ioo</a>]", [{"full_name": "MeasureTheory.integral_Icc_eq_integral_Ico", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [679, 9], "def_end_pos": [679, 37]}, {"full_name": "MeasureTheory.integral_Ico_eq_integral_Ioo", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [687, 9], "def_end_pos": [687, 37]}]], "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 t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : PartialOrder \u03b1\na b : \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u222b (t : \u03b1) in Icc a b, f t \u2202\u03bc = \u222b (t : \u03b1) in Ico a b, f t \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Equiv.piCongrLeft_preimage_univ_pi", "start": [954, 1], "end": [956, 71], "traced_tactics": [{"tactic": "simpa [f.surjective.range_eq] using piCongrLeft_preimage_pi f univ t", "annotated_tactic": ["simpa [f.surjective.range_eq] using <a>piCongrLeft_preimage_pi</a> f <a>univ</a> t", [{"full_name": "Equiv.piCongrLeft_preimage_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [948, 9], "def_end_pos": [948, 32]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\nf : \u03b9' \u2243 \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191(piCongrLeft \u03b1 f) \u207b\u00b9' pi univ t = pi univ fun i => t (\u2191f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.exists_null_frontiers_thickening", "start": [477, 1], "end": [485, 68], "traced_tactics": [{"tactic": "rcases exists_seq_strictAnti_tendsto (0 : \u211d) with \u27e8Rs, \u27e8_, \u27e8Rs_pos, Rs_lim\u27e9\u27e9\u27e9", "annotated_tactic": ["rcases <a>exists_seq_strictAnti_tendsto</a> (0 : \u211d) with \u27e8Rs, \u27e8_, \u27e8Rs_pos, Rs_lim\u27e9\u27e9\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": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\n\u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0", "state_after": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\n\u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0"}, {"tactic": "have obs := fun n : \u2115 => exists_null_frontier_thickening \u03bc s (Rs_pos n)", "annotated_tactic": ["have obs := fun n : \u2115 => <a>exists_null_frontier_thickening</a> \u03bc s (Rs_pos n)", [{"full_name": "MeasureTheory.exists_null_frontier_thickening", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [463, 9], "def_end_pos": [463, 40]}]], "state_before": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\n\u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0", "state_after": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\nobs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0\n\u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0"}, {"tactic": "refine' \u27e8fun n : \u2115 => (obs n).choose, \u27e8_, _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8fun n : \u2115 => (obs n).<a>choose</a>, \u27e8_, _\u27e9\u27e9", [{"full_name": "Exists.choose", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [442, 32], "def_end_pos": [442, 45]}]], "state_before": "case intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\nobs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0\n\u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0", "state_after": "case intro.intro.intro.refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\nobs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0\n\u22a2 Tendsto (fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) atTop (\ud835\udcdd 0)\n\ncase intro.intro.intro.refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\nobs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0\n\u22a2 \u2200 (n : \u2115),\n    0 < (fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) n \u2227\n      \u2191\u2191\u03bc\n          (frontier\n            (Metric.thickening\n              ((fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) n)\n              s)) =\n        0"}, {"tactic": "exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds Rs_lim\n  (fun n => (obs n).choose_spec.1.1.le) fun n => (obs n).choose_spec.1.2.le", "annotated_tactic": ["exact <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> <a>tendsto_const_nhds</a> Rs_lim\n      (fun n => (obs n).<a>choose_spec</a>.1.1.<a>le</a>) fun n => (obs n).<a>choose_spec</a>.1.2.<a>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": "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": "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": "case intro.intro.intro.refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\nobs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0\n\u22a2 Tendsto (fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "exact fun n => \u27e8(obs n).choose_spec.1.1, (obs n).choose_spec.2\u27e9", "annotated_tactic": ["exact fun n => \u27e8(obs n).<a>choose_spec</a>.1.1, (obs n).<a>choose_spec</a>.2\u27e9", [{"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "case intro.intro.intro.refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\nRs : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti Rs\nRs_pos : \u2200 (n : \u2115), 0 < Rs n\nRs_lim : Tendsto Rs atTop (\ud835\udcdd 0)\nobs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0\n\u22a2 \u2200 (n : \u2115),\n    0 < (fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) n \u2227\n      \u2191\u2191\u03bc\n          (frontier\n            (Metric.thickening\n              ((fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) n)\n              s)) =\n        0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.disjUnion_product", "start": [270, 1], "end": [272, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image\u2082_curry", "start": [337, 1], "end": [339, 69], "traced_tactics": [{"tactic": "classical rw [\u2190 image\u2082_mk_eq_product, image_image\u2082]; dsimp [curry]", "annotated_tactic": ["classical rw [\u2190 <a>image\u2082_mk_eq_product</a>, <a>image_image\u2082</a>]; dsimp [<a>curry</a>]", [{"full_name": "Finset.image\u2082_mk_eq_product", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [332, 9], "def_end_pos": [332, 29]}, {"full_name": "Finset.image_image\u2082", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [310, 9], "def_end_pos": [310, 21]}, {"full_name": "Function.curry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [212, 5], "def_end_pos": [212, 10]}]], "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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 image\u2082 (curry f) s t = image f (s \u00d7\u02e2 t)", "state_after": "no goals"}, {"tactic": "rw [\u2190 image\u2082_mk_eq_product, image_image\u2082]", "annotated_tactic": ["rw [\u2190 <a>image\u2082_mk_eq_product</a>, <a>image_image\u2082</a>]", [{"full_name": "Finset.image\u2082_mk_eq_product", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [332, 9], "def_end_pos": [332, 29]}, {"full_name": "Finset.image_image\u2082", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [310, 9], "def_end_pos": [310, 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\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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 image\u2082 (curry f) s t = image f (s \u00d7\u02e2 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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 image\u2082 (curry f) s t = image\u2082 (fun a b => f (a, b)) s t"}, {"tactic": "dsimp [curry]", "annotated_tactic": ["dsimp [<a>curry</a>]", [{"full_name": "Function.curry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [212, 5], "def_end_pos": [212, 10]}]], "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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 image\u2082 (curry f) s t = image\u2082 (fun a b => f (a, b)) s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag", "start": [1176, 1], "end": [1182, 87], "traced_tactics": [{"tactic": "simp_rw [\u2190 Ioi_disjUnion_Iio, prod_disjUnion, prod_mul_distrib]", "annotated_tactic": ["simp_rw [\u2190 <a>Ioi_disjUnion_Iio</a>, <a>prod_disjUnion</a>, <a>prod_mul_distrib</a>]", [{"full_name": "Finset.Ioi_disjUnion_Iio", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [902, 9], "def_end_pos": [902, 26]}, {"full_name": "Finset.prod_disjUnion", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 23]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [658, 9], "def_end_pos": [658, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f i : \u03b9, \u220f j in Ioi i, f j i * f i j = \u220f i : \u03b9, \u220f j in {i}\u1d9c, f j i", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 =\n    (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 =\n    (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x", "state_after": "case e_a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 = \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x"}, {"tactic": "rw [prod_sigma', prod_sigma']", "annotated_tactic": ["rw [<a>prod_sigma'</a>, <a>prod_sigma'</a>]", [{"full_name": "Finset.prod_sigma'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 20]}, {"full_name": "Finset.prod_sigma'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [532, 9], "def_end_pos": [532, 20]}]], "state_before": "case e_a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 = \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x", "state_after": "case e_a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f x in Finset.sigma univ fun x => Ioi x, f x.fst x.snd = \u220f x in Finset.sigma univ fun x => Iio x, f x.snd x.fst"}, {"tactic": "refine' prod_bij' (fun i _ => \u27e8i.2, i.1\u27e9) _ _ (fun i _ => \u27e8i.2, i.1\u27e9) _ _ _ <;> simp", "annotated_tactic": ["refine' <a>prod_bij'</a> (fun i _ => \u27e8i.2, i.1\u27e9) _ _ (fun i _ => \u27e8i.2, i.1\u27e9) _ _ _ <;> simp", [{"full_name": "Finset.prod_bij'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [564, 9], "def_end_pos": [564, 18]}]], "state_before": "case e_a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f x in Finset.sigma univ fun x => Ioi x, f x.fst x.snd = \u220f x in Finset.sigma univ fun x => Iio x, f x.snd x.fst", "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_of_mem_join", "start": [1876, 1], "end": [1881, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.get_set", "start": [961, 1], "end": [963, 48], "traced_tactics": [{"tactic": "if h : m = n then subst m; simp else simp [h]", "annotated_tactic": ["if h : m = n then subst m; simp else simp [h]", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nm n : Nat\nl : List \u03b1\nh : n < length (set l m a)\n\u22a2 get (set l m a) { val := n, isLt := h } = if m = n then a else get l { val := n, isLt := (_ : n < length l) }", "state_after": "no goals"}, {"tactic": "subst m", "annotated_tactic": ["subst m", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nm n : Nat\nl : List \u03b1\nh\u271d : n < length (set l m a)\nh : m = n\n\u22a2 get (set l m a) { val := n, isLt := h\u271d } = if m = n then a else get l { val := n, isLt := (_ : n < length l) }", "state_after": "\u03b1 : Type u_1\na : \u03b1\nn : Nat\nl : List \u03b1\nh : n < length (set l n a)\n\u22a2 get (set l n a) { val := n, isLt := h } = if n = n then a else get l { val := n, isLt := (_ : n < length l) }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nn : Nat\nl : List \u03b1\nh : n < length (set l n a)\n\u22a2 get (set l n a) { val := n, isLt := h } = if n = n then a else get l { val := n, isLt := (_ : n < length l) }", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nm n : Nat\nl : List \u03b1\nh\u271d : n < length (set l m a)\nh : \u00acm = n\n\u22a2 get (set l m a) { val := n, isLt := h\u271d } = if m = n then a else get l { val := n, isLt := (_ : n < length l) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.cauchy_complete_\u2112p", "start": [1664, 1], "end": [1678, 38], "traced_tactics": [{"tactic": "obtain \u27e8f_lim, h_f_lim_meas, h_lim\u27e9 :\n    \u2203 (f_lim : \u03b1 \u2192 E) (_ : StronglyMeasurable f_lim),\n      \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) atTop (nhds (f_lim x)) :=\n  exists_stronglyMeasurable_limit_of_tendsto_ae (fun n => (hf n).1)\n    (ae_tendsto_of_cauchy_snorm (fun n => (hf n).1) hp hB h_cau)", "annotated_tactic": ["obtain \u27e8f_lim, h_f_lim_meas, h_lim\u27e9 :\n      \u2203 (f_lim : \u03b1 \u2192 E) (_ : <a>StronglyMeasurable</a> f_lim),\n        \u2200\u1d50 x \u2202\u03bc, <a>Tendsto</a> (fun n => f n x) <a>atTop</a> (<a>nhds</a> (f_lim x)) :=\n    <a>exists_stronglyMeasurable_limit_of_tendsto_ae</a> (fun n => (hf n).1)\n      (<a>ae_tendsto_of_cauchy_snorm</a> (fun n => (hf n).1) hp hB h_cau)", [{"full_name": "MeasureTheory.StronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [78, 5], "def_end_pos": [78, 23]}, {"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_stronglyMeasurable_limit_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1685, 9], "def_end_pos": [1685, 61]}, {"full_name": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1590, 9], "def_end_pos": [1590, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case intro.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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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\nf_lim : \u03b1 \u2192 E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_tendsto' : atTop.Tendsto (fun n => snorm (f n - f_lim) p \u03bc) (\ud835\udcdd 0) :=\n  cauchy_tendsto_of_tendsto (fun m => (hf m).1) f_lim hB h_cau h_lim", "annotated_tactic": ["have h_tendsto' : atTop.Tendsto (fun n => <a>snorm</a> (f n - f_lim) p \u03bc) (\ud835\udcdd 0) :=\n    <a>cauchy_tendsto_of_tendsto</a> (fun m => (hf m).1) f_lim hB h_cau h_lim", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.Lp.cauchy_tendsto_of_tendsto", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1622, 9], "def_end_pos": [1622, 34]}]], "state_before": "case intro.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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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\nf_lim : \u03b1 \u2192 E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case intro.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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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\nf_lim : \u03b1 \u2192 E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_\u2112p_lim : Mem\u2112p f_lim p \u03bc :=\n  mem\u2112p_of_cauchy_tendsto hp hf f_lim h_f_lim_meas.aestronglyMeasurable h_tendsto'", "annotated_tactic": ["have h_\u2112p_lim : <a>Mem\u2112p</a> f_lim p \u03bc :=\n    <a>mem\u2112p_of_cauchy_tendsto</a> hp hf f_lim h_f_lim_meas.aestronglyMeasurable h_tendsto'", [{"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_of_cauchy_tendsto", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1646, 9], "def_end_pos": [1646, 32]}]], "state_before": "case intro.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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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\nf_lim : \u03b1 \u2192 E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case intro.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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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\nf_lim : \u03b1 \u2192 E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nh_\u2112p_lim : Mem\u2112p f_lim p\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "exact \u27e8f_lim, h_\u2112p_lim, h_tendsto'\u27e9", "annotated_tactic": ["exact \u27e8f_lim, h_\u2112p_lim, h_tendsto'\u27e9", []], "state_before": "case intro.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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nhp : 1 \u2264 p\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), Mem\u2112p (f n) 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\nf_lim : \u03b1 \u2192 E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nh_\u2112p_lim : Mem\u2112p f_lim p\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.sym_eq_empty", "start": [189, 1], "end": [194, 22], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "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 Finset.sym s n = \u2205 \u2194 n \u2260 0 \u2227 s = \u2205", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym \u03b1 Nat.zero\n\u22a2 Finset.sym s Nat.zero = \u2205 \u2194 Nat.zero \u2260 0 \u2227 s = \u2205\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 Finset.sym s (Nat.succ n\u271d) = \u2205 \u2194 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205"}, {"tactic": "exact iff_of_false (singleton_ne_empty _) fun h \u21a6 (h.1 rfl).elim", "annotated_tactic": ["exact <a>iff_of_false</a> (<a>singleton_ne_empty</a> _) fun h \u21a6 (h.1 <a>rfl</a>).<a>elim</a>", [{"full_name": "iff_of_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "Finset.singleton_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [716, 9], "def_end_pos": [716, 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": "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\nm : Sym \u03b1 Nat.zero\n\u22a2 Finset.sym s Nat.zero = \u2205 \u2194 Nat.zero \u2260 0 \u2227 s = \u2205", "state_after": "no goals"}, {"tactic": "refine \u27e8fun h \u21a6 \u27e8Nat.succ_ne_zero _, eq_empty_of_sym_eq_empty h\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 \u27e8<a>Nat.succ_ne_zero</a> _, <a>eq_empty_of_sym_eq_empty</a> h\u27e9, ?_\u27e9", [{"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": "Finset.eq_empty_of_sym_eq_empty", "def_path": "Mathlib/Data/Finset/Sym.lean", "def_pos": [183, 9], "def_end_pos": [183, 33]}]], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 Finset.sym s (Nat.succ n\u271d) = \u2205 \u2194 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205", "state_after": "case succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205 \u2192 Finset.sym s (Nat.succ n\u271d) = \u2205"}, {"tactic": "rintro \u27e8_, rfl\u27e9", "annotated_tactic": ["rintro \u27e8_, rfl\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 : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205 \u2192 Finset.sym s (Nat.succ n\u271d) = \u2205", "state_after": "case succ.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nt : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\nleft\u271d : Nat.succ n\u271d \u2260 0\n\u22a2 Finset.sym \u2205 (Nat.succ n\u271d) = \u2205"}, {"tactic": "exact sym_empty _", "annotated_tactic": ["exact <a>sym_empty</a> _", [{"full_name": "Finset.sym_empty", "def_path": "Mathlib/Data/Finset/Sym.lean", "def_pos": [164, 9], "def_end_pos": [164, 18]}]], "state_before": "case succ.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nt : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\nleft\u271d : Nat.succ n\u271d \u2260 0\n\u22a2 Finset.sym \u2205 (Nat.succ n\u271d) = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.add_ofNat'", "start": [265, 1], "end": [268, 90], "traced_tactics": [{"tactic": "have : \u2200 {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ", "annotated_tactic": ["have : \u2200 {n}, <a>ofNat'</a> n.succ = <a>ofNat'</a> n + 1 := <a>ofNat'_succ</a>", [{"full_name": "Num.ofNat'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [333, 5], "def_end_pos": [333, 11]}, {"full_name": "Num.ofNat'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [333, 5], "def_end_pos": [333, 11]}, {"full_name": "Num.ofNat'_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [252, 9], "def_end_pos": [252, 20]}]], "state_before": "\u03b1 : Type u_1\nm n : \u2115\n\u22a2 ofNat' (m + n) = ofNat' m + ofNat' n", "state_after": "\u03b1 : Type u_1\nm n : \u2115\nthis : \u2200 {n : \u2115}, ofNat' (Nat.succ n) = ofNat' n + 1\n\u22a2 ofNat' (m + n) = ofNat' m + ofNat' n"}, {"tactic": "induction n <;> simp [Nat.add_zero, this, add_zero, Nat.add_succ, add_one, add_succ, *]", "annotated_tactic": ["induction n <;> simp [<a>Nat.add_zero</a>, this, <a>add_zero</a>, <a>Nat.add_succ</a>, <a>add_one</a>, <a>add_succ</a>, *]", [{"full_name": "Nat.add_zero", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [457, 27], "def_end_pos": [457, 39]}, {"full_name": "Num.add_zero", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [208, 9], "def_end_pos": [208, 17]}, {"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": "Num.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 16]}, {"full_name": "Num.add_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [219, 9], "def_end_pos": [219, 17]}]], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nthis : \u2200 {n : \u2115}, ofNat' (Nat.succ n) = ofNat' n + 1\n\u22a2 ofNat' (m + n) = ofNat' m + ofNat' n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.mem_image\u2082_iff", "start": [74, 1], "end": [75, 66], "traced_tactics": [{"tactic": "rw [\u2190 mem_coe, coe_image\u2082, mem_image2_iff hf, mem_coe, mem_coe]", "annotated_tactic": ["rw [\u2190 <a>mem_coe</a>, <a>coe_image\u2082</a>, <a>mem_image2_iff</a> hf, <a>mem_coe</a>, <a>mem_coe</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_image\u2082", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [49, 9], "def_end_pos": [49, 19]}, {"full_name": "Set.mem_image2_iff", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [48, 9], "def_end_pos": [48, 23]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : 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\nhf : Injective2 f\n\u22a2 f a b \u2208 image\u2082 f s t \u2194 a \u2208 s \u2227 b \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "equivalence_of_manyOneEquiv", "start": [176, 1], "end": [177, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_erase_ne_self", "start": [1650, 1], "end": [1655, 50], "traced_tactics": [{"tactic": "convert @max_erase_ne_self \u03b1\u1d52\u1d48 _ (toDual x) (s.map toDual.toEmbedding) using 1", "annotated_tactic": ["convert @<a>max_erase_ne_self</a> \u03b1\u1d52\u1d48 _ (<a>toDual</a> x) (s.map toDual.toEmbedding) using 1", [{"full_name": "Finset.max_erase_ne_self", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1642, 9], "def_end_pos": [1642, 26]}, {"full_name": "OrderDual.toDual", "def_path": "Mathlib/Order/Synonym.lean", "def_pos": [50, 5], "def_end_pos": [50, 11]}]], "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.min (erase s x) \u2260 \u2191x", "state_after": "case h.e'_2.h\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\ne_1\u271d : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\n\u22a2 Finset.min (erase s x) = Finset.max (erase (map (Equiv.toEmbedding toDual) s) (\u2191toDual x))"}, {"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": "case h.e'_2.h\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\ne_1\u271d : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\n\u22a2 Finset.min (erase s x) = Finset.max (erase (map (Equiv.toEmbedding toDual) s) (\u2191toDual x))", "state_after": "case h.e'_2.h.h\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\ne_1\u271d : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\n\u22a2 erase s x = erase (map (Equiv.toEmbedding toDual) s) (\u2191toDual x)"}, {"tactic": "congr!", "annotated_tactic": ["congr!", []], "state_before": "case h.e'_2.h.h\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\ne_1\u271d : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\n\u22a2 erase s x = erase (map (Equiv.toEmbedding toDual) s) (\u2191toDual x)", "state_after": "case h.e'_2.h.h.h.e'_3.h\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\ne_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\ne_1\u271d : \u03b1 = \u03b1\u1d52\u1d48\n\u22a2 s = map (Equiv.toEmbedding toDual) s"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h.e'_2.h.h.h.e'_3.h\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\ne_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\ne_1\u271d : \u03b1 = \u03b1\u1d52\u1d48\n\u22a2 s = map (Equiv.toEmbedding toDual) s", "state_after": "case h.e'_2.h.h.h.e'_3.h.a\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\ne_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\ne_1\u271d : \u03b1 = \u03b1\u1d52\u1d48\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 map (Equiv.toEmbedding toDual) s"}, {"tactic": "simp only [mem_map_equiv]", "annotated_tactic": ["simp only [<a>mem_map_equiv</a>]", [{"full_name": "Finset.mem_map_equiv", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}]], "state_before": "case h.e'_2.h.h.h.e'_3.h.a\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\ne_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\ne_1\u271d : \u03b1 = \u03b1\u1d52\u1d48\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 map (Equiv.toEmbedding toDual) s", "state_after": "case h.e'_2.h.h.h.e'_3.h.a\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\ne_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\ne_1\u271d : \u03b1 = \u03b1\u1d52\u1d48\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 \u2191toDual.symm a\u271d \u2208 s"}, {"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'_2.h.h.h.e'_3.h.a\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\ne_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48\ne_1\u271d : \u03b1 = \u03b1\u1d52\u1d48\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 \u2191toDual.symm a\u271d \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.vector_ofFn", "start": [1321, 1], "end": [1323, 48], "traced_tactics": [{"tactic": "simp [list_ofFn hf]", "annotated_tactic": ["simp [<a>list_ofFn</a> hf]", [{"full_name": "Primrec.list_ofFn", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1314, 9], "def_end_pos": [1314, 18]}]], "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 : Fin n \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin n), Primrec (f i)\n\u22a2 Primrec fun a => Vector.toList (Vector.ofFn fun i => f i a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "volume_regionBetween_eq_lintegral", "start": [553, 1], "end": [577, 87], "traced_tactics": [{"tactic": "have h\u2081 :\n  (fun y => ENNReal.ofReal ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n    ENNReal.ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) :=\n  (hg.ae_eq_mk.sub hf.ae_eq_mk).fun_comp ENNReal.ofReal", "annotated_tactic": ["have h\u2081 :\n    (fun y => <a>ENNReal.ofReal</a> ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n      <a>ENNReal.ofReal</a> ((<a>AEMeasurable.mk</a> g hg - <a>AEMeasurable.mk</a> f hf) y) :=\n    (hg.ae_eq_mk.sub hf.ae_eq_mk).<a>fun_comp</a> <a>ENNReal.ofReal</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": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"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.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\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "have h\u2082 :\n  (\u03bc.restrict s).prod volume (regionBetween f g s) =\n    (\u03bc.restrict s).prod volume\n      (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) := by\n  apply measure_congr\n  apply EventuallyEq.rfl.inter\n  exact\n    ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n      (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", "annotated_tactic": ["have h\u2082 :\n    (\u03bc.restrict s).<a>prod</a> <a>volume</a> (<a>regionBetween</a> f g s) =\n      (\u03bc.restrict s).<a>prod</a> <a>volume</a>\n        (<a>regionBetween</a> (<a>AEMeasurable.mk</a> f hf) (<a>AEMeasurable.mk</a> g hg) s) := by\n    apply <a>measure_congr</a>\n    apply EventuallyEq.rfl.inter\n    exact\n      ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).<a>comp\u2082</a> _ <a>EventuallyEq.rfl</a>).<a>inter</a>\n        (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", [{"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [292, 27], "def_end_pos": [292, 31]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}, {"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [292, 27], "def_end_pos": [292, 31]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "Filter.EventuallyEq.comp\u2082", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1525, 9], "def_end_pos": [1525, 27]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}, {"full_name": "Filter.EventuallyEq.inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1580, 9], "def_end_pos": [1580, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae h\u2081, \u2190\n  volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> h\u2081, \u2190\n    <a>volume_regionBetween_eq_lintegral'</a> hf.measurable_mk hg.measurable_mk hs]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "volume_regionBetween_eq_lintegral'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 43]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "convert h\u2082 using 1", "annotated_tactic": ["convert h\u2082 using 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s)\n\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"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": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)]\n    regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s"}, {"tactic": "apply EventuallyEq.rfl.inter", "annotated_tactic": ["apply EventuallyEq.rfl.inter", []], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)]\n    regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)] fun p =>\n    Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2"}, {"tactic": "exact\n  ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n    (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", "annotated_tactic": ["exact\n      ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).<a>comp\u2082</a> _ <a>EventuallyEq.rfl</a>).<a>inter</a>\n        (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", [{"full_name": "Filter.EventuallyEq.comp\u2082", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1525, 9], "def_end_pos": [1525, 27]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}, {"full_name": "Filter.EventuallyEq.inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1580, 9], "def_end_pos": [1580, 27]}]], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)] fun p =>\n    Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_prod_eq_prod_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_prod_eq_prod_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_prod_eq_prod_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s)", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ)) (regionBetween f g s)"}, {"tactic": "exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm", "annotated_tactic": ["exact (<a>Measure.restrict_eq_self</a> _ (<a>regionBetween_subset</a> f g s)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 25]}, {"full_name": "regionBetween_subset", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [470, 9], "def_end_pos": [470, 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": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ)) (regionBetween f g s)", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_prod_eq_prod_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_prod_eq_prod_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_prod_eq_prod_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ))\n      (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "exact\n  (Measure.restrict_eq_self _\n      (regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm", "annotated_tactic": ["exact\n      (<a>Measure.restrict_eq_self</a> _\n          (<a>regionBetween_subset</a> (<a>AEMeasurable.mk</a> f hf) (<a>AEMeasurable.mk</a> g hg) s)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 25]}, {"full_name": "regionBetween_subset", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [470, 9], "def_end_pos": [470, 29]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"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\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ))\n      (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) 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.condexpInd_disjoint_union_apply", "start": [315, 1], "end": [318, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "full_name": "Sum.map_comp_map", "start": [123, 9], "end": [125, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.add_mul_ediv_right", "start": [141, 1], "end": [170, 61], "traced_tactics": [{"tactic": "rw [\u2190 Int.neg_inj, \u2190 Int.ediv_neg, Int.neg_add, \u2190 Int.ediv_neg, \u2190 Int.neg_mul_neg]", "annotated_tactic": ["rw [\u2190 <a>Int.neg_inj</a>, \u2190 <a>Int.ediv_neg</a>, <a>Int.neg_add</a>, \u2190 <a>Int.ediv_neg</a>, \u2190 <a>Int.neg_mul_neg</a>]", [{"full_name": "Int.neg_inj", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [87, 19], "def_end_pos": [87, 26]}, {"full_name": "Int.ediv_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [71, 27], "def_end_pos": [71, 35]}, {"full_name": "Int.neg_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [334, 33], "def_end_pos": [334, 40]}, {"full_name": "Int.ediv_neg", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [71, 27], "def_end_pos": [71, 35]}, {"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]}]], "state_before": "a b c : Int\nH : c \u2260 0\nthis : \u2200 \u2983a b c : Int\u2984, 0 < c \u2192 ediv (a + b * c) c = ediv a c + b\nhlt : c < 0\n\u22a2 (a + b * c) / c = a / c + b", "state_after": "a b c : Int\nH : c \u2260 0\nthis : \u2200 \u2983a b c : Int\u2984, 0 < c \u2192 ediv (a + b * c) c = ediv a c + b\nhlt : c < 0\n\u22a2 (a + -b * -c) / -c = a / -c + -b"}, {"tactic": "exact this (Int.neg_pos_of_neg hlt)", "annotated_tactic": ["exact this (<a>Int.neg_pos_of_neg</a> hlt)", [{"full_name": "Int.neg_pos_of_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [901, 19], "def_end_pos": [901, 33]}]], "state_before": "a b c : Int\nH : c \u2260 0\nthis : \u2200 \u2983a b c : Int\u2984, 0 < c \u2192 ediv (a + b * c) c = ediv a c + b\nhlt : c < 0\n\u22a2 (a + -b * -c) / -c = a / -c + -b", "state_after": "no goals"}, {"tactic": "rw [\u2190 Int.add_sub_cancel (ediv ..), \u2190 this, Int.sub_add_cancel]", "annotated_tactic": ["rw [\u2190 <a>Int.add_sub_cancel</a> (<a>ediv</a> ..), \u2190 this, <a>Int.sub_add_cancel</a>]", [{"full_name": "Int.add_sub_cancel", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [489, 27], "def_end_pos": [489, 41]}, {"full_name": "Int.ediv", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 9]}, {"full_name": "Int.sub_add_cancel", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [486, 27], "def_end_pos": [486, 41]}]], "state_before": "a\u271d b\u271d c\u271d : Int\nH\u271d : c\u271d \u2260 0\nthis : \u2200 {k n : Nat} {a : Int}, ediv (a + \u2191n * \u2191(succ k)) \u2191(succ k) = ediv a \u2191(succ k) + \u2191n\na b c : Int\nk n : Nat\nH : 0 < \u2191(succ k)\n\u22a2 ediv (a - \u2191(succ n) * \u2191(succ k)) \u2191(succ k) = ediv a \u2191(succ k) - \u2191(succ n)", "state_after": "no goals"}, {"tactic": "show ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat)", "annotated_tactic": ["show ((n * k.succ : <a>Nat</a>) - m.succ : <a>Int</a>).<a>ediv</a> k.succ = n - (m / k.succ + 1 : <a>Nat</a>)", [{"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"full_name": "Int.ediv", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 9]}, {"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}]], "state_before": "a b c : Int\nH : c \u2260 0\nk n m : Nat\n\u22a2 ediv (-[m+1] + \u2191n * \u2191(succ k)) \u2191(succ k) = ediv -[m+1] \u2191(succ k) + \u2191n", "state_after": "a b c : Int\nH : c \u2260 0\nk n m : Nat\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)"}, {"tactic": "if h : m < n * k.succ then\n  rw [\u2190 Int.ofNat_sub h, \u2190 Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)]\n  apply congrArg ofNat\n  rw [Nat.mul_comm, Nat.mul_sub_div]; rwa [Nat.mul_comm]\nelse\n  have h := Nat.not_lt.1 h\n  have H {a b : Nat} (h : a \u2264 b) : (a : Int) + -((b : Int) + 1) = -[b - a +1] := by\n    rw [negSucc_eq, Int.ofNat_sub h]\n    simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc]\n  show ediv (\u2191(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(\u2191(m / succ k) + 1 : Int)\n  rw [H h, H ((Nat.le_div_iff_mul_le k.succ_pos).2 h)]\n  apply congrArg negSucc\n  rw [Nat.mul_comm, Nat.sub_mul_div]; rwa [Nat.mul_comm]", "annotated_tactic": ["if h : m < n * k.succ then\n      rw [\u2190 <a>Int.ofNat_sub</a> h, \u2190 <a>Int.ofNat_sub</a> ((<a>Nat.div_lt_iff_lt_mul</a> k.succ_pos).2 h)]\n      apply <a>congrArg</a> <a>ofNat</a>\n      rw [<a>Nat.mul_comm</a>, <a>Nat.mul_sub_div</a>]; rwa [<a>Nat.mul_comm</a>]\n    else\n      have h := <a>Nat.not_lt</a>.1 h\n      have H {a b : <a>Nat</a>} (h : a \u2264 b) : (a : <a>Int</a>) + -((b : <a>Int</a>) + 1) = -[b - a +1] := by\n        rw [<a>negSucc_eq</a>, <a>Int.ofNat_sub</a> h]\n        simp only [<a>Int.sub_eq_add_neg</a>, <a>Int.neg_add</a>, <a>Int.neg_neg</a>, <a>Int.add_left_comm</a>, <a>Int.add_assoc</a>]\n      show <a>ediv</a> (\u2191(n * <a>succ</a> k) + -((m : <a>Int</a>) + 1)) (<a>succ</a> k) = n + -(\u2191(m / <a>succ</a> k) + 1 : <a>Int</a>)\n      rw [H h, H ((<a>Nat.le_div_iff_mul_le</a> k.succ_pos).2 h)]\n      apply <a>congrArg</a> <a>negSucc</a>\n      rw [<a>Nat.mul_comm</a>, <a>Nat.sub_mul_div</a>]; rwa [<a>Nat.mul_comm</a>]", [{"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}, {"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}, {"full_name": "Nat.div_lt_iff_lt_mul", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [595, 9], "def_end_pos": [595, 26]}, {"full_name": "congrArg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [349, 9], "def_end_pos": [349, 17]}, {"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]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"full_name": "Nat.mul_sub_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [661, 9], "def_end_pos": [661, 20]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"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", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"full_name": "Int.negSucc_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [74, 9], "def_end_pos": [74, 19]}, {"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}, {"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.neg_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [334, 33], "def_end_pos": [334, 40]}, {"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.add_left_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [284, 19], "def_end_pos": [284, 32]}, {"full_name": "Int.add_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [262, 19], "def_end_pos": [262, 28]}, {"full_name": "Int.ediv", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 9]}, {"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": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"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": "congrArg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [349, 9], "def_end_pos": [349, 17]}, {"full_name": "Int.negSucc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 12]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"full_name": "Nat.sub_mul_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [598, 9], "def_end_pos": [598, 20]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "a b c : Int\nH : c \u2260 0\nk n m : Nat\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Int.ofNat_sub h, \u2190 Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)]", "annotated_tactic": ["rw [\u2190 <a>Int.ofNat_sub</a> h, \u2190 <a>Int.ofNat_sub</a> ((<a>Nat.div_lt_iff_lt_mul</a> k.succ_pos).2 h)]", [{"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}, {"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}, {"full_name": "Nat.div_lt_iff_lt_mul", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [595, 9], "def_end_pos": [595, 26]}]], "state_before": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)", "state_after": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 ediv \u2191(n * succ k - succ m) \u2191(succ k) = \u2191(n - succ (m / succ k))"}, {"tactic": "apply congrArg ofNat", "annotated_tactic": ["apply <a>congrArg</a> <a>ofNat</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": "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": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 ediv \u2191(n * succ k - succ m) \u2191(succ k) = \u2191(n - succ (m / succ k))", "state_after": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 (n * succ k - succ m) / succ k = n - succ (m / succ k)"}, {"tactic": "rw [Nat.mul_comm, Nat.mul_sub_div]", "annotated_tactic": ["rw [<a>Nat.mul_comm</a>, <a>Nat.mul_sub_div</a>]", [{"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"full_name": "Nat.mul_sub_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [661, 9], "def_end_pos": [661, 20]}]], "state_before": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 (n * succ k - succ m) / succ k = n - succ (m / succ k)", "state_after": "case h\u2081\na b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 m < succ k * n"}, {"tactic": "rwa [Nat.mul_comm]", "annotated_tactic": ["rwa [<a>Nat.mul_comm</a>]", [{"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "case h\u2081\na b c : Int\nH : c \u2260 0\nk n m : Nat\nh : m < n * succ k\n\u22a2 m < succ k * n", "state_after": "no goals"}, {"tactic": "have h := Nat.not_lt.1 h", "annotated_tactic": ["have h := <a>Nat.not_lt</a>.1 h", [{"full_name": "Nat.not_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [150, 27], "def_end_pos": [150, 33]}]], "state_before": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh : \u00acm < n * succ k\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)", "state_after": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)"}, {"tactic": "have H {a b : Nat} (h : a \u2264 b) : (a : Int) + -((b : Int) + 1) = -[b - a +1] := by\n  rw [negSucc_eq, Int.ofNat_sub h]\n  simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc]", "annotated_tactic": ["have H {a b : <a>Nat</a>} (h : a \u2264 b) : (a : <a>Int</a>) + -((b : <a>Int</a>) + 1) = -[b - a +1] := by\n        rw [<a>negSucc_eq</a>, <a>Int.ofNat_sub</a> h]\n        simp only [<a>Int.sub_eq_add_neg</a>, <a>Int.neg_add</a>, <a>Int.neg_neg</a>, <a>Int.add_left_comm</a>, <a>Int.add_assoc</a>]", [{"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"full_name": "Int.negSucc_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [74, 9], "def_end_pos": [74, 19]}, {"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}, {"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.neg_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [334, 33], "def_end_pos": [334, 40]}, {"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.add_left_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [284, 19], "def_end_pos": [284, 32]}, {"full_name": "Int.add_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [262, 19], "def_end_pos": [262, 28]}]], "state_before": "a b c : Int\nH : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)", "state_after": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)"}, {"tactic": "show ediv (\u2191(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(\u2191(m / succ k) + 1 : Int)", "annotated_tactic": ["show <a>ediv</a> (\u2191(n * <a>succ</a> k) + -((m : <a>Int</a>) + 1)) (<a>succ</a> k) = n + -(\u2191(m / <a>succ</a> k) + 1 : <a>Int</a>)", [{"full_name": "Int.ediv", "def_path": "lake-packages/std/Std/Data/Int/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 9]}, {"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": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}]], "state_before": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1)", "state_after": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 ediv (\u2191(n * succ k) + -(\u2191m + 1)) \u2191(succ k) = \u2191n + -(\u2191(m / succ k) + 1)"}, {"tactic": "rw [H h, H ((Nat.le_div_iff_mul_le k.succ_pos).2 h)]", "annotated_tactic": ["rw [H h, H ((<a>Nat.le_div_iff_mul_le</a> k.succ_pos).2 h)]", [{"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": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 ediv (\u2191(n * succ k) + -(\u2191m + 1)) \u2191(succ k) = \u2191n + -(\u2191(m / succ k) + 1)", "state_after": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 ediv -[m - n * succ k+1] \u2191(succ k) = -[m / succ k - n+1]"}, {"tactic": "apply congrArg negSucc", "annotated_tactic": ["apply <a>congrArg</a> <a>negSucc</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": "Int.negSucc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 12]}]], "state_before": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 ediv -[m - n * succ k+1] \u2191(succ k) = -[m / succ k - n+1]", "state_after": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 (m - n * succ k) / succ k = m / succ k - n"}, {"tactic": "rw [Nat.mul_comm, Nat.sub_mul_div]", "annotated_tactic": ["rw [<a>Nat.mul_comm</a>, <a>Nat.sub_mul_div</a>]", [{"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"full_name": "Nat.sub_mul_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [598, 9], "def_end_pos": [598, 20]}]], "state_before": "a b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 (m - n * succ k) / succ k = m / succ k - n", "state_after": "case h\u2081\na b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 succ k * n \u2264 m"}, {"tactic": "rwa [Nat.mul_comm]", "annotated_tactic": ["rwa [<a>Nat.mul_comm</a>]", [{"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "case h\u2081\na b c : Int\nH\u271d : c \u2260 0\nk n m : Nat\nh\u271d : \u00acm < n * succ k\nh : n * succ k \u2264 m\nH : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1]\n\u22a2 succ k * n \u2264 m", "state_after": "no goals"}, {"tactic": "rw [negSucc_eq, Int.ofNat_sub h]", "annotated_tactic": ["rw [<a>negSucc_eq</a>, <a>Int.ofNat_sub</a> h]", [{"full_name": "Int.negSucc_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [74, 9], "def_end_pos": [74, 19]}, {"full_name": "Int.ofNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [495, 9], "def_end_pos": [495, 18]}]], "state_before": "a\u271d b\u271d c : Int\nH : c \u2260 0\nk n m : Nat\nh\u271d\u00b9 : \u00acm < n * succ k\nh\u271d : n * succ k \u2264 m\na b : Nat\nh : a \u2264 b\n\u22a2 \u2191a + -(\u2191b + 1) = -[b - a+1]", "state_after": "a\u271d b\u271d c : Int\nH : c \u2260 0\nk n m : Nat\nh\u271d\u00b9 : \u00acm < n * succ k\nh\u271d : n * succ k \u2264 m\na b : Nat\nh : a \u2264 b\n\u22a2 \u2191a + -(\u2191b + 1) = -(\u2191b - \u2191a + 1)"}, {"tactic": "simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc]", "annotated_tactic": ["simp only [<a>Int.sub_eq_add_neg</a>, <a>Int.neg_add</a>, <a>Int.neg_neg</a>, <a>Int.add_left_comm</a>, <a>Int.add_assoc</a>]", [{"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.neg_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [334, 33], "def_end_pos": [334, 40]}, {"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.add_left_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [284, 19], "def_end_pos": [284, 32]}, {"full_name": "Int.add_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [262, 19], "def_end_pos": [262, 28]}]], "state_before": "a\u271d b\u271d c : Int\nH : c \u2260 0\nk n m : Nat\nh\u271d\u00b9 : \u00acm < n * succ k\nh\u271d : n * succ k \u2264 m\na b : Nat\nh : a \u2264 b\n\u22a2 \u2191a + -(\u2191b + 1) = -(\u2191b - \u2191a + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_add_le", "start": [321, 1], "end": [322, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.insert_sigma", "start": [146, 1], "end": [148, 10], "traced_tactics": [{"tactic": "rw [insert_eq, union_sigma, singleton_sigma]", "annotated_tactic": ["rw [<a>insert_eq</a>, <a>union_sigma</a>, <a>singleton_sigma</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.union_sigma", "def_path": "Mathlib/Data/Set/Sigma.lean", "def_pos": [132, 9], "def_end_pos": [132, 20]}, {"full_name": "Set.singleton_sigma", "def_path": "Mathlib/Data/Set/Sigma.lean", "def_pos": [109, 9], "def_end_pos": [109, 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 : \u03b1 i\n\u22a2 Set.Sigma (insert i s) t = Sigma.mk i '' t i \u222a Set.Sigma s t", "state_after": "\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 : \u03b1 i\n\u22a2 \u03b1 i"}, {"tactic": "exact a", "annotated_tactic": ["exact a", []], "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 : \u03b1 i\n\u22a2 \u03b1 i", "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_thickening_of_isClosed", "start": [1811, 1], "end": [1815, 28], "traced_tactics": [{"tactic": "convert tendsto_measure_thickening hs", "annotated_tactic": ["convert <a>tendsto_measure_thickening</a> hs", [{"full_name": "tendsto_measure_thickening", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1802, 9], "def_end_pos": [1802, 35]}]], "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 (thickening R s) \u2260 \u22a4\nh's : IsClosed s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (thickening r s)) (\ud835\udcdd[Ioi 0] 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 (thickening 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 (thickening 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/Integral/DivergenceTheorem.lean", "full_name": "MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable", "start": [491, 1], "end": [524, 52], "traced_tactics": [{"tactic": "wlog h\u2081 : a\u2081 \u2264 b\u2081 generalizing a\u2081 b\u2081", "annotated_tactic": ["wlog h\u2081 : a\u2081 \u2264 b\u2081 generalizing a\u2081 b\u2081", []], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nthis :\n  \u2200 (a\u2081 b\u2081 : \u211d),\n    ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n          (\u2200 (x : \u211d \u00d7 \u211d),\n              x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n            IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n              a\u2081 \u2264 b\u2081 \u2192\n                \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                  (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +\n                      \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n                    \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "wlog h\u2082 : a\u2082 \u2264 b\u2082 generalizing a\u2082 b\u2082", "annotated_tactic": ["wlog h\u2082 : a\u2082 \u2264 b\u2082 generalizing a\u2082 b\u2082", []], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nthis :\n  \u2200 (a\u2082 b\u2082 : \u211d),\n    ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n          (\u2200 (x : \u211d \u00d7 \u211d),\n              x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n            IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n              a\u2082 \u2264 b\u2082 \u2192\n                \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                  (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +\n                      \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n                    \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2082 : a\u2082 \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "simp only [uIcc_of_le h\u2081, uIcc_of_le h\u2082, min_eq_left, max_eq_right, h\u2081, h\u2082] at Hcf Hcg Hdf Hdg Hi", "annotated_tactic": ["simp only [<a>uIcc_of_le</a> h\u2081, <a>uIcc_of_le</a> h\u2082, <a>min_eq_left</a>, <a>max_eq_right</a>, h\u2081, h\u2082] at Hcf Hcg Hdf Hdg Hi", [{"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": "Set.uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [69, 7], "def_end_pos": [69, 17]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2082 : a\u2082 \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "calc\n  (\u222b x in a\u2081..b\u2081, \u222b y in a\u2082..b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =\n      \u222b x in Icc a\u2081 b\u2081, \u222b y in Icc a\u2082 b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1) := by\n    simp only [intervalIntegral.integral_of_le, h\u2081, h\u2082,\n      set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]\n  _ = \u222b x in Icc a\u2081 b\u2081 \u00d7\u02e2 Icc a\u2082 b\u2082, f' x (1, 0) + g' x (0, 1) := (set_integral_prod _ Hi).symm\n  _ = (((\u222b x in a\u2081..b\u2081, g (x, b\u2082)) - \u222b x in a\u2081..b\u2081, g (x, a\u2082)) + \u222b y in a\u2082..b\u2082, f (b\u2081, y)) -\n        \u222b y in a\u2082..b\u2082, f (a\u2081, y) := by\n    rw [Icc_prod_Icc] at *\n    apply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g'\n      (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;> assumption", "annotated_tactic": ["calc\n    (\u222b x in a\u2081..b\u2081, \u222b y in a\u2082..b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =\n        \u222b x in <a>Icc</a> a\u2081 b\u2081, \u222b y in <a>Icc</a> a\u2082 b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1) := by\n      simp only [<a>intervalIntegral.integral_of_le</a>, h\u2081, h\u2082,\n        <a>set_integral_congr_set_ae</a> (<a>Ioc_ae_eq_Icc</a> (\u03b1 := \u211d) (\u03bc := <a>volume</a>))]\n    _ = \u222b x in <a>Icc</a> a\u2081 b\u2081 \u00d7\u02e2 <a>Icc</a> a\u2082 b\u2082, f' x (1, 0) + g' x (0, 1) := (<a>set_integral_prod</a> _ Hi).<a>symm</a>\n    _ = (((\u222b x in a\u2081..b\u2081, g (x, b\u2082)) - \u222b x in a\u2081..b\u2081, g (x, a\u2082)) + \u222b y in a\u2082..b\u2082, f (b\u2081, y)) -\n          \u222b y in a\u2082..b\u2082, f (a\u2081, y) := by\n      rw [<a>Icc_prod_Icc</a>] at *\n      apply <a>integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le</a> f g f' g'\n        (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;> assumption", [{"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": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"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.Ioc_ae_eq_Icc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3172, 9], "def_end_pos": [3172, 22]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"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.set_integral_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [505, 9], "def_end_pos": [505, 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": "Set.Icc_prod_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1922, 9], "def_end_pos": [1922, 21]}, {"full_name": "MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le", "def_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "def_pos": [440, 9], "def_end_pos": [440, 78]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "no goals"}, {"tactic": "specialize this b\u2081 a\u2081", "annotated_tactic": ["specialize this b\u2081 a\u2081", []], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nthis :\n  \u2200 (a\u2081 b\u2081 : \u211d),\n    ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n          (\u2200 (x : \u211d \u00d7 \u211d),\n              x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n            IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n              a\u2081 \u2264 b\u2081 \u2192\n                \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                  (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +\n                      \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n                    \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "rw [uIcc_comm b\u2081 a\u2081, min_comm b\u2081 a\u2081, max_comm b\u2081 a\u2081] at this", "annotated_tactic": ["rw [<a>uIcc_comm</a> b\u2081 a\u2081, <a>min_comm</a> b\u2081 a\u2081, <a>max_comm</a> b\u2081 a\u2081] at this", [{"full_name": "Set.uIcc_comm", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "min_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [80, 9], "def_end_pos": [80, 17]}, {"full_name": "max_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [113, 9], "def_end_pos": [113, 17]}]], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "simp only [intervalIntegral.integral_symm b\u2081 a\u2081]", "annotated_tactic": ["simp only [<a>intervalIntegral.integral_symm</a> b\u2081 a\u2081]", [{"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}]], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -\u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h\u2081))).trans _", "annotated_tactic": ["refine' (<a>congr_arg</a> <a>Neg.neg</a> (this Hcf Hcg Hdf Hdg Hi (<a>le_of_not_le</a> h\u2081))).<a>trans</a> _", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Neg.neg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1284, 3], "def_end_pos": [1284, 6]}, {"full_name": "le_of_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}, {"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\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -\u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n        \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) =\n    (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2081 \u2264 a\u2081 \u2192\n              \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n                  \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n        \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) =\n    (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "no goals"}, {"tactic": "specialize this b\u2082 a\u2082", "annotated_tactic": ["specialize this b\u2082 a\u2082", []], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nthis :\n  \u2200 (a\u2082 b\u2082 : \u211d),\n    ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n          (\u2200 (x : \u211d \u00d7 \u211d),\n              x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n            IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n              a\u2082 \u2264 b\u2082 \u2192\n                \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                  (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +\n                      \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n                    \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "rw [uIcc_comm b\u2082 a\u2082, min_comm b\u2082 a\u2082, max_comm b\u2082 a\u2082] at this", "annotated_tactic": ["rw [<a>uIcc_comm</a> b\u2082 a\u2082, <a>min_comm</a> b\u2082 a\u2082, <a>max_comm</a> b\u2082 a\u2082] at this", [{"full_name": "Set.uIcc_comm", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "min_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [80, 9], "def_end_pos": [80, 17]}, {"full_name": "max_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [113, 9], "def_end_pos": [113, 17]}]], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "simp only [intervalIntegral.integral_symm b\u2082 a\u2082, intervalIntegral.integral_neg]", "annotated_tactic": ["simp only [<a>intervalIntegral.integral_symm</a> b\u2082 a\u2082, <a>intervalIntegral.integral_neg</a>]", [{"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_neg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [591, 16], "def_end_pos": [591, 28]}]], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -\u222b (x : \u211d) in a\u2081..b\u2081, \u222b (x_1 : \u211d) in b\u2082..a\u2082, \u2191(f' (x, x_1)) (1, 0) + \u2191(g' (x, x_1)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n      -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)"}, {"tactic": "refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h\u2082))).trans _", "annotated_tactic": ["refine' (<a>congr_arg</a> <a>Neg.neg</a> (this Hcf Hcg Hdf Hdg Hi (<a>le_of_not_le</a> h\u2082))).<a>trans</a> _", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Neg.neg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1284, 3], "def_end_pos": [1284, 6]}, {"full_name": "le_of_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}, {"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\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -\u222b (x : \u211d) in a\u2081..b\u2081, \u222b (x_1 : \u211d) in b\u2082..a\u2082, \u2191(f' (x, x_1)) (1, 0) + \u2191(g' (x, x_1)) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n      -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)", "state_after": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n        \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n      -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n  ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n    ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n      (\u2200 (x : \u211d \u00d7 \u211d),\n          x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n        (\u2200 (x : \u211d \u00d7 \u211d),\n            x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n          IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n            b\u2082 \u2264 a\u2082 \u2192\n              \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n                (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n                  \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n        \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n      -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)", "state_after": "no goals"}, {"tactic": "simp only [intervalIntegral.integral_of_le, h\u2081, h\u2082,\n  set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]", "annotated_tactic": ["simp only [<a>intervalIntegral.integral_of_le</a>, h\u2081, h\u2082,\n        <a>set_integral_congr_set_ae</a> (<a>Ioc_ae_eq_Icc</a> (\u03b1 := \u211d) (\u03bc := <a>volume</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": "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.Ioc_ae_eq_Icc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3172, 9], "def_end_pos": [3172, 22]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n    \u222b (x : \u211d) in Set.Icc a\u2081 b\u2081, \u222b (y : \u211d) in Set.Icc a\u2082 b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1)", "state_after": "no goals"}, {"tactic": "rw [Icc_prod_Icc] at *", "annotated_tactic": ["rw [<a>Icc_prod_Icc</a>] at *", [{"full_name": "Set.Icc_prod_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1922, 9], "def_end_pos": [1922, 21]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082, \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082), \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)"}, {"tactic": "apply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g'\n  (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;> assumption", "annotated_tactic": ["apply <a>integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le</a> f g f' g'\n        (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;> assumption", [{"full_name": "MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le", "def_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "def_pos": [440, 9], "def_end_pos": [440, 78]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082), \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n      \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, 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_union_congr", "start": [1824, 1], "end": [1846, 99], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun h =>\n    \u27e8restrict_congr_mono (subset_union_left _ _) h,\n      restrict_congr_mono (subset_union_right _ _) h\u27e9,\n    _\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun h =>\n      \u27e8<a>restrict_congr_mono</a> (<a>subset_union_left</a> _ _) h,\n        <a>restrict_congr_mono</a> (<a>subset_union_right</a> _ _) h\u27e9,\n      _\u27e9", [{"full_name": "MeasureTheory.Measure.restrict_congr_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 28]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}, {"full_name": "MeasureTheory.Measure.restrict_congr_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 28]}, {"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\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\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t) \u2194 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd 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\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\n\u22a2 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd t \u2192 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t)"}, {"tactic": "rintro \u27e8hs, ht\u27e9", "annotated_tactic": ["rintro \u27e8hs, ht\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\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\n\u22a2 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd t \u2192 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t)", "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\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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t)"}, {"tactic": "ext1 u hu", "annotated_tactic": ["ext1 u hu", []], "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\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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t)", "state_after": "case intro.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 : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc (s \u222a t)) u = \u2191\u2191(restrict \u03bd (s \u222a t)) u"}, {"tactic": "simp only [restrict_apply hu, inter_union_distrib_left]", "annotated_tactic": ["simp only [<a>restrict_apply</a> hu, <a>inter_union_distrib_left</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": "Set.inter_union_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 33]}]], "state_before": "case intro.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 : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc (s \u222a t)) u = \u2191\u2191(restrict \u03bd (s \u222a t)) u", "state_after": "case intro.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 : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t)"}, {"tactic": "rcases exists_measurable_superset\u2082 \u03bc \u03bd (u \u2229 s) with \u27e8US, hsub, hm, h\u03bc, h\u03bd\u27e9", "annotated_tactic": ["rcases <a>exists_measurable_superset\u2082</a> \u03bc \u03bd (u \u2229 s) with \u27e8US, hsub, hm, h\u03bc, h\u03bd\u27e9", [{"full_name": "MeasureTheory.exists_measurable_superset\u2082", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [225, 9], "def_end_pos": [225, 36]}]], "state_before": "case intro.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 : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t)", "state_after": "case intro.h.intro.intro.intro.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\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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t)"}, {"tactic": "calc\n  \u03bc (u \u2229 s \u222a u \u2229 t) = \u03bc (US \u222a u \u2229 t) :=\n    measure_union_congr_of_subset hsub h\u03bc.le Subset.rfl le_rfl\n  _ = \u03bc US + \u03bc ((u \u2229 t) \\ US) := (measure_add_diff hm _).symm\n  _ = restrict \u03bc s u + restrict \u03bc t (u \\ US) := by\n    simp only [restrict_apply, hu, hu.diff hm, h\u03bc, \u2190 inter_comm t, inter_diff_assoc]\n  _ = restrict \u03bd s u + restrict \u03bd t (u \\ US) := by rw [hs, ht]\n  _ = \u03bd US + \u03bd ((u \u2229 t) \\ US) := by\n    simp only [restrict_apply, hu, hu.diff hm, h\u03bd, \u2190 inter_comm t, inter_diff_assoc]\n  _ = \u03bd (US \u222a u \u2229 t) := (measure_add_diff hm _)\n  _ = \u03bd (u \u2229 s \u222a u \u2229 t) := Eq.symm <| measure_union_congr_of_subset hsub h\u03bd.le Subset.rfl le_rfl", "annotated_tactic": ["calc\n    \u03bc (u \u2229 s \u222a u \u2229 t) = \u03bc (US \u222a u \u2229 t) :=\n      <a>measure_union_congr_of_subset</a> hsub h\u03bc.le <a>Subset.rfl</a> <a>le_rfl</a>\n    _ = \u03bc US + \u03bc ((u \u2229 t) \\ US) := (<a>measure_add_diff</a> hm _).<a>symm</a>\n    _ = <a>restrict</a> \u03bc s u + <a>restrict</a> \u03bc t (u \\ US) := by\n      simp only [<a>restrict_apply</a>, hu, hu.diff hm, h\u03bc, \u2190 <a>inter_comm</a> t, <a>inter_diff_assoc</a>]\n    _ = <a>restrict</a> \u03bd s u + <a>restrict</a> \u03bd t (u \\ US) := by rw [hs, ht]\n    _ = \u03bd US + \u03bd ((u \u2229 t) \\ US) := by\n      simp only [<a>restrict_apply</a>, hu, hu.diff hm, h\u03bd, \u2190 <a>inter_comm</a> t, <a>inter_diff_assoc</a>]\n    _ = \u03bd (US \u222a u \u2229 t) := (<a>measure_add_diff</a> hm _)\n    _ = \u03bd (u \u2229 s \u222a u \u2229 t) := <a>Eq.symm</a> <| <a>measure_union_congr_of_subset</a> hsub h\u03bd.le <a>Subset.rfl</a> <a>le_rfl</a>", [{"full_name": "MeasureTheory.measure_union_congr_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [365, 9], "def_end_pos": [365, 38]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.measure_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [243, 9], "def_end_pos": [243, 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.Measure.restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"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_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": "Set.inter_diff_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1879, 9], "def_end_pos": [1879, 25]}, {"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", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1503, 5], "def_end_pos": [1503, 13]}, {"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": "Set.inter_diff_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1879, 9], "def_end_pos": [1879, 25]}, {"full_name": "MeasureTheory.measure_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [243, 9], "def_end_pos": [243, 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.measure_union_congr_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [365, 9], "def_end_pos": [365, 38]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro.h.intro.intro.intro.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\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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t)", "state_after": "no goals"}, {"tactic": "simp only [restrict_apply, hu, hu.diff hm, h\u03bc, \u2190 inter_comm t, inter_diff_assoc]", "annotated_tactic": ["simp only [<a>restrict_apply</a>, hu, hu.diff hm, h\u03bc, \u2190 <a>inter_comm</a> t, <a>inter_diff_assoc</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": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_diff_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1879, 9], "def_end_pos": [1879, 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\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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191\u03bc US + \u2191\u2191\u03bc ((u \u2229 t) \\ US) = \u2191\u2191(restrict \u03bc s) u + \u2191\u2191(restrict \u03bc t) (u \\ US)", "state_after": "no goals"}, {"tactic": "rw [hs, ht]", "annotated_tactic": ["rw [hs, ht]", []], "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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191(restrict \u03bc s) u + \u2191\u2191(restrict \u03bc t) (u \\ US) = \u2191\u2191(restrict \u03bd s) u + \u2191\u2191(restrict \u03bd t) (u \\ US)", "state_after": "no goals"}, {"tactic": "simp only [restrict_apply, hu, hu.diff hm, h\u03bd, \u2190 inter_comm t, inter_diff_assoc]", "annotated_tactic": ["simp only [<a>restrict_apply</a>, hu, hu.diff hm, h\u03bd, \u2190 <a>inter_comm</a> t, <a>inter_diff_assoc</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": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_diff_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1879, 9], "def_end_pos": [1879, 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\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\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191(restrict \u03bd s) u + \u2191\u2191(restrict \u03bd t) (u \\ US) = \u2191\u2191\u03bd US + \u2191\u2191\u03bd ((u \u2229 t) \\ US)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Decidable.not_iff_not", "start": [591, 1], "end": [592, 75], "traced_tactics": [{"tactic": "rw [@iff_def (\u00aca), @iff_def' a]", "annotated_tactic": ["rw [@<a>iff_def</a> (\u00aca), @<a>iff_def'</a> a]", [{"full_name": "iff_def", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [38, 9], "def_end_pos": [38, 16]}, {"full_name": "iff_def'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [40, 9], "def_end_pos": [40, 17]}]], "state_before": "a b : Prop\ninst\u271d\u00b9 : Decidable a\ninst\u271d : Decidable b\n\u22a2 (\u00aca \u2194 \u00acb) \u2194 (a \u2194 b)", "state_after": "a b : Prop\ninst\u271d\u00b9 : Decidable a\ninst\u271d : Decidable b\n\u22a2 (\u00aca \u2192 \u00acb) \u2227 (\u00acb \u2192 \u00aca) \u2194 (b \u2192 a) \u2227 (a \u2192 b)"}, {"tactic": "exact and_congr not_imp_not not_imp_not", "annotated_tactic": ["exact <a>and_congr</a> <a>not_imp_not</a> <a>not_imp_not</a>", [{"full_name": "and_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "Decidable.not_imp_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [564, 9], "def_end_pos": [564, 30]}, {"full_name": "Decidable.not_imp_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [564, 9], "def_end_pos": [564, 30]}]], "state_before": "a b : Prop\ninst\u271d\u00b9 : Decidable a\ninst\u271d : Decidable b\n\u22a2 (\u00aca \u2192 \u00acb) \u2227 (\u00acb \u2192 \u00aca) \u2194 (b \u2192 a) \u2227 (a \u2192 b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec.nat_rec", "start": [470, 1], "end": [477, 19], "traced_tactics": [{"tactic": "cases' e : decode (\u03b1 := \u03b1) n with a <;> simp [e]", "annotated_tactic": ["cases' e : <a>decode</a> (\u03b1 := \u03b1) n with a <;> simp [e]", [{"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\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 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\n\u22a2 (Part.bind (Part.bind \u2191(decode n) fun a => Part.map encode (\u2191f a)) fun n_1 =>\n      Nat.rec (Part.bind \u2191(decode n) fun a => Part.map encode (g a))\n        (fun y IH => do\n          let i \u2190 IH\n          Part.bind \u2191(decode (Nat.pair n (Nat.pair y i))) fun a => Part.map encode ((fun p => h p.1 p.2) a))\n        n_1) =\n    Part.bind \u2191(decode n) fun a =>\n      Part.map encode ((fun a => Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a)) a)", "state_after": "case some\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      (f a) =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a))"}, {"tactic": "induction' f a with m IH <;> simp", "annotated_tactic": ["induction' f a with m IH <;> simp", []], "state_before": "case some\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      (f a) =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a))", "state_after": "case some.succ\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (Part.bind\n      (Nat.rec (Part.map encode (g a))\n        (fun y IH =>\n          Part.bind IH fun i =>\n            Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n        m)\n      fun i => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode i)) fun a => Part.map encode (h a.1 a.2)) =\n    Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y))"}, {"tactic": "rw [IH, Part.bind_map]", "annotated_tactic": ["rw [IH, <a>Part.bind_map</a>]", [{"full_name": "Part.bind_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [549, 9], "def_end_pos": [549, 17]}]], "state_before": "case some.succ\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (Part.bind\n      (Nat.rec (Part.map encode (g a))\n        (fun y IH =>\n          Part.bind IH fun i =>\n            Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n        m)\n      fun i => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode i)) fun a => Part.map encode (h a.1 a.2)) =\n    Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y))", "state_after": "case some.succ\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y =>\n      Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.1 a.2)) =\n    Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case some.succ\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y =>\n      Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.1 a.2)) =\n    Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y))", "state_after": "case some.succ.e_g\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (fun y => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.1 a.2)) =\n    fun y => Part.map encode (h a (m, y))"}, {"tactic": "funext s", "annotated_tactic": ["funext s", []], "state_before": "case some.succ.e_g\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (fun y => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.1 a.2)) =\n    fun y => Part.map encode (h a (m, y))", "state_after": "case some.succ.e_g.h\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\ns : \u03c3\n\u22a2 (Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode s))) fun a => Part.map encode (h a.1 a.2)) =\n    Part.map encode (h a (m, s))"}, {"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.e_g.h\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 : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n  Nat.rec (Part.map encode (g a))\n      (fun y IH =>\n        Part.bind IH fun i =>\n          Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))\n      m =\n    Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\ns : \u03c3\n\u22a2 (Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode s))) fun a => Part.map encode (h a.1 a.2)) =\n    Part.map encode (h a (m, s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "full_name": "aeSeq.measurable", "start": [95, 1], "end": [97, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Measurable.nnreal_tsum", "start": [2151, 1], "end": [2154, 85], "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 : 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 : MeasurableSpace \u03b1\n\u03b9 : Type u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable fun x => \u2211' (i : \u03b9), f i x", "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 : MeasurableSpace \u03b1\n\u03b9 : Type u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable fun x => ENNReal.toNNReal (\u2211' (b : \u03b9), \u2191(f b x))"}, {"tactic": "exact (Measurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal", "annotated_tactic": ["exact (<a>Measurable.ennreal_tsum</a> fun i => (h i).<a>coe_nnreal_ennreal</a>).<a>ennreal_toNNReal</a>", [{"full_name": "Measurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2136, 9], "def_end_pos": [2136, 32]}, {"full_name": "Measurable.coe_nnreal_ennreal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 38]}, {"full_name": "Measurable.ennreal_toNNReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2099, 9], "def_end_pos": [2099, 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 : MeasurableSpace \u03b1\n\u03b9 : Type u_6\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable 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/Group/Arithmetic.lean", "full_name": "Finset.aemeasurable_prod'", "start": [943, 1], "end": [947, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UnionFind.lt_rankMax'", "start": [190, 1], "end": [192, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "full_name": "Besicovitch.isEmpty_satelliteConfig_multiplicity", "start": [538, 1], "end": [547, 98], "traced_tactics": [{"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 SatelliteConfig E (multiplicity E) (good\u03c4 E) \u2192 False", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\na : SatelliteConfig E (multiplicity E) (good\u03c4 E)\n\u22a2 False"}, {"tactic": "let b := a.centerAndRescale", "annotated_tactic": ["let b := a.centerAndRescale", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\na : SatelliteConfig E (multiplicity E) (good\u03c4 E)\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\na : SatelliteConfig E (multiplicity E) (good\u03c4 E)\nb : SatelliteConfig E (multiplicity E) (good\u03c4 E) := SatelliteConfig.centerAndRescale a\n\u22a2 False"}, {"tactic": "rcases b.exists_normalized a.centerAndRescale_center a.centerAndRescale_radius\n    (one_lt_good\u03c4 E).le (good\u03b4 E) le_rfl (good\u03b4_lt_one E).le with\n  \u27e8c', c'_le_two, hc'\u27e9", "annotated_tactic": ["rcases b.exists_normalized a.centerAndRescale_center a.centerAndRescale_radius\n        (<a>one_lt_good\u03c4</a> E).<a>le</a> (<a>good\u03b4</a> E) <a>le_rfl</a> (<a>good\u03b4_lt_one</a> E).<a>le</a> with\n      \u27e8c', c'_le_two, hc'\u27e9", [{"full_name": "Besicovitch.one_lt_good\u03c4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Besicovitch.good\u03b4", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [284, 5], "def_end_pos": [284, 10]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\na : SatelliteConfig E (multiplicity E) (good\u03c4 E)\nb : SatelliteConfig E (multiplicity E) (good\u03c4 E) := SatelliteConfig.centerAndRescale a\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\na : SatelliteConfig E (multiplicity E) (good\u03c4 E)\nb : SatelliteConfig E (multiplicity E) (good\u03c4 E) := SatelliteConfig.centerAndRescale a\nc' : Fin (Nat.succ (multiplicity E)) \u2192 E\nc'_le_two : \u2200 (n : Fin (Nat.succ (multiplicity E))), \u2016c' n\u2016 \u2264 2\nhc' : \u2200 (i j : Fin (Nat.succ (multiplicity E))), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016c' i - c' j\u2016\n\u22a2 False"}, {"tactic": "exact\n  lt_irrefl _ ((Nat.lt_succ_self _).trans_le (le_multiplicity_of_\u03b4_of_fin c' c'_le_two hc'))", "annotated_tactic": ["exact\n      <a>lt_irrefl</a> _ ((<a>Nat.lt_succ_self</a> _).<a>trans_le</a> (<a>le_multiplicity_of_\u03b4_of_fin</a> c' c'_le_two hc'))", [{"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": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "Besicovitch.le_multiplicity_of_\u03b4_of_fin", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [310, 9], "def_end_pos": [310, 36]}]], "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\na : SatelliteConfig E (multiplicity E) (good\u03c4 E)\nb : SatelliteConfig E (multiplicity E) (good\u03c4 E) := SatelliteConfig.centerAndRescale a\nc' : Fin (Nat.succ (multiplicity E)) \u2192 E\nc'_le_two : \u2200 (n : Fin (Nat.succ (multiplicity E))), \u2016c' n\u2016 \u2264 2\nhc' : \u2200 (i j : Fin (Nat.succ (multiplicity E))), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016c' i - c' j\u2016\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.cast_mul", "start": [1341, 1], "end": [1342, 73], "traced_tactics": [{"tactic": "rw [\u2190 cast_to_int, mul_to_int, Int.cast_mul, cast_to_int, cast_to_int]", "annotated_tactic": ["rw [\u2190 <a>cast_to_int</a>, <a>mul_to_int</a>, <a>Int.cast_mul</a>, <a>cast_to_int</a>, <a>cast_to_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.mul_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 19]}, {"full_name": "Int.cast_mul", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}, {"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.cast_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1097, 9], "def_end_pos": [1097, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nm n : ZNum\n\u22a2 \u2191(m * n) = \u2191m * \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.preimage_union", "start": [447, 1], "end": [448, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.div_zero", "start": [1631, 11], "end": [1635, 21], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "n : Num\n\u22a2 div n 0 = 0", "state_after": "case zero\n\n\u22a2 div zero 0 = 0\n\ncase pos\na\u271d : PosNum\n\u22a2 div (pos a\u271d) 0 = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\n\u22a2 div zero 0 = 0", "state_after": "no goals"}, {"tactic": "simp [Num.div]", "annotated_tactic": ["simp [<a>Num.div</a>]", [{"full_name": "Num.div", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [606, 5], "def_end_pos": [606, 8]}]], "state_before": "case pos\na\u271d : PosNum\n\u22a2 div (pos a\u271d) 0 = 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.norm_sq_eq_inner'", "start": [173, 9], "end": [182, 28], "traced_tactics": [{"tactic": "have h_two : (2 : \u211d\u22650\u221e).toReal = 2 := by simp", "annotated_tactic": ["have h_two : (2 : \u211d\u22650\u221e).<a>toReal</a> = 2 := by simp", [{"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\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 : { x // x \u2208 Lp E 2 }\n\u22a2 \u2016f\u2016 ^ 2 = \u2191IsROrC.re (inner f f)", "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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 \u2016f\u2016 ^ 2 = \u2191IsROrC.re (inner f f)"}, {"tactic": "rw [inner_def, integral_inner_eq_sq_snorm, norm_def, \u2190 ENNReal.toReal_pow, IsROrC.ofReal_re,\n  ENNReal.toReal_eq_toReal (ENNReal.pow_ne_top (Lp.snorm_ne_top f)) _]", "annotated_tactic": ["rw [<a>inner_def</a>, <a>integral_inner_eq_sq_snorm</a>, <a>norm_def</a>, \u2190 <a>ENNReal.toReal_pow</a>, <a>IsROrC.ofReal_re</a>,\n    <a>ENNReal.toReal_eq_toReal</a> (<a>ENNReal.pow_ne_top</a> (<a>Lp.snorm_ne_top</a> f)) _]", [{"full_name": "MeasureTheory.L2.inner_def", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [153, 9], "def_end_pos": [153, 18]}, {"full_name": "MeasureTheory.L2.integral_inner_eq_sq_snorm", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [157, 9], "def_end_pos": [157, 35]}, {"full_name": "MeasureTheory.Lp.norm_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [259, 9], "def_end_pos": [259, 17]}, {"full_name": "ENNReal.toReal_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2303, 9], "def_end_pos": [2303, 19]}, {"full_name": "IsROrC.ofReal_re", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [123, 9], "def_end_pos": [123, 18]}, {"full_name": "ENNReal.toReal_eq_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2327, 9], "def_end_pos": [2327, 25]}, {"full_name": "ENNReal.pow_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [662, 9], "def_end_pos": [662, 19]}, {"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\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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 \u2016f\u2016 ^ 2 = \u2191IsROrC.re (inner f f)", "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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 snorm (\u2191\u2191f) 2 \u03bc ^ 2 = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc\n\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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc \u2260 \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 : { x // x \u2208 Lp E 2 }\n\u22a2 ENNReal.toReal 2 = 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.rpow_nat_cast, snorm_eq_snorm' two_ne_zero ENNReal.two_ne_top, snorm', \u2190\n  ENNReal.rpow_mul, one_div, h_two]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.rpow_nat_cast</a>, <a>snorm_eq_snorm'</a> <a>two_ne_zero</a> <a>ENNReal.two_ne_top</a>, <a>snorm'</a>, \u2190\n      <a>ENNReal.rpow_mul</a>, <a>one_div</a>, h_two]", [{"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": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}, {"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": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"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]}]], "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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 snorm (\u2191\u2191f) 2 \u03bc ^ 2 = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc", "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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc) ^ (2\u207b\u00b9 * \u21912) = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc) ^ (2\u207b\u00b9 * \u21912) = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' (lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top zero_lt_two _).ne", "annotated_tactic": ["refine' (<a>lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top</a> <a>zero_lt_two</a> _).<a>ne</a>", [{"full_name": "MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [143, 9], "def_end_pos": [143, 54]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}, {"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\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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc \u2260 \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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 snorm' (fun a => \u2191\u2191f a) 2 \u03bc < \u22a4"}, {"tactic": "rw [\u2190 h_two, \u2190 snorm_eq_snorm' two_ne_zero ENNReal.two_ne_top]", "annotated_tactic": ["rw [\u2190 h_two, \u2190 <a>snorm_eq_snorm'</a> <a>two_ne_zero</a> <a>ENNReal.two_ne_top</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": "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": "\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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 snorm' (fun a => \u2191\u2191f a) 2 \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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 snorm (fun a => \u2191\u2191f a) 2 \u03bc < \u22a4"}, {"tactic": "exact Lp.snorm_lt_top f", "annotated_tactic": ["exact <a>Lp.snorm_lt_top</a> f", [{"full_name": "MeasureTheory.Lp.snorm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [198, 9], "def_end_pos": [198, 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 : { x // x \u2208 Lp E 2 }\nh_two : ENNReal.toReal 2 = 2\n\u22a2 snorm (fun a => \u2191\u2191f a) 2 \u03bc < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "MeasureTheory.Integrable.integral_condKernel", "start": [646, 1], "end": [648, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.predictablePart_add_ae_eq", "start": [156, 1], "end": [162, 38], "traced_tactics": [{"tactic": "filter_upwards [martingalePart_add_ae_eq hf hg hg0 hgint n] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [<a>martingalePart_add_ae_eq</a> hf hg hg0 hgint n] with \u03c9 h\u03c9", [{"full_name": "MeasureTheory.martingalePart_add_ae_eq", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [135, 9], "def_end_pos": [135, 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 predictablePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] g n", "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\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 predictablePart (f + g) \u2131 \u03bc n \u03c9 = g n \u03c9"}, {"tactic": "rw [\u2190 add_right_inj (f n \u03c9)]", "annotated_tactic": ["rw [\u2190 <a>add_right_inj</a> (f n \u03c9)]", [{"full_name": "add_right_inj", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [188, 3], "def_end_pos": [188, 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\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 predictablePart (f + g) \u2131 \u03bc n \u03c9 = g n \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\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 + g n \u03c9"}, {"tactic": "conv_rhs => rw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Pi.add_apply</a>, \u2190 <a>Pi.add_apply</a>, \u2190 <a>martingalePart_add_predictablePart</a> \u2131 \u03bc (f + g)]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "MeasureTheory.martingalePart_add_predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [70, 9], "def_end_pos": [70, 43]}]], "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\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 + g n \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\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = (martingalePart (f + g) \u2131 \u03bc + predictablePart (f + g) \u2131 \u03bc) n \u03c9"}, {"tactic": "rw [Pi.add_apply, Pi.add_apply, h\u03c9]", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>Pi.add_apply</a>, h\u03c9]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 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 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\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = (martingalePart (f + g) \u2131 \u03bc + predictablePart (f + g) \u2131 \u03bc) n \u03c9", "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_measurableSet", "start": [73, 1], "end": [78, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_mul_Ioc_of_neg", "start": [668, 1], "end": [670, 66], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Ioc_of_neg a b h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Ioc_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_Ioc_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [581, 9], "def_end_pos": [581, 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' Ioc a b = Ico (b / c) (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.average_mem_openSegment_compl_self", "start": [379, 1], "end": [384, 61], "traced_tactics": [{"tactic": "simpa only [union_compl_self, restrict_univ] using\n  average_union_mem_openSegment aedisjoint_compl_right hs.compl hs\u2080 hsc\u2080 (measure_ne_top _ _)\n    (measure_ne_top _ _) hfi.integrableOn hfi.integrableOn", "annotated_tactic": ["simpa only [<a>union_compl_self</a>, <a>restrict_univ</a>] using\n    <a>average_union_mem_openSegment</a> <a>aedisjoint_compl_right</a> hs.compl hs\u2080 hsc\u2080 (<a>measure_ne_top</a> _ _)\n      (<a>measure_ne_top</a> _ _) hfi.integrableOn hfi.integrableOn", [{"full_name": "Set.union_compl_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1739, 9], "def_end_pos": [1739, 25]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "MeasureTheory.average_union_mem_openSegment", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [352, 9], "def_end_pos": [352, 38]}, {"full_name": "MeasureTheory.aedisjoint_compl_right", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [161, 9], "def_end_pos": [161, 31]}, {"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": "\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\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : NullMeasurableSet s\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nhsc\u2080 : \u2191\u2191\u03bc s\u1d9c \u2260 0\nhfi : Integrable f\n\u22a2 \u2a0d (x : \u03b1), f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in s\u1d9c, f x \u2202\u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.outerMeasure_exists_open", "start": [292, 1], "end": [297, 28], "traced_tactics": [{"tactic": "rcases inducedOuterMeasure_exists_set _ \u03bc.innerContent_iUnion_nat \u03bc.innerContent_mono hA\n    (ENNReal.coe_ne_zero.2 h\u03b5) with\n  \u27e8U, hU, h2U, h3U\u27e9", "annotated_tactic": ["rcases <a>inducedOuterMeasure_exists_set</a> _ \u03bc.innerContent_iUnion_nat \u03bc.innerContent_mono hA\n      (<a>ENNReal.coe_ne_zero</a>.2 h\u03b5) with\n    \u27e8U, hU, h2U, h3U\u27e9", [{"full_name": "MeasureTheory.inducedOuterMeasure_exists_set", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1495, 9], "def_end_pos": [1495, 39]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\nhA : \u2191(Content.outerMeasure \u03bc) A \u2260 \u22a4\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 U, A \u2286 \u2191U \u2227 \u2191(Content.outerMeasure \u03bc) \u2191U \u2264 \u2191(Content.outerMeasure \u03bc) A + \u2191\u03b5", "state_after": "case intro.intro.intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\nhA : \u2191(Content.outerMeasure \u03bc) A \u2260 \u22a4\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nU : Set G\nhU : IsOpen U\nh2U : A \u2286 U\nh3U :\n  \u2191(inducedOuterMeasure (fun s\u2081 hs\u2081 => innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 }) (_ : IsOpen \u2205)\n          (_ : innerContent \u03bc \u22a5 = 0))\n      U \u2264\n    \u2191(inducedOuterMeasure (fun s\u2081 hs\u2081 => innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 }) (_ : IsOpen \u2205)\n            (_ : innerContent \u03bc \u22a5 = 0))\n        A +\n      \u2191\u03b5\n\u22a2 \u2203 U, A \u2286 \u2191U \u2227 \u2191(Content.outerMeasure \u03bc) \u2191U \u2264 \u2191(Content.outerMeasure \u03bc) A + \u2191\u03b5"}, {"tactic": "exact \u27e8\u27e8U, hU\u27e9, h2U, h3U\u27e9", "annotated_tactic": ["exact \u27e8\u27e8U, hU\u27e9, h2U, h3U\u27e9", []], "state_before": "case intro.intro.intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\nhA : \u2191(Content.outerMeasure \u03bc) A \u2260 \u22a4\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\nU : Set G\nhU : IsOpen U\nh2U : A \u2286 U\nh3U :\n  \u2191(inducedOuterMeasure (fun s\u2081 hs\u2081 => innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 }) (_ : IsOpen \u2205)\n          (_ : innerContent \u03bc \u22a5 = 0))\n      U \u2264\n    \u2191(inducedOuterMeasure (fun s\u2081 hs\u2081 => innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 }) (_ : IsOpen \u2205)\n            (_ : innerContent \u03bc \u22a5 = 0))\n        A +\n      \u2191\u03b5\n\u22a2 \u2203 U, A \u2286 \u2191U \u2227 \u2191(Content.outerMeasure \u03bc) \u2191U \u2264 \u2191(Content.outerMeasure \u03bc) A + \u2191\u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_subset_Ioc_iff", "start": [270, 1], "end": [271, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.shiftLeft_add", "start": [408, 1], "end": [428, 14], "traced_tactics": [{"tactic": "simp [Nat.pow_add, mul_assoc]", "annotated_tactic": ["simp [<a>Nat.pow_add</a>, <a>mul_assoc</a>]", [{"full_name": "Nat.pow_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [816, 19], "def_end_pos": [816, 26]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "m n k : \u2115\n\u22a2 Nat.shiftLeft' false m (n + k) = Nat.shiftLeft' false (Nat.shiftLeft' false m n) k", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "m n\u271d k i n : \u2115\n\u22a2 (fun n k i => \u2191m <<< i = \u2191(Nat.shiftLeft' false m n >>> k)) (n + i) n \u2191i", "state_after": "m n\u271d k i n : \u2115\n\u22a2 \u2191m <<< \u2191i = \u2191(Nat.shiftLeft' false m (n + i) >>> n)"}, {"tactic": "simp [- Nat.shiftLeft_eq, \u2190 Nat.shiftLeft_sub _ , add_tsub_cancel_left]", "annotated_tactic": ["simp [- <a>Nat.shiftLeft_eq</a>, \u2190 <a>Nat.shiftLeft_sub</a> _ , <a>add_tsub_cancel_left</a>]", [{"full_name": "Nat.shiftLeft_eq", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [791, 17], "def_end_pos": [791, 29]}, {"full_name": "Nat.shiftLeft_sub", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [303, 9], "def_end_pos": [303, 22]}, {"full_name": "add_tsub_cancel_left", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [361, 9], "def_end_pos": [361, 29]}]], "state_before": "m n\u271d k i n : \u2115\n\u22a2 \u2191m <<< \u2191i = \u2191(Nat.shiftLeft' false m (n + i) >>> n)", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "m n\u271d k i n : \u2115\n\u22a2 (fun n k i => \u2191m <<< i = \u2191(Nat.shiftLeft' false m n >>> k)) n (n + i + 1) -[i+1]", "state_after": "m n\u271d k i n : \u2115\n\u22a2 \u2191m <<< -[i+1] = \u2191(Nat.shiftLeft' false m n >>> (n + i)) / 2"}, {"tactic": "simp [- Nat.shiftLeft_eq, Nat.shiftLeft_zero, Nat.shiftRight_add, \u2190 Nat.shiftLeft_sub]", "annotated_tactic": ["simp [- <a>Nat.shiftLeft_eq</a>, <a>Nat.shiftLeft_zero</a>, <a>Nat.shiftRight_add</a>, \u2190 <a>Nat.shiftLeft_sub</a>]", [{"full_name": "Nat.shiftLeft_eq", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [791, 17], "def_end_pos": [791, 29]}, {"full_name": "Nat.shiftLeft_zero", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [206, 9], "def_end_pos": [206, 23]}, {"full_name": "Nat.shiftRight_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [985, 9], "def_end_pos": [985, 23]}, {"full_name": "Nat.shiftLeft_sub", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [303, 9], "def_end_pos": [303, 22]}]], "state_before": "m n\u271d k i n : \u2115\n\u22a2 \u2191m <<< -[i+1] = \u2191(Nat.shiftLeft' false m n >>> (n + i)) / 2", "state_after": "m n\u271d k i n : \u2115\n\u22a2 \u2191m <<< -[i+1] = \u2191(m >>> i) / 2"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "m n\u271d k i n : \u2115\n\u22a2 \u2191m <<< -[i+1] = \u2191(m >>> i) / 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 Nat.shiftLeft'_sub, add_tsub_cancel_left]", "annotated_tactic": ["rw [\u2190 <a>Nat.shiftLeft'_sub</a>, <a>add_tsub_cancel_left</a>]", [{"full_name": "Nat.shiftLeft'_sub", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [295, 9], "def_end_pos": [295, 23]}, {"full_name": "add_tsub_cancel_left", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [361, 9], "def_end_pos": [361, 29]}]], "state_before": "m n\u271d k i n : \u2115\n\u22a2 Nat.shiftLeft' true m i = Nat.shiftLeft' true m (n + i) >>> n", "state_after": "case a\nm n\u271d k i n : \u2115\n\u22a2 n \u2264 n + i"}, {"tactic": "apply Nat.le_add_right", "annotated_tactic": ["apply <a>Nat.le_add_right</a>", [{"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 a\nm n\u271d k i n : \u2115\n\u22a2 n \u2264 n + i", "state_after": "no goals"}, {"tactic": "rw [add_assoc, Nat.shiftRight_add, \u2190 Nat.shiftLeft'_sub, tsub_self]\n<;> rfl", "annotated_tactic": ["rw [<a>add_assoc</a>, <a>Nat.shiftRight_add</a>, \u2190 <a>Nat.shiftLeft'_sub</a>, <a>tsub_self</a>]\n      <;> rfl", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "Nat.shiftRight_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [985, 9], "def_end_pos": [985, 23]}, {"full_name": "Nat.shiftLeft'_sub", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [295, 9], "def_end_pos": [295, 23]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}]], "state_before": "m n\u271d k i n : \u2115\n\u22a2 m >>> Nat.succ i = Nat.shiftLeft' true m n >>> (n + i + 1)", "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.pairwise_replaceF", "start": [203, 9], "end": [219, 32], "traced_tactics": [{"tactic": "simp [H]", "annotated_tactic": ["simp [H]", []], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\nH : List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) []\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) [])", "state_after": "no goals"}, {"tactic": "simp at H \u22a2", "annotated_tactic": ["simp at H \u22a2", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (a :: l)\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) (a :: l))", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match bif a.fst == k then some (k, f a) else none with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)"}, {"tactic": "generalize e : cond .. = z", "annotated_tactic": ["generalize e : <a>cond</a> .. = z", [{"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match bif a.fst == k then some (k, f a) else none with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\ne : (bif a.fst == k then some (k, f a) else none) = z\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match z with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)"}, {"tactic": "unfold cond at e", "annotated_tactic": ["unfold <a>cond</a> at e", [{"full_name": "cond", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [971, 21], "def_end_pos": [971, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\ne : (bif a.fst == k then some (k, f a) else none) = z\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match z with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\ne :\n  (match a.fst == k with\n    | true => some (k, f a)\n    | false => none) =\n    z\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match z with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)"}, {"tactic": "revert e", "annotated_tactic": ["revert e", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\ne :\n  (match a.fst == k with\n    | true => some (k, f a)\n    | false => none) =\n    z\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match z with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\n\u22a2 (match a.fst == k with\n      | true => some (k, f a)\n      | false => none) =\n      z \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (match z with\n      | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n      | some a => a :: l)"}, {"tactic": "split <;> (intro h; subst h; simp)", "annotated_tactic": ["split <;> (intro h; subst h; simp)", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\n\u22a2 (match a.fst == k with\n      | true => some (k, f a)\n      | false => none) =\n      z \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (match z with\n      | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n      | some a => a :: l)", "state_after": "case cons.h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\nheq\u271d : (a.fst == k) = true\n\u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(k == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\n\ncase cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\n\u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2),\n      a' \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\n\u22a2 none = z \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (match z with\n      | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n      | some a => a :: l)", "state_after": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\nh : none = z\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match z with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nz : Option (\u03b1 \u00d7 \u03b2)\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\nh : none = z\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match z with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)", "state_after": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match none with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (match none with\n    | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n    | some a => a :: l)", "state_after": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\n\u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2),\n      a' \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)"}, {"tactic": "next e =>\nrefine \u27e8fun a h => ?_, ih H.2\u27e9\nmatch mem_replaceF h with\n| .inl eq => exact eq \u25b8 ne_true_of_eq_false e\n| .inr h => exact H.1 a h", "annotated_tactic": ["next e =>\n      refine \u27e8fun a h => ?_, ih H.2\u27e9\n      match <a>mem_replaceF</a> h with\n      | .inl eq => exact eq \u25b8 <a>ne_true_of_eq_false</a> e\n      | .inr h => exact H.1 a 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": "ne_true_of_eq_false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [655, 9], "def_end_pos": [655, 28]}]], "state_before": "case cons.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\nheq\u271d : (a.fst == k) = false\n\u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2),\n      a' \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)", "state_after": "no goals"}, {"tactic": "refine \u27e8fun a h => ?_, ih H.2\u27e9", "annotated_tactic": ["refine \u27e8fun a h => ?_, ih H.2\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\ne : (a.fst == k) = false\n\u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2),\n      a' \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na\u271d : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\ne : (a\u271d.fst == k) = false\na : \u03b1 \u00d7 \u03b2\nh : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n\u22a2 \u00ac(a\u271d.fst == a.fst) = true"}, {"tactic": "match mem_replaceF h with\n| .inl eq => exact eq \u25b8 ne_true_of_eq_false e\n| .inr h => exact H.1 a h", "annotated_tactic": ["match <a>mem_replaceF</a> h with\n      | .inl eq => exact eq \u25b8 <a>ne_true_of_eq_false</a> e\n      | .inr h => exact H.1 a 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": "ne_true_of_eq_false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [655, 9], "def_end_pos": [655, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na\u271d : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\ne : (a\u271d.fst == k) = false\na : \u03b1 \u00d7 \u03b2\nh : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\n\u22a2 \u00ac(a\u271d.fst == a.fst) = true", "state_after": "no goals"}, {"tactic": "exact eq \u25b8 ne_true_of_eq_false e", "annotated_tactic": ["exact eq \u25b8 <a>ne_true_of_eq_false</a> e", [{"full_name": "ne_true_of_eq_false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [655, 9], "def_end_pos": [655, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na\u271d : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\ne : (a\u271d.fst == k) = false\na : \u03b1 \u00d7 \u03b2\nh : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\neq : a.fst = k\n\u22a2 \u00ac(a\u271d.fst == a.fst) = true", "state_after": "no goals"}, {"tactic": "exact H.1 a h", "annotated_tactic": ["exact H.1 a h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nk : \u03b1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b2\na\u271d : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n      (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l)\nH : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nc\u271d : Bool\ne : (a\u271d.fst == k) = false\na : \u03b1 \u00d7 \u03b2\nh\u271d : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l\nh : a \u2208 l\n\u22a2 \u00ac(a\u271d.fst == a.fst) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_join", "start": [1119, 1], "end": [1121, 36], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "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\nl : List (List \u03b1)\n\u22a2 List.foldr (fun b s => (fun x x_1 => x ++ x_1) (l, b, s).2.1 (l, b, s).2.2) [] (id l) = List.join l", "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\nl : List (List \u03b1)\n\u22a2 List.foldr (fun b s => b ++ s) [] l = List.join l"}, {"tactic": "induction l <;> simp [*]", "annotated_tactic": ["induction l <;> 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\nl : List (List \u03b1)\n\u22a2 List.foldr (fun b s => b ++ s) [] l = List.join l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Count.lean", "full_name": "MeasureTheory.Measure.le_count_apply", "start": [32, 1], "end": [36, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ioc_eq_empty_iff", "start": [87, 1], "end": [88, 53], "traced_tactics": [{"tactic": "rw [\u2190 coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff]", "annotated_tactic": ["rw [\u2190 <a>coe_eq_empty</a>, <a>coe_Ioc</a>, <a>Set.Ioc_eq_empty_iff</a>]", [{"full_name": 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\u2264 ENNReal.ofReal \u03b5\n\u22a2 UniformIntegrable 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "obtain \u27e8C, hC\u27e9 := h 1 one_pos", "annotated_tactic": ["obtain \u27e8C, hC\u27e9 := h 1 <a>one_pos</a>", [{"full_name": "one_pos", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [50, 7], "def_end_pos": [50, 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C"}, {"tactic": "refine' \u27e8((C : \u211d\u22650\u221e) * \u03bc Set.univ ^ p.toReal\u207b\u00b9 + 1).toNNReal, fun i => _\u27e9", "annotated_tactic": ["refine' \u27e8((C : \u211d\u22650\u221e) * \u03bc <a>Set.univ</a> ^ p.toReal\u207b\u00b9 + 1).<a>toNNReal</a>, fun i => _\u27e9", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ENNReal.toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [164, 15], "def_end_pos": [164, 23]}]], "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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C", "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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 \u2191(ENNReal.toNNReal (\u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1))"}, {"tactic": "refine' le_trans (snorm_mono fun x => _) (snorm_add_le\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const)))\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))) hp)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>snorm_mono</a> fun x => _) (<a>snorm_add_le</a>\n        (<a>StronglyMeasurable.aestronglyMeasurable</a>\n          ((hf i).<a>indicator</a> ((hf i).nnnorm.measurableSet_lt <a>stronglyMeasurable_const</a>)))\n        (<a>StronglyMeasurable.aestronglyMeasurable</a>\n          ((hf i).<a>indicator</a> (stronglyMeasurable_const.measurableSet_le (hf i).<a>nnnorm</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_mono", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [434, 9], "def_end_pos": [434, 19]}, {"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_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}, {"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]}]], "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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 snorm (f i) p \u03bc \u2264 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\n\u22a2 \u2016f i x\u2016 \u2264 \u2016(indicator {x | \u2016f i x\u2016\u208a < C} (f i) + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) x\u2016"}, {"tactic": "rw [Pi.add_apply, Set.indicator_apply]", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>Set.indicator_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": "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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\n\u22a2 \u2016f i x\u2016 \u2264 \u2016(indicator {x | \u2016f i x\u2016\u208a < C} (f i) + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\n\u22a2 \u2016f i x\u2016 \u2264 \u2016(if x \u2208 {x | \u2016f i x\u2016\u208a < C} then f i x else 0) + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016"}, {"tactic": "split_ifs with hx", "annotated_tactic": ["split_ifs with hx", []], "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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\n\u22a2 \u2016f i x\u2016 \u2264 \u2016(if x \u2208 {x | \u2016f i x\u2016\u208a < C} then f i x else 0) + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016", "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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : x \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 \u2016f i x\u2016 \u2264 \u2016f i x + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016\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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 \u2016f i x\u2016 \u2264 \u20160 + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016"}, {"tactic": "rw [Set.indicator_of_not_mem, add_zero]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a>, <a>add_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": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : x \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 \u2016f i x\u2016 \u2264 \u2016f i x + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016", "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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : x \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}"}, {"tactic": "simpa using hx", "annotated_tactic": ["simpa using hx", []], "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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : x \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}", "state_after": "no goals"}, {"tactic": "rw [Set.indicator_of_mem, zero_add]", "annotated_tactic": ["rw [<a>Set.indicator_of_mem</a>, <a>zero_add</a>]", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 \u2016f i x\u2016 \u2264 \u20160 + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016", "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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}"}, {"tactic": "simpa using hx", "annotated_tactic": ["simpa using hx", []], "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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 {x | \u2016f i x\u2016\u208a < C}\n\u22a2 x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a}", "state_after": "no goals"}, {"tactic": "have : \u2200\u1d50 x \u2202\u03bc, \u2016{ x : \u03b1 | \u2016f i x\u2016\u208a < C }.indicator (f i) x\u2016\u208a \u2264 C := by\n  refine' eventually_of_forall _\n  simp_rw [nnnorm_indicator_eq_indicator_nnnorm]\n  exact Set.indicator_le fun x (hx : _ < _) => hx.le", "annotated_tactic": ["have : \u2200\u1d50 x \u2202\u03bc, \u2016{ x : \u03b1 | \u2016f i x\u2016\u208a < C }.<a>indicator</a> (f i) x\u2016\u208a \u2264 C := by\n        refine' <a>eventually_of_forall</a> _\n        simp_rw [<a>nnnorm_indicator_eq_indicator_nnnorm</a>]\n        exact <a>Set.indicator_le</a> fun x (hx : _ < _) => hx.le", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"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_le", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [913, 3], "def_end_pos": [913, 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264\n    \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 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\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264\n    \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1"}, {"tactic": "refine' add_le_add (le_trans (snorm_le_of_ae_bound this) _) (ENNReal.ofReal_one \u25b8 hC i)", "annotated_tactic": ["refine' <a>add_le_add</a> (<a>le_trans</a> (<a>snorm_le_of_ae_bound</a> this) _) (<a>ENNReal.ofReal_one</a> \u25b8 hC i)", [{"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_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]}, {"full_name": "ENNReal.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C\n\u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264\n    \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 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\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C\n\u22a2 \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal ((fun a => \u2191a) C) \u2264 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9"}, {"tactic": "simp_rw [NNReal.val_eq_coe, ENNReal.ofReal_coe_nnreal, mul_comm]", "annotated_tactic": ["simp_rw [<a>NNReal.val_eq_coe</a>, <a>ENNReal.ofReal_coe_nnreal</a>, <a>mul_comm</a>]", [{"full_name": "NNReal.val_eq_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [87, 9], "def_end_pos": [87, 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": "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\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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C\n\u22a2 \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal ((fun a => \u2191a) C) \u2264 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9", "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 : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C\n\u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 \u2264 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9"}, {"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": "\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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C\n\u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 \u2264 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall _", "annotated_tactic": ["refine' <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\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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C", "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 : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2200 (x : \u03b1), \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C"}, {"tactic": "simp_rw [nnnorm_indicator_eq_indicator_nnnorm]", "annotated_tactic": ["simp_rw [<a>nnnorm_indicator_eq_indicator_nnnorm</a>]", [{"full_name": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2200 (x : \u03b1), \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C", "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 : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2200 (x : \u03b1), indicator {x | \u2016f i x\u2016\u208a < C} (fun a => \u2016f i a\u2016\u208a) x \u2264 C"}, {"tactic": "exact Set.indicator_le fun x (hx : _ < _) => hx.le", "annotated_tactic": ["exact <a>Set.indicator_le</a> fun x (hx : _ < _) => hx.le", [{"full_name": "Set.indicator_le", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [913, 3], "def_end_pos": [913, 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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2200 (x : \u03b1), indicator {x | \u2016f i x\u2016\u208a < C} (fun a => \u2016f i a\u2016\u208a) x \u2264 C", "state_after": "no goals"}, {"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": "\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 : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 = \u2191(ENNReal.toNNReal (\u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 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\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 \u2260 \u22a4"}, {"tactic": "exact ENNReal.add_ne_top.2\n  \u27e8ENNReal.mul_ne_top ENNReal.coe_ne_top (ENNReal.rpow_ne_top_of_nonneg\n    (inv_nonneg.2 ENNReal.toReal_nonneg) (measure_lt_top _ _).ne),\n  ENNReal.one_ne_top\u27e9", "annotated_tactic": ["exact <a>ENNReal.add_ne_top</a>.2\n        \u27e8<a>ENNReal.mul_ne_top</a> <a>ENNReal.coe_ne_top</a> (<a>ENNReal.rpow_ne_top_of_nonneg</a>\n          (<a>inv_nonneg</a>.2 <a>ENNReal.toReal_nonneg</a>) (<a>measure_lt_top</a> _ _).<a>ne</a>),\n        <a>ENNReal.one_ne_top</a>\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.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": "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": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"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.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\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nC : \u211d\u22650\nhC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1\ni : \u03b9\n\u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 \u2260 \u22a4", "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_Ioc_of_le", "start": [183, 1], "end": [184, 43], "traced_tactics": [{"tactic": "rw [card_fintype_Ioc, toNat_sub_of_le h]", "annotated_tactic": ["rw [<a>card_fintype_Ioc</a>, <a>toNat_sub_of_le</a> h]", [{"full_name": "Int.card_fintype_Ioc", "def_path": "Mathlib/Data/Int/Interval.lean", "def_pos": [162, 9], "def_end_pos": [162, 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.Ioc a b)) = b - a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.lift_comp_castAddHom", "start": [1283, 1], "end": [1284, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "full_name": "MeasureTheory.Submartingale.expected_stoppedValue_mono", "start": [42, 1], "end": [63, 75], "traced_tactics": [{"tactic": "rw [\u2190 sub_nonneg, \u2190 integral_sub', stoppedValue_sub_eq_sum' hle hbdd]", "annotated_tactic": ["rw [\u2190 <a>sub_nonneg</a>, \u2190 <a>integral_sub'</a>, <a>stoppedValue_sub_eq_sum'</a> hle hbdd]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "MeasureTheory.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "MeasureTheory.stoppedValue_sub_eq_sum'", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1071, 9], "def_end_pos": [1071, 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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 0 \u2264\n    \u222b (a : \u03a9),\n      (fun \u03c9 => Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i)) \u03c9)\n        a \u2202\u03bc\n\ncase hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 Integrable fun x => stoppedValue f \u03c0 x\n\ncase hg\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 Integrable fun x => stoppedValue f \u03c4 x"}, {"tactic": "simp only [Finset.sum_apply]", "annotated_tactic": ["simp only [<a>Finset.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": "\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 0 \u2264\n    \u222b (a : \u03a9),\n      (fun \u03c9 => Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i)) \u03c9)\n        a \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 0 \u2264 \u222b (a : \u03a9), Finset.sum (Finset.range (N + 1)) fun c => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 c \u2227 c < \u03c0 \u03c9} (f (c + 1) - f c) a \u2202\u03bc"}, {"tactic": "have : \u2200 i, MeasurableSet[\ud835\udca2 i] {\u03c9 : \u03a9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} := by\n  intro i\n  refine' (h\u03c4 i).inter _\n  convert (h\u03c0 i).compl using 1\n  ext x\n  simp; rfl", "annotated_tactic": ["have : \u2200 i, MeasurableSet[\ud835\udca2 i] {\u03c9 : \u03a9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} := by\n      intro i\n      refine' (h\u03c4 i).<a>inter</a> _\n      convert (h\u03c0 i).<a>compl</a> using 1\n      ext x\n      simp; rfl", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 0 \u2264 \u222b (a : \u03a9), Finset.sum (Finset.range (N + 1)) fun c => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 c \u2227 c < \u03c0 \u03c9} (f (c + 1) - f c) a \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\n\u22a2 0 \u2264 \u222b (a : \u03a9), Finset.sum (Finset.range (N + 1)) fun c => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 c \u2227 c < \u03c0 \u03c9} (f (c + 1) - f c) a \u2202\u03bc"}, {"tactic": "rw [integral_finset_sum]", "annotated_tactic": ["rw [<a>integral_finset_sum</a>]", [{"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\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\n\u22a2 0 \u2264 \u222b (a : \u03a9), Finset.sum (Finset.range (N + 1)) fun c => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 c \u2227 c < \u03c0 \u03c9} (f (c + 1) - f c) a \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\n\u22a2 0 \u2264 Finset.sum (Finset.range (N + 1)) fun i => \u222b (a : \u03a9), Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a \u2202\u03bc\n\ncase hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range (N + 1) \u2192 Integrable fun a => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a"}, {"tactic": "intro i _", "annotated_tactic": ["intro i _", []], "state_before": "case hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range (N + 1) \u2192 Integrable fun a => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a", "state_after": "case hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\na\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 Integrable fun a => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a"}, {"tactic": "exact Integrable.indicator (Integrable.sub (hf.integrable _) (hf.integrable _))\n  (\ud835\udca2.le _ _ (this _))", "annotated_tactic": ["exact <a>Integrable.indicator</a> (<a>Integrable.sub</a> (hf.integrable _) (hf.integrable _))\n      (\ud835\udca2.le _ _ (this _))", [{"full_name": "MeasureTheory.Integrable.indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "state_before": "case hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\na\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 Integrable fun a => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}"}, {"tactic": "refine' (h\u03c4 i).inter _", "annotated_tactic": ["refine' (h\u03c4 i).<a>inter</a> _", [{"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\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\n\u22a2 MeasurableSet fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)"}, {"tactic": "convert (h\u03c0 i).compl using 1", "annotated_tactic": ["convert (h\u03c0 i).<a>compl</a> using 1", [{"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\n\u22a2 MeasurableSet fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)", "state_after": "case h.e'_3\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\n\u22a2 (fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)) = {\u03c9 | \u03c0 \u03c9 \u2264 i}\u1d9c"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_3\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\n\u22a2 (fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)) = {\u03c9 | \u03c0 \u03c9 \u2264 i}\u1d9c", "state_after": "case h.e'_3.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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\nx : \u03a9\n\u22a2 (x \u2208 fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)) \u2194 x \u2208 {\u03c9 | \u03c0 \u03c9 \u2264 i}\u1d9c"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3.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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\nx : \u03a9\n\u22a2 (x \u2208 fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)) \u2194 x \u2208 {\u03c9 | \u03c0 \u03c9 \u2264 i}\u1d9c", "state_after": "case h.e'_3.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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\nx : \u03a9\n\u22a2 (x \u2208 fun \u03c9 => Nat.succ i \u2264 \u03c0 \u03c9) \u2194 i < \u03c0 x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_3.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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\ni : \u2115\nx : \u03a9\n\u22a2 (x \u2208 fun \u03c9 => Nat.succ i \u2264 \u03c0 \u03c9) \u2194 i < \u03c0 x", "state_after": "no goals"}, {"tactic": "refine' Finset.sum_nonneg fun i _ => _", "annotated_tactic": ["refine' <a>Finset.sum_nonneg</a> fun i _ => _", [{"full_name": "Finset.sum_nonneg", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [138, 15], "def_end_pos": [138, 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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\n\u22a2 0 \u2264 Finset.sum (Finset.range (N + 1)) fun i => \u222b (a : \u03a9), Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 0 \u2264 \u222b (a : \u03a9), Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a \u2202\u03bc"}, {"tactic": "rw [integral_indicator (\ud835\udca2.le _ _ (this _)), integral_sub', sub_nonneg]", "annotated_tactic": ["rw [<a>integral_indicator</a> (\ud835\udca2.le _ _ (this _)), <a>integral_sub'</a>, <a>sub_nonneg</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_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"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\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 0 \u2264 \u222b (a : \u03a9), Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a \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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 \u222b (a : \u03a9) in {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}, f i a \u2202\u03bc \u2264 \u222b (a : \u03a9) in {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}, f (i + 1) a \u2202\u03bc\n\ncase hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 Integrable (f (i + 1))\n\ncase hg\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 Integrable (f i)"}, {"tactic": "exact hf.set_integral_le (Nat.le_succ i) (this _)", "annotated_tactic": ["exact hf.set_integral_le (<a>Nat.le_succ</a> i) (this _)", [{"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": "\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 \u222b (a : \u03a9) in {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}, f i a \u2202\u03bc \u2264 \u222b (a : \u03a9) in {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}, f (i + 1) a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact (hf.integrable _).integrableOn", "annotated_tactic": ["exact (hf.integrable _).<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 hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 Integrable (f (i + 1))", "state_after": "no goals"}, {"tactic": "exact (hf.integrable _).integrableOn", "annotated_tactic": ["exact (hf.integrable _).<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 hg\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\nthis : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}\ni : \u2115\nx\u271d : i \u2208 Finset.range (N + 1)\n\u22a2 Integrable (f i)", "state_after": "no goals"}, {"tactic": "exact hf.integrable_stoppedValue h\u03c0 hbdd", "annotated_tactic": ["exact hf.integrable_stoppedValue h\u03c0 hbdd", []], "state_before": "case hf\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 Integrable fun x => stoppedValue f \u03c0 x", "state_after": "no goals"}, {"tactic": "exact hf.integrable_stoppedValue h\u03c4 fun \u03c9 => le_trans (hle \u03c9) (hbdd \u03c9)", "annotated_tactic": ["exact hf.integrable_stoppedValue h\u03c4 fun \u03c9 => <a>le_trans</a> (hle \u03c9) (hbdd \u03c9)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case hg\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 : SigmaFiniteFiltration \u03bc \ud835\udca2\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 Integrable fun x => stoppedValue f \u03c4 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.union", "start": [150, 11], "end": [151, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.neg_le_sub_right_of_le_add", "start": [1027, 11], "end": [1028, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.card_product", "start": [139, 1], "end": [140, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sign.lean", "full_name": "SignType.range_eq", "start": [284, 1], "end": [286, 37], "traced_tactics": [{"tactic": "classical rw [\u2190 Fintype.coe_image_univ, univ_eq]", "annotated_tactic": ["classical rw [\u2190 <a>Fintype.coe_image_univ</a>, <a>univ_eq</a>]", [{"full_name": "Fintype.coe_image_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 31]}, {"full_name": "SignType.univ_eq", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [281, 9], "def_end_pos": [281, 16]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 Set.range f = {f zero, f neg, f pos}", "state_after": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}"}, {"tactic": "classical simp [Finset.coe_insert]", "annotated_tactic": ["classical simp [<a>Finset.coe_insert</a>]", [{"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "state_after": "no goals"}, {"tactic": "rw [\u2190 Fintype.coe_image_univ, univ_eq]", "annotated_tactic": ["rw [\u2190 <a>Fintype.coe_image_univ</a>, <a>univ_eq</a>]", [{"full_name": "Fintype.coe_image_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 31]}, {"full_name": "SignType.univ_eq", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [281, 9], "def_end_pos": [281, 16]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 Set.range f = {f zero, f neg, f pos}", "state_after": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}"}, {"tactic": "simp [Finset.coe_insert]", "annotated_tactic": ["simp [<a>Finset.coe_insert</a>]", [{"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ConditionalExpectation.lean", "full_name": "MeasureTheory.condexp_indep_eq", "start": [40, 1], "end": [77, 46], "traced_tactics": [{"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": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : \u00acIntegrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : \u00acIntegrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc", "state_after": "case neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : \u00acIntegrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc\n\ncase pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc"}, {"tactic": "refine' (ae_eq_condexp_of_forall_set_integral_eq hle\u2082 hfint\n  (fun s _ hs => integrableOn_const.2 (Or.inr hs)) (fun s hms hs => _)\n    stronglyMeasurable_const.aeStronglyMeasurable').symm", "annotated_tactic": ["refine' (<a>ae_eq_condexp_of_forall_set_integral_eq</a> hle\u2082 hfint\n    (fun s _ hs => <a>integrableOn_const</a>.2 (<a>Or.inr</a> hs)) (fun s hms hs => _)\n      stronglyMeasurable_const.aeStronglyMeasurable').<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": "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": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03a9) in s, \u222b (x : \u03a9), f x \u2202\u03bc \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc"}, {"tactic": "rw [set_integral_const]", "annotated_tactic": ["rw [<a>set_integral_const</a>]", [{"full_name": "MeasureTheory.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}]], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03a9) in s, \u222b (x : \u03a9), f x \u2202\u03bc \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc"}, {"tactic": "rw [\u2190 mem\u2112p_one_iff_integrable] at hfint", "annotated_tactic": ["rw [\u2190 <a>mem\u2112p_one_iff_integrable</a>] at hfint", [{"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": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Integrable f\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc"}, {"tactic": "refine' Mem\u2112p.induction_stronglyMeasurable hle\u2081 ENNReal.one_ne_top _ _ _ _ hfint _", "annotated_tactic": ["refine' <a>Mem\u2112p.induction_stronglyMeasurable</a> hle\u2081 <a>ENNReal.one_ne_top</a> _ _ _ _ hfint _", [{"full_name": "MeasureTheory.Mem\u2112p.induction_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [687, 9], "def_end_pos": [687, 43]}, {"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 pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable' m\u2081 f \u03bc\n\ncase pos.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200 (c : E) \u2983s_1 : Set \u03a9\u2984,\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 < \u22a4 \u2192\n        ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator s_1 (fun x => c) x \u2202\u03bc =\n          \u222b (x : \u03a9) in s, Set.indicator s_1 (fun x => c) x \u2202\u03bc\n\ncase pos.refine'_3\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200 \u2983f g : \u03a9 \u2192 E\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f 1 \u2192\n        Mem\u2112p g 1 \u2192\n          StronglyMeasurable f \u2192\n            StronglyMeasurable g \u2192\n              ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc \u2192\n                ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), g x \u2202\u03bc = \u222b (x : \u03a9) in s, g x \u2202\u03bc \u2192\n                  ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (f + g) x \u2202\u03bc = \u222b (x : \u03a9) in s, (f + g) x \u2202\u03bc\n\ncase pos.refine'_4\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc}\n\ncase pos.refine'_5\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200 \u2983f g : \u03a9 \u2192 E\u2984,\n    f =\u1d50[\u03bc] g \u2192\n      Mem\u2112p f 1 \u2192\n        ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc \u2192\n          ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), g x \u2202\u03bc = \u222b (x : \u03a9) in s, g x \u2202\u03bc"}, {"tactic": "rw [condexp_undef hfint, integral_undef hfint]", "annotated_tactic": ["rw [<a>condexp_undef</a> hfint, <a>integral_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]}, {"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\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : \u00acIntegrable f\n\u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc", "state_after": "case neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : \u00acIntegrable f\n\u22a2 0 =\u1d50[\u03bc] fun x => 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : \u00acIntegrable f\n\u22a2 0 =\u1d50[\u03bc] fun x => 0", "state_after": "no goals"}, {"tactic": "exact \u27e8f, hf, EventuallyEq.rfl\u27e9", "annotated_tactic": ["exact \u27e8f, hf, <a>EventuallyEq.rfl</a>\u27e9", [{"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}]], "state_before": "case pos.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable' m\u2081 f \u03bc", "state_after": "no goals"}, {"tactic": "intro c t hmt _", "annotated_tactic": ["intro c t hmt _", []], "state_before": "case pos.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200 (c : E) \u2983s_1 : Set \u03a9\u2984,\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 < \u22a4 \u2192\n        ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator s_1 (fun x => c) x \u2202\u03bc =\n          \u222b (x : \u03a9) in s, Set.indicator s_1 (fun x => c) x \u2202\u03bc", "state_after": "case pos.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nc : E\nt : Set \u03a9\nhmt : MeasurableSet t\na\u271d : \u2191\u2191\u03bc t < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator t (fun x => c) x \u2202\u03bc =\n    \u222b (x : \u03a9) in s, Set.indicator t (fun x => c) x \u2202\u03bc"}, {"tactic": "rw [Indep_iff] at hindp", "annotated_tactic": ["rw [<a>Indep_iff</a>] at hindp", [{"full_name": "ProbabilityTheory.Indep_iff", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [171, 7], "def_end_pos": [171, 16]}]], "state_before": "case pos.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nc : E\nt : Set \u03a9\nhmt : MeasurableSet t\na\u271d : \u2191\u2191\u03bc t < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator t (fun x => c) x \u2202\u03bc =\n    \u222b (x : \u03a9) in s, Set.indicator t (fun x => c) x \u2202\u03bc", "state_after": "case pos.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nc : E\nt : Set \u03a9\nhmt : MeasurableSet t\na\u271d : \u2191\u2191\u03bc t < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator t (fun x => c) x \u2202\u03bc =\n    \u222b (x : \u03a9) in s, Set.indicator t (fun x => c) x \u2202\u03bc"}, {"tactic": "rw [integral_indicator (hle\u2081 _ hmt), set_integral_const, smul_smul, \u2190 ENNReal.toReal_mul,\n  mul_comm, \u2190 hindp _ _ hmt hms, set_integral_indicator (hle\u2081 _ hmt), set_integral_const,\n  Set.inter_comm]", "annotated_tactic": ["rw [<a>integral_indicator</a> (hle\u2081 _ hmt), <a>set_integral_const</a>, <a>smul_smul</a>, \u2190 <a>ENNReal.toReal_mul</a>,\n      <a>mul_comm</a>, \u2190 hindp _ _ hmt hms, <a>set_integral_indicator</a> (hle\u2081 _ hmt), <a>set_integral_const</a>,\n      <a>Set.inter_comm</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.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"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": "MeasureTheory.set_integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [183, 9], "def_end_pos": [183, 31]}, {"full_name": "MeasureTheory.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "case pos.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nc : E\nt : Set \u03a9\nhmt : MeasurableSet t\na\u271d : \u2191\u2191\u03bc t < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator t (fun x => c) x \u2202\u03bc =\n    \u222b (x : \u03a9) in s, Set.indicator t (fun x => c) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro u v _ huint hvint hu hv hu_eq hv_eq", "annotated_tactic": ["intro u v _ huint hvint hu hv hu_eq hv_eq", []], "state_before": "case pos.refine'_3\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200 \u2983f g : \u03a9 \u2192 E\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Mem\u2112p f 1 \u2192\n        Mem\u2112p g 1 \u2192\n          StronglyMeasurable f \u2192\n            StronglyMeasurable g \u2192\n              ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc \u2192\n                ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), g x \u2202\u03bc = \u222b (x : \u03a9) in s, g x \u2202\u03bc \u2192\n                  ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (f + g) x \u2202\u03bc = \u222b (x : \u03a9) in s, (f + g) x \u2202\u03bc", "state_after": "case pos.refine'_3\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\na\u271d : Disjoint (Function.support u) (Function.support v)\nhuint : Mem\u2112p u 1\nhvint : Mem\u2112p v 1\nhu : StronglyMeasurable u\nhv : StronglyMeasurable v\nhu_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\nhv_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (u + v) x \u2202\u03bc = \u222b (x : \u03a9) in s, (u + v) x \u2202\u03bc"}, {"tactic": "rw [mem\u2112p_one_iff_integrable] at huint hvint", "annotated_tactic": ["rw [<a>mem\u2112p_one_iff_integrable</a>] at huint hvint", [{"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": "case pos.refine'_3\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\na\u271d : Disjoint (Function.support u) (Function.support v)\nhuint : Mem\u2112p u 1\nhvint : Mem\u2112p v 1\nhu : StronglyMeasurable u\nhv : StronglyMeasurable v\nhu_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\nhv_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (u + v) x \u2202\u03bc = \u222b (x : \u03a9) in s, (u + v) x \u2202\u03bc", "state_after": "case pos.refine'_3\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\na\u271d : Disjoint (Function.support u) (Function.support v)\nhuint : Integrable u\nhvint : Integrable v\nhu : StronglyMeasurable u\nhv : StronglyMeasurable v\nhu_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\nhv_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (u + v) x \u2202\u03bc = \u222b (x : \u03a9) in s, (u + v) x \u2202\u03bc"}, {"tactic": "rw [integral_add' huint hvint, smul_add, hu_eq, hv_eq,\n  integral_add' huint.integrableOn hvint.integrableOn]", "annotated_tactic": ["rw [<a>integral_add'</a> huint hvint, <a>smul_add</a>, hu_eq, hv_eq,\n      <a>integral_add'</a> huint.integrableOn hvint.integrableOn]", [{"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}, {"full_name": "MeasureTheory.integral_add'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [876, 9], "def_end_pos": [876, 22]}]], "state_before": "case pos.refine'_3\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\na\u271d : Disjoint (Function.support u) (Function.support v)\nhuint : Integrable u\nhvint : Integrable v\nhu : StronglyMeasurable u\nhv : StronglyMeasurable v\nhu_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\nhv_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (u + v) x \u2202\u03bc = \u222b (x : \u03a9) in s, (u + v) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "have heq\u2081 : (fun f : lpMeas E \u211d m\u2081 1 \u03bc => \u222b x, (f : \u03a9 \u2192 E) x \u2202\u03bc) =\n    (fun f : Lp E 1 \u03bc => \u222b x, f x \u2202\u03bc) \u2218 Submodule.subtypeL _ := by\n  refine' funext fun f => integral_congr_ae _\n  simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]; norm_cast", "annotated_tactic": ["have heq\u2081 : (fun f : <a>lpMeas</a> E \u211d m\u2081 1 \u03bc => \u222b x, (f : \u03a9 \u2192 E) x \u2202\u03bc) =\n        (fun f : <a>Lp</a> E 1 \u03bc => \u222b x, f x \u2202\u03bc) \u2218 <a>Submodule.subtypeL</a> _ := by\n      refine' <a>funext</a> fun f => <a>integral_congr_ae</a> _\n      simp_rw [<a>Submodule.coe_subtypeL'</a>, <a>Submodule.coeSubtype</a>]; norm_cast", [{"full_name": "MeasureTheory.lpMeas", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [222, 5], "def_end_pos": [222, 11]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Submodule.subtypeL", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1049, 5], "def_end_pos": [1049, 30]}, {"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.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "Submodule.coe_subtypeL'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 39]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 19]}]], "state_before": "case pos.refine'_4\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc}", "state_after": "case pos.refine'_4\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc}"}, {"tactic": "have heq\u2082 : (fun f : lpMeas E \u211d m\u2081 1 \u03bc => \u222b x in s, (f : \u03a9 \u2192 E) x \u2202\u03bc) =\n    (fun f : Lp E 1 \u03bc => \u222b x in s, f x \u2202\u03bc) \u2218 Submodule.subtypeL _ := by\n  refine' funext fun f => integral_congr_ae (ae_restrict_of_ae _)\n  simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]\n  exact eventually_of_forall fun _ => (by trivial)", "annotated_tactic": ["have heq\u2082 : (fun f : <a>lpMeas</a> E \u211d m\u2081 1 \u03bc => \u222b x in s, (f : \u03a9 \u2192 E) x \u2202\u03bc) =\n        (fun f : <a>Lp</a> E 1 \u03bc => \u222b x in s, f x \u2202\u03bc) \u2218 <a>Submodule.subtypeL</a> _ := by\n      refine' <a>funext</a> fun f => <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> _)\n      simp_rw [<a>Submodule.coe_subtypeL'</a>, <a>Submodule.coeSubtype</a>]\n      exact <a>eventually_of_forall</a> fun _ => (by trivial)", [{"full_name": "MeasureTheory.lpMeas", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [222, 5], "def_end_pos": [222, 11]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Submodule.subtypeL", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1049, 5], "def_end_pos": [1049, 30]}, {"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.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": "Submodule.coe_subtypeL'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 39]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 19]}, {"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.refine'_4\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc}", "state_after": "case pos.refine'_4\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc}"}, {"tactic": "refine' isClosed_eq (Continuous.const_smul _ _) _", "annotated_tactic": ["refine' <a>isClosed_eq</a> (<a>Continuous.const_smul</a> _ _) _", [{"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}, {"full_name": "Continuous.const_smul", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [110, 9], "def_end_pos": [110, 30]}]], "state_before": "case pos.refine'_4\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc}", "state_after": "case pos.refine'_4.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc\n\ncase pos.refine'_4.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc"}, {"tactic": "refine' funext fun f => integral_congr_ae _", "annotated_tactic": ["refine' <a>funext</a> fun f => <a>integral_congr_ae</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.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 (fun x => \u2191\u2191\u2191f x) =\u1d50[\u03bc] fun x => \u2191\u2191(\u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f) x"}, {"tactic": "simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]", "annotated_tactic": ["simp_rw [<a>Submodule.coe_subtypeL'</a>, <a>Submodule.coeSubtype</a>]", [{"full_name": "Submodule.coe_subtypeL'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 39]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 19]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 (fun x => \u2191\u2191\u2191f x) =\u1d50[\u03bc] fun x => \u2191\u2191(\u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f) x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 (fun x => \u2191\u2191\u2191f x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 (fun x => \u2191\u2191\u2191f x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x", "state_after": "no goals"}, {"tactic": "refine' funext fun f => integral_congr_ae (ae_restrict_of_ae _)", "annotated_tactic": ["refine' <a>funext</a> fun f => <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</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.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]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, (fun x => \u2191\u2191\u2191f x) x = (fun x => \u2191\u2191(\u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f) x) x"}, {"tactic": "simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]", "annotated_tactic": ["simp_rw [<a>Submodule.coe_subtypeL'</a>, <a>Submodule.coeSubtype</a>]", [{"full_name": "Submodule.coe_subtypeL'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 39]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 19]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, (fun x => \u2191\u2191\u2191f x) x = (fun x => \u2191\u2191(\u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f) x) x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, True"}, {"tactic": "exact eventually_of_forall fun _ => (by trivial)", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun _ => (by trivial)", [{"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\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, True", "state_after": "no goals"}, {"tactic": "trivial", "annotated_tactic": ["trivial", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\u271d\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f\u271d 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nf : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc }\nx\u271d : \u03a9\n\u22a2 True", "state_after": "no goals"}, {"tactic": "rw [heq\u2081]", "annotated_tactic": ["rw [heq\u2081]", []], "state_before": "case pos.refine'_4.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc", "state_after": "case pos.refine'_4.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous ((fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)))"}, {"tactic": "exact continuous_integral.comp (ContinuousLinearMap.continuous _)", "annotated_tactic": ["exact continuous_integral.comp (<a>ContinuousLinearMap.continuous</a> _)", [{"full_name": "ContinuousLinearMap.continuous", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [448, 19], "def_end_pos": [448, 29]}]], "state_before": "case pos.refine'_4.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous ((fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)))", "state_after": "no goals"}, {"tactic": "rw [heq\u2082]", "annotated_tactic": ["rw [heq\u2082]", []], "state_before": "case pos.refine'_4.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc", "state_after": "case pos.refine'_4.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous ((fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)))"}, {"tactic": "exact (continuous_set_integral _).comp (ContinuousLinearMap.continuous _)", "annotated_tactic": ["exact (<a>continuous_set_integral</a> _).<a>comp</a> (<a>ContinuousLinearMap.continuous</a> _)", [{"full_name": "MeasureTheory.continuous_set_integral", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [968, 9], "def_end_pos": [968, 32]}, {"full_name": "Continuous.comp", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1673, 9], "def_end_pos": [1673, 24]}, {"full_name": "ContinuousLinearMap.continuous", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [448, 19], "def_end_pos": [448, 29]}]], "state_before": "case pos.refine'_4.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nheq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\nheq\u2082 :\n  (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))\n\u22a2 Continuous ((fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)))", "state_after": "no goals"}, {"tactic": "intro u v huv _ hueq", "annotated_tactic": ["intro u v huv _ hueq", []], "state_before": "case pos.refine'_5\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200 \u2983f g : \u03a9 \u2192 E\u2984,\n    f =\u1d50[\u03bc] g \u2192\n      Mem\u2112p f 1 \u2192\n        ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc \u2192\n          ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), g x \u2202\u03bc = \u222b (x : \u03a9) in s, g x \u2202\u03bc", "state_after": "case pos.refine'_5\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\nhuv : u =\u1d50[\u03bc] v\na\u271d : Mem\u2112p u 1\nhueq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc"}, {"tactic": "rwa [\u2190 integral_congr_ae huv, \u2190\n  (set_integral_congr_ae (hle\u2082 _ hms) _ : \u222b x in s, u x \u2202\u03bc = \u222b x in s, v x \u2202\u03bc)]", "annotated_tactic": ["rwa [\u2190 <a>integral_congr_ae</a> huv, \u2190\n      (<a>set_integral_congr_ae</a> (hle\u2082 _ hms) _ : \u222b x in s, u x \u2202\u03bc = \u222b x in s, v x \u2202\u03bc)]", [{"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.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'_5\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\nhuv : u =\u1d50[\u03bc] v\na\u271d : Mem\u2112p u 1\nhueq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\nhuv : u =\u1d50[\u03bc] v\na\u271d : Mem\u2112p u 1\nhueq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 u x = v x"}, {"tactic": "filter_upwards [huv] with x hx _ using hx", "annotated_tactic": ["filter_upwards [huv] with x hx _ using hx", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nm\u2081 m\u2082 m : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 E\nhle\u2081 : m\u2081 \u2264 m\nhle\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082)\nhf : StronglyMeasurable f\nhindp : Indep m\u2081 m\u2082\nhfint : Mem\u2112p f 1\ns : Set \u03a9\nhms : MeasurableSet s\nhs : \u2191\u2191\u03bc s < \u22a4\nu v : \u03a9 \u2192 E\nhuv : u =\u1d50[\u03bc] v\na\u271d : Mem\u2112p u 1\nhueq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 u x = v x", "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.fdiv_nonneg", "start": [96, 1], "end": [98, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable", "start": [1643, 1], "end": [1677, 19], "traced_tactics": [{"tactic": "have : \u222b\u207b x in univ, f x \u2202\u03bc = \u222b\u207b x, f x \u2202\u03bc := by simp only [Measure.restrict_univ]", "annotated_tactic": ["have : \u222b\u207b x in <a>univ</a>, f x \u2202\u03bc = \u222b\u207b x, f x \u2202\u03bc := by simp only [<a>Measure.restrict_univ</a>]", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc \u2264 C"}, {"tactic": "refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _", "annotated_tactic": ["refine' <a>univ_le_of_forall_fin_meas_le</a> hm C hf fun S hS_meas hS_mono => _", [{"full_name": "MeasureTheory.univ_le_of_forall_fin_meas_le", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1625, 9], "def_end_pos": [1625, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (x : \u03b1) in \u22c3 n, S n, f x \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc"}, {"tactic": "rw [\u2190 lintegral_indicator]", "annotated_tactic": ["rw [\u2190 <a>lintegral_indicator</a>]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (x : \u03b1) in \u22c3 n, S n, f x \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc\n\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 MeasurableSet (\u22c3 n, S n)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc\n\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 MeasurableSet (\u22c3 n, S n)", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 MeasurableSet (\u22c3 n, S n)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc"}, {"tactic": "have h_integral_indicator : \u2a06 n, \u222b\u207b x in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b x, (S n).indicator f x \u2202\u03bc := by\n  congr\n  ext1 n\n  rw [lintegral_indicator _ (hm _ (hS_meas n))]", "annotated_tactic": ["have h_integral_indicator : \u2a06 n, \u222b\u207b x in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b x, (S n).<a>indicator</a> f x \u2202\u03bc := by\n    congr\n    ext1 n\n    rw [<a>lintegral_indicator</a> _ (hm _ (hS_meas n))]", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc"}, {"tactic": "rw [h_integral_indicator, \u2190 lintegral_iSup]", "annotated_tactic": ["rw [h_integral_indicator, \u2190 <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\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 n, indicator (S n) f a \u2202\u03bc\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun x => indicator (S n) f x\n\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 Monotone fun n x => indicator (S n) f x"}, {"tactic": "simp only [Measure.restrict_univ]", "annotated_tactic": ["simp only [<a>Measure.restrict_univ</a>]", [{"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact hm (\u22c3 n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)", "annotated_tactic": ["exact hm (\u22c3 n, S n) (@<a>MeasurableSet.iUnion</a> _ _ m _ _ hS_meas)", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 MeasurableSet (\u22c3 n, S n)", "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\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 (fun n => \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc) = fun n => \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 (fun n => \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc) = fun n => \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nn : \u2115\n\u22a2 \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc"}, {"tactic": "rw [lintegral_indicator _ (hm _ (hS_meas n))]", "annotated_tactic": ["rw [<a>lintegral_indicator</a> _ (hm _ (hS_meas n))]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}]], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nn : \u2115\n\u22a2 \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' le_of_eq (lintegral_congr fun x => _)", "annotated_tactic": ["refine' <a>le_of_eq</a> (<a>lintegral_congr</a> fun x => _)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 n, indicator (S n) f 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\n\u22a2 indicator (\u22c3 n, S n) (fun x => f x) x = \u2a06 n, indicator (S n) f x"}, {"tactic": "simp_rw [indicator_apply]", "annotated_tactic": ["simp_rw [<a>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\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\n\u22a2 indicator (\u22c3 n, S n) (fun x => f x) x = \u2a06 n, indicator (S n) f x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\n\u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "by_cases hx_mem : x \u2208 iUnion S", "annotated_tactic": ["by_cases hx_mem : x \u2208 <a>iUnion</a> S", [{"full_name": "Set.iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [103, 5], "def_end_pos": [103, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\n\u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\n\u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00acx \u2208 iUnion S\n\u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "simp only [hx_mem, if_true]", "annotated_tactic": ["simp only [hx_mem, <a>if_true</a>]", [{"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\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\n\u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\n\u22a2 f x = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "obtain \u27e8n, hxn\u27e9 := mem_iUnion.mp hx_mem", "annotated_tactic": ["obtain \u27e8n, hxn\u27e9 := mem_iUnion.mp hx_mem", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\n\u22a2 f x = \u2a06 n, if x \u2208 S n then f x else 0", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\n\u22a2 f x = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "refine' le_antisymm (_root_.trans _ (le_iSup _ n)) (iSup_le fun i => _)", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>_root_.trans</a> _ (<a>le_iSup</a> _ n)) (<a>iSup_le</a> fun i => _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "trans", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [308, 9], "def_end_pos": [308, 14]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\n\u22a2 f x = \u2a06 n, if x \u2208 S n then f x else 0", "state_after": "case pos.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\n\u22a2 f x \u2264 if x \u2208 S n then f x else 0\n\ncase pos.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\ni : \u2115\n\u22a2 (if x \u2208 S i then f x else 0) \u2264 f x"}, {"tactic": "simp only [hxn, le_refl, if_true]", "annotated_tactic": ["simp only [hxn, <a>le_refl</a>, <a>if_true</a>]", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"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.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\n\u22a2 f x \u2264 if x \u2208 S n then f x else 0", "state_after": "no goals"}, {"tactic": "by_cases hxi : x \u2208 S i <;> simp [hxi]", "annotated_tactic": ["by_cases hxi : x \u2208 S i <;> simp [hxi]", []], "state_before": "case pos.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\ni : \u2115\n\u22a2 (if x \u2208 S i then f x else 0) \u2264 f x", "state_after": "no goals"}, {"tactic": "simp only [hx_mem, if_false]", "annotated_tactic": ["simp only [hx_mem, <a>if_false</a>]", [{"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\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00acx \u2208 iUnion S\n\u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00acx \u2208 iUnion S\n\u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "rw [mem_iUnion] at hx_mem", "annotated_tactic": ["rw [<a>mem_iUnion</a>] at hx_mem", [{"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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00acx \u2208 iUnion S\n\u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00ac\u2203 i, x \u2208 S i\n\u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "push_neg at hx_mem", "annotated_tactic": ["push_neg at hx_mem", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00ac\u2203 i, x \u2208 S i\n\u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i\n\u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0"}, {"tactic": "refine' le_antisymm (zero_le _) (iSup_le fun n => _)", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>zero_le</a> _) (<a>iSup_le</a> fun n => _)", [{"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": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 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\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i\n\u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i\nn : \u2115\n\u22a2 (if x \u2208 S n then f x else 0) \u2264 0"}, {"tactic": "simp only [hx_mem n, if_false, nonpos_iff_eq_zero]", "annotated_tactic": ["simp only [hx_mem n, <a>if_false</a>, <a>nonpos_iff_eq_zero</a>]", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"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 neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i\nn : \u2115\n\u22a2 (if x \u2208 S n then f x else 0) \u2264 0", "state_after": "no goals"}, {"tactic": "exact fun n => hf_meas.indicator (hm _ (hS_meas n))", "annotated_tactic": ["exact fun n => hf_meas.indicator (hm _ (hS_meas n))", []], "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun x => indicator (S n) f x", "state_after": "no goals"}, {"tactic": "intro n\u2081 n\u2082 hn\u2081\u2082 a", "annotated_tactic": ["intro n\u2081 n\u2082 hn\u2081\u2082 a", []], "state_before": "case h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 Monotone fun n x => indicator (S n) f x", "state_after": "case h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\n\u22a2 (fun n x => indicator (S n) f x) n\u2081 a \u2264 (fun n x => indicator (S n) f x) n\u2082 a"}, {"tactic": "simp_rw [indicator_apply]", "annotated_tactic": ["simp_rw [<a>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": "case h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\n\u22a2 (fun n x => indicator (S n) f x) n\u2081 a \u2264 (fun n x => indicator (S n) f x) n\u2082 a", "state_after": "case h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\n\u22a2 (if a \u2208 S n\u2081 then f a else 0) \u2264 if a \u2208 S n\u2082 then f a else 0"}, {"tactic": "split_ifs with h h_1", "annotated_tactic": ["split_ifs with h h_1", []], "state_before": "case h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\n\u22a2 (if a \u2208 S n\u2081 then f a else 0) \u2264 if a \u2208 S n\u2082 then f a else 0", "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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : a \u2208 S n\u2081\nh_1 : a \u2208 S n\u2082\n\u22a2 f a \u2264 f 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : a \u2208 S n\u2081\nh_1 : \u00aca \u2208 S n\u2082\n\u22a2 f a \u2264 0\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : \u00aca \u2208 S n\u2081\nh\u271d : a \u2208 S n\u2082\n\u22a2 0 \u2264 f 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : \u00aca \u2208 S n\u2081\nh\u271d : \u00aca \u2208 S n\u2082\n\u22a2 0 \u2264 0"}, {"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 pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : a \u2208 S n\u2081\nh_1 : a \u2208 S n\u2082\n\u22a2 f a \u2264 f a", "state_after": "no goals"}, {"tactic": "exact absurd (mem_of_mem_of_subset h (hS_mono hn\u2081\u2082)) h_1", "annotated_tactic": ["exact <a>absurd</a> (<a>mem_of_mem_of_subset</a> h (hS_mono hn\u2081\u2082)) h_1", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"full_name": "Set.mem_of_mem_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 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\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : a \u2208 S n\u2081\nh_1 : \u00aca \u2208 S n\u2082\n\u22a2 f a \u2264 0", "state_after": "no goals"}, {"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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : \u00aca \u2208 S n\u2081\nh\u271d : a \u2208 S n\u2082\n\u22a2 0 \u2264 f a", "state_after": "no goals"}, {"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\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : \u00aca \u2208 S n\u2081\nh\u271d : \u00aca \u2208 S n\u2082\n\u22a2 0 \u2264 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_condCdf_atBot", "start": [803, 1], "end": [818, 57], "traced_tactics": [{"tactic": "have h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x < q \u2227 \u2191q < x + 1 := fun x => exists_rat_btwn (lt_add_one x)", "annotated_tactic": ["have h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x < q \u2227 \u2191q < x + 1 := fun x => <a>exists_rat_btwn</a> (<a>lt_add_one</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": "lt_add_one", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [20, 7], "def_end_pos": [20, 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)\na : \u03b1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)"}, {"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 < \u2191q \u2227 \u2191q < x + 1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)"}, {"tactic": "have hqs_tendsto : Tendsto qs atBot atBot := by\n  rw [tendsto_atBot_atBot]\n  refine' fun q => \u27e8q - 1, fun y hy => _\u27e9\n  have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 :=\n    (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)\n  rw [sub_add_cancel] at h_le\n  exact_mod_cast h_le", "annotated_tactic": ["have hqs_tendsto : <a>Tendsto</a> qs <a>atBot</a> <a>atBot</a> := by\n    rw [<a>tendsto_atBot_atBot</a>]\n    refine' fun q => \u27e8q - 1, fun y hy => _\u27e9\n    have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 :=\n      (h_exists y).<a>choose_spec</a>.2.le.trans (<a>add_le_add</a> hy <a>le_rfl</a>)\n    rw [<a>sub_add_cancel</a>] at h_le\n    exact_mod_cast 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.atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [49, 5], "def_end_pos": [49, 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_atBot_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1347, 9], "def_end_pos": [1347, 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": "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": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 Tendsto (\u2191(condCdf \u03c1 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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)"}, {"tactic": "refine'\n  tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds\n    ((tendsto_condCdfRat_atBot \u03c1 a).comp hqs_tendsto) (condCdf_nonneg \u03c1 a) fun x => _", "annotated_tactic": ["refine'\n    <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> <a>tendsto_const_nhds</a>\n      ((<a>tendsto_condCdfRat_atBot</a> \u03c1 a).<a>comp</a> hqs_tendsto) (<a>condCdf_nonneg</a> \u03c1 a) fun x => _", [{"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": "ProbabilityTheory.tendsto_condCdfRat_atBot", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [627, 9], "def_end_pos": [627, 33]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\n\u22a2 Tendsto (\u2191(condCdf \u03c1 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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 (condCdfRat \u03c1 a \u2218 qs) 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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 (condCdfRat \u03c1 a \u2218 qs) 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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 \u2191(condCdf \u03c1 a) \u2191(qs x)"}, {"tactic": "exact (condCdf \u03c1 a).mono (h_exists x).choose_spec.1.le", "annotated_tactic": ["exact (<a>condCdf</a> \u03c1 a).<a>mono</a> (h_exists x).<a>choose_spec</a>.1.<a>le</a>", [{"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": "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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 \u2191(condCdf \u03c1 a) \u2191(qs x)", "state_after": "no goals"}, {"tactic": "rw [tendsto_atBot_atBot]", "annotated_tactic": ["rw [<a>tendsto_atBot_atBot</a>]", [{"full_name": "Filter.tendsto_atBot_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1347, 9], "def_end_pos": [1347, 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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 Tendsto qs atBot atBot", "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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), a \u2264 i \u2192 qs a \u2264 b"}, {"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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), a \u2264 i \u2192 qs a \u2264 b", "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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\n\u22a2 qs y \u2264 q"}, {"tactic": "have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 :=\n  (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)", "annotated_tactic": ["have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 :=\n      (h_exists y).<a>choose_spec</a>.2.le.trans (<a>add_le_add</a> hy <a>le_rfl</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": "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\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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\n\u22a2 qs y \u2264 q", "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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\nh_le : \u2191(qs y) \u2264 \u2191q - 1 + 1\n\u22a2 qs y \u2264 q"}, {"tactic": "rw [sub_add_cancel] at h_le", "annotated_tactic": ["rw [<a>sub_add_cancel</a>] at h_le", [{"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 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)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\nh_le : \u2191(qs y) \u2264 \u2191q - 1 + 1\n\u22a2 qs y \u2264 q", "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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\nh_le : \u2191(qs y) \u2264 \u2191q\n\u22a2 qs y \u2264 q"}, {"tactic": "exact_mod_cast h_le", "annotated_tactic": ["exact_mod_cast h_le", []], "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 < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\nh_le : \u2191(qs y) \u2264 \u2191q\n\u22a2 qs y \u2264 q", "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_of_injective", "start": [459, 1], "end": [460, 45], "traced_tactics": [{"tactic": "simp [vars, degrees_map_of_injective _ hf]", "annotated_tactic": ["simp [<a>vars</a>, <a>degrees_map_of_injective</a> _ hf]", [{"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.degrees_map_of_injective", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [252, 9], "def_end_pos": [252, 33]}]], "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\nhf : Injective \u2191f\n\u22a2 vars (\u2191(map f) p) = vars p", "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.prodAssoc_prod", "start": [625, 1], "end": [633, 15], "traced_tactics": [{"tactic": "refine' (prod_eq_generateFrom generateFrom_measurableSet generateFrom_prod\n  isPiSystem_measurableSet isPiSystem_prod \u03bc.toFiniteSpanningSetsIn\n  (\u03bd.toFiniteSpanningSetsIn.prod \u03c4.toFiniteSpanningSetsIn) _).symm", "annotated_tactic": ["refine' (<a>prod_eq_generateFrom</a> <a>generateFrom_measurableSet</a> <a>generateFrom_prod</a>\n    <a>isPiSystem_measurableSet</a> <a>isPiSystem_prod</a> \u03bc.toFiniteSpanningSetsIn\n    (\u03bd.toFiniteSpanningSetsIn.prod \u03c4.toFiniteSpanningSetsIn) _).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.prod_eq_generateFrom", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [551, 9], "def_end_pos": [551, 29]}, {"full_name": "MeasurableSpace.generateFrom_measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [396, 9], "def_end_pos": [396, 35]}, {"full_name": "generateFrom_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [142, 9], "def_end_pos": [142, 26]}, {"full_name": "MeasurableSpace.isPiSystem_measurableSet", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [72, 9], "def_end_pos": [72, 33]}, {"full_name": "isPiSystem_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [150, 9], "def_end_pos": [150, 24]}, {"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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\n\u22a2 map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4) = Measure.prod \u03bc (Measure.prod \u03bd \u03c4)", "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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\n\u22a2 \u2200 (s : Set \u03b1),\n    s \u2208 {s | MeasurableSet s} \u2192\n      \u2200 (t : Set (\u03b2 \u00d7 \u03b3)),\n        t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n          \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 t) =\n            \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) t"}, {"tactic": "rintro s hs _ \u27e8t, u, ht, hu, rfl\u27e9", "annotated_tactic": ["rintro s hs _ \u27e8t, u, ht, hu, rfl\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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\n\u22a2 \u2200 (s : Set \u03b1),\n    s \u2208 {s | MeasurableSet s} \u2192\n      \u2200 (t : Set (\u03b2 \u00d7 \u03b3)),\n        t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n          \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 t) =\n            \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) t", "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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\ns : Set \u03b1\nhs : s \u2208 {s | MeasurableSet s}\nt : Set \u03b2\nu : Set \u03b3\nht : t \u2208 {s | MeasurableSet s}\nhu : u \u2208 {t | MeasurableSet t}\n\u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =\n    \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u)"}, {"tactic": "rw [mem_setOf_eq] at hs ht hu", "annotated_tactic": ["rw [<a>mem_setOf_eq</a>] at hs ht hu", [{"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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\ns : Set \u03b1\nhs : s \u2208 {s | MeasurableSet s}\nt : Set \u03b2\nu : Set \u03b3\nht : t \u2208 {s | MeasurableSet s}\nhu : u \u2208 {t | MeasurableSet t}\n\u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =\n    \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u)", "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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b2\nu : Set \u03b3\nht : MeasurableSet t\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =\n    \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u)"}, {"tactic": "simp_rw [map_apply (MeasurableEquiv.measurable _) (hs.prod (ht.prod hu)),\n  MeasurableEquiv.prodAssoc, MeasurableEquiv.coe_mk, Equiv.prod_assoc_preimage, prod_prod,\n  mul_assoc]", "annotated_tactic": ["simp_rw [<a>map_apply</a> (<a>MeasurableEquiv.measurable</a> _) (hs.prod (ht.prod hu)),\n    <a>MeasurableEquiv.prodAssoc</a>, <a>MeasurableEquiv.coe_mk</a>, <a>Equiv.prod_assoc_preimage</a>, <a>prod_prod</a>,\n    <a>mul_assoc</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": "MeasurableEquiv.measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1298, 19], "def_end_pos": [1298, 29]}, {"full_name": "MeasurableEquiv.prodAssoc", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1507, 5], "def_end_pos": [1507, 14]}, {"full_name": "MeasurableEquiv.coe_mk", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1303, 9], "def_end_pos": [1303, 15]}, {"full_name": "Equiv.prod_assoc_preimage", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [140, 9], "def_end_pos": [140, 28]}, {"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": "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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b2\nu : Set \u03b3\nht : MeasurableSet t\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =\n    \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u)", "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.deleteMin_fst", "start": [269, 1], "end": [272, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.support_rename_of_injective", "start": [341, 1], "end": [345, 73], "traced_tactics": [{"tactic": "rw [rename_eq]", "annotated_tactic": ["rw [<a>rename_eq</a>]", [{"full_name": "MvPolynomial.rename_eq", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [107, 9], "def_end_pos": [107, 18]}]], "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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\ninst\u271d : DecidableEq \u03c4\nh : Injective f\n\u22a2 support (\u2191(rename f) p) = Finset.image (Finsupp.mapDomain f) (support p)", "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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\ninst\u271d : DecidableEq \u03c4\nh : Injective f\n\u22a2 support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) (support p)"}, {"tactic": "exact Finsupp.mapDomain_support_of_injective (mapDomain_injective h) _", "annotated_tactic": ["exact <a>Finsupp.mapDomain_support_of_injective</a> (<a>mapDomain_injective</a> h) _", [{"full_name": "Finsupp.mapDomain_support_of_injective", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [575, 9], "def_end_pos": [575, 39]}, {"full_name": "Finsupp.mapDomain_injective", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [614, 9], "def_end_pos": [614, 28]}]], "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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\np : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\ninst\u271d : DecidableEq \u03c4\nh : Injective f\n\u22a2 support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) (support p)", "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_right", "start": [270, 1], "end": [273, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.matches'_pow", "start": [145, 1], "end": [148, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.ofNat_dvd", "start": [635, 14], "end": [643, 49], "traced_tactics": [{"tactic": "refine \u27e8fun \u27e8a, ae\u27e9 => ?_, fun \u27e8k, e\u27e9 => \u27e8k, by rw [e, Int.ofNat_mul]\u27e9\u27e9", "annotated_tactic": ["refine \u27e8fun \u27e8a, ae\u27e9 => ?_, fun \u27e8k, e\u27e9 => \u27e8k, by rw [e, <a>Int.ofNat_mul</a>]\u27e9\u27e9", [{"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": "m n : Nat\n\u22a2 \u2191m \u2223 \u2191n \u2194 m \u2223 n", "state_after": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\n\u22a2 m \u2223 n"}, {"tactic": "match Int.le_total a 0 with\n| .inl h =>\n  have := ae.symm \u25b8 Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h\n  rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)]\n  apply Nat.dvd_zero\n| .inr h => match a, eq_ofNat_of_zero_le h with\n  | _, \u27e8k, rfl\u27e9 => exact \u27e8k, Int.ofNat.inj ae\u27e9", "annotated_tactic": ["match <a>Int.le_total</a> a 0 with\n  | .inl h =>\n    have := ae.symm \u25b8 <a>Int.mul_nonpos_of_nonneg_of_nonpos</a> (<a>ofNat_zero_le</a> _) h\n    rw [<a>Nat.le_antisymm</a> (<a>ofNat_le</a>.1 this) (<a>Nat.zero_le</a> _)]\n    apply <a>Nat.dvd_zero</a>\n  | .inr h => match a, <a>eq_ofNat_of_zero_le</a> h with\n    | _, \u27e8k, <a>rfl</a>\u27e9 => exact \u27e8k, Int.ofNat.inj ae\u27e9", [{"full_name": "Int.le_total", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [570, 19], "def_end_pos": [570, 27]}, {"full_name": "Int.mul_nonpos_of_nonneg_of_nonpos", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1181, 19], "def_end_pos": [1181, 49]}, {"full_name": "Int.ofNat_zero_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [582, 9], "def_end_pos": [582, 22]}, {"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": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}, {"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.dvd_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [864, 19], "def_end_pos": [864, 27]}, {"full_name": "Int.eq_ofNat_of_zero_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [584, 9], "def_end_pos": [584, 28]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}]], "state_before": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\n\u22a2 m \u2223 n", "state_after": "no goals"}, {"tactic": "rw [e, Int.ofNat_mul]", "annotated_tactic": ["rw [e, <a>Int.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": "m n : Nat\nx\u271d : m \u2223 n\nk : Nat\ne : n = m * k\n\u22a2 \u2191n = \u2191m * \u2191k", "state_after": "no goals"}, {"tactic": "have := ae.symm \u25b8 Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h", "annotated_tactic": ["have := ae.symm \u25b8 <a>Int.mul_nonpos_of_nonneg_of_nonpos</a> (<a>ofNat_zero_le</a> _) h", [{"full_name": "Int.mul_nonpos_of_nonneg_of_nonpos", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1181, 19], "def_end_pos": [1181, 49]}, {"full_name": "Int.ofNat_zero_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [582, 9], "def_end_pos": [582, 22]}]], "state_before": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\nh : a \u2264 0\n\u22a2 m \u2223 n", "state_after": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\nh : a \u2264 0\nthis : \u2191n \u2264 0\n\u22a2 m \u2223 n"}, {"tactic": "rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)]", "annotated_tactic": ["rw [<a>Nat.le_antisymm</a> (<a>ofNat_le</a>.1 this) (<a>Nat.zero_le</a> _)]", [{"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": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}, {"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": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\nh : a \u2264 0\nthis : \u2191n \u2264 0\n\u22a2 m \u2223 n", "state_after": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\nh : a \u2264 0\nthis : \u2191n \u2264 0\n\u22a2 m \u2223 0"}, {"tactic": "apply Nat.dvd_zero", "annotated_tactic": ["apply <a>Nat.dvd_zero</a>", [{"full_name": "Nat.dvd_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [864, 19], "def_end_pos": [864, 27]}]], "state_before": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\nh : a \u2264 0\nthis : \u2191n \u2264 0\n\u22a2 m \u2223 0", "state_after": "no goals"}, {"tactic": "match a, eq_ofNat_of_zero_le h with\n| _, \u27e8k, rfl\u27e9 => exact \u27e8k, Int.ofNat.inj ae\u27e9", "annotated_tactic": ["match a, <a>eq_ofNat_of_zero_le</a> h with\n    | _, \u27e8k, <a>rfl</a>\u27e9 => exact \u27e8k, Int.ofNat.inj ae\u27e9", [{"full_name": "Int.eq_ofNat_of_zero_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [584, 9], "def_end_pos": [584, 28]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}]], "state_before": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nae : \u2191n = \u2191m * a\nh : 0 \u2264 a\n\u22a2 m \u2223 n", "state_after": "no goals"}, {"tactic": "exact \u27e8k, Int.ofNat.inj ae\u27e9", "annotated_tactic": ["exact \u27e8k, Int.ofNat.inj ae\u27e9", []], "state_before": "m n : Nat\nx\u271d : \u2191m \u2223 \u2191n\na : Int\nk : Nat\nae : \u2191n = \u2191m * \u2191k\nh : 0 \u2264 \u2191k\n\u22a2 m \u2223 n", "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_one", "start": [351, 1], "end": [352, 88], "traced_tactics": [{"tactic": "rw [bind\u2082_monomial, f.map_one, one_mul]", "annotated_tactic": ["rw [<a>bind\u2082_monomial</a>, f.map_one, <a>one_mul</a>]", [{"full_name": "MvPolynomial.bind\u2082_monomial", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [344, 9], "def_end_pos": [344, 23]}, {"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\n\u22a2 \u2191(bind\u2082 f) (\u2191(monomial d) 1) = \u2191(monomial d) 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.monotone_map", "start": [140, 1], "end": [140, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.int_cast_cast", "start": [265, 1], "end": [266, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_pi_Ioc_toReal", "start": [240, 1], "end": [242, 93], "traced_tactics": [{"tactic": "simp only [volume_pi_Ioc, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]", "annotated_tactic": ["simp only [<a>volume_pi_Ioc</a>, <a>ENNReal.toReal_prod</a>, <a>ENNReal.toReal_ofReal</a> (<a>sub_nonneg</a>.2 (h _))]", [{"full_name": "Real.volume_pi_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 22]}, {"full_name": "ENNReal.toReal_prod", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2308, 9], "def_end_pos": [2308, 20]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\nh : a \u2264 b\n\u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ioc (a i) (b i))) = \u220f i : \u03b9, (b i - a i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.multiset_prod_subset_multiset_prod", "start": [137, 1], "end": [141, 44], "traced_tactics": [{"tactic": "induction t using Quotient.inductionOn", "annotated_tactic": ["induction t 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": "\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 : Multiset \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 Multiset.prod (Multiset.map f\u2081 t) \u2286 Multiset.prod (Multiset.map f\u2082 t)", "state_after": "case h\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\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 Multiset.prod (Multiset.map f\u2081 (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2286\n    Multiset.prod (Multiset.map f\u2082 (Quotient.mk (List.isSetoid \u03b9) a\u271d))"}, {"tactic": "simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod]", "annotated_tactic": ["simp_rw [<a>Multiset.quot_mk_to_coe</a>, <a>Multiset.coe_map</a>, <a>Multiset.coe_prod</a>]", [{"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.coe_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1166, 9], "def_end_pos": [1166, 16]}, {"full_name": "Multiset.coe_prod", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}]], "state_before": "case h\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\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 Multiset.prod (Multiset.map f\u2081 (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2286\n    Multiset.prod (Multiset.map f\u2082 (Quotient.mk (List.isSetoid \u03b9) a\u271d))", "state_after": "case h\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\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 a\u271d) \u2286 List.prod (List.map f\u2082 a\u271d)"}, {"tactic": "exact list_prod_subset_list_prod _ _ _ hf", "annotated_tactic": ["exact <a>list_prod_subset_list_prod</a> _ _ _ hf", [{"full_name": "Set.list_prod_subset_list_prod", "def_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "def_pos": [108, 9], "def_end_pos": [108, 35]}]], "state_before": "case h\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\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 a\u271d) \u2286 List.prod (List.map f\u2082 a\u271d)", "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", "start": [507, 1], "end": [509, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Subtype.lean", "full_name": "Subtype.equivalence", "start": [247, 1], "end": [248, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_mono", "start": [1397, 1], "end": [1398, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.intervalIntegral_pos_of_pos_on", "start": [1300, 1], "end": [1309, 83], "traced_tactics": [{"tactic": "have hsupp : Ioo a b \u2286 support f \u2229 Ioc a b := fun x hx =>\n  \u27e8mem_support.mpr (hpos x hx).ne', Ioo_subset_Ioc_self hx\u27e9", "annotated_tactic": ["have hsupp : <a>Ioo</a> a b \u2286 <a>support</a> f \u2229 <a>Ioc</a> a b := fun x hx =>\n    \u27e8mem_support.mpr (hpos x hx).<a>ne'</a>, <a>Ioo_subset_Ioc_self</a> hx\u27e9", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [37, 5], "def_end_pos": [37, 12]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Set.Ioo_subset_Ioc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [507, 9], "def_end_pos": [507, 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\u271d g : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\n\u22a2 0 < \u222b (x : \u211d) in a..b, f x", "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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 0 < \u222b (x : \u211d) in a..b, f x"}, {"tactic": "have h\u2080 : 0 \u2264\u1d50[volume.restrict (uIoc a b)] f := by\n  rw [EventuallyLE, uIoc_of_le hab.le]\n  refine' ae_restrict_of_ae_eq_of_ae_restrict Ioo_ae_eq_Ioc _\n  exact (ae_restrict_iff' measurableSet_Ioo).mpr (ae_of_all _ fun x hx => (hpos x hx).le)", "annotated_tactic": ["have h\u2080 : 0 \u2264\u1d50[volume.restrict (<a>uIoc</a> a b)] f := by\n    rw [<a>EventuallyLE</a>, <a>uIoc_of_le</a> hab.le]\n    refine' <a>ae_restrict_of_ae_eq_of_ae_restrict</a> <a>Ioo_ae_eq_Ioc</a> _\n    exact (<a>ae_restrict_iff'</a> <a>measurableSet_Ioo</a>).<a>mpr</a> (<a>ae_of_all</a> _ fun x hx => (hpos x hx).<a>le</a>)", [{"full_name": "Set.uIoc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [279, 5], "def_end_pos": [279, 9]}, {"full_name": "Filter.EventuallyLE", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1638, 5], "def_end_pos": [1638, 17]}, {"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_of_ae_eq_of_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2688, 9], "def_end_pos": [2688, 44]}, {"full_name": "MeasureTheory.Ioo_ae_eq_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3168, 9], "def_end_pos": [3168, 22]}, {"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_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"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.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 0 < \u222b (x : \u211d) in a..b, f x", "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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\nh\u2080 : 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f\n\u22a2 0 < \u222b (x : \u211d) in a..b, f x"}, {"tactic": "rw [integral_pos_iff_support_of_nonneg_ae' h\u2080 hfi]", "annotated_tactic": ["rw [<a>integral_pos_iff_support_of_nonneg_ae'</a> h\u2080 hfi]", [{"full_name": "intervalIntegral.integral_pos_iff_support_of_nonneg_ae'", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1277, 9], "def_end_pos": [1277, 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\nf\u271d g : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\nh\u2080 : 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f\n\u22a2 0 < \u222b (x : \u211d) in a..b, f x", "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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\nh\u2080 : 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f\n\u22a2 a < b \u2227 0 < \u2191\u2191volume (support f \u2229 Ioc a b)"}, {"tactic": "exact \u27e8hab, ((Measure.measure_Ioo_pos _).mpr hab).trans_le (measure_mono hsupp)\u27e9", "annotated_tactic": ["exact \u27e8hab, ((<a>Measure.measure_Ioo_pos</a> _).<a>mpr</a> hab).<a>trans_le</a> (<a>measure_mono</a> hsupp)\u27e9", [{"full_name": "MeasureTheory.Measure.measure_Ioo_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [178, 9], "def_end_pos": [178, 24]}, {"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": "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]}]], "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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\nh\u2080 : 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f\n\u22a2 a < b \u2227 0 < \u2191\u2191volume (support f \u2229 Ioc a b)", "state_after": "no goals"}, {"tactic": "rw [EventuallyLE, uIoc_of_le hab.le]", "annotated_tactic": ["rw [<a>EventuallyLE</a>, <a>uIoc_of_le</a> hab.le]", [{"full_name": "Filter.EventuallyLE", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1638, 5], "def_end_pos": [1638, 17]}, {"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\nf\u271d g : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f", "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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc a b), OfNat.ofNat 0 x \u2264 f x"}, {"tactic": "refine' ae_restrict_of_ae_eq_of_ae_restrict Ioo_ae_eq_Ioc _", "annotated_tactic": ["refine' <a>ae_restrict_of_ae_eq_of_ae_restrict</a> <a>Ioo_ae_eq_Ioc</a> _", [{"full_name": "MeasureTheory.ae_restrict_of_ae_eq_of_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2688, 9], "def_end_pos": [2688, 44]}, {"full_name": "MeasureTheory.Ioo_ae_eq_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3168, 9], "def_end_pos": [3168, 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\u271d g : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc a b), OfNat.ofNat 0 x \u2264 f x", "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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioo a b), OfNat.ofNat 0 x \u2264 f x"}, {"tactic": "exact (ae_restrict_iff' measurableSet_Ioo).mpr (ae_of_all _ fun x hx => (hpos x hx).le)", "annotated_tactic": ["exact (<a>ae_restrict_iff'</a> <a>measurableSet_Ioo</a>).<a>mpr</a> (<a>ae_of_all</a> _ fun x hx => (hpos x hx).<a>le</a>)", [{"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_Ioo", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [584, 9], "def_end_pos": [584, 26]}, {"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.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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 : \u211d \u2192 \u211d\na\u271d b\u271d : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\na b : \u211d\nhfi : IntervalIntegrable f volume a b\nhpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x\nhab : a < b\nhsupp : Ioo a b \u2286 support f \u2229 Ioc a b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioo a b), OfNat.ofNat 0 x \u2264 f 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.erase_append_right", "start": [1193, 1], "end": [1196, 63], "traced_tactics": [{"tactic": "rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right]", "annotated_tactic": ["rw [<a>erase_eq_eraseP</a>, <a>erase_eq_eraseP</a>, <a>eraseP_append_right</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]}, {"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]}, {"full_name": "List.eraseP_append_right", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1100, 9], "def_end_pos": [1100, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u00aca \u2208 l\u2081\n\u22a2 List.erase (l\u2081 ++ l\u2082) a = l\u2081 ++ List.erase l\u2082 a", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u00aca \u2208 l\u2081\n\u22a2 \u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 \u00acdecide (a = b) = true"}, {"tactic": "intros b h' h''", "annotated_tactic": ["intros b h' h''", []], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u00aca \u2208 l\u2081\n\u22a2 \u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 \u00acdecide (a = b) = true", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u00aca \u2208 l\u2081\nb : \u03b1\nh' : b \u2208 l\u2081\nh'' : decide (a = b) = true\n\u22a2 False"}, {"tactic": "rw [of_decide_eq_true h''] at h", "annotated_tactic": ["rw [<a>of_decide_eq_true</a> h''] at h", [{"full_name": "of_decide_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [855, 9], "def_end_pos": [855, 26]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u00aca \u2208 l\u2081\nb : \u03b1\nh' : b \u2208 l\u2081\nh'' : decide (a = b) = true\n\u22a2 False", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nb : \u03b1\nh : \u00acb \u2208 l\u2081\nh' : b \u2208 l\u2081\nh'' : decide (a = b) = true\n\u22a2 False"}, {"tactic": "exact h h'", "annotated_tactic": ["exact h h'", []], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nb : \u03b1\nh : \u00acb \u2208 l\u2081\nh' : b \u2208 l\u2081\nh'' : decide (a = b) = true\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "List.aemeasurable_prod'", "start": [868, 1], "end": [873, 28], "traced_tactics": [{"tactic": "induction' l with f l ihl", "annotated_tactic": ["induction' l with f l ihl", []], "state_before": "M : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod l)", "state_after": "case nil\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod [])\n\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEMeasurable f_1\n\u22a2 AEMeasurable (prod (f :: l))"}, {"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\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEMeasurable f_1\n\u22a2 AEMeasurable (prod (f :: l))", "state_after": "case cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x\n\u22a2 AEMeasurable (prod (f :: l))"}, {"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\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x\n\u22a2 AEMeasurable (prod (f :: l))", "state_after": "case cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x\n\u22a2 AEMeasurable (f * prod l)"}, {"tactic": "exact hl.1.mul (ihl hl.2)", "annotated_tactic": ["exact hl.1.<a>mul</a> (ihl hl.2)", [{"full_name": "AEMeasurable.mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [155, 9], "def_end_pos": [155, 25]}]], "state_before": "case cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x\n\u22a2 AEMeasurable (f * prod l)", "state_after": "no goals"}, {"tactic": "exact aemeasurable_one", "annotated_tactic": ["exact <a>aemeasurable_one</a>", [{"full_name": "aemeasurable_one", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [284, 9], "def_end_pos": [284, 25]}]], "state_before": "case nil\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod [])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.eq_empty_of_ssubset_singleton", "start": [798, 1], "end": [799, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "full_name": "MeasureTheory.Measure.restrict_sub_eq_restrict_sub_restrict", "start": [104, 1], "end": [133, 42], "traced_tactics": [{"tactic": "repeat' rw [sub_def]", "annotated_tactic": ["repeat' rw [<a>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\nh_meas_s : MeasurableSet s\n\u22a2 restrict (\u03bc - \u03bd) s = restrict \u03bc s - restrict \u03bd s", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\n\u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}"}, {"tactic": "have h_nonempty : { d | \u03bc \u2264 d + \u03bd }.Nonempty := \u27e8\u03bc, Measure.le_add_right le_rfl\u27e9", "annotated_tactic": ["have h_nonempty : { d | \u03bc \u2264 d + \u03bd }.<a>Nonempty</a> := \u27e8\u03bc, <a>Measure.le_add_right</a> <a>le_rfl</a>\u27e9", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "MeasureTheory.Measure.le_add_right", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1001, 19], "def_end_pos": [1001, 31]}, {"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\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\n\u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}"}, {"tactic": "rw [restrict_sInf_eq_sInf_restrict h_nonempty h_meas_s]", "annotated_tactic": ["rw [<a>restrict_sInf_eq_sInf_restrict</a> h_nonempty h_meas_s]", [{"full_name": "MeasureTheory.Measure.restrict_sInf_eq_sInf_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1881, 9], "def_end_pos": [1881, 39]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}"}, {"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\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) \u2264 sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}\n\ncase a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2264 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd})"}, {"tactic": "rw [sub_def]", "annotated_tactic": ["rw [<a>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\nh_meas_s : MeasurableSet s\n\u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = restrict \u03bc s - restrict \u03bd s", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\n\u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}"}, {"tactic": "refine' sInf_le_sInf_of_forall_exists_le _", "annotated_tactic": ["refine' <a>sInf_le_sInf_of_forall_exists_le</a> _", [{"full_name": "sInf_le_sInf_of_forall_exists_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [303, 9], "def_end_pos": [303, 41]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) \u2264 sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s}", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 \u2200 (x : Measure \u03b1),\n    x \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2192 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 x"}, {"tactic": "intro \u03bd' h_\u03bd'_in", "annotated_tactic": ["intro \u03bd' h_\u03bd'_in", []], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 \u2200 (x : Measure \u03b1),\n    x \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2192 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 x", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : \u03bd' \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s}\n\u22a2 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 \u03bd'"}, {"tactic": "rw [mem_setOf_eq] at h_\u03bd'_in", "annotated_tactic": ["rw [<a>mem_setOf_eq</a>] at h_\u03bd'_in", [{"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\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : \u03bd' \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s}\n\u22a2 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 \u03bd'", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 \u03bd'"}, {"tactic": "refine' \u27e8\u03bd'.restrict s, _, restrict_le_self\u27e9", "annotated_tactic": ["refine' \u27e8\u03bd'.restrict s, _, <a>restrict_le_self</a>\u27e9", [{"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": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 \u03bd'", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 restrict \u03bd' s \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}"}, {"tactic": "refine' \u27e8\u03bd' + (\u22a4 : Measure \u03b1).restrict s\u1d9c, _, _\u27e9", "annotated_tactic": ["refine' \u27e8\u03bd' + (\u22a4 : <a>Measure</a> \u03b1).<a>restrict</a> s\u1d9c, _, _\u27e9", [{"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]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 restrict \u03bd' s \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}", "state_after": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 \u03bd' + restrict \u22a4 s\u1d9c \u2208 {d | \u03bc \u2264 d + \u03bd}\n\ncase a.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 (fun \u03bc => restrict \u03bc s) (\u03bd' + restrict \u22a4 s\u1d9c) = restrict \u03bd' s"}, {"tactic": "rw [mem_setOf_eq, add_right_comm, Measure.le_iff]", "annotated_tactic": ["rw [<a>mem_setOf_eq</a>, <a>add_right_comm</a>, <a>Measure.le_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": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "MeasureTheory.Measure.le_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [972, 9], "def_end_pos": [972, 15]}]], "state_before": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 \u03bd' + restrict \u22a4 s\u1d9c \u2208 {d | \u03bc \u2264 d + \u03bd}", "state_after": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) s_1"}, {"tactic": "intro t h_meas_t", "annotated_tactic": ["intro t h_meas_t", []], "state_before": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) s_1", "state_after": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) t"}, {"tactic": "repeat' rw [\u2190 measure_inter_add_diff t h_meas_s]", "annotated_tactic": ["repeat' rw [\u2190 <a>measure_inter_add_diff</a> t h_meas_s]", [{"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": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) t", "state_after": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s) + \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s)"}, {"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": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s) + \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s)", "state_after": "case a.refine'_1.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s)\n\ncase a.refine'_1.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s)"}, {"tactic": "rw [\u2190 measure_inter_add_diff t h_meas_s]", "annotated_tactic": ["rw [\u2190 <a>measure_inter_add_diff</a> t h_meas_s]", [{"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": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) t", "state_after": "case a.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s) + \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s)"}, {"tactic": "rw [add_apply, add_apply]", "annotated_tactic": ["rw [<a>add_apply</a>, <a>add_apply</a>]", [{"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]}]], "state_before": "case a.refine'_1.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s)", "state_after": "case a.refine'_1.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191\u03bd (t \u2229 s) + \u2191\u2191(restrict \u22a4 s\u1d9c) (t \u2229 s)"}, {"tactic": "apply le_add_right _", "annotated_tactic": ["apply <a>le_add_right</a> _", [{"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 a.refine'_1.refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191\u03bd (t \u2229 s) + \u2191\u2191(restrict \u22a4 s\u1d9c) (t \u2229 s)", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191\u03bd (t \u2229 s)"}, {"tactic": "rw [\u2190 restrict_eq_self \u03bc (inter_subset_right _ _),\n  \u2190 restrict_eq_self \u03bd (inter_subset_right _ _)]", "annotated_tactic": ["rw [\u2190 <a>restrict_eq_self</a> \u03bc (<a>inter_subset_right</a> _ _),\n          \u2190 <a>restrict_eq_self</a> \u03bd (<a>inter_subset_right</a> _ _)]", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 25]}, {"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.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 25]}, {"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\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191\u03bd (t \u2229 s)", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc s) (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191(restrict \u03bd s) (t \u2229 s)"}, {"tactic": "apply h_\u03bd'_in _ (h_meas_t.inter h_meas_s)", "annotated_tactic": ["apply h_\u03bd'_in _ (h_meas_t.inter h_meas_s)", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc s) (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191(restrict \u03bd s) (t \u2229 s)", "state_after": "no goals"}, {"tactic": "rw [add_apply, restrict_apply (h_meas_t.diff h_meas_s), diff_eq, inter_assoc, inter_self,\n  \u2190 add_apply]", "annotated_tactic": ["rw [<a>add_apply</a>, <a>restrict_apply</a> (h_meas_t.diff h_meas_s), <a>diff_eq</a>, <a>inter_assoc</a>, <a>inter_self</a>,\n          \u2190 <a>add_apply</a>]", [{"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.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"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_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]}, {"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 a.refine'_1.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s)", "state_after": "case a.refine'_1.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s\u1d9c) \u2264 \u2191\u2191(\u03bd' + \u03bd + \u22a4) (t \u2229 s\u1d9c)"}, {"tactic": "have h_mu_le_add_top : \u03bc \u2264 \u03bd' + \u03bd + \u22a4 := by simp only [add_top, le_top]", "annotated_tactic": ["have h_mu_le_add_top : \u03bc \u2264 \u03bd' + \u03bd + \u22a4 := by simp only [<a>add_top</a>, <a>le_top</a>]", [{"full_name": "MeasureTheory.Measure.add_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 16]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case a.refine'_1.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s\u1d9c) \u2264 \u2191\u2191(\u03bd' + \u03bd + \u22a4) (t \u2229 s\u1d9c)", "state_after": "case a.refine'_1.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\nh_mu_le_add_top : \u03bc \u2264 \u03bd' + \u03bd + \u22a4\n\u22a2 \u2191\u2191\u03bc (t \u2229 s\u1d9c) \u2264 \u2191\u2191(\u03bd' + \u03bd + \u22a4) (t \u2229 s\u1d9c)"}, {"tactic": "exact Measure.le_iff'.1 h_mu_le_add_top _", "annotated_tactic": ["exact <a>Measure.le_iff'</a>.1 h_mu_le_add_top _", [{"full_name": "MeasureTheory.Measure.le_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [980, 9], "def_end_pos": [980, 16]}]], "state_before": "case a.refine'_1.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\nh_mu_le_add_top : \u03bc \u2264 \u03bd' + \u03bd + \u22a4\n\u22a2 \u2191\u2191\u03bc (t \u2229 s\u1d9c) \u2264 \u2191\u2191(\u03bd' + \u03bd + \u22a4) (t \u2229 s\u1d9c)", "state_after": "no goals"}, {"tactic": "simp only [add_top, le_top]", "annotated_tactic": ["simp only [<a>add_top</a>, <a>le_top</a>]", [{"full_name": "MeasureTheory.Measure.add_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 16]}, {"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\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u03bc \u2264 \u03bd' + \u03bd + \u22a4", "state_after": "no goals"}, {"tactic": "ext1 t h_meas_t", "annotated_tactic": ["ext1 t h_meas_t", []], "state_before": "case a.refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\n\u22a2 (fun \u03bc => restrict \u03bc s) (\u03bd' + restrict \u22a4 s\u1d9c) = restrict \u03bd' s", "state_after": "case a.refine'_2.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191((fun \u03bc => restrict \u03bc s) (\u03bd' + restrict \u22a4 s\u1d9c)) t = \u2191\u2191(restrict \u03bd' s) t"}, {"tactic": "simp [restrict_apply h_meas_t, restrict_apply (h_meas_t.inter h_meas_s), inter_assoc]", "annotated_tactic": ["simp [<a>restrict_apply</a> h_meas_t, <a>restrict_apply</a> (h_meas_t.inter h_meas_s), <a>inter_assoc</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": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "Set.inter_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [944, 9], "def_end_pos": [944, 20]}]], "state_before": "case a.refine'_2.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u03bd' : Measure \u03b1\nh_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s\nt : Set \u03b1\nh_meas_t : MeasurableSet t\n\u22a2 \u2191\u2191((fun \u03bc => restrict \u03bc s) (\u03bd' + restrict \u22a4 s\u1d9c)) t = \u2191\u2191(restrict \u03bd' s) t", "state_after": "no goals"}, {"tactic": "refine' sInf_le_sInf_of_forall_exists_le _", "annotated_tactic": ["refine' <a>sInf_le_sInf_of_forall_exists_le</a> _", [{"full_name": "sInf_le_sInf_of_forall_exists_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [303, 9], "def_end_pos": [303, 41]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2264 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd})", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 \u2200 (x : Measure \u03b1),\n    x \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2192 \u2203 y, y \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2227 y \u2264 x"}, {"tactic": "refine' ball_image_iff.2 fun t h_t_in => \u27e8t.restrict s, _, le_rfl\u27e9", "annotated_tactic": ["refine' <a>ball_image_iff</a>.2 fun t h_t_in => \u27e8t.restrict s, _, <a>le_rfl</a>\u27e9", [{"full_name": "Set.ball_image_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [249, 9], "def_end_pos": [249, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\n\u22a2 \u2200 (x : Measure \u03b1),\n    x \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2192 \u2203 y, y \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2227 y \u2264 x", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\nt : Measure \u03b1\nh_t_in : t \u2208 {d | \u03bc \u2264 d + \u03bd}\n\u22a2 restrict t s \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s}"}, {"tactic": "rw [Set.mem_setOf_eq, \u2190 restrict_add]", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>, \u2190 <a>restrict_add</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": "MeasureTheory.Measure.restrict_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1621, 9], "def_end_pos": [1621, 21]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\nt : Measure \u03b1\nh_t_in : t \u2208 {d | \u03bc \u2264 d + \u03bd}\n\u22a2 restrict t s \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s}", "state_after": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\nt : Measure \u03b1\nh_t_in : t \u2208 {d | \u03bc \u2264 d + \u03bd}\n\u22a2 restrict \u03bc s \u2264 restrict (t + \u03bd) s"}, {"tactic": "exact restrict_mono Subset.rfl h_t_in", "annotated_tactic": ["exact <a>restrict_mono</a> <a>Subset.rfl</a> h_t_in", [{"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.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "case a\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nh_meas_s : MeasurableSet s\nh_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd}\nt : Measure \u03b1\nh_t_in : t \u2208 {d | \u03bc \u2264 d + \u03bd}\n\u22a2 restrict \u03bc s \u2264 restrict (t + \u03bd) 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_add_measure_compl", "start": [160, 1], "end": [161, 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.find?_eq_zoom", "start": [405, 1], "end": [407, 99], "traced_tactics": [{"tactic": "unfold find? zoom", "annotated_tactic": ["unfold <a>find?</a> <a>zoom</a>", [{"full_name": "Std.RBNode.find?", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [394, 19], "def_end_pos": [394, 24]}, {"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\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nx\u271d : optParam (Path \u03b1) Path.root\n\u22a2 find? cut (node c\u271d l\u271d v\u271d r\u271d) = root? (zoom cut (node c\u271d l\u271d v\u271d r\u271d) x\u271d).fst", "state_after": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nx\u271d : optParam (Path \u03b1) Path.root\n\u22a2 (match cut v\u271d with\n    | Ordering.lt => find? cut l\u271d\n    | Ordering.gt => find? cut r\u271d\n    | Ordering.eq => some v\u271d) =\n    root?\n      (match cut v\u271d with\n        | Ordering.lt => zoom cut l\u271d (Path.left c\u271d x\u271d v\u271d r\u271d)\n        | Ordering.gt => zoom cut r\u271d (Path.right c\u271d l\u271d v\u271d x\u271d)\n        | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, x\u271d)).fst"}, {"tactic": "split <;> [apply find?_eq_zoom; apply find?_eq_zoom; rfl]", "annotated_tactic": ["split <;> [apply find?_eq_zoom; apply find?_eq_zoom; rfl]", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nx\u271d : optParam (Path \u03b1) Path.root\n\u22a2 (match cut v\u271d with\n    | Ordering.lt => find? cut l\u271d\n    | Ordering.gt => find? cut r\u271d\n    | Ordering.eq => some v\u271d) =\n    root?\n      (match cut v\u271d with\n        | Ordering.lt => zoom cut l\u271d (Path.left c\u271d x\u271d v\u271d r\u271d)\n        | Ordering.gt => zoom cut r\u271d (Path.right c\u271d l\u271d v\u271d x\u271d)\n        | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, x\u271d)).fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.powerset_injective", "start": [67, 1], "end": [68, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.pair_eq_mk", "start": [344, 1], "end": [347, 85], "traced_tactics": [{"tactic": "simp only [\u2190 pair_mk_mk, mk_coeFn, f.aestronglyMeasurable, g.aestronglyMeasurable]", "annotated_tactic": ["simp only [\u2190 <a>pair_mk_mk</a>, <a>mk_coeFn</a>, f.aestronglyMeasurable, g.aestronglyMeasurable]", [{"full_name": "MeasureTheory.AEEqFun.pair_mk_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [339, 9], "def_end_pos": [339, 19]}, {"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\ng : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 pair f g = mk (fun x => (\u2191f x, \u2191g x)) (_ : AEStronglyMeasurable (fun x => (\u2191f x, \u2191g x)) \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg", "start": [153, 1], "end": [177, 75], "traced_tactics": [{"tactic": "have M : MeasurableSet (Set.Ioc 0 A) := measurableSet_Ioc", "annotated_tactic": ["have M : <a>MeasurableSet</a> (<a>Set.Ioc</a> 0 A) := <a>measurableSet_Ioc</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": "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\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc"}, {"tactic": "have M' : MeasurableSet (Set.Ioc A 0) := measurableSet_Ioc", "annotated_tactic": ["have M' : <a>MeasurableSet</a> (<a>Set.Ioc</a> A 0) := <a>measurableSet_Ioc</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": "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\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc"}, {"tactic": "rw [truncation_eq_of_nonneg h'f]", "annotated_tactic": ["rw [<a>truncation_eq_of_nonneg</a> h'f]", [{"full_name": "ProbabilityTheory.truncation_eq_of_nonneg", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [116, 9], "def_end_pos": [116, 32]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), (indicator (Set.Ioc 0 A) id \u2218 fun x => f x) x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc"}, {"tactic": "change \u222b x, (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = _", "annotated_tactic": ["change \u222b x, (fun z => <a>indicator</a> (<a>Set.Ioc</a> 0 A) <a>id</a> z ^ n) (f x) \u2202\u03bc = _", [{"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]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), (indicator (Set.Ioc 0 A) id \u2218 fun x => f x) x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc"}, {"tactic": "rcases le_or_lt 0 A with (hA | hA)", "annotated_tactic": ["rcases <a>le_or_lt</a> 0 A 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\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "case inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc"}, {"tactic": "rw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_le hA,\n  \u2190 integral_indicator M]", "annotated_tactic": ["rw [\u2190 <a>integral_map</a> (f := fun z => _ ^ n) hf.aemeasurable, <a>intervalIntegral.integral_of_le</a> hA,\n      \u2190 <a>integral_indicator</a> M]", [{"full_name": "MeasureTheory.integral_map", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1610, 9], "def_end_pos": [1610, 21]}, {"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_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "case inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =\n    \u222b (x : \u211d), indicator (Set.Ioc 0 A) (fun x => x ^ n) x \u2202Measure.map f \u03bc\n\ncase inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc)"}, {"tactic": "simp only [indicator, zero_pow' _ hn, id.def, ite_pow]", "annotated_tactic": ["simp only [<a>indicator</a>, <a>zero_pow'</a> _ hn, <a>id.def</a>, <a>ite_pow</a>]", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "zero_pow'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [37, 9], "def_end_pos": [37, 18]}, {"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": "ite_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [57, 9], "def_end_pos": [57, 16]}]], "state_before": "case inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =\n    \u222b (x : \u211d), indicator (Set.Ioc 0 A) (fun x => x ^ n) x \u2202Measure.map f \u03bc", "state_after": "no goals"}, {"tactic": "exact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable", "annotated_tactic": ["exact ((measurable_id.indicator M).<a>pow_const</a> n).<a>aestronglyMeasurable</a>", [{"full_name": "Measurable.pow_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [228, 9], "def_end_pos": [228, 29]}, {"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1285, 9], "def_end_pos": [1285, 47]}]], "state_before": "case inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc)", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_ge hA.le,\n  \u2190 integral_indicator M']", "annotated_tactic": ["rw [\u2190 <a>integral_map</a> (f := fun z => _ ^ n) hf.aemeasurable, <a>intervalIntegral.integral_of_ge</a> hA.le,\n      \u2190 <a>integral_indicator</a> M']", [{"full_name": "MeasureTheory.integral_map", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1610, 9], "def_end_pos": [1610, 21]}, {"full_name": "intervalIntegral.integral_of_ge", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [478, 9], "def_end_pos": [478, 23]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc", "state_after": "case inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =\n    -\u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc\n\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc)"}, {"tactic": "simp only [Set.Ioc_eq_empty_of_le hA.le, zero_pow' _ hn, Set.indicator_empty, integral_zero,\n  zero_eq_neg]", "annotated_tactic": ["simp only [<a>Set.Ioc_eq_empty_of_le</a> hA.le, <a>zero_pow'</a> _ hn, <a>Set.indicator_empty</a>, <a>integral_zero</a>,\n        <a>zero_eq_neg</a>]", [{"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": "zero_pow'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [37, 9], "def_end_pos": [37, 18]}, {"full_name": "Set.indicator_empty", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [193, 3], "def_end_pos": [193, 14]}, {"full_name": "MeasureTheory.integral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [850, 9], "def_end_pos": [850, 22]}, {"full_name": "zero_eq_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [428, 3], "def_end_pos": [428, 14]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =\n    -\u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc", "state_after": "case inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc = 0"}, {"tactic": "apply integral_eq_zero_of_ae", "annotated_tactic": ["apply <a>integral_eq_zero_of_ae</a>", [{"full_name": "MeasureTheory.integral_eq_zero_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [981, 9], "def_end_pos": [981, 31]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc = 0", "state_after": "case inr.hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0"}, {"tactic": "have : \u2200\u1d50 x \u2202Measure.map f \u03bc, (0 : \u211d) \u2264 x :=\n  (ae_map_iff hf.aemeasurable measurableSet_Ici).2 (eventually_of_forall h'f)", "annotated_tactic": ["have : \u2200\u1d50 x \u2202<a>Measure.map</a> f \u03bc, (0 : \u211d) \u2264 x :=\n        (<a>ae_map_iff</a> hf.aemeasurable <a>measurableSet_Ici</a>).2 (<a>eventually_of_forall</a> h'f)", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.ae_map_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2470, 9], "def_end_pos": [2470, 19]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 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 inr.hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0", "state_after": "case inr.hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\n\u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0"}, {"tactic": "filter_upwards [this] with x hx", "annotated_tactic": ["filter_upwards [this] with x hx", []], "state_before": "case inr.hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\n\u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 indicator (Set.Ioc A 0) (fun x => x ^ n) x = OfNat.ofNat 0 x"}, {"tactic": "simp only [indicator, Set.mem_Ioc, Pi.zero_apply, ite_eq_right_iff, and_imp]", "annotated_tactic": ["simp only [<a>indicator</a>, <a>Set.mem_Ioc</a>, <a>Pi.zero_apply</a>, <a>ite_eq_right_iff</a>, <a>and_imp</a>]", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 indicator (Set.Ioc A 0) (fun x => x ^ n) x = OfNat.ofNat 0 x", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 A < x \u2192 x \u2264 0 \u2192 x ^ n = 0"}, {"tactic": "intro _ h''x", "annotated_tactic": ["intro _ h''x", []], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 A < x \u2192 x \u2264 0 \u2192 x ^ n = 0", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\n\u22a2 x ^ n = 0"}, {"tactic": "have : x = 0 := by linarith", "annotated_tactic": ["have : x = 0 := by linarith", []], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\n\u22a2 x ^ n = 0", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis\u271d : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\nthis : x = 0\n\u22a2 x ^ n = 0"}, {"tactic": "simp [this, zero_pow' _ hn]", "annotated_tactic": ["simp [this, <a>zero_pow'</a> _ hn]", [{"full_name": "zero_pow'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [37, 9], "def_end_pos": [37, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis\u271d : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\nthis : x = 0\n\u22a2 x ^ n = 0", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\n\u22a2 x = 0", "state_after": "no goals"}, {"tactic": "exact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable", "annotated_tactic": ["exact ((measurable_id.indicator M).<a>pow_const</a> n).<a>aestronglyMeasurable</a>", [{"full_name": "Measurable.pow_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [228, 9], "def_end_pos": [228, 29]}, {"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1285, 9], "def_end_pos": [1285, 47]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc)", "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", "start": [341, 1], "end": [347, 85], "traced_tactics": [{"tactic": "by_cases hp_ne_top : p = \u221e", "annotated_tactic": ["by_cases hp_ne_top : p = \u221e", []], "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\nl : Filter \u03b9\nhp_ne_zero : p \u2260 0\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "case pos\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\nl : Filter \u03b9\nhp_ne_zero : p \u2260 0\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nhp_ne_top : p = \u22a4\n\u22a2 TendstoInMeasure \u03bc f l g\n\ncase neg\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\nl : Filter \u03b9\nhp_ne_zero : p \u2260 0\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nhp_ne_top : \u00acp = \u22a4\n\u22a2 TendstoInMeasure \u03bc f l g"}, {"tactic": "subst hp_ne_top", "annotated_tactic": ["subst hp_ne_top", []], "state_before": "case pos\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\nl : Filter \u03b9\nhp_ne_zero : p \u2260 0\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nhp_ne_top : p = \u22a4\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "case pos\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\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp_ne_zero : \u22a4 \u2260 0\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g"}, {"tactic": "exact tendstoInMeasure_of_tendsto_snorm_top hfg", "annotated_tactic": ["exact <a>tendstoInMeasure_of_tendsto_snorm_top</a> hfg", [{"full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_top", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [318, 9], "def_end_pos": [318, 46]}]], "state_before": "case pos\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\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp_ne_zero : \u22a4 \u2260 0\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "no goals"}, {"tactic": "exact tendstoInMeasure_of_tendsto_snorm_of_ne_top hp_ne_zero hp_ne_top hf hg hfg", "annotated_tactic": ["exact <a>tendstoInMeasure_of_tendsto_snorm_of_ne_top</a> hp_ne_zero hp_ne_top hf hg hfg", [{"full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [304, 9], "def_end_pos": [304, 52]}]], "state_before": "case neg\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\nl : Filter \u03b9\nhp_ne_zero : p \u2260 0\nhf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0)\nhp_ne_top : \u00acp = \u22a4\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.AnalyticSet.measurablySeparable", "start": [511, 1], "end": [519, 62], "traced_tactics": [{"tactic": "rw [AnalyticSet] at hs ht", "annotated_tactic": ["rw [<a>AnalyticSet</a>] at hs ht", [{"full_name": "MeasureTheory.AnalyticSet", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [164, 17], "def_end_pos": [164, 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\ns t : Set \u03b1\nhs : AnalyticSet s\nht : AnalyticSet t\nh : Disjoint s t\n\u22a2 MeasurablySeparable s t", "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\ns t : Set \u03b1\nhs : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint s t\n\u22a2 MeasurablySeparable s t"}, {"tactic": "rcases hs with (rfl | \u27e8f, f_cont, rfl\u27e9)", "annotated_tactic": ["rcases hs with (rfl | \u27e8f, f_cont, rfl\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\ns t : Set \u03b1\nhs : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint s t\n\u22a2 MeasurablySeparable s t", "state_after": "case inl\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\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint \u2205 t\n\u22a2 MeasurablySeparable \u2205 t\n\ncase inr.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\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nf : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) t\n\u22a2 MeasurablySeparable (range f) t"}, {"tactic": "rcases ht with (rfl | \u27e8g, g_cont, rfl\u27e9)", "annotated_tactic": ["rcases ht with (rfl | \u27e8g, g_cont, rfl\u27e9)", []], "state_before": "case inr.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\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nf : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) t\n\u22a2 MeasurablySeparable (range f) t", "state_after": "case inr.intro.intro.inl\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 : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) \u2205\n\u22a2 MeasurablySeparable (range f) \u2205\n\ncase inr.intro.intro.inr.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 : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\ng : (\u2115 \u2192 \u2115) \u2192 \u03b1\ng_cont : Continuous g\nh : Disjoint (range f) (range g)\n\u22a2 MeasurablySeparable (range f) (range g)"}, {"tactic": "exact measurablySeparable_range_of_disjoint f_cont g_cont h", "annotated_tactic": ["exact <a>measurablySeparable_range_of_disjoint</a> f_cont g_cont h", [{"full_name": "MeasureTheory.measurablySeparable_range_of_disjoint", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [397, 9], "def_end_pos": [397, 46]}]], "state_before": "case inr.intro.intro.inr.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 : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\ng : (\u2115 \u2192 \u2115) \u2192 \u03b1\ng_cont : Continuous g\nh : Disjoint (range f) (range g)\n\u22a2 MeasurablySeparable (range f) (range g)", "state_after": "no goals"}, {"tactic": "refine' \u27e8\u2205, Subset.refl _, by simp, MeasurableSet.empty\u27e9", "annotated_tactic": ["refine' \u27e8\u2205, <a>Subset.refl</a> _, by simp, <a>MeasurableSet.empty</a>\u27e9", [{"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"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\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\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint \u2205 t\n\u22a2 MeasurablySeparable \u2205 t", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint \u2205 t\n\u22a2 Disjoint t \u2205", "state_after": "no goals"}, {"tactic": "exact \u27e8univ, subset_univ _, by simp, MeasurableSet.univ\u27e9", "annotated_tactic": ["exact \u27e8<a>univ</a>, <a>subset_univ</a> _, by simp, <a>MeasurableSet.univ</a>\u27e9", [{"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": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "case inr.intro.intro.inl\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 : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) \u2205\n\u22a2 MeasurablySeparable (range f) \u2205", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) \u2205\n\u22a2 Disjoint \u2205 univ", "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.add.aux", "start": [181, 1], "end": [207, 70], "traced_tactics": [{"tactic": "intro den num", "annotated_tactic": ["intro den num", []], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\n\u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den"}, {"tactic": "have ae : ad * g = a.den := had \u25b8 Nat.div_mul_cancel (hg \u25b8 Nat.gcd_dvd_left ..)", "annotated_tactic": ["have ae : ad * g = a.den := had \u25b8 <a>Nat.div_mul_cancel</a> (hg \u25b8 <a>Nat.gcd_dvd_left</a> ..)", [{"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.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}]], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\n\u22a2 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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\n\u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den"}, {"tactic": "have be : bd * g = b.den := hbd \u25b8 Nat.div_mul_cancel (hg \u25b8 Nat.gcd_dvd_right ..)", "annotated_tactic": ["have be : bd * g = b.den := hbd \u25b8 <a>Nat.div_mul_cancel</a> (hg \u25b8 <a>Nat.gcd_dvd_right</a> ..)", [{"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.gcd_dvd_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}]], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\n\u22a2 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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\n\u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den"}, {"tactic": "have hden : den = ad * bd * g := by rw [Nat.mul_assoc, be]", "annotated_tactic": ["have hden : den = ad * bd * g := by rw [<a>Nat.mul_assoc</a>, be]", [{"full_name": "Nat.mul_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [198, 19], "def_end_pos": [198, 28]}]], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\n\u22a2 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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n\u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den"}, {"tactic": "rw [hden, Nat.Coprime.gcd_mul_left_cancel_right]", "annotated_tactic": ["rw [hden, <a>Nat.Coprime.gcd_mul_left_cancel_right</a>]", [{"full_name": "Nat.Coprime.gcd_mul_left_cancel_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [280, 9], "def_end_pos": [280, 42]}]], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n\u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den", "state_after": "case H\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n\u22a2 Nat.Coprime (ad * bd) (Int.natAbs num)"}, {"tactic": "have cop : ad.Coprime bd := had \u25b8 hbd \u25b8 hg \u25b8\n  Nat.coprime_div_gcd_div_gcd (Nat.gcd_pos_of_pos_left _ a.den_pos)", "annotated_tactic": ["have cop : ad.Coprime bd := had \u25b8 hbd \u25b8 hg \u25b8\n    <a>Nat.coprime_div_gcd_div_gcd</a> (<a>Nat.gcd_pos_of_pos_left</a> _ a.den_pos)", [{"full_name": "Nat.coprime_div_gcd_div_gcd", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [288, 9], "def_end_pos": [288, 32]}, {"full_name": "Nat.gcd_pos_of_pos_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [92, 9], "def_end_pos": [92, 28]}]], "state_before": "case H\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n\u22a2 Nat.Coprime (ad * bd) (Int.natAbs num)", "state_after": "case H\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\n\u22a2 Nat.Coprime (ad * bd) (Int.natAbs num)"}, {"tactic": "have H1 (d : Nat) :\n    d.gcd num.natAbs \u2223 a.num.natAbs * bd \u2194 d.gcd num.natAbs \u2223 b.num.natAbs * ad := by\n  have := d.gcd_dvd_right num.natAbs\n  rw [\u2190 Int.ofNat_dvd, Int.dvd_natAbs] at this\n  have := Int.dvd_iff_dvd_of_dvd_add this\n  rwa [\u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul,\n    \u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul] at this", "annotated_tactic": ["have H1 (d : <a>Nat</a>) :\n      d.gcd num.natAbs \u2223 a.num.natAbs * bd \u2194 d.gcd num.natAbs \u2223 b.num.natAbs * ad := by\n    have := d.gcd_dvd_right num.natAbs\n    rw [\u2190 <a>Int.ofNat_dvd</a>, <a>Int.dvd_natAbs</a>] at this\n    have := <a>Int.dvd_iff_dvd_of_dvd_add</a> this\n    rwa [\u2190 <a>Int.dvd_natAbs</a>, <a>Int.ofNat_dvd</a>, <a>Int.natAbs_mul</a>,\n      \u2190 <a>Int.dvd_natAbs</a>, <a>Int.ofNat_dvd</a>, <a>Int.natAbs_mul</a>] at this", [{"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"full_name": "Int.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}, {"full_name": "Int.dvd_iff_dvd_of_dvd_add", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [632, 19], "def_end_pos": [632, 41]}, {"full_name": "Int.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}, {"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"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.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}, {"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"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 H\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\n\u22a2 Nat.Coprime (ad * bd) (Int.natAbs num)", "state_after": "case H\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime (ad * bd) (Int.natAbs num)"}, {"tactic": "apply Nat.Coprime.mul", "annotated_tactic": ["apply <a>Nat.Coprime.mul</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]}]], "state_before": "case H\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime (ad * bd) (Int.natAbs num)", "state_after": "case H.H1\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime ad (Int.natAbs num)\n\ncase H.H2\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime bd (Int.natAbs num)"}, {"tactic": "rw [Nat.mul_assoc, be]", "annotated_tactic": ["rw [<a>Nat.mul_assoc</a>, be]", [{"full_name": "Nat.mul_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [198, 19], "def_end_pos": [198, 28]}]], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\n\u22a2 den = ad * bd * g", "state_after": "no goals"}, {"tactic": "have := d.gcd_dvd_right num.natAbs", "annotated_tactic": ["have := d.gcd_dvd_right num.natAbs", []], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad", "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\nthis : Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs num\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad"}, {"tactic": "rw [\u2190 Int.ofNat_dvd, Int.dvd_natAbs] at this", "annotated_tactic": ["rw [\u2190 <a>Int.ofNat_dvd</a>, <a>Int.dvd_natAbs</a>] at this", [{"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"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": "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\nthis : Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs num\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad", "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\nthis : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 num\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad"}, {"tactic": "have := Int.dvd_iff_dvd_of_dvd_add this", "annotated_tactic": ["have := <a>Int.dvd_iff_dvd_of_dvd_add</a> this", [{"full_name": "Int.dvd_iff_dvd_of_dvd_add", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [632, 19], "def_end_pos": [632, 41]}]], "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\nthis : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 num\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad", "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\nthis\u271d : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 num\nthis : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 a.num * \u2191bd \u2194 \u2191(Nat.gcd d (Int.natAbs num)) \u2223 b.num * \u2191ad\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad"}, {"tactic": "rwa [\u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul,\n  \u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul] at this", "annotated_tactic": ["rwa [\u2190 <a>Int.dvd_natAbs</a>, <a>Int.ofNat_dvd</a>, <a>Int.natAbs_mul</a>,\n      \u2190 <a>Int.dvd_natAbs</a>, <a>Int.ofNat_dvd</a>, <a>Int.natAbs_mul</a>] at this", [{"full_name": "Int.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}, {"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"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.dvd_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [657, 9], "def_end_pos": [657, 19]}, {"full_name": "Int.ofNat_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [635, 22], "def_end_pos": [635, 31]}, {"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": "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\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nd : Nat\nthis\u271d : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 num\nthis : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 a.num * \u2191bd \u2194 \u2191(Nat.gcd d (Int.natAbs num)) \u2223 b.num * \u2191ad\n\u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad", "state_after": "no goals"}, {"tactic": "have := (H1 ad).2 <| Nat.dvd_trans (Nat.gcd_dvd_left ..) (Nat.dvd_mul_left ..)", "annotated_tactic": ["have := (H1 ad).2 <| <a>Nat.dvd_trans</a> (<a>Nat.gcd_dvd_left</a> ..) (<a>Nat.dvd_mul_left</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_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"full_name": "Nat.dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [866, 19], "def_end_pos": [866, 31]}]], "state_before": "case H.H1\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime ad (Int.natAbs num)", "state_after": "case H.H1\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num * bd\n\u22a2 Nat.Coprime ad (Int.natAbs num)"}, {"tactic": "have := (cop.coprime_dvd_left <| Nat.gcd_dvd_left ..).dvd_of_dvd_mul_right this", "annotated_tactic": ["have := (cop.coprime_dvd_left <| <a>Nat.gcd_dvd_left</a> ..).<a>dvd_of_dvd_mul_right</a> this", [{"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"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 H.H1\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num * bd\n\u22a2 Nat.Coprime ad (Int.natAbs num)", "state_after": "case H.H1\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis\u271d : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num * bd\nthis : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num\n\u22a2 Nat.Coprime ad (Int.natAbs num)"}, {"tactic": "exact Nat.eq_one_of_dvd_one <| a.reduced.gcd_eq_one \u25b8 Nat.dvd_gcd this <|\n  Nat.dvd_trans (Nat.gcd_dvd_left ..) (ae \u25b8 Nat.dvd_mul_right ..)", "annotated_tactic": ["exact <a>Nat.eq_one_of_dvd_one</a> <| a.reduced.gcd_eq_one \u25b8 <a>Nat.dvd_gcd</a> this <|\n      <a>Nat.dvd_trans</a> (<a>Nat.gcd_dvd_left</a> ..) (ae \u25b8 <a>Nat.dvd_mul_right</a> ..)", [{"full_name": "Nat.eq_one_of_dvd_one", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [926, 9], "def_end_pos": [926, 26]}, {"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.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_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"full_name": "Nat.dvd_mul_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [868, 19], "def_end_pos": [868, 32]}]], "state_before": "case H.H1\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis\u271d : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num * bd\nthis : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num\n\u22a2 Nat.Coprime ad (Int.natAbs num)", "state_after": "no goals"}, {"tactic": "have := (H1 bd).1 <| Nat.dvd_trans (Nat.gcd_dvd_left ..) (Nat.dvd_mul_left ..)", "annotated_tactic": ["have := (H1 bd).1 <| <a>Nat.dvd_trans</a> (<a>Nat.gcd_dvd_left</a> ..) (<a>Nat.dvd_mul_left</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_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"full_name": "Nat.dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [866, 19], "def_end_pos": [866, 31]}]], "state_before": "case H.H2\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime bd (Int.natAbs num)", "state_after": "case H.H2\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime bd (Int.natAbs num)"}, {"tactic": "have := (cop.symm.coprime_dvd_left <| Nat.gcd_dvd_left ..).dvd_of_dvd_mul_right this", "annotated_tactic": ["have := (cop.symm.coprime_dvd_left <| <a>Nat.gcd_dvd_left</a> ..).<a>dvd_of_dvd_mul_right</a> this", [{"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"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 H.H2\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num * ad\n\u22a2 Nat.Coprime bd (Int.natAbs num)", "state_after": "case H.H2\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis\u271d : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num\n\u22a2 Nat.Coprime bd (Int.natAbs num)"}, {"tactic": "exact Nat.eq_one_of_dvd_one <| b.reduced.gcd_eq_one \u25b8 Nat.dvd_gcd this <|\n  Nat.dvd_trans (Nat.gcd_dvd_left ..) (be \u25b8 Nat.dvd_mul_right ..)", "annotated_tactic": ["exact <a>Nat.eq_one_of_dvd_one</a> <| b.reduced.gcd_eq_one \u25b8 <a>Nat.dvd_gcd</a> this <|\n      <a>Nat.dvd_trans</a> (<a>Nat.gcd_dvd_left</a> ..) (be \u25b8 <a>Nat.dvd_mul_right</a> ..)", [{"full_name": "Nat.eq_one_of_dvd_one", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [926, 9], "def_end_pos": [926, 26]}, {"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.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_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"full_name": "Nat.dvd_mul_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [868, 19], "def_end_pos": [868, 32]}]], "state_before": "case H.H2\na b : Rat\ng ad bd : Nat\nhg : g = Nat.gcd a.den b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * \u2191bd + b.num * \u2191ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : Nat.Coprime ad bd\nH1 :\n  \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis\u271d : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num * ad\nthis : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num\n\u22a2 Nat.Coprime bd (Int.natAbs num)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_const_lt_top_iff", "start": [325, 1], "end": [341, 68], "traced_tactics": [{"tactic": "have hp : 0 < p.toReal := ENNReal.toReal_pos hp_ne_zero hp_ne_top", "annotated_tactic": ["have hp : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp_ne_zero 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\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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4", "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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4"}, {"tactic": "by_cases h\u03bc : \u03bc = 0", "annotated_tactic": ["by_cases h\u03bc : \u03bc = 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 : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4"}, {"tactic": "by_cases hc : c = 0", "annotated_tactic": ["by_cases hc : c = 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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : c = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4"}, {"tactic": "rw [snorm_const' c hp_ne_zero hp_ne_top]", "annotated_tactic": ["rw [<a>snorm_const'</a> c hp_ne_zero hp_ne_top]", [{"full_name": "MeasureTheory.snorm_const'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [320, 9], "def_end_pos": [320, 21]}]], "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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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 : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4"}, {"tactic": "by_cases h\u03bc_top : \u03bc Set.univ = \u221e", "annotated_tactic": ["by_cases h\u03bc_top : \u03bc <a>Set.univ</a> = \u221e", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4"}, {"tactic": "rw [ENNReal.mul_lt_top_iff]", "annotated_tactic": ["rw [<a>ENNReal.mul_lt_top_iff</a>]", [{"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 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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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 : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a < \u22a4 \u2227 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2228 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) = 0 \u2194\n    c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4"}, {"tactic": "simp only [true_and_iff, one_div, ENNReal.rpow_eq_zero_iff, h\u03bc, false_or_iff, or_false_iff,\n  ENNReal.coe_lt_top, nnnorm_eq_zero, ENNReal.coe_eq_zero,\n  MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, and_false_iff, false_and_iff,\n  _root_.inv_pos, or_self_iff, h\u03bc_top, Ne.lt_top h\u03bc_top, iff_true_iff]", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>one_div</a>, <a>ENNReal.rpow_eq_zero_iff</a>, h\u03bc, <a>false_or_iff</a>, <a>or_false_iff</a>,\n    <a>ENNReal.coe_lt_top</a>, <a>nnnorm_eq_zero</a>, <a>ENNReal.coe_eq_zero</a>,\n    <a>MeasureTheory.Measure.measure_univ_eq_zero</a>, hp, <a>inv_lt_zero</a>, hc, <a>and_false_iff</a>, <a>false_and_iff</a>,\n    <a>_root_.inv_pos</a>, <a>or_self_iff</a>, h\u03bc_top, <a>Ne.lt_top</a> h\u03bc_top, <a>iff_true_iff</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"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_zero_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [465, 9], "def_end_pos": [465, 25]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "nnnorm_eq_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2065, 30], "def_end_pos": [2065, 44]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}, {"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": "inv_lt_zero", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"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": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}, {"full_name": "Ne.lt_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [186, 9], "def_end_pos": [186, 18]}, {"full_name": "iff_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [196, 9], "def_end_pos": [196, 21]}]], "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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a < \u22a4 \u2227 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2228 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) = 0 \u2194\n    c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \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 : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4"}, {"tactic": "exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) h\u03bc_top", "annotated_tactic": ["exact <a>ENNReal.rpow_lt_top_of_nonneg</a> (inv_nonneg.mpr hp.le) h\u03bc_top", [{"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]}]], "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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [h\u03bc, Measure.coe_zero, Pi.zero_apply, or_true_iff, WithTop.zero_lt_top,\n  snorm_measure_zero]", "annotated_tactic": ["simp only [h\u03bc, <a>Measure.coe_zero</a>, <a>Pi.zero_apply</a>, <a>or_true_iff</a>, <a>WithTop.zero_lt_top</a>,\n      <a>snorm_measure_zero</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]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"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": "MeasureTheory.snorm_measure_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [243, 9], "def_end_pos": [243, 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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [hc, true_or_iff, eq_self_iff_true, WithTop.zero_lt_top, snorm_zero']", "annotated_tactic": ["simp only [hc, <a>true_or_iff</a>, <a>eq_self_iff_true</a>, <a>WithTop.zero_lt_top</a>, <a>snorm_zero'</a>]", [{"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"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": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}, {"full_name": "MeasureTheory.snorm_zero'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [211, 9], "def_end_pos": [211, 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\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : c = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4", "state_after": "no goals"}, {"tactic": "simp [hc, h\u03bc_top, hp]", "annotated_tactic": ["simp [hc, h\u03bc_top, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.add_self_eq_zero_iff_eq_zero", "start": [842, 1], "end": [845, 95], "traced_tactics": [{"tactic": "rw [Nat.odd_iff, \u2190 Nat.two_dvd_ne_zero, \u2190 Nat.prime_two.coprime_iff_not_dvd] at hn", "annotated_tactic": ["rw [<a>Nat.odd_iff</a>, \u2190 <a>Nat.two_dvd_ne_zero</a>, \u2190 Nat.prime_two.coprime_iff_not_dvd] at hn", [{"full_name": "Nat.odd_iff", "def_path": "Mathlib/Data/Nat/Parity.lean", "def_pos": [44, 9], "def_end_pos": [44, 16]}, {"full_name": "Nat.two_dvd_ne_zero", "def_path": "Mathlib/Data/Nat/Parity.lean", "def_pos": [89, 17], "def_end_pos": [89, 32]}]], "state_before": "n : \u2115\nhn : Odd n\na : ZMod n\n\u22a2 a + a = 0 \u2194 a = 0", "state_after": "n : \u2115\nhn : Nat.Coprime 2 n\na : ZMod n\n\u22a2 a + a = 0 \u2194 a = 0"}, {"tactic": "rw [\u2190mul_two, \u2190@Nat.cast_two (ZMod n), \u2190ZMod.coe_unitOfCoprime 2 hn, Units.mul_left_eq_zero]", "annotated_tactic": ["rw [\u2190<a>mul_two</a>, \u2190@<a>Nat.cast_two</a> (<a>ZMod</a> n), \u2190<a>ZMod.coe_unitOfCoprime</a> 2 hn, <a>Units.mul_left_eq_zero</a>]", [{"full_name": "mul_two", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 16]}, {"full_name": "Nat.cast_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [193, 9], "def_end_pos": [193, 17]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod.coe_unitOfCoprime", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [739, 9], "def_end_pos": [739, 26]}, {"full_name": "Units.mul_left_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [39, 9], "def_end_pos": [39, 25]}]], "state_before": "n : \u2115\nhn : Nat.Coprime 2 n\na : ZMod n\n\u22a2 a + a = 0 \u2194 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict", "start": [202, 1], "end": [214, 50], "traced_tactics": [{"tactic": "refine' \u27e8fun hf x => _, locallyIntegrableOn_of_locallyIntegrable_restrict\u27e9", "annotated_tactic": ["refine' \u27e8fun hf x => _, <a>locallyIntegrableOn_of_locallyIntegrable_restrict</a>\u27e9", [{"full_name": "MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [189, 9], "def_end_pos": [189, 58]}]], "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 : OpensMeasurableSpace X\nhs : IsClosed s\n\u22a2 LocallyIntegrableOn f s \u2194 LocallyIntegrable f", "state_after": "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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "by_cases h : x \u2208 s", "annotated_tactic": ["by_cases h : x \u2208 s", []], "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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case pos\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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n\ncase neg\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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "obtain \u27e8t, ht_nhds, ht_int\u27e9 := hf x h", "annotated_tactic": ["obtain \u27e8t, ht_nhds, ht_int\u27e9 := hf x h", []], "state_before": "case pos\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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case pos.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "obtain \u27e8u, hu_o, hu_x, hu_sub\u27e9 := mem_nhdsWithin.mp ht_nhds", "annotated_tactic": ["obtain \u27e8u, hu_o, hu_x, hu_sub\u27e9 := mem_nhdsWithin.mp ht_nhds", []], "state_before": "case pos.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case pos.intro.intro.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "refine' \u27e8u, hu_o.mem_nhds hu_x, _\u27e9", "annotated_tactic": ["refine' \u27e8u, hu_o.mem_nhds hu_x, _\u27e9", []], "state_before": "case pos.intro.intro.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case pos.intro.intro.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 IntegrableOn f u"}, {"tactic": "rw [IntegrableOn, restrict_restrict hu_o.measurableSet]", "annotated_tactic": ["rw [<a>IntegrableOn</a>, <a>restrict_restrict</a> hu_o.measurableSet]", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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.intro.intro.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 IntegrableOn f u", "state_after": "case pos.intro.intro.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 Integrable f"}, {"tactic": "exact ht_int.mono_set hu_sub", "annotated_tactic": ["exact ht_int.mono_set hu_sub", []], "state_before": "case pos.intro.intro.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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 Integrable f", "state_after": "no goals"}, {"tactic": "rw [\u2190 isOpen_compl_iff] at hs", "annotated_tactic": ["rw [\u2190 <a>isOpen_compl_iff</a>] at hs", [{"full_name": "isOpen_compl_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [205, 17], "def_end_pos": [205, 33]}]], "state_before": "case neg\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 : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)"}, {"tactic": "refine' \u27e8s\u1d9c, hs.mem_nhds h, _\u27e9", "annotated_tactic": ["refine' \u27e8s\u1d9c, hs.mem_nhds h, _\u27e9", []], "state_before": "case neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)", "state_after": "case neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableOn f s\u1d9c"}, {"tactic": "rw [IntegrableOn, restrict_restrict, inter_comm, inter_compl_self, \u2190 IntegrableOn]", "annotated_tactic": ["rw [<a>IntegrableOn</a>, <a>restrict_restrict</a>, <a>inter_comm</a>, <a>inter_compl_self</a>, \u2190 <a>IntegrableOn</a>]", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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_compl_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1667, 9], "def_end_pos": [1667, 25]}, {"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "state_before": "case neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableOn f s\u1d9c", "state_after": "case neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableOn f \u2205\n\ncase neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 MeasurableSet s\u1d9c"}, {"tactic": "exacts [integrableOn_empty, hs.measurableSet]", "annotated_tactic": ["exacts [<a>integrableOn_empty</a>, hs.measurableSet]", [{"full_name": "MeasureTheory.integrableOn_empty", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [106, 9], "def_end_pos": [106, 27]}]], "state_before": "case neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableOn f \u2205\n\ncase neg\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 : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 MeasurableSet s\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.insert_entries_of_neg", "start": [485, 1], "end": [489, 78], "traced_tactics": [{"tactic": "simp [AList.insert_entries_of_neg (mt mem_toFinmap.1 h), -insert_entries]", "annotated_tactic": ["simp [<a>AList.insert_entries_of_neg</a> (<a>mt</a> <a>mem_toFinmap</a>.1 h), -<a>insert_entries</a>]", [{"full_name": "AList.insert_entries_of_neg", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [279, 9], "def_end_pos": [279, 30]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Finmap.mem_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [181, 9], "def_end_pos": [181, 21]}, {"full_name": "AList.insert_entries", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [274, 9], "def_end_pos": [274, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u271d : Finmap \u03b2\ns : AList \u03b2\nh : \u00aca \u2208 \u27e6s\u27e7\n\u22a2 (insert a b \u27e6s\u27e7).entries = { fst := a, snd := b } ::\u2098 \u27e6s\u27e7.entries", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.one_add_kstar_mul_self_eq_kstar", "start": [275, 1], "end": [276, 60], "traced_tactics": [{"tactic": "rw [mul_self_kstar_comm, one_add_self_mul_kstar_eq_kstar]", "annotated_tactic": ["rw [<a>mul_self_kstar_comm</a>, <a>one_add_self_mul_kstar_eq_kstar</a>]", [{"full_name": "Language.mul_self_kstar_comm", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [264, 9], "def_end_pos": [264, 28]}, {"full_name": "Language.one_add_self_mul_kstar_eq_kstar", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [269, 9], "def_end_pos": [269, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x : List \u03b1\nl : Language \u03b1\n\u22a2 1 + l\u2217 * l = l\u2217", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.card_le_diff_of_interleaved", "start": [1700, 1], "end": [1707, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommProd_coe", "start": [124, 1], "end": [131, 71], "traced_tactics": [{"tactic": "rw [noncommProd]", "annotated_tactic": ["rw [<a>noncommProd</a>]", [{"full_name": "Multiset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [118, 5], "def_end_pos": [118, 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 : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 noncommProd (\u2191l) comm = List.prod 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\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 noncommFold (fun x x_1 => x * x_1) (\u2191l) comm 1 = List.prod l"}, {"tactic": "simp only [noncommFold_coe]", "annotated_tactic": ["simp only [<a>noncommFold_coe</a>]", [{"full_name": "Multiset.noncommFold_coe", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "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\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 noncommFold (fun x x_1 => x * x_1) (\u2191l) comm 1 = List.prod 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\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 l = List.prod l"}, {"tactic": "induction' l with hd tl hl", "annotated_tactic": ["induction' l with hd tl hl", []], "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\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 l = List.prod l", "state_after": "case nil\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\ncomm : Set.Pairwise {x | x \u2208 \u2191[]} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 [] = List.prod []\n\ncase cons\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 (hd :: tl) = List.prod (hd :: tl)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\ncomm : Set.Pairwise {x | x \u2208 \u2191[]} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 [] = List.prod []", "state_after": "no goals"}, {"tactic": "rw [List.prod_cons, List.foldr, hl]", "annotated_tactic": ["rw [<a>List.prod_cons</a>, <a>List.foldr</a>, hl]", [{"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "List.foldr", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [514, 19], "def_end_pos": [514, 24]}]], "state_before": "case cons\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 (hd :: tl) = List.prod (hd :: tl)", "state_after": "case cons\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\n\u22a2 Set.Pairwise {x | x \u2208 \u2191tl} Commute"}, {"tactic": "intro x hx y hy", "annotated_tactic": ["intro x hx y hy", []], "state_before": "case cons\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\n\u22a2 Set.Pairwise {x | x \u2208 \u2191tl} Commute", "state_after": "case cons\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\nx : \u03b1\nhx : x \u2208 {x | x \u2208 \u2191tl}\ny : \u03b1\nhy : y \u2208 {x | x \u2208 \u2191tl}\n\u22a2 x \u2260 y \u2192 Commute x y"}, {"tactic": "exact comm (List.mem_cons_of_mem _ hx) (List.mem_cons_of_mem _ hy)", "annotated_tactic": ["exact comm (<a>List.mem_cons_of_mem</a> _ hx) (<a>List.mem_cons_of_mem</a> _ hy)", [{"full_name": "List.mem_cons_of_mem", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [68, 9], "def_end_pos": [68, 24]}, {"full_name": "List.mem_cons_of_mem", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [68, 9], "def_end_pos": [68, 24]}]], "state_before": "case cons\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\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\nx : \u03b1\nhx : x \u2208 {x | x \u2208 \u2191tl}\ny : \u03b1\nhy : y \u2208 {x | x \u2208 \u2191tl}\n\u22a2 x \u2260 y \u2192 Commute x y", "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_mod_eq_zero", "start": [685, 1], "end": [686, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image3_congr'", "start": [255, 1], "end": [256, 42], "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", "start": [702, 1], "end": [704, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_lintegral_nn_filter_of_le_const", "start": [565, 1], "end": [574, 71], "traced_tactics": [{"tactic": "refine tendsto_lintegral_filter_of_dominated_convergence (fun _ => c)\n  (eventually_of_forall fun i => (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_\n  (@lintegral_const_lt_top _ _ \u03bc _ _ (@ENNReal.coe_ne_top c)).ne ?_", "annotated_tactic": ["refine <a>tendsto_lintegral_filter_of_dominated_convergence</a> (fun _ => c)\n    (<a>eventually_of_forall</a> fun i => (ENNReal.continuous_coe.comp (fs i).<a>continuous</a>).<a>measurable</a>) ?_\n    (@<a>lintegral_const_lt_top</a> _ _ \u03bc _ _ (@<a>ENNReal.coe_ne_top</a> c)).<a>ne</a> ?_", [{"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": "BoundedContinuousFunction.continuous", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [107, 19], "def_end_pos": [107, 29]}, {"full_name": "Continuous.measurable", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 30]}, {"full_name": "MeasureTheory.lintegral_const_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [165, 9], "def_end_pos": [165, 31]}, {"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": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs i) \u03c9) \u2202\u03bc) L (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u03bc))", "state_after": "case refine_1\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 \u2200\u1da0 (n : \u03b9) in L, \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2191(\u2191(fs n) a) \u2264 (fun x => \u2191c) a\n\ncase refine_2\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 \u2200\u1d50 (a : \u03a9) \u2202\u03bc, Tendsto (fun n => \u2191(\u2191(fs n) a)) L (\ud835\udcdd \u2191(f a))"}, {"tactic": "simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const", "annotated_tactic": ["simpa only [<a>Function.comp_apply</a>, <a>ENNReal.coe_le_coe</a>] using fs_le_const", [{"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.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\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 \u2200\u1da0 (n : \u03b9) in L, \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2191(\u2191(fs n) a) \u2264 (fun x => \u2191c) a", "state_after": "no goals"}, {"tactic": "simpa only [Function.comp_apply, ENNReal.tendsto_coe] using fs_lim", "annotated_tactic": ["simpa only [<a>Function.comp_apply</a>, <a>ENNReal.tendsto_coe</a>] using fs_lim", [{"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.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}]], "state_before": "case refine_2\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 \u2200\u1d50 (a : \u03a9) \u2202\u03bc, Tendsto (fun n => \u2191(\u2191(fs n) a)) L (\ud835\udcdd \u2191(f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "full_name": "MeasureTheory.AEDisjoint.mono_ae", "start": [79, 1], "end": [80, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasurableSet.isClopenable", "start": [299, 1], "end": [305, 49], "traced_tactics": [{"tactic": "revert s", "annotated_tactic": ["revert s", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 IsClopenable s", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 IsClopenable s"}, {"tactic": "apply MeasurableSet.induction_on_open", "annotated_tactic": ["apply <a>MeasurableSet.induction_on_open</a>", [{"full_name": "MeasurableSet.induction_on_open", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 IsClopenable s", "state_after": "case h_open\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 IsClopenable U\n\ncase h_compl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 IsClopenable t \u2192 IsClopenable t\u1d9c\n\ncase h_union\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), IsClopenable (f i)) \u2192 IsClopenable (\u22c3 i, f i)"}, {"tactic": "exact fun u hu => hu.isClopenable", "annotated_tactic": ["exact fun u hu => hu.isClopenable", []], "state_before": "case h_open\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 IsClopenable U", "state_after": "no goals"}, {"tactic": "exact fun u _ h'u => h'u.compl", "annotated_tactic": ["exact fun u _ h'u => h'u.compl", []], "state_before": "case h_compl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 IsClopenable t \u2192 IsClopenable t\u1d9c", "state_after": "no goals"}, {"tactic": "exact fun f _ _ hf => IsClopenable.iUnion hf", "annotated_tactic": ["exact fun f _ _ hf => <a>IsClopenable.iUnion</a> hf", [{"full_name": "PolishSpace.IsClopenable.iUnion", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [425, 9], "def_end_pos": [425, 28]}]], "state_before": "case h_union\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), IsClopenable (f i)) \u2192 IsClopenable (\u22c3 i, f i)", "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.findMin_val", "start": [11, 1], "end": [14, 73], "traced_tactics": [{"tactic": "rw [findMin, headD]", "annotated_tactic": ["rw [<a>findMin</a>, <a>headD</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.findMin", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [183, 19], "def_end_pos": [183, 31]}, {"full_name": "Std.BinomialHeap.Imp.Heap.headD", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [156, 19], "def_end_pos": [156, 29]}]], "state_before": "\u03b1 : Type u_1\ns\u271d : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nk : Heap \u03b1 \u2192 Heap \u03b1\nres : FindMin \u03b1\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 (findMin le k (cons r a c s) res).val = headD le res.val (cons r a c s)", "state_after": "\u03b1 : Type u_1\ns\u271d : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nk : Heap \u03b1 \u2192 Heap \u03b1\nres : FindMin \u03b1\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 (findMin le (k \u2218 cons r a c) s\n        (if le res.val a = true then res else { before := k, val := a, node := c, next := s })).val =\n    headD le (if le res.val a = true then res.val else a) s"}, {"tactic": "split <;> apply findMin_val", "annotated_tactic": ["split <;> apply findMin_val", []], "state_before": "\u03b1 : Type u_1\ns\u271d : Heap \u03b1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nk : Heap \u03b1 \u2192 Heap \u03b1\nres : FindMin \u03b1\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 (findMin le (k \u2218 cons r a c) s\n        (if le res.val a = true then res else { before := k, val := a, node := c, next := s })).val =\n    headD le (if le res.val a = true then res.val else a) s", "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.filterMap", "start": [308, 1], "end": [347, 43], "traced_tactics": [{"tactic": "have H1 (l n acc) : filterMap.go f acc l n =\n    (((g\u2081 l).reverse ++ acc.toList).toAssocList, \u27e8n.1 + (g\u2081 l).length\u27e9) := by\n  induction l generalizing n acc with simp [filterMap.go, *]\n  | cons a b l => match f a b with\n    | none => rfl\n    | some c => simp; rw [Nat.add_right_comm]; rfl", "annotated_tactic": ["have H1 (l n acc) : <a>filterMap.go</a> f acc l n =\n      (((g\u2081 l).<a>reverse</a> ++ acc.toList).<a>toAssocList</a>, \u27e8n.1 + (g\u2081 l).<a>length</a>\u27e9) := by\n    induction l generalizing n acc with simp [<a>filterMap.go</a>, *]\n    | <a>cons</a> a b l => match f a b with\n      | <a>none</a> => rfl\n      | <a>some</a> c => simp; rw [<a>Nat.add_right_comm</a>]; rfl", [{"full_name": "Std.HashMap.Imp.filterMap.go", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [210, 17], "def_end_pos": [210, 19]}, {"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]}, {"full_name": "List.toAssocList", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [230, 13], "def_end_pos": [230, 36]}, {"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": "Std.HashMap.Imp.filterMap.go", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [210, 17], "def_end_pos": [210, 19]}, {"full_name": "Std.AssocList.cons", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [20, 5], "def_end_pos": [20, 9]}, {"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": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"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\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n\u22a2 WF (Imp.filterMap f m)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\n\u22a2 WF (Imp.filterMap f m)"}, {"tactic": "let g l := (g\u2081 l).reverse.toAssocList", "annotated_tactic": ["let g l := (g\u2081 l).reverse.toAssocList", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\n\u22a2 WF (Imp.filterMap f m)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\n\u22a2 WF (Imp.filterMap f m)"}, {"tactic": "let M := StateT (ULift Nat) Id", "annotated_tactic": ["let M := <a>StateT</a> (<a>ULift</a> <a>Nat</a>) <a>Id</a>", [{"full_name": "StateT", "def_path": "lake-packages/lean4/src/lean/Init/Control/State.lean", "def_pos": [14, 5], "def_end_pos": [14, 11]}, {"full_name": "ULift", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [786, 11], "def_end_pos": [786, 16]}, {"full_name": "Nat", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1038, 11], "def_end_pos": [1038, 14]}, {"full_name": "Id", "def_path": "lake-packages/lean4/src/lean/Init/Control/Id.lean", "def_pos": [13, 5], "def_end_pos": [13, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\n\u22a2 WF (Imp.filterMap f m)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type ?u.115304 \u2192 Type ?u.115304 := StateT (ULift Nat) Id\n\u22a2 WF (Imp.filterMap f m)"}, {"tactic": "suffices \u2200 bk sz (h : 0 < bk.length),\n  m.buckets.val.mapM (m := M) (filterMap.go f .nil) \u27e80\u27e9 = (\u27e8bk\u27e9, \u27e8sz\u27e9) \u2192\n  WF \u27e8sz, \u27e8bk\u27e9, h\u27e9 from this _ _ _ rfl", "annotated_tactic": ["suffices \u2200 bk sz (h : 0 < bk.length),\n    m.buckets.val.mapM (m := M) (<a>filterMap.go</a> f .nil) \u27e80\u27e9 = (\u27e8bk\u27e9, \u27e8sz\u27e9) \u2192\n    <a>WF</a> \u27e8sz, \u27e8bk\u27e9, h\u27e9 from this _ _ _ <a>rfl</a>", [{"full_name": "Std.HashMap.Imp.filterMap.go", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [210, 17], "def_end_pos": [210, 19]}, {"full_name": "Std.HashMap.Imp.WF", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [225, 11], "def_end_pos": [225, 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\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\n\u22a2 WF (Imp.filterMap f m)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\n\u22a2 \u2200 (bk : List (AssocList \u03b1 \u03b3)) (sz : Nat) (h : 0 < List.length bk),\n    Array.mapM (filterMap.go f AssocList.nil) m.buckets.val { down := 0 } = ({ data := bk }, { down := sz }) \u2192\n      WF { size := sz, buckets := { val := { data := bk }, property := h } }"}, {"tactic": "simp [Array.mapM_eq_mapM_data, bind, StateT.bind, H2]", "annotated_tactic": ["simp [<a>Array.mapM_eq_mapM_data</a>, <a>bind</a>, <a>StateT.bind</a>, H2]", [{"full_name": "Array.mapM_eq_mapM_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [163, 9], "def_end_pos": [163, 26]}, {"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": "StateT.bind", "def_path": "lake-packages/lean4/src/lean/Init/Control/State.lean", "def_pos": [46, 15], "def_end_pos": [46, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\n\u22a2 \u2200 (bk : List (AssocList \u03b1 \u03b3)) (sz : Nat) (h : 0 < List.length bk),\n    Array.mapM (filterMap.go f AssocList.nil) m.buckets.val { down := 0 } = ({ data := bk }, { down := sz }) \u2192\n      WF { size := sz, buckets := { val := { data := bk }, property := h } }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\n\u22a2 \u2200 (bk : List (AssocList \u03b1 \u03b3)) (sz : Nat) (h : 0 < List.length bk),\n    pure\n          {\n            data :=\n              List.map\n                (fun l =>\n                  List.toAssocList\n                    (List.reverse\n                      (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                        (AssocList.toList l))))\n                m.buckets.val.data }\n          {\n            down :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data) } =\n        ({ data := bk }, { down := sz }) \u2192\n      WF { size := sz, buckets := { val := { data := bk }, property := h } }"}, {"tactic": "intro bk sz h e'", "annotated_tactic": ["intro bk sz h e'", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\n\u22a2 \u2200 (bk : List (AssocList \u03b1 \u03b3)) (sz : Nat) (h : 0 < List.length bk),\n    pure\n          {\n            data :=\n              List.map\n                (fun l =>\n                  List.toAssocList\n                    (List.reverse\n                      (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                        (AssocList.toList l))))\n                m.buckets.val.data }\n          {\n            down :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data) } =\n        ({ data := bk }, { down := sz }) \u2192\n      WF { size := sz, buckets := { val := { data := bk }, property := h } }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nbk : List (AssocList \u03b1 \u03b3)\nsz : Nat\nh : 0 < List.length bk\ne' :\n  pure\n      {\n        data :=\n          List.map\n            (fun l =>\n              List.toAssocList\n                (List.reverse\n                  (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l))))\n            m.buckets.val.data }\n      {\n        down :=\n          Nat.sum\n            (List.map\n              ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                List.toAssocList\n                  (List.reverse\n                    (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                      (AssocList.toList l))))\n              m.buckets.val.data) } =\n    ({ data := bk }, { down := sz })\n\u22a2 WF { size := sz, buckets := { val := { data := bk }, property := h } }"}, {"tactic": "cases e'", "annotated_tactic": ["cases e'", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nbk : List (AssocList \u03b1 \u03b3)\nsz : Nat\nh : 0 < List.length bk\ne' :\n  pure\n      {\n        data :=\n          List.map\n            (fun l =>\n              List.toAssocList\n                (List.reverse\n                  (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l))))\n            m.buckets.val.data }\n      {\n        down :=\n          Nat.sum\n            (List.map\n              ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                List.toAssocList\n                  (List.reverse\n                    (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                      (AssocList.toList l))))\n              m.buckets.val.data) } =\n    ({ data := bk }, { down := sz })\n\u22a2 WF { size := sz, buckets := { val := { data := bk }, property := h } }", "state_after": "case refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\n\u22a2 WF\n    {\n      size :=\n        Nat.sum\n          (List.map\n            ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n              List.toAssocList\n                (List.reverse\n                  (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l))))\n            m.buckets.val.data),\n      buckets :=\n        {\n          val :=\n            {\n              data :=\n                List.map\n                  (fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data },\n          property := h } }"}, {"tactic": "refine .mk (by simp [Buckets.size]) \u27e8?_, fun i h => ?_\u27e9", "annotated_tactic": ["refine .mk (by simp [<a>Buckets.size</a>]) \u27e8?_, fun i h => ?_\u27e9", [{"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 refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\n\u22a2 WF\n    {\n      size :=\n        Nat.sum\n          (List.map\n            ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n              List.toAssocList\n                (List.reverse\n                  (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l))))\n            m.buckets.val.data),\n      buckets :=\n        {\n          val :=\n            {\n              data :=\n                List.map\n                  (fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data },\n          property := h } }", "state_after": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b3),\n    bucket \u2208\n        {\n                size :=\n                  Nat.sum\n                    (List.map\n                      ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                        List.toAssocList\n                          (List.reverse\n                            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                              (AssocList.toList l))))\n                      m.buckets.val.data),\n                buckets :=\n                  {\n                    val :=\n                      {\n                        data :=\n                          List.map\n                            (fun l =>\n                              List.toAssocList\n                                (List.reverse\n                                  (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                    (AssocList.toList l))))\n                            m.buckets.val.data },\n                    property := h } }.buckets.val.data \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\n\ncase refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d } }.buckets.val\n\u22a2 AssocList.All\n    (fun k x =>\n      USize.toNat\n          (UInt64.toUSize (hash k) %\n            Array.size\n              {\n                    size :=\n                      Nat.sum\n                        (List.map\n                          ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                            List.toAssocList\n                              (List.reverse\n                                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                  (AssocList.toList l))))\n                          m.buckets.val.data),\n                    buckets :=\n                      {\n                        val :=\n                          {\n                            data :=\n                              List.map\n                                (fun l =>\n                                  List.toAssocList\n                                    (List.reverse\n                                      (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                        (AssocList.toList l))))\n                                m.buckets.val.data },\n                        property := h\u271d } }.buckets.val) =\n        i)\n    {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d } }.buckets.val[i]"}, {"tactic": "induction l generalizing n acc with simp [filterMap.go, *]\n| cons a b l => match f a b with\n  | none => rfl\n  | some c => simp; rw [Nat.add_right_comm]; rfl", "annotated_tactic": ["induction l generalizing n acc with simp [<a>filterMap.go</a>, *]\n    | <a>cons</a> a b l => match f a b with\n      | <a>none</a> => rfl\n      | <a>some</a> c => simp; rw [<a>Nat.add_right_comm</a>]; rfl", [{"full_name": "Std.HashMap.Imp.filterMap.go", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [210, 17], "def_end_pos": [210, 19]}, {"full_name": "Std.AssocList.cons", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [20, 5], "def_end_pos": [20, 9]}, {"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": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"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\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nl : AssocList \u03b1 \u03b2\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\n\u22a2 filterMap.go f acc l n =\n    (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })", "state_after": "no goals"}, {"tactic": "match f a b with\n| none => rfl\n| some c => simp; rw [Nat.add_right_comm]; rfl", "annotated_tactic": ["match f a b with\n      | <a>none</a> => rfl\n      | <a>some</a> c => simp; rw [<a>Nat.add_right_comm</a>]; 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": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"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 cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\n\u22a2 (match f a b with\n    | none =>\n      (List.toAssocList\n          (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n            AssocList.toList acc),\n        {\n          down :=\n            n.down +\n              List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) })\n    | some c =>\n      (List.toAssocList\n          (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n            (a, c) :: AssocList.toList acc),\n        {\n          down :=\n            n.down + 1 +\n              List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                  (AssocList.toList l)) })) =\n    (List.toAssocList\n        (List.reverse\n            (match Option.map (fun x => (a, x)) (f a b) with\n            | none =>\n              List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n            | some b =>\n              b ::\n                List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n          AssocList.toList acc),\n      {\n        down :=\n          n.down +\n            List.length\n              (match Option.map (fun x => (a, x)) (f a b) with\n              | none =>\n                List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n              | some b =>\n                b ::\n                  List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l)) })", "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\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\n\u22a2 (match none with\n    | none =>\n      (List.toAssocList\n          (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n            AssocList.toList acc),\n        {\n          down :=\n            n.down +\n              List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) })\n    | some c =>\n      (List.toAssocList\n          (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n            (a, c) :: AssocList.toList acc),\n        {\n          down :=\n            n.down + 1 +\n              List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                  (AssocList.toList l)) })) =\n    (List.toAssocList\n        (List.reverse\n            (match Option.map (fun x => (a, x)) none with\n            | none =>\n              List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n            | some b =>\n              b ::\n                List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n          AssocList.toList acc),\n      {\n        down :=\n          n.down +\n            List.length\n              (match Option.map (fun x => (a, x)) none with\n              | none =>\n                List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n              | some b =>\n                b ::\n                  List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l)) })", "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\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\nc : \u03b3\n\u22a2 (match some c with\n    | none =>\n      (List.toAssocList\n          (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n            AssocList.toList acc),\n        {\n          down :=\n            n.down +\n              List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) })\n    | some c =>\n      (List.toAssocList\n          (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n            (a, c) :: AssocList.toList acc),\n        {\n          down :=\n            n.down + 1 +\n              List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                  (AssocList.toList l)) })) =\n    (List.toAssocList\n        (List.reverse\n            (match Option.map (fun x => (a, x)) (some c) with\n            | none =>\n              List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n            | some b =>\n              b ::\n                List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++\n          AssocList.toList acc),\n      {\n        down :=\n          n.down +\n            List.length\n              (match Option.map (fun x => (a, x)) (some c) with\n              | none =>\n                List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\n              | some b =>\n                b ::\n                  List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                    (AssocList.toList l)) })", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\nc : \u03b3\n\u22a2 {\n      down :=\n        n.down + 1 +\n          List.length\n            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) } =\n    {\n      down :=\n        n.down +\n          Nat.succ\n            (List.length\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) }"}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\nc : \u03b3\n\u22a2 {\n      down :=\n        n.down + 1 +\n          List.length\n            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) } =\n    {\n      down :=\n        n.down +\n          Nat.succ\n            (List.length\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) }", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\nc : \u03b3\n\u22a2 {\n      down :=\n        n.down +\n            List.length\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +\n          1 } =\n    {\n      down :=\n        n.down +\n          Nat.succ\n            (List.length\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\na : \u03b1\nb : \u03b2\nl : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\nn : ULift Nat\nacc : AssocList \u03b1 \u03b3\nc : \u03b3\n\u22a2 {\n      down :=\n        n.down +\n            List.length\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +\n          1 } =\n    {\n      down :=\n        n.down +\n          Nat.succ\n            (List.length\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) }", "state_after": "no goals"}, {"tactic": "induction l generalizing n with\n| nil => rfl\n| cons l L IH => simp [bind, StateT.bind, IH, H1, Nat.add_assoc]; rfl", "annotated_tactic": ["induction l generalizing n with\n    | <a>nil</a> => rfl\n    | <a>cons</a> l L IH => simp [<a>bind</a>, <a>StateT.bind</a>, IH, H1, <a>Nat.add_assoc</a>]; rfl", [{"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": "Bind.bind", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2662, 3], "def_end_pos": [2662, 7]}, {"full_name": "StateT.bind", "def_path": "lake-packages/lean4/src/lean/Init/Control/State.lean", "def_pos": [46, 15], "def_end_pos": [46, 19]}, {"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\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nl : List (AssocList \u03b1 \u03b2)\nn : ULift Nat\n\u22a2 List.mapM (filterMap.go f AssocList.nil) l n =\n    (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nn : ULift Nat\n\u22a2 List.mapM (filterMap.go f AssocList.nil) [] n =\n    (List.map g [], { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g [])) })", "state_after": "no goals"}, {"tactic": "simp [bind, StateT.bind, IH, H1, Nat.add_assoc]", "annotated_tactic": ["simp [<a>bind</a>, <a>StateT.bind</a>, IH, H1, <a>Nat.add_assoc</a>]", [{"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": "StateT.bind", "def_path": "lake-packages/lean4/src/lean/Init/Control/State.lean", "def_pos": [46, 15], "def_end_pos": [46, 19]}, {"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 cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nl : AssocList \u03b1 \u03b2\nL : List (AssocList \u03b1 \u03b2)\nIH :\n  \u2200 (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) L n =\n      (List.map g L, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g L)) })\nn : ULift Nat\n\u22a2 List.mapM (filterMap.go f AssocList.nil) (l :: L) n =\n    (List.map g (l :: L),\n      { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g (l :: L))) })", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nl : AssocList \u03b1 \u03b2\nL : List (AssocList \u03b1 \u03b2)\nIH :\n  \u2200 (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) L n =\n      (List.map g L, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g L)) })\nn : ULift Nat\n\u22a2 pure\n      (List.toAssocList\n          (List.reverse\n            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ::\n        List.map\n          (fun l =>\n            List.toAssocList\n              (List.reverse\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n          L)\n      {\n        down :=\n          n.down +\n            (List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  L)) } =\n    (List.toAssocList\n          (List.reverse\n            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ::\n        List.map\n          (fun l =>\n            List.toAssocList\n              (List.reverse\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n          L,\n      {\n        down :=\n          n.down +\n            (List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  L)) })"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nl : AssocList \u03b1 \u03b2\nL : List (AssocList \u03b1 \u03b2)\nIH :\n  \u2200 (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) L n =\n      (List.map g L, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g L)) })\nn : ULift Nat\n\u22a2 pure\n      (List.toAssocList\n          (List.reverse\n            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ::\n        List.map\n          (fun l =>\n            List.toAssocList\n              (List.reverse\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n          L)\n      {\n        down :=\n          n.down +\n            (List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  L)) } =\n    (List.toAssocList\n          (List.reverse\n            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ::\n        List.map\n          (fun l =>\n            List.toAssocList\n              (List.reverse\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n          L,\n      {\n        down :=\n          n.down +\n            (List.length\n                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  L)) })", "state_after": "no goals"}, {"tactic": "induction l with\n| nil => exact .slnil\n| cons a l ih =>\n  simp; exact match f a.1 a.2 with\n  | none => .cons _ ih\n  | some b => .cons\u2082 _ ih", "annotated_tactic": ["induction l with\n    | <a>nil</a> => exact .slnil\n    | <a>cons</a> a l ih =>\n      simp; exact match f a.1 a.2 with\n      | <a>none</a> => .cons _ ih\n      | <a>some</a> b => .cons\u2082 _ 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]}, {"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": "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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 List.Sublist\n    (List.map (fun a => a.fst)\n      (List.filterMap\n        (fun x =>\n          match x with\n          | (a, b) => Option.map (fun x => (a, x)) (f a b))\n        l))\n    (List.map (fun x => x.fst) l)", "state_after": "no goals"}, {"tactic": "exact .slnil", "annotated_tactic": ["exact .slnil", []], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\n\u22a2 List.Sublist\n    (List.map (fun a => a.fst)\n      (List.filterMap\n        (fun x =>\n          match x with\n          | (a, b) => Option.map (fun x => (a, x)) (f a b))\n        []))\n    (List.map (fun x => x.fst) [])", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Sublist\n    (List.map (fun a => a.fst)\n      (List.filterMap\n        (fun x =>\n          match x with\n          | (a, b) => Option.map (fun x => (a, x)) (f a b))\n        l))\n    (List.map (fun x => x.fst) l)\n\u22a2 List.Sublist\n    (List.map (fun a => a.fst)\n      (List.filterMap\n        (fun x =>\n          match x with\n          | (a, b) => Option.map (fun x => (a, x)) (f a b))\n        (a :: l)))\n    (List.map (fun x => x.fst) (a :: l))", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Sublist\n    (List.map (fun a => a.fst)\n      (List.filterMap\n        (fun x =>\n          match x with\n          | (a, b) => Option.map (fun x => (a, x)) (f a b))\n        l))\n    (List.map (fun x => x.fst) l)\n\u22a2 List.Sublist\n    (List.map (fun a => a.fst)\n      (match Option.map (fun x => (a.fst, x)) (f a.fst a.snd) with\n      | none => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l\n      | some b => b :: List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l))\n    (a.fst :: List.map (fun x => x.fst) l)"}, {"tactic": "exact match f a.1 a.2 with\n| none => .cons _ ih\n| some b => .cons\u2082 _ ih", "annotated_tactic": ["exact match f a.1 a.2 with\n      | <a>none</a> => .cons _ ih\n      | <a>some</a> b => .cons\u2082 _ ih", [{"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": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\na : \u03b1 \u00d7 \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\nih :\n  List.Sublist\n    (List.map (fun a => a.fst)\n      (List.filterMap\n        (fun x =>\n          match x with\n          | (a, b) => Option.map (fun x => (a, x)) (f a b))\n        l))\n    (List.map (fun x => x.fst) l)\n\u22a2 List.Sublist\n    (List.map (fun a => a.fst)\n      (match Option.map (fun x => (a.fst, x)) (f a.fst a.snd) with\n      | none => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l\n      | some b => b :: List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l))\n    (a.fst :: List.map (fun x => x.fst) l)", "state_after": "no goals"}, {"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\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\n\u22a2 {\n        size :=\n          Nat.sum\n            (List.map\n              ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                List.toAssocList\n                  (List.reverse\n                    (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                      (AssocList.toList l))))\n              m.buckets.val.data),\n        buckets :=\n          {\n            val :=\n              {\n                data :=\n                  List.map\n                    (fun l =>\n                      List.toAssocList\n                        (List.reverse\n                          (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                            (AssocList.toList l))))\n                    m.buckets.val.data },\n            property := h } }.size =\n    Buckets.size\n      {\n          size :=\n            Nat.sum\n              (List.map\n                ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                  List.toAssocList\n                    (List.reverse\n                      (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                        (AssocList.toList l))))\n                m.buckets.val.data),\n          buckets :=\n            {\n              val :=\n                {\n                  data :=\n                    List.map\n                      (fun l =>\n                        List.toAssocList\n                          (List.reverse\n                            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                              (AssocList.toList l))))\n                      m.buckets.val.data },\n              property := h } }.buckets", "state_after": "no goals"}, {"tactic": "simp only [List.forall_mem_map_iff, List.toAssocList_toList]", "annotated_tactic": ["simp only [<a>List.forall_mem_map_iff</a>, <a>List.toAssocList_toList</a>]", [{"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]}, {"full_name": "List.toAssocList_toList", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [234, 17], "def_end_pos": [234, 47]}]], "state_before": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\n\u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b3),\n    bucket \u2208\n        {\n                size :=\n                  Nat.sum\n                    (List.map\n                      ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                        List.toAssocList\n                          (List.reverse\n                            (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                              (AssocList.toList l))))\n                      m.buckets.val.data),\n                buckets :=\n                  {\n                    val :=\n                      {\n                        data :=\n                          List.map\n                            (fun l =>\n                              List.toAssocList\n                                (List.reverse\n                                  (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                    (AssocList.toList l))))\n                            m.buckets.val.data },\n                    property := h } }.buckets.val.data \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)", "state_after": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\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)\n        (List.reverse\n          (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList j)))"}, {"tactic": "refine fun l h => (List.pairwise_reverse.2 ?_).imp (mt PartialEquivBEq.symm)", "annotated_tactic": ["refine fun l h => (<a>List.pairwise_reverse</a>.2 ?_).<a>imp</a> (<a>mt</a> <a>PartialEquivBEq.symm</a>)", [{"full_name": "List.pairwise_reverse", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1461, 9], "def_end_pos": [1461, 25]}, {"full_name": "List.Pairwise.imp", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1465, 9], "def_end_pos": [1465, 21]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"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 refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\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)\n        (List.reverse\n          (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList j)))", "state_after": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nh : l \u2208 m.buckets.val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))"}, {"tactic": "have := H.out.2.1 _ h", "annotated_tactic": ["have := H.out.2.1 _ h", []], "state_before": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nh : l \u2208 m.buckets.val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))", "state_after": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nh : l \u2208 m.buckets.val.data\nthis : List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList l)\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)\n    (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))"}, {"tactic": "exact this.sublist (H3 l.toList)", "annotated_tactic": ["exact this.sublist (H3 l.toList)", []], "state_before": "case refl.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nh : l \u2208 m.buckets.val.data\nthis : List.Pairwise (fun x x_1 => \u00ac(x == x_1) = true) (List.map (fun a => a.fst) (AssocList.toList l))\n\u22a2 List.Pairwise (fun x x_1 => \u00ac(x == x_1) = true)\n    (List.map (fun a => a.fst)\n      (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)))", "state_after": "no goals"}, {"tactic": "simp [Array.getElem_eq_data_get] at h \u22a2", "annotated_tactic": ["simp [<a>Array.getElem_eq_data_get</a>] at h \u22a2", [{"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 refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d } }.buckets.val\n\u22a2 AssocList.All\n    (fun k x =>\n      USize.toNat\n          (UInt64.toUSize (hash k) %\n            Array.size\n              {\n                    size :=\n                      Nat.sum\n                        (List.map\n                          ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                            List.toAssocList\n                              (List.reverse\n                                (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                  (AssocList.toList l))))\n                          m.buckets.val.data),\n                    buckets :=\n                      {\n                        val :=\n                          {\n                            data :=\n                              List.map\n                                (fun l =>\n                                  List.toAssocList\n                                    (List.reverse\n                                      (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                        (AssocList.toList l))))\n                                m.buckets.val.data },\n                        property := h\u271d } }.buckets.val) =\n        i)\n    {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d } }.buckets.val[i]", "state_after": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % List.length m.buckets.val.data) = i)\n    (List.toAssocList\n      (List.reverse\n        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n          (AssocList.toList\n            (List.get m.buckets.val.data\n              { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) })))))"}, {"tactic": "have := H.out.2.2 _ h", "annotated_tactic": ["have := H.out.2.2 _ h", []], "state_before": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % List.length m.buckets.val.data) = i)\n    (List.toAssocList\n      (List.reverse\n        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n          (AssocList.toList\n            (List.get m.buckets.val.data\n              { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) })))))", "state_after": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\nthis : AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size m.buckets.val) = i) m.buckets.val[i]\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % List.length m.buckets.val.data) = i)\n    (List.toAssocList\n      (List.reverse\n        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n          (AssocList.toList\n            (List.get m.buckets.val.data\n              { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) })))))"}, {"tactic": "simp [AssocList.All] at this \u22a2", "annotated_tactic": ["simp [<a>AssocList.All</a>] at this \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 refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\nthis : AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size m.buckets.val) = i) m.buckets.val[i]\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % List.length m.buckets.val.data) = i)\n    (List.toAssocList\n      (List.reverse\n        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n          (AssocList.toList\n            (List.get m.buckets.val.data\n              { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) })))))", "state_after": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\nthis :\n  \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\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b3) (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          (List.get m.buckets.val.data\n            { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) }) \u2192\n      \u2200 (x_1 : \u03b3),\n        f x.fst x.snd = some x_1 \u2192\n          (x.fst, x_1) = a \u2192 USize.toNat (UInt64.toUSize (hash a.fst) % List.length m.buckets.val.data) = i"}, {"tactic": "rintro _ _ h' _ _ rfl", "annotated_tactic": ["rintro _ _ h' _ _ rfl", []], "state_before": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\nthis :\n  \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\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b3) (x : \u03b1 \u00d7 \u03b2),\n    x \u2208\n        AssocList.toList\n          (List.get m.buckets.val.data\n            { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) }) \u2192\n      \u2200 (x_1 : \u03b3),\n        f x.fst x.snd = some x_1 \u2192\n          (x.fst, x_1) = a \u2192 USize.toNat (UInt64.toUSize (hash a.fst) % List.length m.buckets.val.data) = i", "state_after": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\nthis :\n  \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\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nh' :\n  x\u271d\u00b9 \u2208\n    AssocList.toList\n      (List.get m.buckets.val.data\n        { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) })\nx\u271d : \u03b3\na\u271d : f x\u271d\u00b9.fst x\u271d\u00b9.snd = some x\u271d\n\u22a2 USize.toNat (UInt64.toUSize (hash (x\u271d\u00b9.fst, x\u271d).fst) % List.length m.buckets.val.data) = i"}, {"tactic": "exact this _ h'", "annotated_tactic": ["exact this _ h'", []], "state_before": "case refl.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nH : WF m\ng\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=\n  fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)\nH1 :\n  \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),\n    filterMap.go f acc l n =\n      (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) })\ng : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l))\nM : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id\nH2 :\n  \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),\n    List.mapM (filterMap.go f AssocList.nil) l n =\n      (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) })\nH3 :\n  \u2200 (l : List (\u03b1 \u00d7 \u03b2)),\n    List.Sublist\n      (List.map (fun a => a.fst)\n        (List.filterMap\n          (fun x =>\n            match x with\n            | (a, b) => Option.map (fun x => (a, x)) (f a b))\n          l))\n      (List.map (fun x => x.fst) l)\nh\u271d\u00b9 :\n  0 <\n    List.length\n      (List.map\n        (fun l =>\n          List.toAssocList\n            (List.reverse\n              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))\n        m.buckets.val.data)\ni : Nat\nh\u271d :\n  i <\n    Array.size\n      {\n            size :=\n              Nat.sum\n                (List.map\n                  ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>\n                    List.toAssocList\n                      (List.reverse\n                        (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                          (AssocList.toList l))))\n                  m.buckets.val.data),\n            buckets :=\n              {\n                val :=\n                  {\n                    data :=\n                      List.map\n                        (fun l =>\n                          List.toAssocList\n                            (List.reverse\n                              (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))\n                                (AssocList.toList l))))\n                        m.buckets.val.data },\n                property := h\u271d\u00b9 } }.buckets.val\nh : i < List.length m.buckets.val.data\nthis :\n  \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\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nh' :\n  x\u271d\u00b9 \u2208\n    AssocList.toList\n      (List.get m.buckets.val.data\n        { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) })\nx\u271d : \u03b3\na\u271d : f x\u271d\u00b9.fst x\u271d\u00b9.snd = some x\u271d\n\u22a2 USize.toNat (UInt64.toUSize (hash (x\u271d\u00b9.fst, x\u271d).fst) % List.length m.buckets.val.data) = i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.monomial_one_dvd_iff_modMonomial_eq_zero", "start": [147, 1], "end": [149, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pi.lean", "full_name": "Finset.pi_empty", "start": [87, 1], "end": [88, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.firstDiffPos_eq", "start": [395, 1], "end": [398, 93], "traced_tactics": [{"tactic": "simpa [firstDiffPos] using\n  firstDiffPos_loop_eq [] [] a.1 b.1 ((utf8Len a.1).min (utf8Len b.1)) 0 rfl rfl (by simp)", "annotated_tactic": ["simpa [<a>firstDiffPos</a>] using\n    <a>firstDiffPos_loop_eq</a> [] [] a.1 b.1 ((<a>utf8Len</a> a.1).<a>min</a> (<a>utf8Len</a> b.1)) 0 <a>rfl</a> <a>rfl</a> (by simp)", [{"full_name": "String.firstDiffPos", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [207, 5], "def_end_pos": [207, 17]}, {"full_name": "String.firstDiffPos_loop_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [368, 9], "def_end_pos": [368, 29]}, {"full_name": "String.utf8Len", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [54, 15], "def_end_pos": [54, 22]}, {"full_name": "Nat.min", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [504, 18], "def_end_pos": [504, 21]}, {"full_name": "String.utf8Len", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [54, 15], "def_end_pos": [54, 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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "a b : String\n\u22a2 firstDiffPos a b = { byteIdx := utf8Len (List.takeWhile\u2082 (fun x x_1 => decide (x = x_1)) a.data b.data).fst }", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b : String\n\u22a2 Nat.min (utf8Len a.data) (utf8Len b.data) = min (utf8Len [] + utf8Len a.data) (utf8Len [] + utf8Len b.data)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "toNat_manyOneReducible", "start": [337, 1], "end": [339, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.trNormal_respects", "start": [1612, 1], "end": [1644, 28], "traced_tactics": [{"tactic": "induction c generalizing k v s", "annotated_tactic": ["induction c generalizing k v s", []], "state_before": "c : Code\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal c k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal c (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case zero'\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.zero' k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase succ\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.succ k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase tail\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "case zero' => refine' \u27e8_, \u27e8s, rfl\u27e9, TransGen.single _\u27e9; simp", "annotated_tactic": ["case zero' => refine' \u27e8_, \u27e8s, <a>rfl</a>\u27e9, <a>TransGen.single</a> _\u27e9; simp", [{"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 zero'\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.zero' k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase succ\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.succ k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase tail\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case succ\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.succ k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase tail\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "case succ => refine' \u27e8_, \u27e8none, rfl\u27e9, head_main_ok.trans succ_ok\u27e9", "annotated_tactic": ["case succ => refine' \u27e8_, \u27e8<a>none</a>, <a>rfl</a>\u27e9, head_main_ok.trans <a>succ_ok</a>\u27e9", [{"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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Turing.PartrecToTM2.succ_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 16]}]], "state_before": "case succ\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.succ k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase tail\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case tail\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "case tail =>\n  let o : Option \u0393' := List.casesOn v none fun _ _ => some \u0393'.cons\n  refine' \u27e8_, \u27e8o, rfl\u27e9, _\u27e9; convert clear_ok _ using 2; simp; rfl; swap\n  refine' splitAtPred_eq _ _ (trNat v.headI) _ _ (trNat_natEnd _) _\n  cases v <;> simp", "annotated_tactic": ["case tail =>\n    let o : <a>Option</a> <a>\u0393'</a> := <a>List.casesOn</a> v <a>none</a> fun _ _ => <a>some</a> <a>\u0393'.cons</a>\n    refine' \u27e8_, \u27e8o, <a>rfl</a>\u27e9, _\u27e9; convert <a>clear_ok</a> _ using 2; simp; rfl; swap\n    refine' <a>splitAtPred_eq</a> _ _ (<a>trNat</a> v.headI) _ _ (<a>trNat_natEnd</a> _) _\n    cases v <;> simp", [{"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Turing.PartrecToTM2.\u0393'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [867, 11], "def_end_pos": [867, 13]}, {"full_name": "List.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"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": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Turing.PartrecToTM2.\u0393'.cons", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [869, 5], "def_end_pos": [869, 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": "Turing.PartrecToTM2.clear_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 17]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"full_name": "Turing.PartrecToTM2.trNat", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1194, 5], "def_end_pos": [1194, 10]}, {"full_name": "Turing.PartrecToTM2.trNat_natEnd", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1465, 9], "def_end_pos": [1465, 21]}]], "state_before": "case tail\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "case\n  cons f fs IHf _ =>\n  obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf (Cont.cons\u2081 fs v k) v none\n  refine' \u27e8c, h\u2081, TransGen.head rfl <| (move_ok (by decide) (splitAtPred_false _)).trans _\u27e9\n  simp only [TM2.step, Option.mem_def, elim_stack, elim_update_stack, elim_update_main, ne_eq,\n    Function.update_noteq, elim_main, elim_rev, elim_update_rev]\n  refine' (copy_ok _ none [] (trList v).reverse _ _).trans _\n  convert h\u2082 using 2\n  simp [List.reverseAux_eq, trContStack]", "annotated_tactic": ["case\n    cons f fs IHf _ =>\n    obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf (<a>Cont.cons\u2081</a> fs v k) v <a>none</a>\n    refine' \u27e8c, h\u2081, <a>TransGen.head</a> <a>rfl</a> <| (<a>move_ok</a> (by decide) (<a>splitAtPred_false</a> _)).<a>trans</a> _\u27e9\n    simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>elim_stack</a>, <a>elim_update_stack</a>, <a>elim_update_main</a>, <a>ne_eq</a>,\n      <a>Function.update_noteq</a>, <a>elim_main</a>, <a>elim_rev</a>, <a>elim_update_rev</a>]\n    refine' (<a>copy_ok</a> _ <a>none</a> [] (<a>trList</a> v).<a>reverse</a> _ _).<a>trans</a> _\n    convert h\u2082 using 2\n    simp [<a>List.reverseAux_eq</a>, <a>trContStack</a>]", [{"full_name": "Turing.ToPartrec.Cont.cons\u2081", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [434, 5], "def_end_pos": [434, 10]}, {"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": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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.PartrecToTM2.move_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 16]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_false", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1345, 9], "def_end_pos": [1345, 26]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}, {"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.PartrecToTM2.K'.elim_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 22]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 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": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 20]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1284, 9], "def_end_pos": [1284, 27]}, {"full_name": "Turing.PartrecToTM2.copy_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1440, 9], "def_end_pos": [1440, 16]}, {"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": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"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]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}, {"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}, {"full_name": "Turing.PartrecToTM2.trContStack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1249, 5], "def_end_pos": [1249, 16]}]], "state_before": "case cons\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "case comp f _ _ IHg => exact IHg (Cont.comp f k) v s", "annotated_tactic": ["case comp f _ _ IHg => exact IHg (<a>Cont.comp</a> f k) v s", [{"full_name": "Turing.ToPartrec.Cont.comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [436, 5], "def_end_pos": [436, 9]}]], "state_before": "case comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case case\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082\n\ncase fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "case fix f IH => apply IH", "annotated_tactic": ["case fix f IH => apply IH", []], "state_before": "case fix\na\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "no goals"}, {"tactic": "refine' \u27e8_, \u27e8s, rfl\u27e9, TransGen.single _\u27e9", "annotated_tactic": ["refine' \u27e8_, \u27e8s, <a>rfl</a>\u27e9, <a>TransGen.single</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": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}]], "state_before": "k : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.zero' k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "k : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList (0 :: v)) [] [] (trContStack k) } \u2208\n    TM2.step tr { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "k : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList (0 :: v)) [] [] (trContStack k) } \u2208\n    TM2.step tr { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }", "state_after": "no goals"}, {"tactic": "refine' \u27e8_, \u27e8none, rfl\u27e9, head_main_ok.trans succ_ok\u27e9", "annotated_tactic": ["refine' \u27e8_, \u27e8<a>none</a>, <a>rfl</a>\u27e9, head_main_ok.trans <a>succ_ok</a>\u27e9", [{"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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Turing.PartrecToTM2.succ_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 16]}]], "state_before": "k : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.succ k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "no goals"}, {"tactic": "let o : Option \u0393' := List.casesOn v none fun _ _ => some \u0393'.cons", "annotated_tactic": ["let o : <a>Option</a> <a>\u0393'</a> := <a>List.casesOn</a> v <a>none</a> fun _ _ => <a>some</a> <a>\u0393'.cons</a>", [{"full_name": "Option", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2139, 11], "def_end_pos": [2139, 17]}, {"full_name": "Turing.PartrecToTM2.\u0393'", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [867, 11], "def_end_pos": [867, 13]}, {"full_name": "List.casesOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"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": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Turing.PartrecToTM2.\u0393'.cons", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [869, 5], "def_end_pos": [869, 9]}]], "state_before": "k : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "k : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "refine' \u27e8_, \u27e8o, rfl\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8_, \u27e8o, <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": "k : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal Code.tail k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "k : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n    { l := some (\u039b'.ret (trCont k)), var := o, stk := elim (trList (List.tail v)) [] [] (trContStack k) }"}, {"tactic": "convert clear_ok _ using 2", "annotated_tactic": ["convert <a>clear_ok</a> _ using 2", [{"full_name": "Turing.PartrecToTM2.clear_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 17]}]], "state_before": "k : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n    { l := some (\u039b'.ret (trCont k)), var := o, stk := elim (trList (List.tail v)) [] [] (trContStack k) }", "state_after": "case h.e'_4.h.e'_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack k) = update (elim (trList v) [] [] (trContStack k)) main ?convert_7\n\ncase convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, ?convert_7)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_4.h.e'_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack k) = update (elim (trList v) [] [] (trContStack k)) main ?convert_7\n\ncase convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, ?convert_7)", "state_after": "case h.e'_4.h.e'_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack k) = elim ?convert_7 [] [] (trContStack k)\n\ncase convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, ?convert_7)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_4.h.e'_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack k) = elim ?convert_7 [] [] (trContStack k)\n\ncase convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_7\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, ?convert_7)", "state_after": "case convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, trList (List.tail v))"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'\n\ncase convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, trList (List.tail v))", "state_after": "case convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, trList (List.tail v))\n\ncase convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'"}, {"tactic": "refine' splitAtPred_eq _ _ (trNat v.headI) _ _ (trNat_natEnd _) _", "annotated_tactic": ["refine' <a>splitAtPred_eq</a> _ _ (<a>trNat</a> v.headI) _ _ (<a>trNat_natEnd</a> _) _", [{"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"full_name": "Turing.PartrecToTM2.trNat", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1194, 5], "def_end_pos": [1194, 10]}, {"full_name": "Turing.PartrecToTM2.trNat_natEnd", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1465, 9], "def_end_pos": [1465, 21]}]], "state_before": "case convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, trList (List.tail v))\n\ncase convert_5\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 List \u0393'", "state_after": "case convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 Option.elim' (elim (trList v) [] [] (trContStack k) main = trNat (List.headI v) \u2227 trList (List.tail v) = [])\n    (fun a =>\n      natEnd a = true \u2227 elim (trList v) [] [] (trContStack k) main = trNat (List.headI v) ++ a :: trList (List.tail v))\n    o"}, {"tactic": "cases v <;> simp", "annotated_tactic": ["cases v <;> simp", []], "state_before": "case convert_9\nk : Cont\nv : List \u2115\ns : Option \u0393'\no : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons\n\u22a2 Option.elim' (elim (trList v) [] [] (trContStack k) main = trNat (List.headI v) \u2227 trList (List.tail v) = [])\n    (fun a =>\n      natEnd a = true \u2227 elim (trList v) [] [] (trContStack k) main = trNat (List.headI v) ++ a :: trList (List.tail v))\n    o", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf (Cont.cons\u2081 fs v k) v none", "annotated_tactic": ["obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf (<a>Cont.cons\u2081</a> fs v k) v <a>none</a>", [{"full_name": "Turing.ToPartrec.Cont.cons\u2081", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [434, 5], "def_end_pos": [434, 10]}, {"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": "f fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons f fs) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons f fs) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons f fs) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons f fs) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "refine' \u27e8c, h\u2081, TransGen.head rfl <| (move_ok (by decide) (splitAtPred_false _)).trans _\u27e9", "annotated_tactic": ["refine' \u27e8c, h\u2081, <a>TransGen.head</a> <a>rfl</a> <| (<a>move_ok</a> (by decide) (<a>splitAtPred_false</a> _)).<a>trans</a> _\u27e9", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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.PartrecToTM2.move_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 16]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_false", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1345, 9], "def_end_pos": [1345, 26]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}]], "state_before": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.cons f fs) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.cons f fs) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (\u039b'.copy (trNormal f (Cont'.cons\u2081 fs (trCont k)))), var := none,\n      stk :=\n        update\n          (update\n            (update (elim (trList v) [] [] (trContStack k)) stack\n              ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack))\n            main [])\n          rev\n          (List.reverseAux\n            (update (elim (trList v) [] [] (trContStack k)) stack\n              ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack)\n              main)\n            (update (elim (trList v) [] [] (trContStack k)) stack\n              ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack)\n              rev)) }\n    c"}, {"tactic": "simp only [TM2.step, Option.mem_def, elim_stack, elim_update_stack, elim_update_main, ne_eq,\n  Function.update_noteq, elim_main, elim_rev, elim_update_rev]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>elim_stack</a>, <a>elim_update_stack</a>, <a>elim_update_main</a>, <a>ne_eq</a>,\n      <a>Function.update_noteq</a>, <a>elim_main</a>, <a>elim_rev</a>, <a>elim_update_rev</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.PartrecToTM2.K'.elim_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 22]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 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": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 20]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1284, 9], "def_end_pos": [1284, 27]}]], "state_before": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (\u039b'.copy (trNormal f (Cont'.cons\u2081 fs (trCont k)))), var := none,\n      stk :=\n        update\n          (update\n            (update (elim (trList v) [] [] (trContStack k)) stack\n              ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack))\n            main [])\n          rev\n          (List.reverseAux\n            (update (elim (trList v) [] [] (trContStack k)) stack\n              ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack)\n              main)\n            (update (elim (trList v) [] [] (trContStack k)) stack\n              ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack)\n              rev)) }\n    c", "state_after": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (\u039b'.copy (trNormal f (Cont'.cons\u2081 fs (trCont k)))), var := none,\n      stk := elim [] (List.reverseAux (trList v) []) [] (Option.iget (some \u0393'.cons\u2097) :: trContStack k) }\n    c"}, {"tactic": "refine' (copy_ok _ none [] (trList v).reverse _ _).trans _", "annotated_tactic": ["refine' (<a>copy_ok</a> _ <a>none</a> [] (<a>trList</a> v).<a>reverse</a> _ _).<a>trans</a> _", [{"full_name": "Turing.PartrecToTM2.copy_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1440, 9], "def_end_pos": [1440, 16]}, {"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": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"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]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}]], "state_before": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (\u039b'.copy (trNormal f (Cont'.cons\u2081 fs (trCont k)))), var := none,\n      stk := elim [] (List.reverseAux (trList v) []) [] (Option.iget (some \u0393'.cons\u2097) :: trContStack k) }\n    c", "state_after": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (trNormal f (Cont'.cons\u2081 fs (trCont k))), var := none,\n      stk :=\n        elim (List.reverseAux (List.reverse (trList v)) []) [] []\n          (List.reverseAux (List.reverse (trList v)) (Option.iget (some \u0393'.cons\u2097) :: trContStack k)) }\n    c"}, {"tactic": "convert h\u2082 using 2", "annotated_tactic": ["convert h\u2082 using 2", []], "state_before": "case intro.intro\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (trNormal f (Cont'.cons\u2081 fs (trCont k))), var := none,\n      stk :=\n        elim (List.reverseAux (List.reverse (trList v)) []) [] []\n          (List.reverseAux (List.reverse (trList v)) (Option.iget (some \u0393'.cons\u2097) :: trContStack k)) }\n    c", "state_after": "case h.e'_1.h.e'_7\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 elim (List.reverseAux (List.reverse (trList v)) []) [] []\n      (List.reverseAux (List.reverse (trList v)) (Option.iget (some \u0393'.cons\u2097) :: trContStack k)) =\n    elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k))"}, {"tactic": "simp [List.reverseAux_eq, trContStack]", "annotated_tactic": ["simp [<a>List.reverseAux_eq</a>, <a>trContStack</a>]", [{"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}, {"full_name": "Turing.PartrecToTM2.trContStack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1249, 5], "def_end_pos": [1249, 16]}]], "state_before": "case h.e'_1.h.e'_7\nf fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 elim (List.reverseAux (List.reverse (trList v)) []) [] []\n      (List.reverseAux (List.reverse (trList v)) (Option.iget (some \u0393'.cons\u2097) :: trContStack k)) =\n    elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k))", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "f fs : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal fs k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,\n      stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }\n    c\n\u22a2 main \u2260 rev", "state_after": "no goals"}, {"tactic": "exact IHg (Cont.comp f k) v s", "annotated_tactic": ["exact IHg (<a>Cont.comp</a> f k) v s", [{"full_name": "Turing.ToPartrec.Cont.comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [436, 5], "def_end_pos": [436, 9]}]], "state_before": "f a\u271d : Code\na_ih\u271d :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal a\u271d k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.comp f a\u271d) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.comp f a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "no goals"}, {"tactic": "rw [stepNormal]", "annotated_tactic": ["rw [<a>stepNormal</a>]", [{"full_name": "Turing.ToPartrec.stepNormal", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [489, 5], "def_end_pos": [489, 15]}]], "state_before": "f g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.case f g) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "f g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg\n        ((fun k v =>\n            Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v))\n          k v)\n        b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "state_before": "f g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg\n        ((fun k v =>\n            Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v))\n          k v)\n        b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "f g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "obtain \u27e8s', h\u27e9 := pred_ok _ _ s v _ _", "annotated_tactic": ["obtain \u27e8s', h\u27e9 := <a>pred_ok</a> _ _ s v _ _", [{"full_name": "Turing.PartrecToTM2.pred_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1565, 9], "def_end_pos": [1565, 16]}]], "state_before": "f g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      (List.headI v))\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "revert h", "annotated_tactic": ["revert h", []], "state_before": "case intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      (List.headI v))\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr)\n      { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n      (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n        (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n        (List.headI v)) \u2192\n    \u2203 b\u2082,\n      TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "cases' v.headI with n <;> intro h", "annotated_tactic": ["cases' v.headI with n <;> intro h", []], "state_before": "case intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr)\n      { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n      (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n        (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n        (List.headI v)) \u2192\n    \u2203 b\u2082,\n      TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro.zero\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      Nat.zero)\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) Nat.zero) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\n\ncase intro.succ\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nn : \u2115\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      (Nat.succ n))\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (Nat.succ n)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf k _ s'", "annotated_tactic": ["obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf k _ s'", []], "state_before": "case intro.zero\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      Nat.zero)\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) Nat.zero) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro.zero.intro.intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      Nat.zero)\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f k ?m.552695) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont k)), var := s', stk := elim (trList ?m.552695) [] [] (trContStack k) } c\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) Nat.zero) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "exact \u27e8_, h\u2081, h.trans h\u2082\u27e9", "annotated_tactic": ["exact \u27e8_, h\u2081, h.trans h\u2082\u27e9", []], "state_before": "case intro.zero.intro.intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }\n    (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }\n      (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })\n      Nat.zero)\nc : Cfg'\nh\u2081 : TrCfg (stepNormal f k ?m.552695) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont k)), var := s', stk := elim (trList ?m.552695) [] [] (trContStack k) } c\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) Nat.zero) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHg k _ s'", "annotated_tactic": ["obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHg k _ s'", []], "state_before": "case intro.succ\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nn : \u2115\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred (trNormal f (trCont k)) (trNormal g (trCont k))), var := s,\n      stk := elim (trList v) [] [] (trContStack k) }\n    (Nat.rec { l := some (trNormal f (trCont k)), var := s', stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n      (fun n x =>\n        { l := some (trNormal g (trCont k)), var := s', stk := elim (trList (n :: List.tail v)) [] [] (trContStack k) })\n      (Nat.succ n))\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (Nat.succ n)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "case intro.succ.intro.intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nn : \u2115\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred (trNormal f (trCont k)) (trNormal g (trCont k))), var := s,\n      stk := elim (trList v) [] [] (trContStack k) }\n    (Nat.rec { l := some (trNormal f (trCont k)), var := s', stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n      (fun n x =>\n        { l := some (trNormal g (trCont k)), var := s', stk := elim (trList (n :: List.tail v)) [] [] (trContStack k) })\n      (Nat.succ n))\nc : Cfg'\nh\u2081 : TrCfg (stepNormal g k ?m.552819) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal g (trCont k)), var := s', stk := elim (trList ?m.552819) [] [] (trContStack k) } c\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (Nat.succ n)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082"}, {"tactic": "exact \u27e8_, h\u2081, h.trans h\u2082\u27e9", "annotated_tactic": ["exact \u27e8_, h\u2081, h.trans h\u2082\u27e9", []], "state_before": "case intro.succ.intro.intro\nf g : Code\nIHf :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nIHg :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal g k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns s' : Option \u0393'\nn : \u2115\nh :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.pred (trNormal f (trCont k)) (trNormal g (trCont k))), var := s,\n      stk := elim (trList v) [] [] (trContStack k) }\n    (Nat.rec { l := some (trNormal f (trCont k)), var := s', stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n      (fun n x =>\n        { l := some (trNormal g (trCont k)), var := s', stk := elim (trList (n :: List.tail v)) [] [] (trContStack k) })\n      (Nat.succ n))\nc : Cfg'\nh\u2081 : TrCfg (stepNormal g k ?m.552819) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal g (trCont k)), var := s', stk := elim (trList ?m.552819) [] [] (trContStack k) } c\n\u22a2 \u2203 b\u2082,\n    TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (Nat.succ n)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "no goals"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "f : Code\nIH :\n  \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepNormal f k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082\nk : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal (Code.fix f) k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (trNormal (Code.fix f) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.assert_pos", "start": [472, 1], "end": [478, 10], "traced_tactics": [{"tactic": "dsimp [assert]", "annotated_tactic": ["dsimp [<a>assert</a>]", [{"full_name": "Part.assert", "def_path": "Mathlib/Data/Part.lean", "def_pos": [422, 5], "def_end_pos": [422, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\n\u22a2 assert p f = f h", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\n\u22a2 { Dom := \u2203 h, (f h).Dom, get := fun ha => get (f (_ : p)) (_ : (f (_ : p)).Dom) } = f h"}, {"tactic": "cases h' : f h", "annotated_tactic": ["cases h' : f h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\n\u22a2 { Dom := \u2203 h, (f h).Dom, get := fun ha => get (f (_ : p)) (_ : (f (_ : p)).Dom) } = f h", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 { Dom := \u2203 h, (f h).Dom, get := fun ha => get (f (_ : p)) (_ : (f (_ : p)).Dom) } = { Dom := Dom\u271d, get := get\u271d }"}, {"tactic": "simp only [h', mk.injEq, h, exists_prop_of_true, true_and]", "annotated_tactic": ["simp only [h', mk.injEq, h, <a>exists_prop_of_true</a>, <a>true_and</a>]", [{"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [879, 9], "def_end_pos": [879, 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 mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 { Dom := \u2203 h, (f h).Dom, get := fun ha => get (f (_ : p)) (_ : (f (_ : p)).Dom) } = { Dom := Dom\u271d, get := get\u271d }", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 HEq (fun ha => get\u271d (_ : { Dom := Dom\u271d, get := get\u271d }.Dom)) get\u271d"}, {"tactic": "apply Function.hfunext", "annotated_tactic": ["apply <a>Function.hfunext</a>", [{"full_name": "Function.hfunext", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [70, 7], "def_end_pos": [70, 14]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 HEq (fun ha => get\u271d (_ : { Dom := Dom\u271d, get := get\u271d }.Dom)) get\u271d", "state_after": "case mk.h\u03b1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 (\u2203 h, (f h).Dom) = Dom\u271d\n\ncase mk.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 \u2200 (a : \u2203 h, (f h).Dom) (a' : Dom\u271d), HEq a a' \u2192 HEq (get\u271d (_ : { Dom := Dom\u271d, get := get\u271d }.Dom)) (get\u271d a')"}, {"tactic": "simp only [h, h', exists_prop_of_true]", "annotated_tactic": ["simp only [h, h', <a>exists_prop_of_true</a>]", [{"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [879, 9], "def_end_pos": [879, 28]}]], "state_before": "case mk.h\u03b1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 (\u2203 h, (f h).Dom) = Dom\u271d", "state_after": "no goals"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "case mk.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Prop\nf : p \u2192 Part \u03b1\nh : p\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\nh' : f h = { Dom := Dom\u271d, get := get\u271d }\n\u22a2 \u2200 (a : \u2203 h, (f h).Dom) (a' : Dom\u271d), HEq a a' \u2192 HEq (get\u271d (_ : { Dom := Dom\u271d, get := get\u271d }.Dom)) (get\u271d a')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.mul_le_addHaar_image_of_lt_det", "start": [392, 1], "end": [460, 51], "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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\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 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0"}, {"tactic": "rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> (<a>zero_le</a> m) with (rfl | mpos)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}, {"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 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0", "state_after": "case a.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\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\nhm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n\ncase a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0"}, {"tactic": "have hA : A.det \u2260 0 := by\n  intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm", "annotated_tactic": ["have hA : A.det \u2260 0 := by\n    intro h; simp only [h, <a>ENNReal.not_lt_zero</a>, <a>ENNReal.ofReal_zero</a>, <a>abs_zero</a>] at hm", [{"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}, {"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 a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0", "state_after": "case a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0"}, {"tactic": "let B := A.toContinuousLinearEquivOfDetNeZero hA", "annotated_tactic": ["let B := A.toContinuousLinearEquivOfDetNeZero hA", []], "state_before": "case a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0", "state_after": "case a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0"}, {"tactic": "have I : ENNReal.ofReal |(B.symm : E \u2192L[\u211d] E).det| < (m\u207b\u00b9 : \u211d\u22650) := by\n  simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,\n    ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm \u22a2\n  exact NNReal.inv_lt_inv mpos.ne' hm", "annotated_tactic": ["have I : <a>ENNReal.ofReal</a> |(B.symm : E \u2192L[\u211d] E).<a>det</a>| < (m\u207b\u00b9 : \u211d\u22650) := by\n    simp only [<a>ENNReal.ofReal</a>, <a>abs_inv</a>, <a>Real.toNNReal_inv</a>, <a>ContinuousLinearEquiv.det_coe_symm</a>,\n      <a>ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero</a>, <a>ENNReal.coe_lt_coe</a>] at hm \u22a2\n    exact <a>NNReal.inv_lt_inv</a> mpos.ne' hm", [{"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": "abs_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 16]}, {"full_name": "Real.toNNReal_inv", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [897, 9], "def_end_pos": [897, 33]}, {"full_name": "ContinuousLinearEquiv.det_coe_symm", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [32, 9], "def_end_pos": [32, 21]}, {"full_name": "ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [486, 9], "def_end_pos": [486, 47]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "NNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [922, 9], "def_end_pos": [922, 19]}]], "state_before": "case a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0", "state_after": "case a.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0"}, {"tactic": "obtain \u27e8\u03b4\u2080, \u03b4\u2080pos, h\u03b4\u2080\u27e9 :\n  \u2203 \u03b4 : \u211d\u22650,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t := by\n  have :\n    \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n    addHaar_image_le_mul_of_det_lt \u03bc B.symm I\n  rcases (this.and self_mem_nhdsWithin).exists with \u27e8\u03b4\u2080, h, h'\u27e9\n  exact \u27e8\u03b4\u2080, h', h\u27e9", "annotated_tactic": ["obtain \u27e8\u03b4\u2080, \u03b4\u2080pos, h\u03b4\u2080\u27e9 :\n    \u2203 \u03b4 : \u211d\u22650,\n      0 < \u03b4 \u2227\n        \u2200 (t : <a>Set</a> E) (g : E \u2192 E),\n          <a>ApproximatesLinearOn</a> g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t := by\n    have :\n      \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n        \u2200 (t : <a>Set</a> E) (g : E \u2192 E),\n          <a>ApproximatesLinearOn</a> g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n      <a>addHaar_image_le_mul_of_det_lt</a> \u03bc B.symm I\n    rcases (this.and <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8\u03b4\u2080, h, h'\u27e9\n    exact \u27e8\u03b4\u2080, h', h\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": "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": "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": "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.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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0", "state_after": "case a.inr.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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0"}, {"tactic": "filter_upwards [L1, L2]", "annotated_tactic": ["filter_upwards [L1, L2]", []], "state_before": "case a.inr.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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u22a2 \u2200 (a : \u211d\u22650),\n    Subsingleton E \u2228 a < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 \u2192\n      \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - a)\u207b\u00b9 * a < \u03b4\u2080 \u2192\n        \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "intro \u03b4 h1\u03b4 h2\u03b4 s f hf", "annotated_tactic": ["intro \u03b4 h1\u03b4 h2\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u22a2 \u2200 (a : \u211d\u22650),\n    Subsingleton E \u2228 a < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 \u2192\n      \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - a)\u207b\u00b9 * a < \u03b4\u2080 \u2192\n        \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "have hf' : ApproximatesLinearOn f (B : E \u2192L[\u211d] E) s \u03b4 := by convert hf", "annotated_tactic": ["have hf' : <a>ApproximatesLinearOn</a> f (B : E \u2192L[\u211d] E) s \u03b4 := by convert hf", [{"full_name": "ApproximatesLinearOn", "def_path": "Mathlib/Analysis/Calculus/Inverse.lean", "def_pos": [116, 5], "def_end_pos": [116, 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "let F := hf'.toLocalEquiv h1\u03b4", "annotated_tactic": ["let F := hf'.toLocalEquiv h1\u03b4", []], "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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)", "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "suffices H : \u03bc (F.symm '' F.target) \u2264 (m\u207b\u00b9 : \u211d\u22650) * \u03bc F.target", "annotated_tactic": ["suffices H : \u03bc (F.symm '' F.target) \u2264 (m\u207b\u00b9 : \u211d\u22650) * \u03bc F.target", []], "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)", "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\ncase 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\n\u22a2 \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target"}, {"tactic": "exact h\u03b4\u2080 _ _ ((hf'.to_inv h1\u03b4).mono_num h2\u03b4.le)", "annotated_tactic": ["exact h\u03b4\u2080 _ _ ((hf'.to_inv h1\u03b4).<a>mono_num</a> h2\u03b4.le)", [{"full_name": "ApproximatesLinearOn.mono_num", "def_path": "Mathlib/Analysis/Calculus/Inverse.lean", "def_pos": [137, 9], "def_end_pos": [137, 17]}]], "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\n\u22a2 \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target", "state_after": "no goals"}, {"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 a.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\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\nhm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0", "state_after": "case a.inl.hp\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\nhm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2200 (x : \u211d\u22650) (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s x \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)"}, {"tactic": "simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]", "annotated_tactic": ["simp only [<a>forall_const</a>, <a>zero_mul</a>, <a>imp_true_iff</a>, <a>zero_le</a>, <a>ENNReal.coe_zero</a>]", [{"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"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.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}]], "state_before": "case a.inl.hp\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\nhm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2200 (x : \u211d\u22650) (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s x \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\n\u22a2 ContinuousLinearMap.det A \u2260 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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nh : ContinuousLinearMap.det A = 0\n\u22a2 False"}, {"tactic": "simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm", "annotated_tactic": ["simp only [h, <a>ENNReal.not_lt_zero</a>, <a>ENNReal.ofReal_zero</a>, <a>abs_zero</a>] at hm", [{"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}, {"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": "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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nh : ContinuousLinearMap.det A = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,\n  ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm \u22a2", "annotated_tactic": ["simp only [<a>ENNReal.ofReal</a>, <a>abs_inv</a>, <a>Real.toNNReal_inv</a>, <a>ContinuousLinearEquiv.det_coe_symm</a>,\n      <a>ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero</a>, <a>ENNReal.coe_lt_coe</a>] at hm \u22a2", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "abs_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 16]}, {"full_name": "Real.toNNReal_inv", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [897, 9], "def_end_pos": [897, 33]}, {"full_name": "ContinuousLinearEquiv.det_coe_symm", "def_path": "Mathlib/Topology/Algebra/Module/Determinant.lean", "def_pos": [32, 9], "def_end_pos": [32, 21]}, {"full_name": "ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero", "def_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "def_pos": [486, 9], "def_end_pos": [486, 47]}, {"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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9", "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\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nhm : m < Real.toNNReal |ContinuousLinearMap.det A|\n\u22a2 (Real.toNNReal |ContinuousLinearMap.det A|)\u207b\u00b9 < m\u207b\u00b9"}, {"tactic": "exact NNReal.inv_lt_inv mpos.ne' hm", "annotated_tactic": ["exact <a>NNReal.inv_lt_inv</a> mpos.ne' hm", [{"full_name": "NNReal.inv_lt_inv", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [922, 9], "def_end_pos": [922, 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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nhm : m < Real.toNNReal |ContinuousLinearMap.det A|\n\u22a2 (Real.toNNReal |ContinuousLinearMap.det A|)\u207b\u00b9 < m\u207b\u00b9", "state_after": "no goals"}, {"tactic": "have :\n  \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n    \u2200 (t : Set E) (g : E \u2192 E),\n      ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n  addHaar_image_le_mul_of_det_lt \u03bc B.symm I", "annotated_tactic": ["have :\n      \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n        \u2200 (t : <a>Set</a> E) (g : E \u2192 E),\n          <a>ApproximatesLinearOn</a> g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n      <a>addHaar_image_le_mul_of_det_lt</a> \u03bc B.symm I", [{"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": "MeasureTheory.addHaar_image_le_mul_of_det_lt", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [285, 9], "def_end_pos": [285, 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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\nthis :\n  \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (t : Set E) (g : E \u2192 E),\n      ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t"}, {"tactic": "rcases (this.and self_mem_nhdsWithin).exists with \u27e8\u03b4\u2080, h, h'\u27e9", "annotated_tactic": ["rcases (this.and <a>self_mem_nhdsWithin</a>).<a>exists</a> with \u27e8\u03b4\u2080, h, h'\u27e9", [{"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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\nthis :\n  \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (t : Set E) (g : E \u2192 E),\n      ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\nthis :\n  \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (t : Set E) (g : E \u2192 E),\n      ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u03b4\u2080 : \u211d\u22650\nh :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh' : 0 < \u03b4\u2080\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t"}, {"tactic": "exact \u27e8\u03b4\u2080, h', h\u27e9", "annotated_tactic": ["exact \u27e8\u03b4\u2080, h', h\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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\nthis :\n  \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (t : Set E) (g : E \u2192 E),\n      ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u03b4\u2080 : \u211d\u22650\nh :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh' : 0 < \u03b4\u2080\n\u22a2 \u2203 \u03b4,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t", "state_after": "no goals"}, {"tactic": "by_cases Subsingleton E", "annotated_tactic": ["by_cases <a>Subsingleton</a> E", [{"full_name": "Subsingleton", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [867, 7], "def_end_pos": [867, 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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9", "state_after": "case pos\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : Subsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n\ncase neg\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9"}, {"tactic": "simp only [h, false_or_iff]", "annotated_tactic": ["simp only [h, <a>false_or_iff</a>]", [{"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "case neg\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9", "state_after": "case neg\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0,\n    \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9"}, {"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": "case neg\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0,\n    \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9", "state_after": "case neg.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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 0 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9"}, {"tactic": "simpa only [h, false_or_iff, inv_pos] using B.subsingleton_or_nnnorm_symm_pos", "annotated_tactic": ["simpa only [h, <a>false_or_iff</a>, <a>inv_pos</a>] using B.subsingleton_or_nnnorm_symm_pos", [{"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "case neg.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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 0 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simp only [h, true_or_iff, eventually_const]", "annotated_tactic": ["simp only [h, <a>true_or_iff</a>, <a>eventually_const</a>]", [{"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"full_name": "Filter.eventually_const", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 25]}]], "state_before": "case pos\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : Subsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simp only [mul_zero] at this", "annotated_tactic": ["simp only [<a>mul_zero</a>] at this", [{"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\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nthis :\n  Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080", "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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nthis :\n  Tendsto\n    (fun \u03b4 =>\n      \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n          (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n        \u03b4)\n    (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080"}, {"tactic": "exact (tendsto_order.1 this).2 \u03b4\u2080 \u03b4\u2080pos", "annotated_tactic": ["exact (<a>tendsto_order</a>.1 this).2 \u03b4\u2080 \u03b4\u2080pos", [{"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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nthis :\n  Tendsto\n    (fun \u03b4 =>\n      \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n          (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n        \u03b4)\n    (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080", "state_after": "no goals"}, {"tactic": "rcases eq_or_ne \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a 0 with (H | H)", "annotated_tactic": ["rcases <a>eq_or_ne</a> \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a 0 with (H | H)", [{"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\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))", "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a = 0\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))\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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))"}, {"tactic": "refine' Tendsto.mul (tendsto_const_nhds.mul _) tendsto_id", "annotated_tactic": ["refine' <a>Tendsto.mul</a> (tendsto_const_nhds.mul _) <a>tendsto_id</a>", [{"full_name": "Filter.Tendsto.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}]], "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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))", "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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 Tendsto (fun \u03b4 => (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9) (\ud835\udcdd 0) (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9)"}, {"tactic": "refine' (Tendsto.sub tendsto_const_nhds tendsto_id).inv\u2080 _", "annotated_tactic": ["refine' (<a>Tendsto.sub</a> <a>tendsto_const_nhds</a> <a>tendsto_id</a>).<a>inv\u2080</a> _", [{"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Filter.Tendsto.inv\u2080", "def_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "def_pos": [114, 9], "def_end_pos": [114, 28]}]], "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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 Tendsto (fun \u03b4 => (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9) (\ud835\udcdd 0) (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9)", "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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0 \u2260 0"}, {"tactic": "simpa only [tsub_zero, inv_eq_zero, Ne.def] using H", "annotated_tactic": ["simpa only [<a>tsub_zero</a>, <a>inv_eq_zero</a>, <a>Ne.def</a>] using H", [{"full_name": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 18]}, {"full_name": "inv_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 20]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "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\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0 \u2260 0", "state_after": "no goals"}, {"tactic": "simpa only [H, zero_mul] using tendsto_const_nhds", "annotated_tactic": ["simpa only [H, <a>zero_mul</a>] using <a>tendsto_const_nhds</a>", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a = 0\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n    (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))", "state_after": "no goals"}, {"tactic": "convert hf", "annotated_tactic": ["convert hf", []], "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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 ApproximatesLinearOn f (\u2191B) s \u03b4", "state_after": "no goals"}, {"tactic": "change (m : \u211d\u22650\u221e) * \u03bc F.source \u2264 \u03bc F.target", "annotated_tactic": ["change (m : \u211d\u22650\u221e) * \u03bc F.source \u2264 \u03bc F.target", []], "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)", "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m * \u2191\u2191\u03bc F.source \u2264 \u2191\u2191\u03bc F.target"}, {"tactic": "rwa [\u2190 F.symm_image_target_eq_source, mul_comm, \u2190 ENNReal.le_div_iff_mul_le, div_eq_mul_inv,\n  mul_comm, \u2190 ENNReal.coe_inv mpos.ne']", "annotated_tactic": ["rwa [\u2190 F.symm_image_target_eq_source, <a>mul_comm</a>, \u2190 <a>ENNReal.le_div_iff_mul_le</a>, <a>div_eq_mul_inv</a>,\n      <a>mul_comm</a>, \u2190 <a>ENNReal.coe_inv</a> mpos.ne']", [{"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": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.coe_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1387, 9], "def_end_pos": [1387, 16]}]], "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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m * \u2191\u2191\u03bc F.source \u2264 \u2191\u2191\u03bc F.target", "state_after": "case h.h0\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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 0 \u2228 \u2191\u2191\u03bc F.target \u2260 0\n\ncase h.ht\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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 \u22a4 \u2228 \u2191\u2191\u03bc F.target \u2260 \u22a4"}, {"tactic": "apply Or.inl", "annotated_tactic": ["apply <a>Or.inl</a>", [{"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 h.h0\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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 0 \u2228 \u2191\u2191\u03bc F.target \u2260 0", "state_after": "case h.h0.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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 0"}, {"tactic": "simpa only [ENNReal.coe_eq_zero, Ne.def] using mpos.ne'", "annotated_tactic": ["simpa only [<a>ENNReal.coe_eq_zero</a>, <a>Ne.def</a>] using mpos.ne'", [{"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]}]], "state_before": "case h.h0.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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.coe_ne_top, true_or_iff, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [<a>ENNReal.coe_ne_top</a>, <a>true_or_iff</a>, <a>Ne.def</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": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"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.ht\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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n  \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nh2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 \u22a4 \u2228 \u2191\u2191\u03bc F.target \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ext_of_generate_finite", "start": [3986, 1], "end": [3989, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.Nonempty.to_subtype", "start": [516, 1], "end": [517, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.X_pow_eq_monomial", "start": [326, 1], "end": [327, 25], "traced_tactics": [{"tactic": "simp [X, monomial_pow]", "annotated_tactic": ["simp [<a>X</a>, <a>monomial_pow</a>]", [{"full_name": "MvPolynomial.X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [193, 5], "def_end_pos": [193, 6]}, {"full_name": "MvPolynomial.monomial_pow", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 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\n\u22a2 X n ^ e = \u2191(monomial fun\u2080 | n => e) 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "full_name": "SatisfiesM_StateRefT_eq", "start": [199, 1], "end": [200, 80], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "m : Type \u2192 Type\n\u03c9 \u03c3 \u03b1\u271d : Type\np : \u03b1\u271d \u2192 Prop\nx : StateRefT' \u03c9 \u03c3 m \u03b1\u271d\ninst\u271d : Monad m\n\u22a2 SatisfiesM p x \u2194 \u2200 (s : ST.Ref \u03c9 \u03c3), SatisfiesM p (x s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_pi_ball", "start": [681, 1], "end": [684, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to0.tr_supports", "start": [1535, 1], "end": [1577, 26], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 TM0.Supports (tr M) \u2191(trStmts M S)", "state_after": "case left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default \u2208 \u2191(trStmts M S)\n\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 \u2200 {q : \u039b'\u2081\u2080} {a : \u0393} {q' : \u039b'\u2081\u2080} {s : Stmt\u2080}, (q', s) \u2208 tr M q a \u2192 q \u2208 \u2191(trStmts M S) \u2192 q' \u2208 \u2191(trStmts M S)"}, {"tactic": "apply Finset.mem_product.2", "annotated_tactic": ["apply <a>Finset.mem_product</a>.2", [{"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [53, 9], "def_end_pos": [53, 20]}]], "state_before": "case left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default \u2208 \u2191(trStmts M S)", "state_after": "case left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.1 \u2208 TM1.stmts M S \u2227 default.2 \u2208 Finset.univ"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.1 \u2208 TM1.stmts M S \u2227 default.2 \u2208 Finset.univ", "state_after": "case left.left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.1 \u2208 TM1.stmts M S\n\ncase left.right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.2 \u2208 Finset.univ"}, {"tactic": "simp only [default, TM1.stmts, Finset.mem_insertNone, Option.mem_def, Option.some_inj,\n  forall_eq', Finset.mem_biUnion]", "annotated_tactic": ["simp only [<a>default</a>, <a>TM1.stmts</a>, <a>Finset.mem_insertNone</a>, <a>Option.mem_def</a>, <a>Option.some_inj</a>,\n        <a>forall_eq'</a>, <a>Finset.mem_biUnion</a>]", [{"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": "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": "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.some_inj", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 17]}, {"full_name": "forall_eq'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [453, 17], "def_end_pos": [453, 27]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}]], "state_before": "case left.left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.1 \u2208 TM1.stmts M S", "state_after": "case left.left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 \u2203 a, a \u2208 S \u2227 M default \u2208 TM1.stmts\u2081 (M a)"}, {"tactic": "exact \u27e8_, ss.1, TM1.stmts\u2081_self\u27e9", "annotated_tactic": ["exact \u27e8_, ss.1, <a>TM1.stmts\u2081_self</a>\u27e9", [{"full_name": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case left.left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 \u2203 a, a \u2208 S \u2227 M default \u2208 TM1.stmts\u2081 (M a)", "state_after": "no goals"}, {"tactic": "apply Finset.mem_univ", "annotated_tactic": ["apply <a>Finset.mem_univ</a>", [{"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case left.right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.2 \u2208 Finset.univ", "state_after": "no goals"}, {"tactic": "intro q a q' s h\u2081 h\u2082", "annotated_tactic": ["intro q a q' s h\u2081 h\u2082", []], "state_before": "case right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 \u2200 {q : \u039b'\u2081\u2080} {a : \u0393} {q' : \u039b'\u2081\u2080} {s : Stmt\u2080}, (q', s) \u2208 tr M q a \u2192 q \u2208 \u2191(trStmts M S) \u2192 q' \u2208 \u2191(trStmts M S)", "state_after": "case right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\nq : \u039b'\u2081\u2080\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nh\u2081 : (q', s) \u2208 tr M q a\nh\u2082 : q \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)"}, {"tactic": "rcases q with \u27e8_ | q, v\u27e9", "annotated_tactic": ["rcases q with \u27e8_ | q, v\u27e9", []], "state_before": "case right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\nq : \u039b'\u2081\u2080\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nh\u2081 : (q', s) \u2208 tr M q a\nh\u2082 : q \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)", "state_after": "case right.mk.none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nv : \u03c3\nh\u2081 : (q', s) \u2208 tr M (none, v) a\nh\u2082 : (none, v) \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)\n\ncase right.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nh\u2081 : (q', s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v) \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)"}, {"tactic": "cases' q' with q' v'", "annotated_tactic": ["cases' q' with q' v'", []], "state_before": "case right.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nh\u2081 : (q', s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v) \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)", "state_after": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nh\u2082 : (some q, v) \u2208 \u2191(trStmts M S)\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\n\u22a2 (q', v') \u2208 \u2191(trStmts M S)"}, {"tactic": "simp only [trStmts, Finset.mem_coe] at h\u2082 \u22a2", "annotated_tactic": ["simp only [<a>trStmts</a>, <a>Finset.mem_coe</a>] at h\u2082 \u22a2", [{"full_name": "Turing.TM1to0.trStmts", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1527, 19], "def_end_pos": [1527, 26]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nh\u2082 : (some q, v) \u2208 \u2191(trStmts M S)\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\n\u22a2 (q', v') \u2208 \u2191(trStmts M S)", "state_after": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v) \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ\n\u22a2 (q', v') \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ"}, {"tactic": "rw [Finset.mem_product] at h\u2082 \u22a2", "annotated_tactic": ["rw [<a>Finset.mem_product</a>] at h\u2082 \u22a2", [{"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [53, 9], "def_end_pos": [53, 20]}]], "state_before": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v) \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ\n\u22a2 (q', v') \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ", "state_after": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v).1 \u2208 TM1.stmts M S \u2227 (some q, v).2 \u2208 Finset.univ\n\u22a2 (q', v').1 \u2208 TM1.stmts M S \u2227 (q', v').2 \u2208 Finset.univ"}, {"tactic": "simp only [Finset.mem_univ, and_true_iff] at h\u2082 \u22a2", "annotated_tactic": ["simp only [<a>Finset.mem_univ</a>, <a>and_true_iff</a>] at h\u2082 \u22a2", [{"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}]], "state_before": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v).1 \u2208 TM1.stmts M S \u2227 (some q, v).2 \u2208 Finset.univ\n\u22a2 (q', v').1 \u2208 TM1.stmts M S \u2227 (q', v').2 \u2208 Finset.univ", "state_after": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : some q \u2208 TM1.stmts M S\n\u22a2 q' \u2208 TM1.stmts M S"}, {"tactic": "cases q'", "annotated_tactic": ["cases q'", []], "state_before": "case right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : some q \u2208 TM1.stmts M S\n\u22a2 q' \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nh\u2081 : ((none, v'), s) \u2208 tr M (some q, v) a\n\u22a2 none \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : ((some val\u271d, v'), s) \u2208 tr M (some q, v) a\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "simp only [tr, Option.mem_def] at h\u2081", "annotated_tactic": ["simp only [<a>tr</a>, <a>Option.mem_def</a>] at h\u2081", [{"full_name": "Turing.TM1to0.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1489, 5], "def_end_pos": [1489, 7]}, {"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 right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : ((some val\u271d, v'), s) \u2208 tr M (some q, v) a\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "have := TM1.stmts_supportsStmt ss h\u2082", "annotated_tactic": ["have := <a>TM1.stmts_supportsStmt</a> ss h\u2082", [{"full_name": "Turing.TM1.stmts_supportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1374, 9], "def_end_pos": [1374, 27]}]], "state_before": "case right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\nthis : TM1.SupportsStmt S q\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "case right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\nthis : TM1.SupportsStmt S q\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "induction q generalizing v <;> intro hs", "annotated_tactic": ["induction q generalizing v <;> intro hs", []], "state_before": "case right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "case move d q =>\n  cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n  unfold TM1.stmts\u2081\n  exact Finset.mem_insert_of_mem TM1.stmts\u2081_self", "annotated_tactic": ["case move d q =>\n      cases h\u2081; refine' <a>TM1.stmts_trans</a> _ h\u2082\n      unfold <a>TM1.stmts\u2081</a>\n      exact <a>Finset.mem_insert_of_mem</a> <a>TM1.stmts\u2081_self</a>", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}, {"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}, {"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": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case right.mk.some.mk.some.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "case write b q =>\n  cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n  unfold TM1.stmts\u2081\n  exact Finset.mem_insert_of_mem TM1.stmts\u2081_self", "annotated_tactic": ["case write b q =>\n      cases h\u2081; refine' <a>TM1.stmts_trans</a> _ h\u2082\n      unfold <a>TM1.stmts\u2081</a>\n      exact <a>Finset.mem_insert_of_mem</a> <a>TM1.stmts\u2081_self</a>", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}, {"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}, {"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": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "case load b q IH =>\n  refine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs\n  unfold TM1.stmts\u2081\n  exact Finset.mem_insert_of_mem TM1.stmts\u2081_self", "annotated_tactic": ["case load b q IH =>\n      refine' IH _ (<a>TM1.stmts_trans</a> _ h\u2082) h\u2081 hs\n      unfold <a>TM1.stmts\u2081</a>\n      exact <a>Finset.mem_insert_of_mem</a> <a>TM1.stmts\u2081_self</a>", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}, {"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}, {"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": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3),\n    some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n  \u2200 (v : \u03c3),\n    some a\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "case goto l =>\n  cases h\u2081\n  exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9)", "annotated_tactic": ["case goto l =>\n      cases h\u2081\n      exact <a>Finset.some_mem_insertNone</a>.2 (<a>Finset.mem_biUnion</a>.2 \u27e8_, hs _ _, <a>TM1.stmts\u2081_self</a>\u27e9)", [{"full_name": "Finset.some_mem_insertNone", "def_path": "Mathlib/Data/Finset/Option.lean", "def_pos": [75, 9], "def_end_pos": [75, 28]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "case halt => cases h\u2081", "annotated_tactic": ["case halt => cases h\u2081", []], "state_before": "case right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "no goals"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "case right.mk.none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nv : \u03c3\nh\u2081 : (q', s) \u2208 tr M (none, v) a\nh\u2082 : (none, v) \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)", "state_after": "no goals"}, {"tactic": "exact Multiset.mem_cons_self _ _", "annotated_tactic": ["exact <a>Multiset.mem_cons_self</a> _ _", [{"full_name": "Multiset.mem_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 22]}]], "state_before": "case right.mk.some.mk.none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nh\u2081 : ((none, v'), s) \u2208 tr M (some q, v) a\n\u22a2 none \u2208 TM1.stmts M S", "state_after": "no goals"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nd : Stmt\u2081\nq :\n  \u2200 (v : \u03c3),\n    some d \u2208 TM1.stmts M S \u2192\n      some (trAux M a d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "refine' TM1.stmts_trans _ h\u2082", "annotated_tactic": ["refine' <a>TM1.stmts_trans</a> _ h\u2082", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.move a\u271d val\u271d)"}, {"tactic": "unfold TM1.stmts\u2081", "annotated_tactic": ["unfold <a>TM1.stmts\u2081</a>", [{"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.move a\u271d val\u271d)", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 insert (TM1.Stmt.move a\u271d val\u271d) (TM1.stmts\u2081 val\u271d)"}, {"tactic": "exact Finset.mem_insert_of_mem TM1.stmts\u2081_self", "annotated_tactic": ["exact <a>Finset.mem_insert_of_mem</a> <a>TM1.stmts\u2081_self</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": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 insert (TM1.Stmt.move a\u271d val\u271d) (TM1.stmts\u2081 val\u271d)", "state_after": "no goals"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nb : Stmt\u2081\nq :\n  \u2200 (v : \u03c3),\n    some b \u2208 TM1.stmts M S \u2192\n      some (trAux M a b v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S b \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d b) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d b) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d b)\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n        TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "refine' TM1.stmts_trans _ h\u2082", "annotated_tactic": ["refine' <a>TM1.stmts_trans</a> _ h\u2082", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n        TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n        TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.write a\u271d val\u271d)"}, {"tactic": "unfold TM1.stmts\u2081", "annotated_tactic": ["unfold <a>TM1.stmts\u2081</a>", [{"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n        TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.write a\u271d val\u271d)", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n        TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 insert (TM1.Stmt.write a\u271d val\u271d) (TM1.stmts\u2081 val\u271d)"}, {"tactic": "exact Finset.mem_insert_of_mem TM1.stmts\u2081_self", "annotated_tactic": ["exact <a>Finset.mem_insert_of_mem</a> <a>TM1.stmts\u2081_self</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": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n  \u2200 (v : \u03c3),\n    some val\u271d \u2208 TM1.stmts M S \u2192\n      some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n        TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 insert (TM1.Stmt.write a\u271d val\u271d) (TM1.stmts\u2081 val\u271d)", "state_after": "no goals"}, {"tactic": "refine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs", "annotated_tactic": ["refine' IH _ (<a>TM1.stmts_trans</a> _ h\u2082) h\u2081 hs", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3),\n    some q \u2208 TM1.stmts M S \u2192\n      some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3),\n    some q \u2208 TM1.stmts M S \u2192\n      some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 q \u2208 TM1.stmts\u2081 (TM1.Stmt.load b q)"}, {"tactic": "unfold TM1.stmts\u2081", "annotated_tactic": ["unfold <a>TM1.stmts\u2081</a>", [{"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3),\n    some q \u2208 TM1.stmts M S \u2192\n      some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 q \u2208 TM1.stmts\u2081 (TM1.Stmt.load b q)", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3),\n    some q \u2208 TM1.stmts M S \u2192\n      some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 q \u2208 insert (TM1.Stmt.load b q) (TM1.stmts\u2081 q)"}, {"tactic": "exact Finset.mem_insert_of_mem TM1.stmts\u2081_self", "annotated_tactic": ["exact <a>Finset.mem_insert_of_mem</a> <a>TM1.stmts\u2081_self</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": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3),\n    some q \u2208 TM1.stmts M S \u2192\n      some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 q \u2208 insert (TM1.Stmt.load b q) (TM1.stmts\u2081 q)", "state_after": "no goals"}, {"tactic": "cases h : p a v <;> rw [trAux, h] at h\u2081", "annotated_tactic": ["cases h : p a v <;> rw [<a>trAux</a>, h] at h\u2081", [{"full_name": "Turing.TM1to0.trAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1477, 5], "def_end_pos": [1477, 10]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch p q\u2081 q\u2082) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 some val\u271d \u2208 TM1.stmts M S"}, {"tactic": "refine' IH\u2082 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.2", "annotated_tactic": ["refine' IH\u2082 _ (<a>TM1.stmts_trans</a> _ h\u2082) h\u2081 hs.2", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}]], "state_before": "case false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 q\u2082 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082)"}, {"tactic": "unfold TM1.stmts\u2081", "annotated_tactic": ["unfold <a>TM1.stmts\u2081</a>", [{"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}]], "state_before": "case false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 q\u2082 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082)", "state_after": "case false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 q\u2082 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082)"}, {"tactic": "exact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts\u2081_self)", "annotated_tactic": ["exact <a>Finset.mem_insert_of_mem</a> (<a>Finset.mem_union_right</a> _ <a>TM1.stmts\u2081_self</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_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}, {"full_name": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 q\u2082 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082)", "state_after": "no goals"}, {"tactic": "refine' IH\u2081 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.1", "annotated_tactic": ["refine' IH\u2081 _ (<a>TM1.stmts_trans</a> _ h\u2082) h\u2081 hs.1", [{"full_name": "Turing.TM1.stmts_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 20]}]], "state_before": "case true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 q\u2081 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082)"}, {"tactic": "unfold TM1.stmts\u2081", "annotated_tactic": ["unfold <a>TM1.stmts\u2081</a>", [{"full_name": "Turing.TM1.stmts\u2081", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1310, 19], "def_end_pos": [1310, 25]}]], "state_before": "case true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 q\u2081 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082)", "state_after": "case true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 q\u2081 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082)"}, {"tactic": "exact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts\u2081_self)", "annotated_tactic": ["exact <a>Finset.mem_insert_of_mem</a> (<a>Finset.mem_union_left</a> _ <a>TM1.stmts\u2081_self</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_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1382, 9], "def_end_pos": [1382, 23]}, {"full_name": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3),\n    some q\u2081 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n  \u2200 (v : \u03c3),\n    some q\u2082 \u2208 TM1.stmts M S \u2192\n      some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 q\u2081 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082)", "state_after": "no goals"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto l) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto l)\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.goto l)\n\u22a2 some (M (l a v')) \u2208 TM1.stmts M S"}, {"tactic": "exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9)", "annotated_tactic": ["exact <a>Finset.some_mem_insertNone</a>.2 (<a>Finset.mem_biUnion</a>.2 \u27e8_, hs _ _, <a>TM1.stmts\u2081_self</a>\u27e9)", [{"full_name": "Finset.some_mem_insertNone", "def_path": "Mathlib/Data/Finset/Option.lean", "def_pos": [75, 9], "def_end_pos": [75, 28]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Turing.TM1.stmts\u2081_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 20]}]], "state_before": "case refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.goto l)\n\u22a2 some (M (l a v')) \u2208 TM1.stmts M S", "state_after": "no goals"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Fin.sort_univ", "start": [265, 1], "end": [270, 34], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\n\u22a2 List.toFinset (Finset.sort (fun x y => x \u2264 y) Finset.univ) = List.toFinset (List.finRange n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max'_insert", "start": [1564, 1], "end": [1568, 41], "traced_tactics": [{"tactic": "rw [coe_insert, max_comm]", "annotated_tactic": ["rw [<a>coe_insert</a>, <a>max_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": "max_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [113, 9], "def_end_pos": [113, 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 IsGreatest (\u2191(insert a s)) (max (max' 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 IsGreatest (insert a \u2191s) (max a (max' s H))"}, {"tactic": "exact (isGreatest_max' _ _).insert _", "annotated_tactic": ["exact (<a>isGreatest_max'</a> _ _).<a>insert</a> _", [{"full_name": "Finset.isGreatest_max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1453, 9], "def_end_pos": [1453, 24]}, {"full_name": "IsGreatest.insert", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [958, 19], "def_end_pos": [958, 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 : LinearOrder \u03b1\ns\u271d : Finset \u03b1\nH\u271d : Finset.Nonempty s\u271d\nx a : \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\n\u22a2 IsGreatest (insert a \u2191s) (max a (max' s H))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Filtration.lean", "full_name": "MeasureTheory.Filtration.const_apply", "start": [83, 1], "end": [84, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_const", "start": [1409, 1], "end": [1419, 31], "traced_tactics": [{"tactic": "cases' (@le_top _ _ _ (\u03bc univ)).lt_or_eq with h\u03bc h\u03bc", "annotated_tactic": ["cases' (@<a>le_top</a> _ _ _ (\u03bc <a>univ</a>)).<a>lt_or_eq</a> with h\u03bc h\u03bc", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "LE.le.lt_or_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [138, 7], "def_end_pos": [138, 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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "state_after": "case inl\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ < \u22a4\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c\n\ncase inr\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c"}, {"tactic": "haveI : IsFiniteMeasure \u03bc := \u27e8h\u03bc\u27e9", "annotated_tactic": ["haveI : <a>IsFiniteMeasure</a> \u03bc := \u27e8h\u03bc\u27e9", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}]], "state_before": "case inl\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ < \u22a4\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "state_after": "case inl\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ < \u22a4\nthis : IsFiniteMeasure \u03bc\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c"}, {"tactic": "simp only [integral, hE, L1.integral]", "annotated_tactic": ["simp only [<a>integral</a>, hE, <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 inl\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ < \u22a4\nthis : IsFiniteMeasure \u03bc\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "state_after": "case inl\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ < \u22a4\nthis : IsFiniteMeasure \u03bc\n\u22a2 (if h : True then if hf : Integrable fun x => c then \u2191L1.integralCLM (Integrable.toL1 (fun x => c) hf) else 0\n    else 0) =\n    ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c"}, {"tactic": "exact setToFun_const (dominatedFinMeasAdditive_weightedSMul _) _", "annotated_tactic": ["exact <a>setToFun_const</a> (<a>dominatedFinMeasAdditive_weightedSMul</a> _) _", [{"full_name": "MeasureTheory.setToFun_const", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 23]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}]], "state_before": "case inl\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ < \u22a4\nthis : IsFiniteMeasure \u03bc\n\u22a2 (if h : True then if hf : Integrable fun x => c then \u2191L1.integralCLM (Integrable.toL1 (fun x => c) hf) else 0\n    else 0) =\n    ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "state_after": "no goals"}, {"tactic": "by_cases hc : c = 0", "annotated_tactic": ["by_cases hc : c = 0", []], "state_before": "case inr\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : c = 0\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c"}, {"tactic": "simp [hc, integral_zero]", "annotated_tactic": ["simp [hc, <a>integral_zero</a>]", [{"full_name": "MeasureTheory.integral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [850, 9], "def_end_pos": [850, 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\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : c = 0\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "state_after": "no goals"}, {"tactic": "have : \u00acIntegrable (fun _ : \u03b1 => c) \u03bc := by\n  simp only [integrable_const_iff, not_or]\n  exact \u27e8hc, h\u03bc.not_lt\u27e9", "annotated_tactic": ["have : \u00ac<a>Integrable</a> (fun _ : \u03b1 => c) \u03bc := by\n        simp only [<a>integrable_const_iff</a>, <a>not_or</a>]\n        exact \u27e8hc, h\u03bc.not_lt\u27e9", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integrable_const_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [500, 9], "def_end_pos": [500, 29]}, {"full_name": "not_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [340, 9], "def_end_pos": [340, 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\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\nthis : \u00acIntegrable fun x => c\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c"}, {"tactic": "simp [integral_undef, *]", "annotated_tactic": ["simp [<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\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\nthis : \u00acIntegrable fun x => c\n\u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c", "state_after": "no goals"}, {"tactic": "simp only [integrable_const_iff, not_or]", "annotated_tactic": ["simp only [<a>integrable_const_iff</a>, <a>not_or</a>]", [{"full_name": "MeasureTheory.integrable_const_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [500, 9], "def_end_pos": [500, 29]}, {"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\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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\n\u22a2 \u00acIntegrable fun x => c", "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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\n\u22a2 \u00acc = 0 \u2227 \u00ac\u2191\u2191\u03bc univ < \u22a4"}, {"tactic": "exact \u27e8hc, h\u03bc.not_lt\u27e9", "annotated_tactic": ["exact \u27e8hc, h\u03bc.not_lt\u27e9", []], "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 g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nc : E\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\nhc : \u00acc = 0\n\u22a2 \u00acc = 0 \u2227 \u00ac\u2191\u2191\u03bc univ < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.degreeOf_rename_of_injective", "start": [577, 1], "end": [580, 98], "traced_tactics": [{"tactic": "classical\nsimp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees h]", "annotated_tactic": ["classical\n  simp only [<a>degreeOf</a>, <a>degrees_rename_of_injective</a> h, <a>Multiset.count_map_eq_count'</a> f p.degrees h]", [{"full_name": "MvPolynomial.degreeOf", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [485, 5], "def_end_pos": [485, 13]}, {"full_name": "MvPolynomial.degrees_rename_of_injective", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [257, 9], "def_end_pos": [257, 36]}, {"full_name": "Multiset.count_map_eq_count'", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2559, 9], "def_end_pos": [2559, 28]}]], "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 p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\ni : \u03c3\n\u22a2 degreeOf (f i) (\u2191(rename f) p) = degreeOf i p", "state_after": "no goals"}, {"tactic": "simp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees h]", "annotated_tactic": ["simp only [<a>degreeOf</a>, <a>degrees_rename_of_injective</a> h, <a>Multiset.count_map_eq_count'</a> f p.degrees h]", [{"full_name": "MvPolynomial.degreeOf", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [485, 5], "def_end_pos": [485, 13]}, {"full_name": "MvPolynomial.degrees_rename_of_injective", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [257, 9], "def_end_pos": [257, 36]}, {"full_name": "Multiset.count_map_eq_count'", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2559, 9], "def_end_pos": [2559, 28]}]], "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 p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\ni : \u03c3\n\u22a2 degreeOf (f i) (\u2191(rename f) p) = degreeOf i p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.measurePreserving_piEquivPiSubtypeProd", "start": [734, 1], "end": [744, 67], "traced_tactics": [{"tactic": "set e := (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p).symm", "annotated_tactic": ["set e := (<a>MeasurableEquiv.piEquivPiSubtypeProd</a> \u03b1 p).<a>symm</a>", [{"full_name": "MeasurableEquiv.piEquivPiSubtypeProd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1670, 5], "def_end_pos": [1670, 25]}, {"full_name": "MeasurableEquiv.symm", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1325, 5], "def_end_pos": [1325, 9]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)"}, {"tactic": "refine' MeasurePreserving.symm e _", "annotated_tactic": ["refine' <a>MeasurePreserving.symm</a> e _", [{"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 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\n\u22a2 MeasurePreserving \u2191(MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\n\u22a2 MeasurePreserving \u2191e"}, {"tactic": "refine' \u27e8e.measurable, (pi_eq fun s _ => _).symm\u27e9", "annotated_tactic": ["refine' \u27e8e.measurable, (<a>pi_eq</a> fun s _ => _).<a>symm</a>\u27e9", [{"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\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\n\u22a2 MeasurePreserving \u2191e", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.map (\u2191e) (Measure.prod (Measure.pi fun i => \u03bc \u2191i) (Measure.pi fun i => \u03bc \u2191i))) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "have : e \u207b\u00b9' pi univ s =\n    (pi univ fun i : { i // p i } => s i) \u00d7\u02e2 pi univ fun i : { i // \u00acp i } => s i :=\n  Equiv.preimage_piEquivPiSubtypeProd_symm_pi p s", "annotated_tactic": ["have : e \u207b\u00b9' <a>pi</a> <a>univ</a> s =\n      (<a>pi</a> <a>univ</a> fun i : { i // p i } => s i) \u00d7\u02e2 <a>pi</a> <a>univ</a> fun i : { i // \u00acp i } => s i :=\n    <a>Equiv.preimage_piEquivPiSubtypeProd_symm_pi</a> p s", [{"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.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.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": "Equiv.preimage_piEquivPiSubtypeProd_symm_pi", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [656, 9], "def_end_pos": [656, 46]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.map (\u2191e) (Measure.prod (Measure.pi fun i => \u03bc \u2191i) (Measure.pi fun i => \u03bc \u2191i))) (Set.pi univ s) =\n    \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\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : \u2191e \u207b\u00b9' Set.pi univ s = (Set.pi univ fun i => s \u2191i) \u00d7\u02e2 Set.pi univ fun i => s \u2191i\n\u22a2 \u2191\u2191(Measure.map (\u2191e) (Measure.prod (Measure.pi fun i => \u03bc \u2191i) (Measure.pi fun i => \u03bc \u2191i))) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "rw [e.map_apply, this, prod_prod, pi_pi, pi_pi]", "annotated_tactic": ["rw [e.map_apply, this, <a>prod_prod</a>, <a>pi_pi</a>, <a>pi_pi</a>]", [{"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": "MeasureTheory.Measure.pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [394, 9], "def_end_pos": [394, 14]}, {"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\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : \u2191e \u207b\u00b9' Set.pi univ s = (Set.pi univ fun i => s \u2191i) \u00d7\u02e2 Set.pi univ fun i => s \u2191i\n\u22a2 \u2191\u2191(Measure.map (\u2191e) (Measure.prod (Measure.pi fun i => \u03bc \u2191i) (Measure.pi fun i => \u03bc \u2191i))) (Set.pi univ s) =\n    \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\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : \u2191e \u207b\u00b9' Set.pi univ s = (Set.pi univ fun i => s \u2191i) \u00d7\u02e2 Set.pi univ fun i => s \u2191i\n\u22a2 (\u220f i : Subtype p, \u2191\u2191(\u03bc \u2191i) (s \u2191i)) * \u220f i : { i // \u00acp i }, \u2191\u2191(\u03bc \u2191i) (s \u2191i) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "exact Fintype.prod_subtype_mul_prod_subtype p fun i => \u03bc i (s i)", "annotated_tactic": ["exact <a>Fintype.prod_subtype_mul_prod_subtype</a> p fun i => \u03bc i (s i)", [{"full_name": "Fintype.prod_subtype_mul_prod_subtype", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2033, 9], "def_end_pos": [2033, 38]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : Fintype \u03b9'\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ne : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=\n  MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : \u2191e \u207b\u00b9' Set.pi univ s = (Set.pi univ fun i => s \u2191i) \u00d7\u02e2 Set.pi univ fun i => s \u2191i\n\u22a2 (\u220f i : Subtype p, \u2191\u2191(\u03bc \u2191i) (s \u2191i)) * \u220f i : { i // \u00acp i }, \u2191\u2191(\u03bc \u2191i) (s \u2191i) = \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/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.encode_lt_pair", "start": [202, 1], "end": [208, 98], "traced_tactics": [{"tactic": "simp only [encodeCode_eq, encodeCode]", "annotated_tactic": ["simp only [<a>encodeCode_eq</a>, <a>encodeCode</a>]", [{"full_name": "Nat.Partrec.Code.encodeCode_eq", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [194, 9], "def_end_pos": [194, 22]}, {"full_name": "Nat.Partrec.Code.encodeCode", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [129, 5], "def_end_pos": [129, 15]}]], "state_before": "cf cg : Code\n\u22a2 encode cf < encode (pair cf cg) \u2227 encode cg < encode (pair cf cg)", "state_after": "cf cg : Code\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4"}, {"tactic": "have := Nat.mul_le_mul_right (Nat.pair cf.encodeCode cg.encodeCode) (by decide : 1 \u2264 2 * 2)", "annotated_tactic": ["have := <a>Nat.mul_le_mul_right</a> (<a>Nat.pair</a> cf.encodeCode cg.encodeCode) (by decide : 1 \u2264 2 * 2)", [{"full_name": "Nat.mul_le_mul_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}, {"full_name": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}]], "state_before": "cf cg : Code\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4", "state_after": "cf cg : Code\nthis : 1 * Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * 2 * Nat.pair (encodeCode cf) (encodeCode cg)\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4"}, {"tactic": "rw [one_mul, mul_assoc] at this", "annotated_tactic": ["rw [<a>one_mul</a>, <a>mul_assoc</a>] at this", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "cf cg : Code\nthis : 1 * Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * 2 * Nat.pair (encodeCode cf) (encodeCode cg)\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4", "state_after": "cf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4"}, {"tactic": "have := lt_of_le_of_lt this (lt_add_of_pos_right _ (by decide : 0 < 4))", "annotated_tactic": ["have := <a>lt_of_le_of_lt</a> this (<a>lt_add_of_pos_right</a> _ (by decide : 0 < 4))", [{"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": "lt_add_of_pos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [495, 15], "def_end_pos": [495, 34]}]], "state_before": "cf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4", "state_after": "cf cg : Code\nthis\u271d : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\nthis : Nat.pair (encodeCode cf) (encodeCode cg) < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4"}, {"tactic": "exact \u27e8lt_of_le_of_lt (Nat.left_le_pair _ _) this, lt_of_le_of_lt (Nat.right_le_pair _ _) this\u27e9", "annotated_tactic": ["exact \u27e8<a>lt_of_le_of_lt</a> (<a>Nat.left_le_pair</a> _ _) this, <a>lt_of_le_of_lt</a> (<a>Nat.right_le_pair</a> _ _) this\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": "Nat.left_le_pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [115, 9], "def_end_pos": [115, 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": "Nat.right_le_pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}]], "state_before": "cf cg : Code\nthis\u271d : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\nthis : Nat.pair (encodeCode cf) (encodeCode cg) < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "cf cg : Code\n\u22a2 1 \u2264 2 * 2", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "cf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n\u22a2 0 < 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_getD", "start": [1089, 1], "end": [1091, 46], "traced_tactics": [{"tactic": "simp only [List.getD_eq_getD_get?]", "annotated_tactic": ["simp only [<a>List.getD_eq_getD_get?</a>]", [{"full_name": "List.getD_eq_getD_get?", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [4362, 9], "def_end_pos": [4362, 26]}]], "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\nd : \u03b1\n\u22a2 Primrec\u2082 fun l n => List.getD l n d", "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\nd : \u03b1\n\u22a2 Primrec\u2082 fun l n => Option.getD (List.get? l n) d"}, {"tactic": "exact option_getD.comp\u2082 list_get? (const _)", "annotated_tactic": ["exact option_getD.comp\u2082 <a>list_get?</a> (<a>const</a> _)", [{"full_name": "Primrec.list_get?", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 18]}, {"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\nd : \u03b1\n\u22a2 Primrec\u2082 fun l n => Option.getD (List.get? l n) d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/PProd.lean", "full_name": "PProd.forall'", "start": [36, 1], "end": [37, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_coe_nnreal_real_iff", "start": [2011, 1], "end": [2013, 97], "traced_tactics": [{"tactic": "simpa only [Real.toNNReal_coe] using h.real_toNNReal", "annotated_tactic": ["simpa only [<a>Real.toNNReal_coe</a>] using h.real_toNNReal", [{"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 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 t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nh : Measurable fun x => \u2191(f x)\n\u22a2 Measurable f", "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_filterMap", "start": [128, 1], "end": [142, 75], "traced_tactics": [{"tactic": "let _S (a a' : \u03b2) := \u2200 b \u2208 f a, \u2200 b' \u2208 f a', R b b'", "annotated_tactic": ["let _S (a a' : \u03b2) := \u2200 b \u2208 f a, \u2200 b' \u2208 f a', R b b'", []], "state_before": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l", "state_after": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l"}, {"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": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l", "state_after": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l"}, {"tactic": "induction l with\n| nil => simp only [filterMap, Pairwise.nil]\n| cons a l IH => ?_", "annotated_tactic": ["induction l with\n  | <a>nil</a> => simp only [<a>filterMap</a>, <a>Pairwise.nil</a>]\n  | <a>cons</a> a 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.filterMap", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [214, 19], "def_end_pos": [214, 28]}, {"full_name": "List.Pairwise.nil", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1120, 5], "def_end_pos": [1120, 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": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l", "state_after": "case cons\n\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)"}, {"tactic": "match e : f a with\n| none =>\n  rw [filterMap_cons_none _ _ e, pairwise_cons]\n  simp only [e, false_implies, implies_true, true_and, IH]\n| some b =>\n  rw [filterMap_cons_some _ _ _ e]\n  simpa [IH, e] using fun _ =>\n    \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9", "annotated_tactic": ["match e : f a with\n  | <a>none</a> =>\n    rw [<a>filterMap_cons_none</a> _ _ e, <a>pairwise_cons</a>]\n    simp only [e, <a>false_implies</a>, <a>implies_true</a>, <a>true_and</a>, IH]\n  | <a>some</a> b =>\n    rw [<a>filterMap_cons_some</a> _ _ _ e]\n    simpa [IH, e] using fun _ =>\n      \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9", [{"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.filterMap_cons_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1280, 9], "def_end_pos": [1280, 28]}, {"full_name": "List.pairwise_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1129, 17], "def_end_pos": [1129, 30]}, {"full_name": "false_implies", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [97, 17], "def_end_pos": [97, 30]}, {"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": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}, {"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.filterMap_cons_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 28]}]], "state_before": "case cons\n\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)", "state_after": "no goals"}, {"tactic": "simp only [filterMap, Pairwise.nil]", "annotated_tactic": ["simp only [<a>filterMap</a>, <a>Pairwise.nil</a>]", [{"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]}, {"full_name": "List.Pairwise.nil", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1120, 5], "def_end_pos": [1120, 8]}]], "state_before": "case nil\n\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f []) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') []", "state_after": "no goals"}, {"tactic": "rw [filterMap_cons_none _ _ e, pairwise_cons]", "annotated_tactic": ["rw [<a>filterMap_cons_none</a> _ _ e, <a>pairwise_cons</a>]", [{"full_name": "List.filterMap_cons_none", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1280, 9], "def_end_pos": [1280, 28]}, {"full_name": "List.pairwise_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1129, 17], "def_end_pos": [1129, 30]}]], "state_before": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\ne : f a = none\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)", "state_after": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\ne : f a = none\n\u22a2 Pairwise R (filterMap f l) \u2194\n    (\u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') \u2227\n      Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l"}, {"tactic": "simp only [e, false_implies, implies_true, true_and, IH]", "annotated_tactic": ["simp only [e, <a>false_implies</a>, <a>implies_true</a>, <a>true_and</a>, IH]", [{"full_name": "false_implies", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [97, 17], "def_end_pos": [97, 30]}, {"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": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\ne : f a = none\n\u22a2 Pairwise R (filterMap f l) \u2194\n    (\u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') \u2227\n      Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l", "state_after": "no goals"}, {"tactic": "rw [filterMap_cons_some _ _ _ e]", "annotated_tactic": ["rw [<a>filterMap_cons_some</a> _ _ _ e]", [{"full_name": "List.filterMap_cons_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 28]}]], "state_before": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)", "state_after": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\n\u22a2 Pairwise R (b :: filterMap f l) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)"}, {"tactic": "simpa [IH, e] using fun _ =>\n  \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9", "annotated_tactic": ["simpa [IH, e] using fun _ =>\n      \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9", []], "state_before": "\u03b2 : Type u_1\n\u03b1 : Type u_2\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\n\u22a2 Pairwise R (b :: filterMap f l) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)", "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_of_measure_ne_top", "start": [404, 1], "end": [424, 36], "traced_tactics": [{"tactic": "refine' @Lp.induction _ _ _ _ _ _ _ ENNReal.one_ne_top\n  (fun f : \u03b1 \u2192\u2081[\u03bc] F' => \u222b x in s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc = \u222b x in s, f x \u2202\u03bc) _ _\n  (isClosed_eq _ _) f", "annotated_tactic": ["refine' @<a>Lp.induction</a> _ _ _ _ _ _ _ <a>ENNReal.one_ne_top</a>\n    (fun f : \u03b1 \u2192\u2081[\u03bc] F' => \u222b x in s, <a>condexpL1Clm</a> F' hm \u03bc f x \u2202\u03bc = \u222b x in s, f x \u2202\u03bc) _ _\n    (<a>isClosed_eq</a> _ _) 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.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"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\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\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \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": "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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (c : F') {s_1 : Set \u03b1} (hs : MeasurableSet s_1) (h\u03bcs : \u2191\u2191\u03bc s_1 < \u22a4),\n    (fun f => \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)\n      \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s_1 \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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\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 => \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) (Mem\u2112p.toLp f hf) \u2192\n        (fun f => \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) (Mem\u2112p.toLp g hg) \u2192\n          (fun f => \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)\n            (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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc\n\ncase refine'_4\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "intro x t ht h\u03bct", "annotated_tactic": ["intro x t ht h\u03bct", []], "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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (c : F') {s_1 : Set \u03b1} (hs : MeasurableSet s_1) (h\u03bcs : \u2191\u2191\u03bc s_1 < \u22a4),\n    (fun f => \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)\n      \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s_1 \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 : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : F'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x)) x_1 \u2202\u03bc =\n    \u222b (x_1 : \u03b1) in s, \u2191\u2191\u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x) x_1 \u2202\u03bc"}, {"tactic": "simp_rw [condexpL1Clm_indicatorConst ht h\u03bct.ne x]", "annotated_tactic": ["simp_rw [<a>condexpL1Clm_indicatorConst</a> ht h\u03bct.ne x]", [{"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 : Set \u03b1\nf : { x // x \u2208 Lp F' 1 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : F'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x)) x_1 \u2202\u03bc =\n    \u222b (x_1 : \u03b1) in s, \u2191\u2191\u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x) x_1 \u2202\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : F'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpInd F' hm \u03bc t) x) x_1 \u2202\u03bc =\n    \u222b (x_1 : \u03b1) in s, \u2191\u2191\u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x) x_1 \u2202\u03bc"}, {"tactic": "rw [Lp.simpleFunc.coe_indicatorConst, set_integral_indicatorConstLp (hm _ hs)]", "annotated_tactic": ["rw [<a>Lp.simpleFunc.coe_indicatorConst</a>, <a>set_integral_indicatorConstLp</a> (hm _ hs)]", [{"full_name": "MeasureTheory.Lp.simpleFunc.coe_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [700, 9], "def_end_pos": [700, 27]}, {"full_name": "MeasureTheory.set_integral_indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [490, 9], "def_end_pos": [490, 38]}]], "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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : F'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpInd F' hm \u03bc t) x) x_1 \u2202\u03bc =\n    \u222b (x_1 : \u03b1) in s, \u2191\u2191\u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x) x_1 \u2202\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : F'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpInd F' hm \u03bc t) x) x_1 \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc (t \u2229 s)) \u2022 x"}, {"tactic": "exact set_integral_condexpInd hs ht h\u03bcs h\u03bct.ne x", "annotated_tactic": ["exact <a>set_integral_condexpInd</a> hs ht h\u03bcs h\u03bct.ne x", [{"full_name": "MeasureTheory.set_integral_condexpInd", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [337, 9], "def_end_pos": [337, 32]}]], "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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : F'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpInd F' hm \u03bc t) x) x_1 \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc (t \u2229 s)) \u2022 x", "state_after": "no goals"}, {"tactic": "intro f g hf_Lp hg_Lp _ hf hg", "annotated_tactic": ["intro f g hf_Lp hg_Lp _ hf hg", []], "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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\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 => \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) (Mem\u2112p.toLp f hf) \u2192\n        (fun f => \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) (Mem\u2112p.toLp g hg) \u2192\n          (fun f => \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)\n            (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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc"}, {"tactic": "simp_rw [(condexpL1Clm F' hm \u03bc).map_add]", "annotated_tactic": ["simp_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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp) + \u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc"}, {"tactic": "rw [set_integral_congr_ae (hm s hs) ((Lp.coeFn_add (condexpL1Clm F' hm \u03bc (hf_Lp.toLp f))\n  (condexpL1Clm F' hm \u03bc (hg_Lp.toLp g))).mono fun x hx _ => hx)]", "annotated_tactic": ["rw [<a>set_integral_congr_ae</a> (hm s hs) ((<a>Lp.coeFn_add</a> (<a>condexpL1Clm</a> F' hm \u03bc (hf_Lp.toLp f))\n      (<a>condexpL1Clm</a> F' hm \u03bc (hg_Lp.toLp g))).<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.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"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": "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\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp) + \u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s,\n      (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc"}, {"tactic": "rw [set_integral_congr_ae (hm s hs)\n  ((Lp.coeFn_add (hf_Lp.toLp f) (hg_Lp.toLp g)).mono fun x hx _ => hx)]", "annotated_tactic": ["rw [<a>set_integral_congr_ae</a> (hm s hs)\n      ((<a>Lp.coeFn_add</a> (hf_Lp.toLp f) (hg_Lp.toLp g)).<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.Lp.coeFn_add", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [232, 9], "def_end_pos": [232, 18]}, {"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\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s,\n      (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s,\n      (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, (\u2191\u2191(Mem\u2112p.toLp f hf_Lp) + \u2191\u2191(Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc"}, {"tactic": "simp_rw [Pi.add_apply]", "annotated_tactic": ["simp_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 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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s,\n      (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, (\u2191\u2191(Mem\u2112p.toLp f hf_Lp) + \u2191\u2191(Mem\u2112p.toLp g hg_Lp)) x \u2202\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x + \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc"}, {"tactic": "rw [integral_add (L1.integrable_coeFn _).integrableOn (L1.integrable_coeFn _).integrableOn,\n  integral_add (L1.integrable_coeFn _).integrableOn (L1.integrable_coeFn _).integrableOn, hf,\n  hg]", "annotated_tactic": ["rw [<a>integral_add</a> (<a>L1.integrable_coeFn</a> _).<a>integrableOn</a> (<a>L1.integrable_coeFn</a> _).<a>integrableOn</a>,\n      <a>integral_add</a> (<a>L1.integrable_coeFn</a> _).<a>integrableOn</a> (<a>L1.integrable_coeFn</a> _).<a>integrableOn</a>, hf,\n      hg]", [{"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}, {"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]}, {"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]}, {"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}, {"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]}, {"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": "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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf g : \u03b1 \u2192 F'\nhf_Lp : Mem\u2112p f 1\nhg_Lp : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc\nhg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x + \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact (continuous_set_integral s).comp (condexpL1Clm F' hm \u03bc).continuous", "annotated_tactic": ["exact (<a>continuous_set_integral</a> s).<a>comp</a> (<a>condexpL1Clm</a> F' hm \u03bc).<a>continuous</a>", [{"full_name": "MeasureTheory.continuous_set_integral", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [968, 9], "def_end_pos": [968, 32]}, {"full_name": "Continuous.comp", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1673, 9], "def_end_pos": [1673, 24]}, {"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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact continuous_set_integral s", "annotated_tactic": ["exact <a>continuous_set_integral</a> s", [{"full_name": "MeasureTheory.continuous_set_integral", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [968, 9], "def_end_pos": [968, 32]}]], "state_before": "case refine'_4\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 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\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.map_iInf_le", "start": [1215, 1], "end": [1217, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "full_name": "MeasureTheory.integral_norm_le_of_forall_fin_meas_integral_eq", "start": [170, 1], "end": [196, 63], "traced_tactics": [{"tactic": "rw [integral_norm_eq_pos_sub_neg hgi, integral_norm_eq_pos_sub_neg hfi]", "annotated_tactic": ["rw [<a>integral_norm_eq_pos_sub_neg</a> hgi, <a>integral_norm_eq_pos_sub_neg</a> hfi]", [{"full_name": "MeasureTheory.integral_norm_eq_pos_sub_neg", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [449, 9], "def_end_pos": [449, 37]}, {"full_name": "MeasureTheory.integral_norm_eq_pos_sub_neg", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [449, 9], "def_end_pos": [449, 37]}]], "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": "\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 {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "have h_meas_nonneg_g : MeasurableSet[m] {x | 0 \u2264 g x} :=\n  (@stronglyMeasurable_const _ _ m _ _).measurableSet_le hg", "annotated_tactic": ["have h_meas_nonneg_g : MeasurableSet[m] {x | 0 \u2264 g x} :=\n    (@<a>stronglyMeasurable_const</a> _ _ m _ _).<a>measurableSet_le</a> hg", [{"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}, {"full_name": "MeasureTheory.StronglyMeasurable.measurableSet_le", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [876, 9], "def_end_pos": [876, 25]}]], "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 {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s", "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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "have h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} :=\n  stronglyMeasurable_const.measurableSet_le hf", "annotated_tactic": ["have h_meas_nonneg_f : <a>MeasurableSet</a> {x | 0 \u2264 f x} :=\n    stronglyMeasurable_const.measurableSet_le hf", [{"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\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s", "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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "have h_meas_nonpos_g : MeasurableSet[m] {x | g x \u2264 0} :=\n  hg.measurableSet_le (@stronglyMeasurable_const _ _ m _ _)", "annotated_tactic": ["have h_meas_nonpos_g : MeasurableSet[m] {x | g x \u2264 0} :=\n    hg.measurableSet_le (@<a>stronglyMeasurable_const</a> _ _ m _ _)", [{"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\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s", "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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "have h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} :=\n  hf.measurableSet_le stronglyMeasurable_const", "annotated_tactic": ["have h_meas_nonpos_f : <a>MeasurableSet</a> {x | f x \u2264 0} :=\n    hf.measurableSet_le <a>stronglyMeasurable_const</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"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\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s", "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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "refine' sub_le_sub _ _", "annotated_tactic": ["refine' <a>sub_le_sub</a> _ _", [{"full_name": "sub_le_sub", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [850, 15], "def_end_pos": [850, 25]}]], "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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264\n    \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc 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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc 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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s"}, {"tactic": "rw [Measure.restrict_restrict (hm _ h_meas_nonneg_g), Measure.restrict_restrict h_meas_nonneg_f,\n  hgf _ (@MeasurableSet.inter \u03b1 m _ _ h_meas_nonneg_g hs)\n    ((measure_mono (Set.inter_subset_right _ _)).trans_lt (lt_top_iff_ne_top.mpr h\u03bcs)),\n  \u2190 Measure.restrict_restrict (hm _ h_meas_nonneg_g), \u2190\n  Measure.restrict_restrict h_meas_nonneg_f]", "annotated_tactic": ["rw [<a>Measure.restrict_restrict</a> (hm _ h_meas_nonneg_g), <a>Measure.restrict_restrict</a> h_meas_nonneg_f,\n      hgf _ (@<a>MeasurableSet.inter</a> \u03b1 m _ _ h_meas_nonneg_g hs)\n        ((<a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)).<a>trans_lt</a> (lt_top_iff_ne_top.mpr h\u03bcs)),\n      \u2190 <a>Measure.restrict_restrict</a> (hm _ h_meas_nonneg_g), \u2190\n      <a>Measure.restrict_restrict</a> h_meas_nonneg_f]", [{"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": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"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.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]}]], "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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc 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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "exact set_integral_le_nonneg (hm _ h_meas_nonneg_g) hf hfi", "annotated_tactic": ["exact <a>set_integral_le_nonneg</a> (hm _ h_meas_nonneg_g) hf hfi", [{"full_name": "MeasureTheory.set_integral_le_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [776, 9], "def_end_pos": [776, 31]}]], "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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_restrict (hm _ h_meas_nonpos_g), Measure.restrict_restrict h_meas_nonpos_f,\n  hgf _ (@MeasurableSet.inter \u03b1 m _ _ h_meas_nonpos_g hs)\n    ((measure_mono (Set.inter_subset_right _ _)).trans_lt (lt_top_iff_ne_top.mpr h\u03bcs)),\n  \u2190 Measure.restrict_restrict (hm _ h_meas_nonpos_g), \u2190\n  Measure.restrict_restrict h_meas_nonpos_f]", "annotated_tactic": ["rw [<a>Measure.restrict_restrict</a> (hm _ h_meas_nonpos_g), <a>Measure.restrict_restrict</a> h_meas_nonpos_f,\n      hgf _ (@<a>MeasurableSet.inter</a> \u03b1 m _ _ h_meas_nonpos_g hs)\n        ((<a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)).<a>trans_lt</a> (lt_top_iff_ne_top.mpr h\u03bcs)),\n      \u2190 <a>Measure.restrict_restrict</a> (hm _ h_meas_nonpos_g), \u2190\n      <a>Measure.restrict_restrict</a> h_meas_nonpos_f]", [{"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": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"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.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]}]], "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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s", "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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | g x \u2264 0}, f x \u2202Measure.restrict \u03bc s"}, {"tactic": "exact set_integral_nonpos_le (hm _ h_meas_nonpos_g) hf hfi", "annotated_tactic": ["exact <a>set_integral_nonpos_le</a> (hm _ h_meas_nonpos_g) hf hfi", [{"full_name": "MeasureTheory.set_integral_nonpos_le", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [804, 9], "def_end_pos": [804, 31]}]], "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'\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\nh_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x}\nh_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x}\nh_meas_nonpos_g : MeasurableSet {x | g x \u2264 0}\nh_meas_nonpos_f : MeasurableSet {x | f x \u2264 0}\n\u22a2 \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | g x \u2264 0}, f x \u2202Measure.restrict \u03bc 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.dominatedFinMeasAdditive_condexpInd", "start": [329, 1], "end": [332, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.singleton_mul_inter", "start": [1890, 1], "end": [1891, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.FinStronglyMeasurable.const_smul", "start": [1099, 11], "end": [1104, 92], "traced_tactics": [{"tactic": "refine' \u27e8fun n => c \u2022 hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).const_smul c\u27e9", "annotated_tactic": ["refine' \u27e8fun n => c \u2022 hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).<a>const_smul</a> c\u27e9", [{"full_name": "Filter.Tendsto.const_smul", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [81, 9], "def_end_pos": [81, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2076 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b2\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : AddMonoid \u03b2\ninst\u271d\u00b2 : Monoid \ud835\udd5c\ninst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2\ninst\u271d : ContinuousSMul \ud835\udd5c \u03b2\nhf : FinStronglyMeasurable f \u03bc\nc : \ud835\udd5c\n\u22a2 FinStronglyMeasurable (c \u2022 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\u2076 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b2\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : AddMonoid \u03b2\ninst\u271d\u00b2 : Monoid \ud835\udd5c\ninst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2\ninst\u271d : ContinuousSMul \ud835\udd5c \u03b2\nhf : FinStronglyMeasurable f \u03bc\nc : \ud835\udd5c\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => c \u2022 FinStronglyMeasurable.approx hf n) n)) < \u22a4"}, {"tactic": "rw [SimpleFunc.coe_smul]", "annotated_tactic": ["rw [<a>SimpleFunc.coe_smul</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_smul", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [579, 9], "def_end_pos": [579, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2076 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b2\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : AddMonoid \u03b2\ninst\u271d\u00b2 : Monoid \ud835\udd5c\ninst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2\ninst\u271d : ContinuousSMul \ud835\udd5c \u03b2\nhf : FinStronglyMeasurable f \u03bc\nc : \ud835\udd5c\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support \u2191((fun n => c \u2022 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\u2076 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b2\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : AddMonoid \u03b2\ninst\u271d\u00b2 : Monoid \ud835\udd5c\ninst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2\ninst\u271d : ContinuousSMul \ud835\udd5c \u03b2\nhf : FinStronglyMeasurable f \u03bc\nc : \ud835\udd5c\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support (c \u2022 \u2191(FinStronglyMeasurable.approx hf n))) < \u22a4"}, {"tactic": "refine' (measure_mono (support_smul_subset_right c _)).trans_lt (hf.fin_support_approx n)", "annotated_tactic": ["refine' (<a>measure_mono</a> (<a>support_smul_subset_right</a> c _)).<a>trans_lt</a> (hf.fin_support_approx n)", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Function.support_smul_subset_right", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [372, 9], "def_end_pos": [372, 34]}, {"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\u2076 : Countable \u03b9\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u2075 : TopologicalSpace \u03b2\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : AddMonoid \u03b2\ninst\u271d\u00b2 : Monoid \ud835\udd5c\ninst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2\ninst\u271d : ContinuousSMul \ud835\udd5c \u03b2\nhf : FinStronglyMeasurable f \u03bc\nc : \ud835\udd5c\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (support (c \u2022 \u2191(FinStronglyMeasurable.approx hf n))) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/NFA.lean", "full_name": "NFA.eval_append_singleton", "start": [103, 1], "end": [104, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.choice_eq", "start": [429, 1], "end": [431, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.set_integral_condexpInd", "start": [337, 1], "end": [343, 78], "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", "start": [484, 1], "end": [486, 90], "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\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\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 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\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 \u00d7 \u03b3)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) a) s"}, {"tactic": "simp_rw [kernel.sum_apply' _ a hs]", "annotated_tactic": ["simp_rw [<a>kernel.sum_apply'</a> _ a hs]", [{"full_name": "ProbabilityTheory.kernel.sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 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\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 \u00d7 \u03b3)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 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\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 \u00d7 \u03b3)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = \u2211' (n : \u2115) (n_1 : \u2115), \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 seq \u03b7 n_1) a) s"}, {"tactic": "rw [compProd_eq_tsum_compProd \u03ba \u03b7 a hs]", "annotated_tactic": ["rw [<a>compProd_eq_tsum_compProd</a> \u03ba \u03b7 a hs]", [{"full_name": "ProbabilityTheory.kernel.compProd_eq_tsum_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [478, 9], "def_end_pos": [478, 34]}]], "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 : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = \u2211' (n : \u2115) (n_1 : \u2115), \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 seq \u03b7 n_1) a) s", "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_piiUnionInter_of_disjoint", "start": [400, 1], "end": [437, 28], "traced_tactics": [{"tactic": "rintro t1 t2 \u27e8p1, hp1, f1, ht1_m, ht1_eq\u27e9 \u27e8p2, hp2, f2, ht2_m, ht2_eq\u27e9", "annotated_tactic": ["rintro t1 t2 \u27e8p1, hp1, f1, ht1_m, ht1_eq\u27e9 \u27e8p2, hp2, f2, ht2_m, ht2_eq\u27e9", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\n\u22a2 IndepSets (piiUnionInter s S) (piiUnionInter s T) \u03ba", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\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": "let g i := ite (i \u2208 p1) (f1 i) Set.univ \u2229 ite (i \u2208 p2) (f2 i) Set.univ", "annotated_tactic": ["let g i := <a>ite</a> (i \u2208 p1) (f1 i) <a>Set.univ</a> \u2229 <a>ite</a> (i \u2208 p2) (f2 i) <a>Set.univ</a>", [{"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\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 intro.intro.intro.intro.intro.intro.intro.intro\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\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": "filter_upwards [h_P_inter, h_indep p1 ht1_m, h_indep p2 ht2_m] with a h_P_inter ha1 ha2", "annotated_tactic": ["filter_upwards [h_P_inter, h_indep p1 ht1_m, h_indep p2 ht2_m] with a h_P_inter ha1 ha2", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\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 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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t1 * \u2191\u2191(\u2191\u03ba a) t2"}, {"tactic": "simp_rw [h_P_inter, h_\u03bcg, Finset.prod_mul_distrib,\n  Finset.prod_ite_mem (p1 \u222a p2) p1 (fun x \u21a6 \u03ba a (f1 x)), Finset.union_inter_cancel_left,\n  Finset.prod_ite_mem (p1 \u222a p2) p2 (fun x => \u03ba a (f2 x)), Finset.union_inter_cancel_right, ht1_eq,\n    \u2190 ha1, ht2_eq, \u2190 ha2]", "annotated_tactic": ["simp_rw [h_P_inter, h_\u03bcg, <a>Finset.prod_mul_distrib</a>,\n    <a>Finset.prod_ite_mem</a> (p1 \u222a p2) p1 (fun x \u21a6 \u03ba a (f1 x)), <a>Finset.union_inter_cancel_left</a>,\n    <a>Finset.prod_ite_mem</a> (p1 \u222a p2) p2 (fun x => \u03ba a (f2 x)), <a>Finset.union_inter_cancel_right</a>, ht1_eq,\n      \u2190 ha1, ht2_eq, \u2190 ha2]", [{"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_ite_mem", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 21]}, {"full_name": "Finset.union_inter_cancel_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1633, 9], "def_end_pos": [1633, 32]}, {"full_name": "Finset.prod_ite_mem", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 21]}, {"full_name": "Finset.union_inter_cancel_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1638, 9], "def_end_pos": [1638, 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\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nh_\u03bcg : \u2200 (n : \u03b9), \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t1 * \u2191\u2191(\u2191\u03ba a) t2", "state_after": "no goals"}, {"tactic": "have h_p1_inter_p2 :\n  ((\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x) =\n    \u22c2 i \u2208 p1 \u222a p2, ite (i \u2208 p1) (f1 i) Set.univ \u2229 ite (i \u2208 p2) (f2 i) Set.univ := by\n  ext1 x\n  simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]\n  exact\n    \u27e8fun h i _ => \u27e8h.1 i, h.2 i\u27e9, fun h =>\n      \u27e8fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi\u27e9\u27e9", "annotated_tactic": ["have h_p1_inter_p2 :\n      ((\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x) =\n        \u22c2 i \u2208 p1 \u222a p2, <a>ite</a> (i \u2208 p1) (f1 i) <a>Set.univ</a> \u2229 <a>ite</a> (i \u2208 p2) (f2 i) <a>Set.univ</a> := by\n      ext1 x\n      simp only [<a>Set.mem_ite_univ_right</a>, <a>Set.mem_inter_iff</a>, <a>Set.mem_iInter</a>, <a>Finset.mem_union</a>]\n      exact\n        \u27e8fun h i _ => \u27e8h.1 i, h.2 i\u27e9, fun h =>\n          \u27e8fun i hi => (h i (<a>Or.inl</a> hi)).1 hi, fun i hi => (h i (<a>Or.inr</a> hi)).2 hi\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": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.mem_ite_univ_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2220, 9], "def_end_pos": [2220, 27]}, {"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_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"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]}]], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nh_p1_inter_p2 :\n  (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =\n    \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)"}, {"tactic": "filter_upwards [h_indep _ hgm] with a ha", "annotated_tactic": ["filter_upwards [h_indep _ hgm] with a ha", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nh_p1_inter_p2 :\n  (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =\n    \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nh_p1_inter_p2 :\n  (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =\n    \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1 \u222a p2, g i) = \u220f i in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g i)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)"}, {"tactic": "rw [ht1_eq, ht2_eq, h_p1_inter_p2, \u2190 ha]", "annotated_tactic": ["rw [ht1_eq, ht2_eq, h_p1_inter_p2, \u2190 ha]", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nh_p1_inter_p2 :\n  (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =\n    \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\na : \u03b1\nha : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1 \u222a p2, g i) = \u220f i in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g i)\n\u22a2 \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)", "state_after": "no goals"}, {"tactic": "intro i hi_mem_union", "annotated_tactic": ["intro i hi_mem_union", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\n\u22a2 \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi_mem_union : i \u2208 p1 \u222a p2\n\u22a2 g i \u2208 s i"}, {"tactic": "rw [Finset.mem_union] at hi_mem_union", "annotated_tactic": ["rw [<a>Finset.mem_union</a>] at hi_mem_union", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi_mem_union : i \u2208 p1 \u222a p2\n\u22a2 g i \u2208 s i", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi_mem_union : i \u2208 p1 \u2228 i \u2208 p2\n\u22a2 g i \u2208 s i"}, {"tactic": "cases' hi_mem_union with hi1 hi2", "annotated_tactic": ["cases' hi_mem_union with hi1 hi2", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi_mem_union : i \u2208 p1 \u2228 i \u2208 p2\n\u22a2 g i \u2208 s i", "state_after": "case inl\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi1 : i \u2208 p1\n\u22a2 g i \u2208 s i\n\ncase inr\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi2 : i \u2208 p2\n\u22a2 g i \u2208 s i"}, {"tactic": "have hi2 : i \u2209 p2 := fun hip2 => Set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)", "annotated_tactic": ["have hi2 : i \u2209 p2 := fun hip2 => Set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)", []], "state_before": "case inl\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi1 : i \u2208 p1\n\u22a2 g i \u2208 s i", "state_after": "case inl\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi1 : i \u2208 p1\nhi2 : \u00aci \u2208 p2\n\u22a2 g i \u2208 s i"}, {"tactic": "simp_rw [if_pos hi1, if_neg hi2, Set.inter_univ]", "annotated_tactic": ["simp_rw [<a>if_pos</a> hi1, <a>if_neg</a> hi2, <a>Set.inter_univ</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": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "case inl\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi1 : i \u2208 p1\nhi2 : \u00aci \u2208 p2\n\u22a2 g i \u2208 s i", "state_after": "case inl\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi1 : i \u2208 p1\nhi2 : \u00aci \u2208 p2\n\u22a2 f1 i \u2208 s i"}, {"tactic": "exact ht1_m i hi1", "annotated_tactic": ["exact ht1_m i hi1", []], "state_before": "case inl\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi1 : i \u2208 p1\nhi2 : \u00aci \u2208 p2\n\u22a2 f1 i \u2208 s i", "state_after": "no goals"}, {"tactic": "have hi1 : i \u2209 p1 := fun hip1 => Set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)", "annotated_tactic": ["have hi1 : i \u2209 p1 := fun hip1 => Set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)", []], "state_before": "case inr\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi2 : i \u2208 p2\n\u22a2 g i \u2208 s i", "state_after": "case inr\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi2 : i \u2208 p2\nhi1 : \u00aci \u2208 p1\n\u22a2 g i \u2208 s i"}, {"tactic": "simp_rw [if_neg hi1, if_pos hi2, Set.univ_inter]", "annotated_tactic": ["simp_rw [<a>if_neg</a> hi1, <a>if_pos</a> hi2, <a>Set.univ_inter</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": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case inr\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi2 : i \u2208 p2\nhi1 : \u00aci \u2208 p1\n\u22a2 g i \u2208 s i", "state_after": "case inr\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi2 : i \u2208 p2\nhi1 : \u00aci \u2208 p1\n\u22a2 f2 i \u2208 s i"}, {"tactic": "exact ht2_m i hi2", "annotated_tactic": ["exact ht2_m i hi2", []], "state_before": "case inr\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\ni : \u03b9\nhi2 : i \u2208 p2\nhi1 : \u00aci \u2208 p1\n\u22a2 f2 i \u2208 s i", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\n\u22a2 (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =\n    \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nx : \u03a9\n\u22a2 x \u2208 (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x \u2194\n    x \u2208 \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ"}, {"tactic": "simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]", "annotated_tactic": ["simp only [<a>Set.mem_ite_univ_right</a>, <a>Set.mem_inter_iff</a>, <a>Set.mem_iInter</a>, <a>Finset.mem_union</a>]", [{"full_name": "Set.mem_ite_univ_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2220, 9], "def_end_pos": [2220, 27]}, {"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_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nx : \u03a9\n\u22a2 x \u2208 (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x \u2194\n    x \u2208 \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nx : \u03a9\n\u22a2 ((\u2200 (i : \u03b9), i \u2208 p1 \u2192 x \u2208 f1 i) \u2227 \u2200 (i : \u03b9), i \u2208 p2 \u2192 x \u2208 f2 i) \u2194\n    \u2200 (i : \u03b9), i \u2208 p1 \u2228 i \u2208 p2 \u2192 (i \u2208 p1 \u2192 x \u2208 f1 i) \u2227 (i \u2208 p2 \u2192 x \u2208 f2 i)"}, {"tactic": "exact\n  \u27e8fun h i _ => \u27e8h.1 i, h.2 i\u27e9, fun h =>\n    \u27e8fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi\u27e9\u27e9", "annotated_tactic": ["exact\n        \u27e8fun h i _ => \u27e8h.1 i, h.2 i\u27e9, fun h =>\n          \u27e8fun i hi => (h i (<a>Or.inl</a> hi)).1 hi, fun i hi => (h i (<a>Or.inr</a> hi)).2 hi\u27e9\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": "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\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nhgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i\nx : \u03a9\n\u22a2 ((\u2200 (i : \u03b9), i \u2208 p1 \u2192 x \u2208 f1 i) \u2227 \u2200 (i : \u03b9), i \u2208 p2 \u2192 x \u2208 f2 i) \u2194\n    \u2200 (i : \u03b9), i \u2208 p1 \u2228 i \u2208 p2 \u2192 (i \u2208 p1 \u2192 x \u2208 f1 i) \u2227 (i \u2208 p2 \u2192 x \u2208 f2 i)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro 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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\n\u22a2 \u2200 (n : \u03b9), \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\n\u22a2 \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\n\u22a2 \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1", "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if n \u2208 p1 then f1 n else Set.univ) \u2229 if n \u2208 p2 then f2 n else Set.univ) =\n    (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1"}, {"tactic": "split_ifs with h1 h2", "annotated_tactic": ["split_ifs with h1 h2", []], "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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\n\u22a2 \u2191\u2191(\u2191\u03ba a) ((if n \u2208 p1 then f1 n else Set.univ) \u2229 if n \u2208 p2 then f2 n else Set.univ) =\n    (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1", "state_after": "case pos\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : n \u2208 p1\nh2 : n \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (f1 n \u2229 f2 n) = \u2191\u2191(\u2191\u03ba a) (f1 n) * \u2191\u2191(\u2191\u03ba a) (f2 n)\n\ncase neg\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : n \u2208 p1\nh2 : \u00acn \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (f1 n \u2229 Set.univ) = \u2191\u2191(\u2191\u03ba a) (f1 n) * 1\n\ncase pos\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : \u00acn \u2208 p1\nh\u271d : n \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 f2 n) = 1 * \u2191\u2191(\u2191\u03ba a) (f2 n)\n\ncase neg\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : \u00acn \u2208 p1\nh\u271d : \u00acn \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 Set.univ) = 1 * 1"}, {"tactic": "all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]", "annotated_tactic": ["all_goals simp only [<a>measure_univ</a>, <a>one_mul</a>, <a>mul_one</a>, <a>Set.inter_univ</a>, <a>Set.univ_inter</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": "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]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case neg\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : n \u2208 p1\nh2 : \u00acn \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (f1 n \u2229 Set.univ) = \u2191\u2191(\u2191\u03ba a) (f1 n) * 1\n\ncase pos\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : \u00acn \u2208 p1\nh\u271d : n \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 f2 n) = 1 * \u2191\u2191(\u2191\u03ba a) (f2 n)\n\ncase neg\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : \u00acn \u2208 p1\nh\u271d : \u00acn \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 Set.univ) = 1 * 1", "state_after": "no goals"}, {"tactic": "exact absurd rfl (Set.disjoint_iff_forall_ne.mp hST (hp1 h1) (hp2 h2))", "annotated_tactic": ["exact <a>absurd</a> <a>rfl</a> (Set.disjoint_iff_forall_ne.mp hST (hp1 h1) (hp2 h2))", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"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\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : n \u2208 p1\nh2 : n \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (f1 n \u2229 f2 n) = \u2191\u2191(\u2191\u03ba a) (f1 n) * \u2191\u2191(\u2191\u03ba a) (f2 n)", "state_after": "no goals"}, {"tactic": "simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]", "annotated_tactic": ["simp only [<a>measure_univ</a>, <a>one_mul</a>, <a>mul_one</a>, <a>Set.inter_univ</a>, <a>Set.univ_inter</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": "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]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "state_before": "case neg\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\ns : \u03b9 \u2192 Set (Set \u03a9)\nS T : Set \u03b9\nh_indep : iIndepSets s \u03ba\nhST : Disjoint S T\nt1 t2 : Set \u03a9\np1 : Finset \u03b9\nhp1 : \u2191p1 \u2286 S\nf1 : \u03b9 \u2192 Set \u03a9\nht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x\nht1_eq : t1 = \u22c2 x \u2208 p1, f1 x\np2 : Finset \u03b9\nhp2 : \u2191p2 \u2286 T\nf2 : \u03b9 \u2192 Set \u03a9\nht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x\nht2_eq : t2 = \u22c2 x \u2208 p2, f2 x\ng : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ\nh_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\na : \u03b1\nh_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n)\nha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i)\nha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i)\nn : \u03b9\nh1 : \u00acn \u2208 p1\nh\u271d : \u00acn \u2208 p2\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 Set.univ) = 1 * 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpInd_empty", "start": [292, 1], "end": [299, 6], "traced_tactics": [{"tactic": "ext1 x", "annotated_tactic": ["ext1 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\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)\n\u22a2 condexpInd G hm \u03bc \u2205 = 0", "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)\nx : G\n\u22a2 \u2191(condexpInd G hm \u03bc \u2205) x = \u21910 x"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "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)\nx : G\n\u22a2 \u2191(condexpInd G hm \u03bc \u2205) x = \u21910 x", "state_after": "case h.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)\nx : G\n\u22a2 \u2191\u2191(\u2191(condexpInd G hm \u03bc \u2205) x) =\u1d50[\u03bc] \u2191\u2191(\u21910 x)"}, {"tactic": "refine' (condexpInd_ae_eq_condexpIndSMul hm MeasurableSet.empty (by simp) x).trans _", "annotated_tactic": ["refine' (<a>condexpInd_ae_eq_condexpIndSMul</a> hm <a>MeasurableSet.empty</a> (by simp) x).<a>trans</a> _", [{"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": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"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.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)\nx : G\n\u22a2 \u2191\u2191(\u2191(condexpInd G hm \u03bc \u2205) x) =\u1d50[\u03bc] \u2191\u2191(\u21910 x)", "state_after": "case h.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)\nx : G\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) x) =\u1d50[\u03bc] \u2191\u2191(\u21910 x)"}, {"tactic": "rw [condexpIndSMul_empty]", "annotated_tactic": ["rw [<a>condexpIndSMul_empty</a>]", [{"full_name": "MeasureTheory.condexpIndSMul_empty", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [461, 9], "def_end_pos": [461, 29]}]], "state_before": "case h.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)\nx : G\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) x) =\u1d50[\u03bc] \u2191\u2191(\u21910 x)", "state_after": "case h.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)\nx : G\n\u22a2 \u2191\u21910 =\u1d50[\u03bc] \u2191\u2191(\u21910 x)"}, {"tactic": "refine' (Lp.coeFn_zero G 2 \u03bc).trans _", "annotated_tactic": ["refine' (<a>Lp.coeFn_zero</a> G 2 \u03bc).<a>trans</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.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case h.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)\nx : G\n\u22a2 \u2191\u21910 =\u1d50[\u03bc] \u2191\u2191(\u21910 x)", "state_after": "case h.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)\nx : G\n\u22a2 0 =\u1d50[\u03bc] \u2191\u2191(\u21910 x)"}, {"tactic": "refine' EventuallyEq.trans _ (Lp.coeFn_zero G 1 \u03bc).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>Lp.coeFn_zero</a> G 1 \u03bc).<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_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 h.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)\nx : G\n\u22a2 0 =\u1d50[\u03bc] \u2191\u2191(\u21910 x)", "state_after": "case h.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)\nx : G\n\u22a2 0 =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.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)\nx : G\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 \u2191\u2191\u03bc \u2205 \u2260 \u22a4", "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.dirac_prod_dirac", "start": [663, 1], "end": [664, 54], "traced_tactics": [{"tactic": "rw [prod_dirac, map_dirac measurable_prod_mk_right]", "annotated_tactic": ["rw [<a>prod_dirac</a>, <a>map_dirac</a> <a>measurable_prod_mk_right</a>]", [{"full_name": "MeasureTheory.Measure.prod_dirac", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [651, 9], "def_end_pos": [651, 19]}, {"full_name": "MeasureTheory.Measure.map_dirac", "def_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "def_pos": [62, 9], "def_end_pos": [62, 18]}, {"full_name": "measurable_prod_mk_right", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [740, 9], "def_end_pos": [740, 33]}]], "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\nx : \u03b1\ny : \u03b2\n\u22a2 Measure.prod (dirac x) (dirac y) = dirac (x, y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Nat.gcd_eq_gcd_ab", "start": [152, 1], "end": [154, 38], "traced_tactics": [{"tactic": "have := @xgcdAux_P x y x y 1 0 0 1 (by simp [P]) (by simp [P])", "annotated_tactic": ["have := @<a>xgcdAux_P</a> x y x y 1 0 0 1 (by simp [<a>P</a>]) (by simp [<a>P</a>])", [{"full_name": "Nat.xgcdAux_P", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [136, 9], "def_end_pos": [136, 18]}, {"full_name": "_private.Mathlib.Data.Int.GCD.0.Nat.P", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [133, 13], "def_end_pos": [133, 14]}, {"full_name": "_private.Mathlib.Data.Int.GCD.0.Nat.P", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [133, 13], "def_end_pos": [133, 14]}]], "state_before": "x y : \u2115\n\u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y", "state_after": "x y : \u2115\nthis : Nat.P x y (xgcdAux x 1 0 y 0 1)\n\u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y"}, {"tactic": "rwa [xgcdAux_val, xgcd_val] at this", "annotated_tactic": ["rwa [<a>xgcdAux_val</a>, <a>xgcd_val</a>] at this", [{"full_name": "Nat.xgcdAux_val", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [121, 9], "def_end_pos": [121, 20]}, {"full_name": "Nat.xgcd_val", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [125, 9], "def_end_pos": [125, 17]}]], "state_before": "x y : \u2115\nthis : Nat.P x y (xgcdAux x 1 0 y 0 1)\n\u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y", "state_after": "no goals"}, {"tactic": "simp [P]", "annotated_tactic": ["simp [<a>P</a>]", [{"full_name": "_private.Mathlib.Data.Int.GCD.0.Nat.P", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [133, 13], "def_end_pos": [133, 14]}]], "state_before": "x y : \u2115\n\u22a2 Nat.P x y (x, 1, 0)", "state_after": "no goals"}, {"tactic": "simp [P]", "annotated_tactic": ["simp [<a>P</a>]", [{"full_name": "_private.Mathlib.Data.Int.GCD.0.Nat.P", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [133, 13], "def_end_pos": [133, 14]}]], "state_before": "x y : \u2115\n\u22a2 Nat.P x y (y, 0, 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_sInf", "start": [1406, 1], "end": [1409, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_smul", "start": [303, 1], "end": [313, 46], "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 : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \u03bc[f|m]"}, {"tactic": "refine' (condexp_ae_eq_condexpL1 hm _).trans _", "annotated_tactic": ["refine' (<a>condexp_ae_eq_condexpL1</a> hm _).<a>trans</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]}]], "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (c \u2022 f)) =\u1d50[\u03bc] c \u2022 \u03bc[f|m]"}, {"tactic": "rw [condexpL1_smul c f]", "annotated_tactic": ["rw [<a>condexpL1_smul</a> c f]", [{"full_name": "MeasureTheory.condexpL1_smul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [574, 9], "def_end_pos": [574, 23]}]], "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (c \u2022 f)) =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) =\u1d50[\u03bc] c \u2022 \u03bc[f|m]"}, {"tactic": "refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _", "annotated_tactic": ["refine' (@<a>condexp_ae_eq_condexpL1</a> _ _ _ _ _ m _ _ hm _ f).<a>mp</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.Eventually.mp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}]], "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u03bc[f|m]) x = \u2191\u2191(condexpL1 hm \u03bc f) x \u2192 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u03bc[f|m]) x"}, {"tactic": "refine' (coeFn_smul c (condexpL1 hm \u03bc f)).mono fun x hx1 hx2 => _", "annotated_tactic": ["refine' (<a>coeFn_smul</a> c (<a>condexpL1</a> hm \u03bc f)).<a>mono</a> fun x hx1 hx2 => _", [{"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.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [514, 5], "def_end_pos": [514, 14]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u03bc[f|m]) x = \u2191\u2191(condexpL1 hm \u03bc f) x \u2192 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u03bc[f|m]) x", "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nx : \u03b1\nhx1 : \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u2191\u2191(condexpL1 hm \u03bc f)) x\nhx2 : (\u03bc[f|m]) x = \u2191\u2191(condexpL1 hm \u03bc f) x\n\u22a2 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u03bc[f|m]) x"}, {"tactic": "rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]", "annotated_tactic": ["rw [hx1, <a>Pi.smul_apply</a>, <a>Pi.smul_apply</a>, hx2]", [{"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 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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\nx : \u03b1\nhx1 : \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u2191\u2191(condexpL1 hm \u03bc f)) x\nhx2 : (\u03bc[f|m]) x = \u2191\u2191(condexpL1 hm \u03bc f) x\n\u22a2 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u03bc[f|m]) x", "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 : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] c \u2022 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] c \u2022 0", "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\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 : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\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 : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] c \u2022 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] c \u2022 0", "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\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\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 : \u03b1 \u2192 F'\ns : Set \u03b1\nc : \ud835\udd5c\nf : \u03b1 \u2192 F'\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_insert_eq_ite", "start": [122, 1], "end": [125, 44], "traced_tactics": [{"tactic": "by_cases h : a \u2208 s", "annotated_tactic": ["by_cases h : a \u2208 s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1"}, {"tactic": "rw [card_insert_of_mem h, if_pos h]", "annotated_tactic": ["rw [<a>card_insert_of_mem</a> h, <a>if_pos</a> h]", [{"full_name": "Finset.card_insert_of_mem", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [110, 9], "def_end_pos": [110, 27]}, {"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\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1", "state_after": "no goals"}, {"tactic": "rw [card_insert_of_not_mem h, if_neg h]", "annotated_tactic": ["rw [<a>card_insert_of_not_mem</a> h, <a>if_neg</a> 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": "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\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.subset_image_symmDiff", "start": [457, 1], "end": [459, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.erase_cons", "start": [1155, 1], "end": [1158, 62], "traced_tactics": [{"tactic": "simp [List.erase, h]", "annotated_tactic": ["simp [<a>List.erase</a>, h]", [{"full_name": "List.erase", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [120, 15], "def_end_pos": [120, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl : List \u03b1\nh : b = a\n\u22a2 List.erase (b :: l) a = if b = a then l else b :: List.erase l a", "state_after": "no goals"}, {"tactic": "simp [List.erase, h, (beq_eq_false_iff_ne _ _).2 h]", "annotated_tactic": ["simp [<a>List.erase</a>, h, (<a>beq_eq_false_iff_ne</a> _ _).2 h]", [{"full_name": "List.erase", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [120, 15], "def_end_pos": [120, 20]}, {"full_name": "beq_eq_false_iff_ne", "def_path": "lake-packages/std/Std/Classes/BEq.lean", "def_pos": [20, 17], "def_end_pos": [20, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl : List \u03b1\nh : \u00acb = a\n\u22a2 List.erase (b :: l) a = if b = a then l else b :: List.erase l a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_of_fintype", "start": [511, 1], "end": [514, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurable.congr", "start": [431, 1], "end": [434, 45], "traced_tactics": [{"tactic": "rw [mem_preimage, mem_preimage, hx]", "annotated_tactic": ["rw [<a>mem_preimage</a>, <a>mem_preimage</a>, 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_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : NullMeasurable f\nhg : f =\u1d50[\u03bc] g\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b1\nhx : f x = g x\n\u22a2 x \u2208 f \u207b\u00b9' s \u2194 x \u2208 g \u207b\u00b9' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.neg_smul", "start": [1113, 11], "end": [1115, 40], "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 : Ring \u03b1\ninst\u271d\u00b9 : AddCommGroup \u03b2\ninst\u271d : Module \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 : Ring \u03b1\ninst\u271d\u00b9 : AddCommGroup \u03b2\ninst\u271d : Module \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Neg.neg '' s \u2022 t = Neg.neg '' (s \u2022 t)"}, {"tactic": "exact image2_image_left_comm neg_smul", "annotated_tactic": ["exact <a>image2_image_left_comm</a> <a>neg_smul</a>", [{"full_name": "Set.image2_image_left_comm", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [370, 9], "def_end_pos": [370, 31]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 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 : Ring \u03b1\ninst\u271d\u00b9 : AddCommGroup \u03b2\ninst\u271d : Module \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Neg.neg '' s \u2022 t = Neg.neg '' (s \u2022 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "full_name": "MeasurableSpace.le_invariants_iterate", "start": [50, 1], "end": [54, 82], "traced_tactics": [{"tactic": "induction n with\n| zero => simp [invariants_le]\n| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)", "annotated_tactic": ["induction n with\n  | <a>zero</a> => simp [<a>invariants_le</a>]\n  | <a>succ</a> n ihn => exact <a>le_trans</a> (<a>le_inf</a> ihn <a>le_rfl</a>) (<a>inf_le_invariants_comp</a> _ _)", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "MeasurableSpace.invariants_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}, {"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": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_inf", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [409, 9], "def_end_pos": [409, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasurableSpace.inf_le_invariants_comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [46, 9], "def_end_pos": [46, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\nn : \u2115\n\u22a2 invariants f \u2264 invariants f^[n]", "state_after": "no goals"}, {"tactic": "simp [invariants_le]", "annotated_tactic": ["simp [<a>invariants_le</a>]", [{"full_name": "MeasurableSpace.invariants_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\n\u22a2 invariants f \u2264 invariants f^[Nat.zero]", "state_after": "no goals"}, {"tactic": "exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)", "annotated_tactic": ["exact <a>le_trans</a> (<a>le_inf</a> ihn <a>le_rfl</a>) (<a>inf_le_invariants_comp</a> _ _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_inf", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [409, 9], "def_end_pos": [409, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasurableSpace.inf_le_invariants_comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [46, 9], "def_end_pos": [46, 31]}]], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nihn : invariants f \u2264 invariants f^[n]\n\u22a2 invariants f \u2264 invariants f^[Nat.succ n]", "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.regular_of_isMulLeftInvariant", "start": [718, 1], "end": [720, 67], "traced_tactics": [{"tactic": "rw [haarMeasure_unique \u03bc \u27e8\u27e8K, hK\u27e9, h2K\u27e9]", "annotated_tactic": ["rw [<a>haarMeasure_unique</a> \u03bc \u27e8\u27e8K, hK\u27e9, h2K\u27e9]", [{"full_name": "MeasureTheory.Measure.haarMeasure_unique", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}]], "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 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nh2K : Set.Nonempty (interior K)\nh\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 Regular \u03bc", "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 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nh2K : Set.Nonempty (interior K)\nh\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 Regular\n    (\u2191\u2191\u03bc \u2191{ toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K } \u2022\n      haarMeasure { toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K })"}, {"tactic": "exact Regular.smul h\u03bcK", "annotated_tactic": ["exact <a>Regular.smul</a> h\u03bcK", [{"full_name": "MeasureTheory.Measure.Regular.smul", "def_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "def_pos": [541, 19], "def_end_pos": [541, 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 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nh2K : Set.Nonempty (interior K)\nh\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 Regular\n    (\u2191\u2191\u03bc \u2191{ toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K } \u2022\n      haarMeasure { toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.orderEmbOfFin_last", "start": [198, 1], "end": [201, 53], "traced_tactics": [{"tactic": "simp [orderEmbOfFin_apply, max'_eq_sorted_last, h]", "annotated_tactic": ["simp [<a>orderEmbOfFin_apply</a>, <a>max'_eq_sorted_last</a>, h]", [{"full_name": "Finset.orderEmbOfFin_apply", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [170, 9], "def_end_pos": [170, 28]}, {"full_name": "Finset.max'_eq_sorted_last", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [135, 9], "def_end_pos": [135, 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 := k - 1, isLt := (_ : k - 1 < k) } = max' s (_ : Finset.Nonempty s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Constructions.lean", "full_name": "FiniteInter.finiteInterClosure_finiteInter", "start": [46, 1], "end": [48, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.Adapted.stoppedProcess", "start": [1027, 1], "end": [1030, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "full_name": "MeasureTheory.Measure.integral_comp_inv_mul_right", "start": [122, 1], "end": [124, 61], "traced_tactics": [{"tactic": "simpa only [mul_comm] using integral_comp_inv_mul_left g a", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>integral_comp_inv_mul_left</a> g 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.integral_comp_inv_mul_left", "def_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "def_pos": [113, 9], "def_end_pos": [113, 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 : Set E\ng : \u211d \u2192 F\na : \u211d\n\u22a2 \u222b (x : \u211d), g (x * a\u207b\u00b9) = |a| \u2022 \u222b (y : \u211d), g y", "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_closedBall_center", "start": [432, 1], "end": [436, 34], "traced_tactics": [{"tactic": "rw [this, measure_preimage_add]", "annotated_tactic": ["rw [this, <a>measure_preimage_add</a>]", [{"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\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\nthis : closedBall 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' closedBall x r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (closedBall 0 r)", "state_after": "no goals"}, {"tactic": "simp [preimage_add_closedBall]", "annotated_tactic": ["simp [<a>preimage_add_closedBall</a>]", [{"full_name": "preimage_add_closedBall", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1586, 3], "def_end_pos": [1586, 14]}]], "state_before": "E\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\n\u22a2 closedBall 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' closedBall x r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.exists_measurableEmbedding_real", "start": [1072, 1], "end": [1074, 83], "traced_tactics": [{"tactic": "obtain \u27e8s, hs, \u27e8e\u27e9\u27e9 := exists_subset_real_measurableEquiv \u03b1", "annotated_tactic": ["obtain \u27e8s, hs, \u27e8e\u27e9\u27e9 := <a>exists_subset_real_measurableEquiv</a> \u03b1", [{"full_name": "MeasureTheory.exists_subset_real_measurableEquiv", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\n\u22a2 \u2203 f, MeasurableEmbedding f", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\ns : Set \u211d\nhs : MeasurableSet s\ne : \u03b1 \u2243\u1d50 \u2191s\n\u22a2 \u2203 f, MeasurableEmbedding f"}, {"tactic": "exact \u27e8(\u2191) \u2218 e, (MeasurableEmbedding.subtype_coe hs).comp e.measurableEmbedding\u27e9", "annotated_tactic": ["exact \u27e8(\u2191) \u2218 e, (<a>MeasurableEmbedding.subtype_coe</a> hs).<a>comp</a> e.measurableEmbedding\u27e9", [{"full_name": "MeasurableEmbedding.subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 20]}, {"full_name": "MeasurableEmbedding.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1202, 9], "def_end_pos": [1202, 13]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : StandardBorelSpace \u03b1\ns : Set \u211d\nhs : MeasurableSet s\ne : \u03b1 \u2243\u1d50 \u2191s\n\u22a2 \u2203 f, MeasurableEmbedding f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.integral_truncation_le_integral_of_nonneg", "start": [190, 1], "end": [198, 39], "traced_tactics": [{"tactic": "apply integral_mono_of_nonneg\n  (eventually_of_forall fun x => ?_) hf (eventually_of_forall fun x => ?_)", "annotated_tactic": ["apply <a>integral_mono_of_nonneg</a>\n    (<a>eventually_of_forall</a> fun x => ?_) hf (<a>eventually_of_forall</a> fun x => ?_)", [{"full_name": "MeasureTheory.integral_mono_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1356, 9], "def_end_pos": [1356, 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": "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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\n\u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc \u2264 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\nx : \u03b1\n\u22a2 OfNat.ofNat 0 x \u2264 truncation f A x\n\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\nx : \u03b1\n\u22a2 truncation f A x \u2264 f x"}, {"tactic": "exact truncation_nonneg _ (h'f x)", "annotated_tactic": ["exact <a>truncation_nonneg</a> _ (h'f x)", [{"full_name": "ProbabilityTheory.truncation_nonneg", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [128, 9], "def_end_pos": [128, 26]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\nx : \u03b1\n\u22a2 OfNat.ofNat 0 x \u2264 truncation f A x", "state_after": "no goals"}, {"tactic": "calc\n  truncation f A x \u2264 |truncation f A x| := le_abs_self _\n  _ \u2264 |f x| := (abs_truncation_le_abs_self _ _ _)\n  _ = f x := abs_of_nonneg (h'f x)", "annotated_tactic": ["calc\n      <a>truncation</a> f A x \u2264 |<a>truncation</a> f A x| := <a>le_abs_self</a> _\n      _ \u2264 |f x| := (<a>abs_truncation_le_abs_self</a> _ _ _)\n      _ = f x := <a>abs_of_nonneg</a> (h'f x)", [{"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}, {"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}, {"full_name": "ProbabilityTheory.abs_truncation_le_abs_self", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [101, 9], "def_end_pos": [101, 35]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\nx : \u03b1\n\u22a2 truncation f A x \u2264 f x", "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.ediv_eq_iff_eq_mul_left", "start": [768, 11], "end": [770, 61], "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": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 a / b = c \u2194 a = c * b", "state_after": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 a / b = c \u2194 a = b * c"}, {"tactic": "exact Int.ediv_eq_iff_eq_mul_right H H'", "annotated_tactic": ["exact <a>Int.ediv_eq_iff_eq_mul_right</a> H H'", [{"full_name": "Int.ediv_eq_iff_eq_mul_right", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [760, 19], "def_end_pos": [760, 43]}]], "state_before": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 a / b = c \u2194 a = b * c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.intervalIntegral_pos_of_pos", "start": [1315, 1], "end": [1318, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.zero_mem_smul_iff", "start": [2154, 1], "end": [2157, 6], "traced_tactics": [{"tactic": "rw [\u2190 mem_coe, coe_smul, Set.zero_mem_smul_iff]", "annotated_tactic": ["rw [\u2190 <a>mem_coe</a>, <a>coe_smul</a>, <a>Set.zero_mem_smul_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.coe_smul", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 17]}, {"full_name": "Set.zero_mem_smul_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [830, 9], "def_end_pos": [830, 26]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\n\u22a2 0 \u2208 s \u2022 t \u2194 0 \u2208 s \u2227 Finset.Nonempty t \u2228 0 \u2208 t \u2227 Finset.Nonempty s", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\n\u22a2 0 \u2208 \u2191s \u2227 Set.Nonempty \u2191t \u2228 0 \u2208 \u2191t \u2227 Set.Nonempty \u2191s \u2194 0 \u2208 s \u2227 Finset.Nonempty t \u2228 0 \u2208 t \u2227 Finset.Nonempty s"}, {"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\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\n\u22a2 0 \u2208 \u2191s \u2227 Set.Nonempty \u2191t \u2228 0 \u2208 \u2191t \u2227 Set.Nonempty \u2191s \u2194 0 \u2208 s \u2227 Finset.Nonempty t \u2228 0 \u2208 t \u2227 Finset.Nonempty s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.liftOn\u2082_mk", "start": [338, 1], "end": [341, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.ae_le_trim_iff", "start": [1928, 1], "end": [1931, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.self_eq_mass_mul_normalize", "start": [341, 1], "end": [351, 39], "traced_tactics": [{"tactic": "obtain rfl | h := eq_or_ne \u03bc 0", "annotated_tactic": ["obtain rfl | h := <a>eq_or_ne</a> \u03bc 0", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s", "state_after": "case inl\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u21910 s)) s = mass 0 * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize 0) s)) s\n\ncase inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s"}, {"tactic": "have mass_nonzero : \u03bc.mass \u2260 0 := by rwa [\u03bc.mass_nonzero_iff]", "annotated_tactic": ["have mass_nonzero : \u03bc.mass \u2260 0 := by rwa [\u03bc.mass_nonzero_iff]", []], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s"}, {"tactic": "simp only [normalize, dif_neg mass_nonzero]", "annotated_tactic": ["simp only [<a>normalize</a>, <a>dif_neg</a> mass_nonzero]", [{"full_name": "MeasureTheory.FiniteMeasure.normalize", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [326, 5], "def_end_pos": [326, 14]}, {"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 inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s) =\n    mass \u03bc *\n      ENNReal.toNNReal (\u2191\u2191\u2191{ val := \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc), property := (_ : IsProbabilityMeasure \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)) } s)"}, {"tactic": "change \u03bc s = mass \u03bc * ((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s", "annotated_tactic": ["change \u03bc s = <a>mass</a> \u03bc * ((<a>mass</a> \u03bc)\u207b\u00b9 \u2022 \u03bc) s", [{"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.mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [163, 5], "def_end_pos": [163, 9]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s) =\n    mass \u03bc *\n      ENNReal.toNNReal (\u2191\u2191\u2191{ val := \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc), property := (_ : IsProbabilityMeasure \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)) } s)", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s)) s"}, {"tactic": "simp only [toMeasure_smul, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n  Measure.nnreal_smul_coe_apply, ne_eq, mass_zero_iff, ENNReal.toNNReal_mul, ENNReal.toNNReal_coe,\n  mul_inv_cancel_left\u2080 mass_nonzero]", "annotated_tactic": ["simp only [<a>toMeasure_smul</a>, <a>Measure.smul_toOuterMeasure</a>, <a>OuterMeasure.coe_smul</a>, <a>Pi.smul_apply</a>,\n    <a>Measure.nnreal_smul_coe_apply</a>, <a>ne_eq</a>, <a>mass_zero_iff</a>, <a>ENNReal.toNNReal_mul</a>, <a>ENNReal.toNNReal_coe</a>,\n    <a>mul_inv_cancel_left\u2080</a> mass_nonzero]", [{"full_name": "MeasureTheory.FiniteMeasure.toMeasure_smul", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [235, 9], "def_end_pos": [235, 23]}, {"full_name": "MeasureTheory.Measure.smul_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [830, 9], "def_end_pos": [830, 28]}, {"full_name": "MeasureTheory.OuterMeasure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [306, 9], "def_end_pos": [306, 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.Measure.nnreal_smul_coe_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [911, 9], "def_end_pos": [911, 30]}, {"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": "MeasureTheory.FiniteMeasure.mass_zero_iff", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [184, 9], "def_end_pos": [184, 22]}, {"full_name": "ENNReal.toNNReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2254, 9], "def_end_pos": [2254, 21]}, {"full_name": "ENNReal.toNNReal_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [176, 9], "def_end_pos": [176, 21]}, {"full_name": "mul_inv_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [217, 9], "def_end_pos": [217, 29]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s)) s", "state_after": "no goals"}, {"tactic": "simp only [zero_mass, coeFn_zero, Pi.zero_apply, zero_mul]", "annotated_tactic": ["simp only [<a>zero_mass</a>, <a>coeFn_zero</a>, <a>Pi.zero_apply</a>, <a>zero_mul</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.zero_mass", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [179, 9], "def_end_pos": [179, 18]}, {"full_name": "MeasureTheory.FiniteMeasure.coeFn_zero", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [240, 9], "def_end_pos": [240, 19]}, {"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]}]], "state_before": "case inl\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u21910 s)) s = mass 0 * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize 0) s)) s", "state_after": "case inl\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u21910 s) = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case inl\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u21910 s) = 0", "state_after": "no goals"}, {"tactic": "rwa [\u03bc.mass_nonzero_iff]", "annotated_tactic": ["rwa [\u03bc.mass_nonzero_iff]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\n\u22a2 mass \u03bc \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.integrableOn_add_measure", "start": [227, 1], "end": [231, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.set_biInter_singleton", "start": [2099, 1], "end": [2100, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_comp_mul_right", "start": [693, 1], "end": [703, 45], "traced_tactics": [{"tactic": "have A : MeasurableEmbedding fun x => x * c :=\n  (Homeomorph.mulRight\u2080 c hc).closedEmbedding.measurableEmbedding", "annotated_tactic": ["have A : <a>MeasurableEmbedding</a> fun x => x * c :=\n    (<a>Homeomorph.mulRight\u2080</a> c hc).closedEmbedding.measurableEmbedding", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1178, 11], "def_end_pos": [1178, 30]}, {"full_name": "Homeomorph.mulRight\u2080", "def_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "def_pos": [248, 15], "def_end_pos": [248, 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\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\n\u22a2 \u222b (x : \u211d) in a..b, f (x * c) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\n\u22a2 \u222b (x : \u211d) in a..b, f (x * c) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x"}, {"tactic": "conv_rhs => rw [\u2190 Real.smul_map_volume_mul_right hc]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Real.smul_map_volume_mul_right</a> hc]", [{"full_name": "Real.smul_map_volume_mul_right", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 34]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\n\u22a2 \u222b (x : \u211d) in a..b, f (x * c) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\n\u22a2 \u222b (x : \u211d) in a..b, f (x * c) =\n    c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x \u2202ENNReal.ofReal |c| \u2022 Measure.map (fun x => x * c) volume"}, {"tactic": "simp_rw [integral_smul_measure, intervalIntegral, A.set_integral_map,\n  ENNReal.toReal_ofReal (abs_nonneg c)]", "annotated_tactic": ["simp_rw [<a>integral_smul_measure</a>, <a>intervalIntegral</a>, A.set_integral_map,\n    <a>ENNReal.toReal_ofReal</a> (<a>abs_nonneg</a> c)]", [{"full_name": "intervalIntegral.integral_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [643, 16], "def_end_pos": [643, 37]}, {"full_name": "intervalIntegral", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\n\u22a2 \u222b (x : \u211d) in a..b, f (x * c) =\n    c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x \u2202ENNReal.ofReal |c| \u2022 Measure.map (fun x => x * c) volume", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\n\u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =\n    c\u207b\u00b9 \u2022\n      |c| \u2022\n        ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -\n          \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c))"}, {"tactic": "cases' hc.lt_or_lt with h h", "annotated_tactic": ["cases' hc.lt_or_lt with h h", []], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\n\u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =\n    c\u207b\u00b9 \u2022\n      |c| \u2022\n        ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -\n          \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c))", "state_after": "case inl\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\nh : c < 0\n\u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =\n    c\u207b\u00b9 \u2022\n      |c| \u2022\n        ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -\n          \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c))\n\ncase inr\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\nh : 0 < c\n\u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =\n    c\u207b\u00b9 \u2022\n      |c| \u2022\n        ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -\n          \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c))"}, {"tactic": "simp [h, mul_div_cancel, hc, abs_of_neg,\n  Measure.restrict_congr_set (\u03b1 := \u211d) (\u03bc := volume) Ico_ae_eq_Ioc]", "annotated_tactic": ["simp [h, <a>mul_div_cancel</a>, hc, <a>abs_of_neg</a>,\n      <a>Measure.restrict_congr_set</a> (\u03b1 := \u211d) (\u03bc := <a>volume</a>) <a>Ico_ae_eq_Ioc</a>]", [{"full_name": "mul_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"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.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.Ico_ae_eq_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3188, 9], "def_end_pos": [3188, 22]}]], "state_before": "case inl\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\nh : c < 0\n\u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =\n    c\u207b\u00b9 \u2022\n      |c| \u2022\n        ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -\n          \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c))", "state_after": "no goals"}, {"tactic": "simp [h, mul_div_cancel, hc, abs_of_pos]", "annotated_tactic": ["simp [h, <a>mul_div_cancel</a>, hc, <a>abs_of_pos</a>]", [{"full_name": "mul_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}]], "state_before": "case inr\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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nA : MeasurableEmbedding fun x => x * c\nh : 0 < c\n\u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =\n    c\u207b\u00b9 \u2022\n      |c| \u2022\n        ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -\n          \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "ENNReal.measurable_toReal", "start": [2068, 1], "end": [2069, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.Adapted.measurable_upcrossingsBefore", "start": [800, 1], "end": [808, 78], "traced_tactics": [{"tactic": "have : upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n \u03c9 < N}.indicator 1 i := by\n  ext \u03c9\n  exact upcrossingsBefore_eq_sum hab", "annotated_tactic": ["have : <a>upcrossingsBefore</a> a b f N = fun \u03c9 =>\n      \u2211 i in <a>Finset.Ico</a> 1 (N + 1), {n | <a>upperCrossingTime</a> a b f N n \u03c9 < N}.<a>indicator</a> 1 i := by\n    ext \u03c9\n    exact <a>upcrossingsBefore_eq_sum</a> hab", [{"full_name": "MeasureTheory.upcrossingsBefore", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [450, 19], "def_end_pos": [450, 36]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [309, 5], "def_end_pos": [309, 8]}, {"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.upcrossingsBefore_eq_sum", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [774, 9], "def_end_pos": [774, 33]}]], "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 : Adapted \u2131 f\nhab : a < b\n\u22a2 Measurable (upcrossingsBefore a b f 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\nhf : Adapted \u2131 f\nhab : a < b\nthis :\n  upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i\n\u22a2 Measurable (upcrossingsBefore a b f N)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "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 : Adapted \u2131 f\nhab : a < b\nthis :\n  upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i\n\u22a2 Measurable (upcrossingsBefore a b f 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\nhf : Adapted \u2131 f\nhab : a < b\nthis :\n  upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i\n\u22a2 Measurable fun \u03c9 => \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i"}, {"tactic": "exact Finset.measurable_sum _ fun i _ => Measurable.indicator measurable_const <|\n  \u2131.le N _ (hf.isStoppingTime_upperCrossingTime.measurableSet_lt_of_pred N)", "annotated_tactic": ["exact <a>Finset.measurable_sum</a> _ fun i _ => <a>Measurable.indicator</a> <a>measurable_const</a> <|\n    \u2131.le N _ (hf.isStoppingTime_upperCrossingTime.measurableSet_lt_of_pred N)", [{"full_name": "Finset.measurable_sum", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [935, 3], "def_end_pos": [935, 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": "\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 : Adapted \u2131 f\nhab : a < b\nthis :\n  upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i\n\u22a2 Measurable fun \u03c9 => \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i", "state_after": "no goals"}, {"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\nhf : Adapted \u2131 f\nhab : a < b\n\u22a2 upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i", "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\nhf : Adapted \u2131 f\nhab : a < b\n\u03c9 : \u03a9\n\u22a2 upcrossingsBefore a b f N \u03c9 = \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i"}, {"tactic": "exact upcrossingsBefore_eq_sum hab", "annotated_tactic": ["exact <a>upcrossingsBefore_eq_sum</a> hab", [{"full_name": "MeasureTheory.upcrossingsBefore_eq_sum", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [774, 9], "def_end_pos": [774, 33]}]], "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\nhf : Adapted \u2131 f\nhab : a < b\n\u03c9 : \u03a9\n\u22a2 upcrossingsBefore a b f N \u03c9 = \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_segment", "start": [1173, 1], "end": [1175, 66], "traced_tactics": [{"tactic": "rw [\u2190 affineSegment_eq_segment, hausdorffMeasure_affineSegment]", "annotated_tactic": ["rw [\u2190 <a>affineSegment_eq_segment</a>, <a>hausdorffMeasure_affineSegment</a>]", [{"full_name": "affineSegment_eq_segment", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [48, 9], "def_end_pos": [48, 33]}, {"full_name": "MeasureTheory.hausdorffMeasure_affineSegment", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1163, 9], "def_end_pos": [1163, 39]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE\u271d : Type u_5\nP : Type u_6\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nx y : E\n\u22a2 \u2191\u2191\u03bcH[1] (segment \u211d x y) = edist x y", "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.lcm_dvd", "start": [220, 1], "end": [226, 75], "traced_tactics": [{"tactic": "match eq_zero_or_pos k with\n| .inl h => rw [h]; exact Nat.dvd_zero _\n| .inr kpos =>\n  apply Nat.dvd_of_mul_dvd_mul_left (gcd_pos_of_pos_left n (pos_of_dvd_of_pos H1 kpos))\n  rw [gcd_mul_lcm, \u2190 gcd_mul_right, Nat.mul_comm n k]\n  exact dvd_gcd (Nat.mul_dvd_mul_left _ H2) (Nat.mul_dvd_mul_right H1 _)", "annotated_tactic": ["match <a>eq_zero_or_pos</a> k with\n  | .inl h => rw [h]; exact <a>Nat.dvd_zero</a> _\n  | .inr kpos =>\n    apply <a>Nat.dvd_of_mul_dvd_mul_left</a> (<a>gcd_pos_of_pos_left</a> n (<a>pos_of_dvd_of_pos</a> H1 kpos))\n    rw [<a>gcd_mul_lcm</a>, \u2190 <a>gcd_mul_right</a>, <a>Nat.mul_comm</a> n k]\n    exact <a>dvd_gcd</a> (<a>Nat.mul_dvd_mul_left</a> _ H2) (<a>Nat.mul_dvd_mul_right</a> H1 _)", [{"full_name": "Nat.eq_zero_or_pos", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}, {"full_name": "Nat.dvd_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [864, 19], "def_end_pos": [864, 27]}, {"full_name": "Nat.dvd_of_mul_dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [957, 19], "def_end_pos": [957, 42]}, {"full_name": "Nat.gcd_pos_of_pos_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [92, 9], "def_end_pos": [92, 28]}, {"full_name": "Nat.pos_of_dvd_of_pos", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}, {"full_name": "Nat.gcd_mul_lcm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [217, 9], "def_end_pos": [217, 20]}, {"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.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"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.mul_dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [897, 19], "def_end_pos": [897, 35]}, {"full_name": "Nat.mul_dvd_mul_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [900, 19], "def_end_pos": [900, 36]}]], "state_before": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\n\u22a2 lcm m n \u2223 k", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nh : k = 0\n\u22a2 lcm m n \u2223 k", "state_after": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nh : k = 0\n\u22a2 lcm m n \u2223 0"}, {"tactic": "exact Nat.dvd_zero _", "annotated_tactic": ["exact <a>Nat.dvd_zero</a> _", [{"full_name": "Nat.dvd_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [864, 19], "def_end_pos": [864, 27]}]], "state_before": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nh : k = 0\n\u22a2 lcm m n \u2223 0", "state_after": "no goals"}, {"tactic": "apply Nat.dvd_of_mul_dvd_mul_left (gcd_pos_of_pos_left n (pos_of_dvd_of_pos H1 kpos))", "annotated_tactic": ["apply <a>Nat.dvd_of_mul_dvd_mul_left</a> (<a>gcd_pos_of_pos_left</a> n (<a>pos_of_dvd_of_pos</a> H1 kpos))", [{"full_name": "Nat.dvd_of_mul_dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [957, 19], "def_end_pos": [957, 42]}, {"full_name": "Nat.gcd_pos_of_pos_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [92, 9], "def_end_pos": [92, 28]}, {"full_name": "Nat.pos_of_dvd_of_pos", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}]], "state_before": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nkpos : k > 0\n\u22a2 lcm m n \u2223 k", "state_after": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nkpos : k > 0\n\u22a2 gcd m n * lcm m n \u2223 gcd m n * k"}, {"tactic": "rw [gcd_mul_lcm, \u2190 gcd_mul_right, Nat.mul_comm n k]", "annotated_tactic": ["rw [<a>gcd_mul_lcm</a>, \u2190 <a>gcd_mul_right</a>, <a>Nat.mul_comm</a> n k]", [{"full_name": "Nat.gcd_mul_lcm", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [217, 9], "def_end_pos": [217, 20]}, {"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.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nkpos : k > 0\n\u22a2 gcd m n * lcm m n \u2223 gcd m n * k", "state_after": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nkpos : k > 0\n\u22a2 m * n \u2223 gcd (m * k) (k * n)"}, {"tactic": "exact dvd_gcd (Nat.mul_dvd_mul_left _ H2) (Nat.mul_dvd_mul_right H1 _)", "annotated_tactic": ["exact <a>dvd_gcd</a> (<a>Nat.mul_dvd_mul_left</a> _ H2) (<a>Nat.mul_dvd_mul_right</a> H1 _)", [{"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.mul_dvd_mul_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [897, 19], "def_end_pos": [897, 35]}, {"full_name": "Nat.mul_dvd_mul_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [900, 19], "def_end_pos": [900, 36]}]], "state_before": "m n k : Nat\nH1 : m \u2223 k\nH2 : n \u2223 k\nkpos : k > 0\n\u22a2 m * n \u2223 gcd (m * k) (k * n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq_of_countable", "start": [1177, 1], "end": [1181, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_isTop", "start": [248, 1], "end": [249, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "integrableOn_Ico_iff_integrableOn_Ioo", "start": [708, 1], "end": [710, 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 {a} \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/Set/Intervals/SurjOn.lean", "full_name": "surjOn_Ioc_of_monotone_surjective", "start": [47, 1], "end": [49, 89], "traced_tactics": [{"tactic": "simpa using surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a)", "annotated_tactic": ["simpa using <a>surjOn_Ico_of_monotone_surjective</a> h_mono.dual h_surj (<a>toDual</a> b) (<a>toDual</a> a)", [{"full_name": "surjOn_Ico_of_monotone_surjective", "def_path": "Mathlib/Data/Set/Intervals/SurjOn.lean", "def_pos": [35, 9], "def_end_pos": [35, 42]}, {"full_name": "OrderDual.toDual", "def_path": "Mathlib/Order/Synonym.lean", "def_pos": [50, 5], "def_end_pos": [50, 11]}, {"full_name": "OrderDual.toDual", "def_path": "Mathlib/Order/Synonym.lean", "def_pos": [50, 5], "def_end_pos": [50, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\nh_surj : Surjective f\na b : \u03b1\n\u22a2 SurjOn f (Ioc a b) (Ioc (f a) (f b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.continuous_L1_toL1", "start": [1560, 1], "end": [1602, 33], "traced_tactics": [{"tactic": "by_cases hc'0 : c' = 0", "annotated_tactic": ["by_cases hc'0 : 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 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)", "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)"}, {"tactic": "rw [Metric.continuous_iff]", "annotated_tactic": ["rw [<a>Metric.continuous_iff</a>]", [{"full_name": "Metric.continuous_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1077, 9], "def_end_pos": [1077, 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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)", "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\n\u22a2 \u2200 (b : { x // x \u2208 Lp G 1 }) (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 \u03b4,\n        \u03b4 > 0 \u2227\n          \u2200 (a : { x // x \u2208 Lp G 1 }),\n            dist a b < \u03b4 \u2192\n              dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191b (_ : Integrable \u2191\u2191b)) < \u03b5"}, {"tactic": "intro f \u03b5 h\u03b5_pos", "annotated_tactic": ["intro f \u03b5 h\u03b5_pos", []], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\n\u22a2 \u2200 (b : { x // x \u2208 Lp G 1 }) (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 \u03b4,\n        \u03b4 > 0 \u2227\n          \u2200 (a : { x // x \u2208 Lp G 1 }),\n            dist a b < \u03b4 \u2192\n              dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191b (_ : Integrable \u2191\u2191b)) < \u03b5", "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 \u03b4,\n    \u03b4 > 0 \u2227\n      \u2200 (a : { x // x \u2208 Lp G 1 }),\n        dist a f < \u03b4 \u2192 dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5"}, {"tactic": "use \u03b5 / 2 / c'.toReal", "annotated_tactic": ["use \u03b5 / 2 / c'.toReal", []], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 \u03b4,\n    \u03b4 > 0 \u2227\n      \u2200 (a : { x // x \u2208 Lp G 1 }),\n        dist a f < \u03b4 \u2192 dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5", "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\n\u22a2 \u03b5 / 2 / ENNReal.toReal c' > 0 \u2227\n    \u2200 (a : { x // x \u2208 Lp G 1 }),\n      dist a f < \u03b5 / 2 / ENNReal.toReal c' \u2192\n        dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5"}, {"tactic": "refine' \u27e8div_pos (half_pos h\u03b5_pos) (toReal_pos hc'0 hc'), _\u27e9", "annotated_tactic": ["refine' \u27e8<a>div_pos</a> (<a>half_pos</a> h\u03b5_pos) (<a>toReal_pos</a> hc'0 hc'), _\u27e9", [{"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\n\u22a2 \u03b5 / 2 / ENNReal.toReal c' > 0 \u2227\n    \u2200 (a : { x // x \u2208 Lp G 1 }),\n      dist a f < \u03b5 / 2 / ENNReal.toReal c' \u2192\n        dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5", "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\n\u22a2 \u2200 (a : { x // x \u2208 Lp G 1 }),\n    dist a f < \u03b5 / 2 / ENNReal.toReal c' \u2192\n      dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5"}, {"tactic": "intro g hfg", "annotated_tactic": ["intro g hfg", []], "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\n\u22a2 \u2200 (a : { x // x \u2208 Lp G 1 }),\n    dist a f < \u03b5 / 2 / ENNReal.toReal c' \u2192\n      dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : dist g f < \u03b5 / 2 / ENNReal.toReal c'\n\u22a2 dist (Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5"}, {"tactic": "rw [Lp.dist_def] at hfg \u22a2", "annotated_tactic": ["rw [<a>Lp.dist_def</a>] at hfg \u22a2", [{"full_name": "MeasureTheory.Lp.dist_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [290, 9], "def_end_pos": [290, 17]}]], "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : dist g f < \u03b5 / 2 / ENNReal.toReal c'\n\u22a2 dist (Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\n\u22a2 ENNReal.toReal\n      (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <\n    \u03b5"}, {"tactic": "let h_int := fun f' : \u03b1 \u2192\u2081[\u03bc] G => (L1.integrable_coeFn f').of_measure_le_smul c' hc' h\u03bc'_le", "annotated_tactic": ["let h_int := fun f' : \u03b1 \u2192\u2081[\u03bc] G => (<a>L1.integrable_coeFn</a> f').<a>of_measure_le_smul</a> c' hc' h\u03bc'_le", [{"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.of_measure_le_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [535, 9], "def_end_pos": [535, 38]}]], "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\n\u22a2 ENNReal.toReal\n      (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <\n    \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\n\u22a2 ENNReal.toReal\n      (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <\n    \u03b5"}, {"tactic": "have :\n  snorm (\u21d1(Integrable.toL1 g (h_int g)) - \u21d1(Integrable.toL1 f (h_int f))) 1 \u03bc' =\n    snorm (\u21d1g - \u21d1f) 1 \u03bc' :=\n  snorm_congr_ae ((Integrable.coeFn_toL1 _).sub (Integrable.coeFn_toL1 _))", "annotated_tactic": ["have :\n    <a>snorm</a> (\u21d1(<a>Integrable.toL1</a> g (h_int g)) - \u21d1(<a>Integrable.toL1</a> f (h_int f))) 1 \u03bc' =\n      <a>snorm</a> (\u21d1g - \u21d1f) 1 \u03bc' :=\n    <a>snorm_congr_ae</a> ((<a>Integrable.coeFn_toL1</a> _).<a>sub</a> (<a>Integrable.coeFn_toL1</a> _))", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.Integrable.toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1402, 5], "def_end_pos": [1402, 9]}, {"full_name": "MeasureTheory.Integrable.toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1402, 5], "def_end_pos": [1402, 9]}, {"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": "MeasureTheory.Integrable.coeFn_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1411, 9], "def_end_pos": [1411, 19]}, {"full_name": "Filter.EventuallyEq.sub", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1544, 3], "def_end_pos": [1544, 14]}, {"full_name": "MeasureTheory.Integrable.coeFn_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1411, 9], "def_end_pos": [1411, 19]}]], "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\n\u22a2 ENNReal.toReal\n      (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <\n    \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 ENNReal.toReal\n      (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <\n    \u03b5"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 ENNReal.toReal\n      (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <\n    \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5"}, {"tactic": "have h_snorm_ne_top : snorm (\u21d1g - \u21d1f) 1 \u03bc \u2260 \u221e := by\n  rw [\u2190 snorm_congr_ae (Lp.coeFn_sub _ _)]; exact Lp.snorm_ne_top _", "annotated_tactic": ["have h_snorm_ne_top : <a>snorm</a> (\u21d1g - \u21d1f) 1 \u03bc \u2260 \u221e := by\n    rw [\u2190 <a>snorm_congr_ae</a> (<a>Lp.coeFn_sub</a> _ _)]; exact <a>Lp.snorm_ne_top</a> _", [{"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": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}, {"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": "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5"}, {"tactic": "have h_snorm_ne_top' : snorm (\u21d1g - \u21d1f) 1 \u03bc' \u2260 \u221e := by\n  refine' ((snorm_mono_measure _ h\u03bc'_le).trans_lt _).ne\n  rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]\n  refine' ENNReal.mul_lt_top _ h_snorm_ne_top\n  simp [hc', hc'0]", "annotated_tactic": ["have h_snorm_ne_top' : <a>snorm</a> (\u21d1g - \u21d1f) 1 \u03bc' \u2260 \u221e := by\n    refine' ((<a>snorm_mono_measure</a> _ h\u03bc'_le).<a>trans_lt</a> _).<a>ne</a>\n    rw [<a>snorm_smul_measure_of_ne_zero</a> hc'0, <a>smul_eq_mul</a>]\n    refine' <a>ENNReal.mul_lt_top</a> _ h_snorm_ne_top\n    simp [hc', hc'0]", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [612, 9], "def_end_pos": [612, 27]}, {"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.snorm_smul_measure_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [651, 9], "def_end_pos": [651, 38]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"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 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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5", "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5"}, {"tactic": "calc\n  (snorm (\u21d1g - \u21d1f) 1 \u03bc').toReal \u2264 (c' * snorm (\u21d1g - \u21d1f) 1 \u03bc).toReal := by\n    rw [toReal_le_toReal h_snorm_ne_top' (ENNReal.mul_ne_top hc' h_snorm_ne_top)]\n    refine' (snorm_mono_measure (\u21d1g - \u21d1f) h\u03bc'_le).trans _\n    rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]\n    simp\n  _ = c'.toReal * (snorm (\u21d1g - \u21d1f) 1 \u03bc).toReal := toReal_mul\n  _ \u2264 c'.toReal * (\u03b5 / 2 / c'.toReal) :=\n    (mul_le_mul le_rfl hfg.le toReal_nonneg toReal_nonneg)\n  _ = \u03b5 / 2 := by\n    refine' mul_div_cancel' (\u03b5 / 2) _; rw [Ne.def, toReal_eq_zero_iff]; simp [hc', hc'0]\n  _ < \u03b5 := half_lt_self h\u03b5_pos", "annotated_tactic": ["calc\n    (<a>snorm</a> (\u21d1g - \u21d1f) 1 \u03bc').<a>toReal</a> \u2264 (c' * <a>snorm</a> (\u21d1g - \u21d1f) 1 \u03bc).<a>toReal</a> := by\n      rw [<a>toReal_le_toReal</a> h_snorm_ne_top' (<a>ENNReal.mul_ne_top</a> hc' h_snorm_ne_top)]\n      refine' (<a>snorm_mono_measure</a> (\u21d1g - \u21d1f) h\u03bc'_le).<a>trans</a> _\n      rw [<a>snorm_smul_measure_of_ne_zero</a> hc'0, <a>smul_eq_mul</a>]\n      simp\n    _ = c'.toReal * (<a>snorm</a> (\u21d1g - \u21d1f) 1 \u03bc).<a>toReal</a> := <a>toReal_mul</a>\n    _ \u2264 c'.toReal * (\u03b5 / 2 / c'.toReal) :=\n      (<a>mul_le_mul</a> <a>le_rfl</a> hfg.le <a>toReal_nonneg</a> <a>toReal_nonneg</a>)\n    _ = \u03b5 / 2 := by\n      refine' <a>mul_div_cancel'</a> (\u03b5 / 2) _; rw [<a>Ne.def</a>, <a>toReal_eq_zero_iff</a>]; simp [hc', hc'0]\n    _ < \u03b5 := <a>half_lt_self</a> h\u03b5_pos", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"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.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "MeasureTheory.snorm_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [612, 9], "def_end_pos": [612, 27]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.snorm_smul_measure_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [651, 9], "def_end_pos": [651, 38]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"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_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"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": "half_lt_self", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [525, 11], "def_end_pos": [525, 23]}]], "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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5", "state_after": "no goals"}, {"tactic": "have h\u03bc'0 : \u03bc' = 0 := by rw [\u2190 Measure.nonpos_iff_eq_zero']; refine' h\u03bc'_le.trans _; simp [hc'0]", "annotated_tactic": ["have h\u03bc'0 : \u03bc' = 0 := by rw [\u2190 <a>Measure.nonpos_iff_eq_zero'</a>]; refine' h\u03bc'_le.trans _; simp [hc'0]", [{"full_name": "MeasureTheory.Measure.nonpos_iff_eq_zero'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 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 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)", "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)"}, {"tactic": "have h_im_zero :\n  (fun f : \u03b1 \u2192\u2081[\u03bc] G =>\n      (Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f)).toL1 f) =\n    0 := by\n  ext1 f; ext1; simp_rw [h\u03bc'0]; simp only [ae_zero, EventuallyEq, eventually_bot]", "annotated_tactic": ["have h_im_zero :\n      (fun f : \u03b1 \u2192\u2081[\u03bc] G =>\n          (<a>Integrable.of_measure_le_smul</a> c' hc' h\u03bc'_le (<a>L1.integrable_coeFn</a> f)).<a>toL1</a> f) =\n        0 := by\n      ext1 f; ext1; simp_rw [h\u03bc'0]; simp only [<a>ae_zero</a>, <a>EventuallyEq</a>, <a>eventually_bot</a>]", [{"full_name": "MeasureTheory.Integrable.of_measure_le_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [535, 9], "def_end_pos": [535, 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.toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1402, 5], "def_end_pos": [1402, 9]}, {"full_name": "MeasureTheory.ae_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2451, 9], "def_end_pos": [2451, 16]}, {"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "Filter.eventually_bot", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1213, 9], "def_end_pos": [1213, 23]}]], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)", "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nh_im_zero : (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)"}, {"tactic": "rw [h_im_zero]", "annotated_tactic": ["rw [h_im_zero]", []], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nh_im_zero : (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0\n\u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)", "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nh_im_zero : (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0\n\u22a2 Continuous 0"}, {"tactic": "exact continuous_zero", "annotated_tactic": ["exact <a>continuous_zero</a>", [{"full_name": "continuous_zero", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [31, 3], "def_end_pos": [31, 14]}]], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nh_im_zero : (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0\n\u22a2 Continuous 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 Measure.nonpos_iff_eq_zero']", "annotated_tactic": ["rw [\u2190 <a>Measure.nonpos_iff_eq_zero'</a>]", [{"full_name": "MeasureTheory.Measure.nonpos_iff_eq_zero'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 \u03bc' = 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 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 \u03bc' \u2264 0"}, {"tactic": "refine' h\u03bc'_le.trans _", "annotated_tactic": ["refine' h\u03bc'_le.trans _", []], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 \u03bc' \u2264 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 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 c' \u2022 \u03bc \u2264 0"}, {"tactic": "simp [hc'0]", "annotated_tactic": ["simp [hc'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 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\n\u22a2 c' \u2022 \u03bc \u2264 0", "state_after": "no goals"}, {"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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\n\u22a2 (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0", "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u22a2 Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) = OfNat.ofNat 0 f"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u22a2 Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) = OfNat.ofNat 0 f", "state_after": "case h.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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u22a2 \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) =\u1d50[\u03bc'] \u2191\u2191(OfNat.ofNat 0 f)"}, {"tactic": "simp_rw [h\u03bc'0]", "annotated_tactic": ["simp_rw [h\u03bc'0]", []], "state_before": "case h.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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u22a2 \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) =\u1d50[\u03bc'] \u2191\u2191(OfNat.ofNat 0 f)", "state_after": "case h.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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u22a2 \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) =\u1d50[0] \u2191\u2191(OfNat.ofNat 0 f)"}, {"tactic": "simp only [ae_zero, EventuallyEq, eventually_bot]", "annotated_tactic": ["simp only [<a>ae_zero</a>, <a>EventuallyEq</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.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "Filter.eventually_bot", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1213, 9], "def_end_pos": [1213, 23]}]], "state_before": "case h.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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : c' = 0\nh\u03bc'0 : \u03bc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u22a2 \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) =\u1d50[0] \u2191\u2191(OfNat.ofNat 0 f)", "state_after": "no goals"}, {"tactic": "rw [\u2190 snorm_congr_ae (Lp.coeFn_sub _ _)]", "annotated_tactic": ["rw [\u2190 <a>snorm_congr_ae</a> (<a>Lp.coeFn_sub</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]}]], "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 snorm (\u2191\u2191(g - f)) 1 \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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\n\u22a2 snorm (\u2191\u2191(g - f)) 1 \u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "refine' ((snorm_mono_measure _ h\u03bc'_le).trans_lt _).ne", "annotated_tactic": ["refine' ((<a>snorm_mono_measure</a> _ h\u03bc'_le).<a>trans_lt</a> _).<a>ne</a>", [{"full_name": "MeasureTheory.snorm_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [612, 9], "def_end_pos": [612, 27]}, {"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\nE : Type u_2\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 (c' \u2022 \u03bc) < \u22a4"}, {"tactic": "rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]", "annotated_tactic": ["rw [<a>snorm_smul_measure_of_ne_zero</a> hc'0, <a>smul_eq_mul</a>]", [{"full_name": "MeasureTheory.snorm_smul_measure_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [651, 9], "def_end_pos": [651, 38]}, {"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\nE : Type u_2\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 (c' \u2022 \u03bc) < \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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 c' ^ ENNReal.toReal (1 / 1) * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc < \u22a4"}, {"tactic": "refine' ENNReal.mul_lt_top _ h_snorm_ne_top", "annotated_tactic": ["refine' <a>ENNReal.mul_lt_top</a> _ h_snorm_ne_top", [{"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 c' ^ ENNReal.toReal (1 / 1) * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc < \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\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 c' ^ ENNReal.toReal (1 / 1) \u2260 \u22a4"}, {"tactic": "simp [hc', hc'0]", "annotated_tactic": ["simp [hc', hc'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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\n\u22a2 c' ^ ENNReal.toReal (1 / 1) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [toReal_le_toReal h_snorm_ne_top' (ENNReal.mul_ne_top hc' h_snorm_ne_top)]", "annotated_tactic": ["rw [<a>toReal_le_toReal</a> h_snorm_ne_top' (<a>ENNReal.mul_ne_top</a> hc' h_snorm_ne_top)]", [{"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.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') \u2264 ENNReal.toReal (c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc)", "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc"}, {"tactic": "refine' (snorm_mono_measure (\u21d1g - \u21d1f) h\u03bc'_le).trans _", "annotated_tactic": ["refine' (<a>snorm_mono_measure</a> (\u21d1g - \u21d1f) h\u03bc'_le).<a>trans</a> _", [{"full_name": "MeasureTheory.snorm_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [612, 9], "def_end_pos": [612, 27]}, {"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\nE : Type u_2\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc", "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 (c' \u2022 \u03bc) \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc"}, {"tactic": "rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]", "annotated_tactic": ["rw [<a>snorm_smul_measure_of_ne_zero</a> hc'0, <a>smul_eq_mul</a>]", [{"full_name": "MeasureTheory.snorm_smul_measure_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [651, 9], "def_end_pos": [651, 38]}, {"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\nE : Type u_2\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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 (c' \u2022 \u03bc) \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc", "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 c' ^ ENNReal.toReal (1 / 1) * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 c' ^ ENNReal.toReal (1 / 1) * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc", "state_after": "no goals"}, {"tactic": "refine' mul_div_cancel' (\u03b5 / 2) _", "annotated_tactic": ["refine' <a>mul_div_cancel'</a> (\u03b5 / 2) _", [{"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 ENNReal.toReal c' * (\u03b5 / 2 / ENNReal.toReal c') = \u03b5 / 2", "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 ENNReal.toReal c' \u2260 0"}, {"tactic": "rw [Ne.def, toReal_eq_zero_iff]", "annotated_tactic": ["rw [<a>Ne.def</a>, <a>toReal_eq_zero_iff</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]}]], "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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 ENNReal.toReal c' \u2260 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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 \u00ac(c' = 0 \u2228 c' = \u22a4)"}, {"tactic": "simp [hc', hc'0]", "annotated_tactic": ["simp [hc', hc'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\u271d : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhc'0 : \u00acc' = 0\nf : { x // x \u2208 Lp G 1 }\n\u03b5 : \u211d\nh\u03b5_pos : \u03b5 > 0\ng : { x // x \u2208 Lp G 1 }\nhfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c'\nh_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f')\nthis :\n  snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =\n    snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc'\nh_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4\nh_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4\n\u22a2 \u00ac(c' = 0 \u2228 c' = \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_inv", "start": [158, 1], "end": [169, 20], "traced_tactics": [{"tactic": "refine' \u27e8measurable_inv, AbsolutelyContinuous.mk fun s hsm h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>measurable_inv</a>, <a>AbsolutelyContinuous.mk</a> fun s hsm h\u03bcs => _\u27e9", [{"full_name": "MeasurableInv.measurable_inv", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [427, 3], "def_end_pos": [427, 17]}, {"full_name": "MeasureTheory.Measure.AbsolutelyContinuous.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 11]}]], "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 : IsMulLeftInvariant \u03bc\n\u22a2 QuasiMeasurePreserving Inv.inv", "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(map Inv.inv \u03bc) s = 0"}, {"tactic": "rw [map_apply measurable_inv hsm, inv_preimage]", "annotated_tactic": ["rw [<a>map_apply</a> <a>measurable_inv</a> hsm, <a>inv_preimage</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": "MeasurableInv.measurable_inv", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [427, 3], "def_end_pos": [427, 17]}, {"full_name": "Set.inv_preimage", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(map Inv.inv \u03bc) s = 0", "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0"}, {"tactic": "have hf : Measurable fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9) :=\n  (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv", "annotated_tactic": ["have hf : <a>Measurable</a> fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9) :=\n    (measurable_snd.mul <a>measurable_fst</a>).<a>prod_mk</a> measurable_fst.inv", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"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": "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0", "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0"}, {"tactic": "suffices map (fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9)) (\u03bc.prod \u03bc) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 by\n  simpa only [(measurePreserving_mul_prod_inv \u03bc \u03bc).map_eq, prod_prod, mul_eq_zero (M\u2080 := \u211d\u22650\u221e),\n    or_self_iff] using this", "annotated_tactic": ["suffices <a>map</a> (fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9)) (\u03bc.prod \u03bc) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 by\n    simpa only [(<a>measurePreserving_mul_prod_inv</a> \u03bc \u03bc).<a>map_eq</a>, <a>prod_prod</a>, <a>mul_eq_zero</a> (M\u2080 := \u211d\u22650\u221e),\n      <a>or_self_iff</a>] using this", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.measurePreserving_mul_prod_inv", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [148, 9], "def_end_pos": [148, 39]}, {"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.Measure.prod_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 18]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0", "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\n\u22a2 \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0"}, {"tactic": "have hsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) := hsm.inv.prod hsm.inv", "annotated_tactic": ["have hsm' : <a>MeasurableSet</a> (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) := hsm.inv.prod hsm.inv", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\n\u22a2 \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0", "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\nhsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9)\n\u22a2 \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0"}, {"tactic": "simp_rw [map_apply hf hsm', prod_apply_symm (\u03bc := \u03bc) (\u03bd := \u03bc) (hf hsm'), preimage_preimage,\n  mk_preimage_prod, inv_preimage, inv_inv, measure_mono_null (inter_subset_right _ _) h\u03bcs,\n  lintegral_zero]", "annotated_tactic": ["simp_rw [<a>map_apply</a> hf hsm', <a>prod_apply_symm</a> (\u03bc := \u03bc) (\u03bd := \u03bc) (hf hsm'), <a>preimage_preimage</a>,\n    <a>mk_preimage_prod</a>, <a>inv_preimage</a>, <a>inv_inv</a>, <a>measure_mono_null</a> (<a>inter_subset_right</a> _ _) h\u03bcs,\n    <a>lintegral_zero</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.prod_apply_symm", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 24]}, {"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [176, 9], "def_end_pos": [176, 26]}, {"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.inv_preimage", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}, {"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": "MeasureTheory.lintegral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [141, 9], "def_end_pos": [141, 23]}]], "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\nhsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9)\n\u22a2 \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0", "state_after": "no goals"}, {"tactic": "simpa only [(measurePreserving_mul_prod_inv \u03bc \u03bc).map_eq, prod_prod, mul_eq_zero (M\u2080 := \u211d\u22650\u221e),\n  or_self_iff] using this", "annotated_tactic": ["simpa only [(<a>measurePreserving_mul_prod_inv</a> \u03bc \u03bc).<a>map_eq</a>, <a>prod_prod</a>, <a>mul_eq_zero</a> (M\u2080 := \u211d\u22650\u221e),\n      <a>or_self_iff</a>] using this", [{"full_name": "MeasureTheory.measurePreserving_mul_prod_inv", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [148, 9], "def_end_pos": [148, 39]}, {"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.Measure.prod_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 18]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "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\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9)\nthis : \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 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_empty_left", "start": [2255, 1], "end": [2257, 57], "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 \u00acp \u2227 x \u2208 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_id_set_eq_sInter", "start": [739, 1], "end": [740, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.Coprime.gcd_both", "start": [361, 1], "end": [362, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image\u2082_right", "start": [362, 1], "end": [365, 25], "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\nh : Finset.Nonempty s\n\u22a2 \u2191(image\u2082 (fun x y => y) s t) = \u2191t", "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\nh : Finset.Nonempty s\n\u22a2 image2 (fun x y => y) \u2191s \u2191t = \u2191t"}, {"tactic": "exact image2_right h", "annotated_tactic": ["exact <a>image2_right</a> h", [{"full_name": "Set.image2_right", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [310, 9], "def_end_pos": [310, 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\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\nh : Finset.Nonempty s\n\u22a2 image2 (fun x y => y) \u2191s \u2191t = \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "AddCircle.integral_preimage", "start": [169, 11], "end": [180, 6], "traced_tactics": [{"tactic": "have m : MeasurableSet (Ioc t (t + T)) := measurableSet_Ioc", "annotated_tactic": ["have m : <a>MeasurableSet</a> (<a>Ioc</a> t (t + T)) := <a>measurableSet_Ioc</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": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b"}, {"tactic": "have := integral_map_equiv (\u03bc := volume) (measurableEquivIoc T t).symm f", "annotated_tactic": ["have := <a>integral_map_equiv</a> (\u03bc := <a>volume</a>) (<a>measurableEquivIoc</a> T t).<a>symm</a> f", [{"full_name": "MeasureTheory.integral_map_equiv", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1637, 9], "def_end_pos": [1637, 27]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "AddCircle.measurableEquivIoc", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [127, 19], "def_end_pos": [127, 37]}, {"full_name": "MeasurableEquiv.symm", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1325, 5], "def_end_pos": [1325, 9]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis :\n  \u222b (y : AddCircle T), f y \u2202Measure.map (\u2191(MeasurableEquiv.symm (measurableEquivIoc T t))) volume =\n    \u222b (x : \u2191(Ioc t (t + T))), f (\u2191(MeasurableEquiv.symm (measurableEquivIoc T t)) x)\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b"}, {"tactic": "simp only [measurableEquivIoc, equivIoc, QuotientAddGroup.equivIocMod, MeasurableEquiv.symm_mk,\n  MeasurableEquiv.coe_mk, Equiv.coe_fn_symm_mk] at this", "annotated_tactic": ["simp only [<a>measurableEquivIoc</a>, <a>equivIoc</a>, <a>QuotientAddGroup.equivIocMod</a>, <a>MeasurableEquiv.symm_mk</a>,\n    <a>MeasurableEquiv.coe_mk</a>, <a>Equiv.coe_fn_symm_mk</a>] at this", [{"full_name": "AddCircle.measurableEquivIoc", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [127, 19], "def_end_pos": [127, 37]}, {"full_name": "AddCircle.equivIoc", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [197, 5], "def_end_pos": [197, 13]}, {"full_name": "QuotientAddGroup.equivIocMod", "def_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "def_pos": [819, 5], "def_end_pos": [819, 33]}, {"full_name": "MeasurableEquiv.symm_mk", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1351, 9], "def_end_pos": [1351, 16]}, {"full_name": "MeasurableEquiv.coe_mk", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1303, 9], "def_end_pos": [1303, 15]}, {"full_name": "Equiv.coe_fn_symm_mk", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [255, 17], "def_end_pos": [255, 31]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis :\n  \u222b (y : AddCircle T), f y \u2202Measure.map (\u2191(MeasurableEquiv.symm (measurableEquivIoc T t))) volume =\n    \u222b (x : \u2191(Ioc t (t + T))), f (\u2191(MeasurableEquiv.symm (measurableEquivIoc T t)) x)\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b"}, {"tactic": "rw [\u2190 (AddCircle.measurePreserving_mk T t).map_eq, set_integral_eq_subtype m, \u2190 this]", "annotated_tactic": ["rw [\u2190 (<a>AddCircle.measurePreserving_mk</a> T t).<a>map_eq</a>, <a>set_integral_eq_subtype</a> m, \u2190 this]", [{"full_name": "AddCircle.measurePreserving_mk", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [86, 19], "def_end_pos": [86, 39]}, {"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.set_integral_eq_subtype", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1654, 9], "def_end_pos": [1654, 32]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\n\u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume =\n    \u222b (b : AddCircle T), f b \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T)))"}, {"tactic": "have : ((\u2191) : Ioc t (t + T) \u2192 AddCircle T) = ((\u2191) : \u211d \u2192 AddCircle T) \u2218 ((\u2191) : _ \u2192 \u211d) := by\n  ext1 x; rfl", "annotated_tactic": ["have : ((\u2191) : <a>Ioc</a> t (t + T) \u2192 <a>AddCircle</a> T) = ((\u2191) : \u211d \u2192 <a>AddCircle</a> T) \u2218 ((\u2191) : _ \u2192 \u211d) := by\n    ext1 x; rfl", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [124, 8], "def_end_pos": [124, 17]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [124, 8], "def_end_pos": [124, 17]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume =\n    \u222b (b : AddCircle T), f b \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T)))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nthis : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume =\n    \u222b (b : AddCircle T), f b \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T)))"}, {"tactic": "simp_rw [this]", "annotated_tactic": ["simp_rw [this]", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nthis : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume =\n    \u222b (b : AddCircle T), f b \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T)))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nthis : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (QuotientAddGroup.mk \u2218 Subtype.val) volume =\n    \u222b (y : AddCircle T), f y \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T)))"}, {"tactic": "rw [\u2190 map_map AddCircle.measurable_mk' measurable_subtype_coe, \u2190 map_comap_subtype_coe m]", "annotated_tactic": ["rw [\u2190 <a>map_map</a> <a>AddCircle.measurable_mk'</a> <a>measurable_subtype_coe</a>, \u2190 <a>map_comap_subtype_coe</a> m]", [{"full_name": "MeasureTheory.Measure.map_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 16]}, {"full_name": "AddCircle.measurable_mk'", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [34, 19], "def_end_pos": [34, 43]}, {"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]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nthis : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (QuotientAddGroup.mk \u2218 Subtype.val) volume =\n    \u222b (y : AddCircle T), f y \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T)))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nthis : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map QuotientAddGroup.mk (Measure.map Subtype.val volume) =\n    \u222b (y : AddCircle T),\n      f y \u2202Measure.map QuotientAddGroup.mk (Measure.map Subtype.val (Measure.comap Subtype.val volume))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nthis : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val\n\u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map QuotientAddGroup.mk (Measure.map Subtype.val volume) =\n    \u222b (y : AddCircle T),\n      f y \u2202Measure.map QuotientAddGroup.mk (Measure.map Subtype.val (Measure.comap Subtype.val volume))", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\n\u22a2 (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val", "state_after": "case h\nT : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nx : \u2191(Ioc t (t + T))\n\u22a2 \u2191\u2191x = (QuotientAddGroup.mk \u2218 Subtype.val) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nT : \u211d\nhT : Fact (0 < T)\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nt : \u211d\nf : AddCircle T \u2192 E\nm : MeasurableSet (Ioc t (t + T))\nthis : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x\nx : \u2191(Ioc t (t + T))\n\u22a2 \u2191\u2191x = (QuotientAddGroup.mk \u2218 Subtype.val) x", "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_zero", "start": [249, 1], "end": [252, 69], "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 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u22a2 (kernel.sum fun x => 0) = 0", "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 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun x => 0) a) s = \u2191\u2191(\u21910 a) s"}, {"tactic": "rw [sum_apply' _ a hs]", "annotated_tactic": ["rw [<a>sum_apply'</a> _ a hs]", [{"full_name": "ProbabilityTheory.kernel.sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 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 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun x => 0) a) s = \u2191\u2191(\u21910 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 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2211' (n : \u03b9), \u2191\u2191(\u21910 a) s = \u2191\u2191(\u21910 a) s"}, {"tactic": "simp only [zero_apply, Measure.coe_zero, Pi.zero_apply, tsum_zero]", "annotated_tactic": ["simp only [<a>zero_apply</a>, <a>Measure.coe_zero</a>, <a>Pi.zero_apply</a>, <a>tsum_zero</a>]", [{"full_name": "ProbabilityTheory.kernel.zero_apply", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"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": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 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 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2211' (n : \u03b9), \u2191\u2191(\u21910 a) s = \u2191\u2191(\u21910 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.MeasurableSet.nullMeasurableSet_subtype_coe", "start": [1398, 1], "end": [1414, 35], "traced_tactics": [{"tactic": "refine'\n  generateFrom_induction (p := fun t : Set s => NullMeasurableSet ((\u2191) '' t) \u03bc)\n    { t : Set s | \u2203 s' : Set \u03b1, MeasurableSet s' \u2227 (\u2191) \u207b\u00b9' s' = t } _ _ _ _ ht", "annotated_tactic": ["refine'\n    <a>generateFrom_induction</a> (p := fun t : <a>Set</a> s => <a>NullMeasurableSet</a> ((\u2191) '' t) \u03bc)\n      { t : <a>Set</a> s | \u2203 s' : <a>Set</a> \u03b1, <a>MeasurableSet</a> s' \u2227 (\u2191) \u207b\u00b9' s' = t } _ _ _ _ ht", [{"full_name": "MeasurableSpace.generateFrom_induction", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [376, 9], "def_end_pos": [376, 31]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.NullMeasurableSet", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [97, 5], "def_end_pos": [97, 22]}, {"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": "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 NullMeasurableSet (Subtype.val '' t)", "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s),\n    t \u2208 {t | \u2203 s', MeasurableSet s' \u2227 Subtype.val \u207b\u00b9' s' = t} \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t\n\ncase 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\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) \u2205\n\ncase 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\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s), (fun t => NullMeasurableSet (Subtype.val '' t)) t \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t\u1d9c\n\ncase refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u2191s),\n    (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192\n      (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 i, f i)"}, {"tactic": "rintro t' \u27e8s', hs', rfl\u27e9", "annotated_tactic": ["rintro t' \u27e8s', hs', rfl\u27e9", []], "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s),\n    t \u2208 {t | \u2203 s', MeasurableSet s' \u2227 Subtype.val \u207b\u00b9' s' = t} \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t", "state_after": "case refine'_1.intro.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\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'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (Subtype.val '' (Subtype.val \u207b\u00b9' s'))"}, {"tactic": "rw [Subtype.image_preimage_coe]", "annotated_tactic": ["rw [<a>Subtype.image_preimage_coe</a>]", [{"full_name": "Subtype.image_preimage_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1448, 9], "def_end_pos": [1448, 27]}]], "state_before": "case refine'_1.intro.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\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'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (Subtype.val '' (Subtype.val \u207b\u00b9' s'))", "state_after": "case refine'_1.intro.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\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'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (s' \u2229 s)"}, {"tactic": "exact hs'.nullMeasurableSet.inter hs", "annotated_tactic": ["exact hs'.nullMeasurableSet.inter hs", []], "state_before": "case refine'_1.intro.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\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'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (s' \u2229 s)", "state_after": "no goals"}, {"tactic": "simp only [image_empty, nullMeasurableSet_empty]", "annotated_tactic": ["simp only [<a>image_empty</a>, <a>nullMeasurableSet_empty</a>]", [{"full_name": "Set.image_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [337, 9], "def_end_pos": [337, 20]}, {"full_name": "MeasureTheory.nullMeasurableSet_empty", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [108, 9], "def_end_pos": [108, 32]}]], "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) \u2205", "state_after": "no goals"}, {"tactic": "intro t'", "annotated_tactic": ["intro t'", []], "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s), (fun t => NullMeasurableSet (Subtype.val '' t)) t \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t\u1d9c", "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) t' \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t'\u1d9c"}, {"tactic": "simp only [\u2190 range_diff_image Subtype.coe_injective, Subtype.range_coe_subtype, setOf_mem_eq]", "annotated_tactic": ["simp only [\u2190 <a>range_diff_image</a> <a>Subtype.coe_injective</a>, <a>Subtype.range_coe_subtype</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Set.range_diff_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1136, 9], "def_end_pos": [1136, 25]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}, {"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": "case 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\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) t' \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t'\u1d9c", "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 NullMeasurableSet (Subtype.val '' t') \u2192 NullMeasurableSet (s \\ (fun a => \u2191a) '' t')"}, {"tactic": "exact hs.diff", "annotated_tactic": ["exact hs.diff", []], "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\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 NullMeasurableSet (Subtype.val '' t') \u2192 NullMeasurableSet (s \\ (fun a => \u2191a) '' t')", "state_after": "no goals"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "state_before": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u2191s),\n    (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192\n      (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 i, f i)", "state_after": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192\n    (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 i, f i)"}, {"tactic": "dsimp only []", "annotated_tactic": ["dsimp only []", []], "state_before": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192\n    (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 i, f i)", "state_after": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (Subtype.val '' \u22c3 i, f i)"}, {"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": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (Subtype.val '' \u22c3 i, f i)", "state_after": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (\u22c3 i, Subtype.val '' f i)"}, {"tactic": "exact NullMeasurableSet.iUnion", "annotated_tactic": ["exact <a>NullMeasurableSet.iUnion</a>", [{"full_name": "MeasureTheory.NullMeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [146, 19], "def_end_pos": [146, 25]}]], "state_before": "case refine'_4\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 : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (\u22c3 i, Subtype.val '' f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.sub_to_nat", "start": [1298, 1], "end": [1300, 82], "traced_tactics": [{"tactic": "rw [ofZNum_toNat, cast_sub', \u2190 to_nat_to_int, \u2190 to_nat_to_int, Int.toNat_sub]", "annotated_tactic": ["rw [<a>ofZNum_toNat</a>, <a>cast_sub'</a>, \u2190 <a>to_nat_to_int</a>, \u2190 <a>to_nat_to_int</a>, <a>Int.toNat_sub</a>]", [{"full_name": "Num.ofZNum_toNat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1284, 9], "def_end_pos": [1284, 21]}, {"full_name": "Num.cast_sub'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1218, 9], "def_end_pos": [1218, 18]}, {"full_name": "Num.to_nat_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [479, 9], "def_end_pos": [479, 22]}, {"full_name": "Num.to_nat_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [479, 9], "def_end_pos": [479, 22]}, {"full_name": "Int.toNat_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [515, 9], "def_end_pos": [515, 18]}]], "state_before": "\u03b1 : Type u_1\nm n : Num\n\u22a2 \u2191(ofZNum (sub' m n)) = \u2191m - \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/PProd.lean", "full_name": "Function.Injective.pprod_map", "start": [46, 1], "end": [50, 36], "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.expand_WF", "start": [143, 1], "end": [172, 54], "traced_tactics": [{"tactic": "simp_all [Buckets.mk, List.mem_replicate]", "annotated_tactic": ["simp_all [<a>Buckets.mk</a>, <a>List.mem_replicate</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.mk", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [28, 5], "def_end_pos": [28, 7]}, {"full_name": "List.mem_replicate", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [325, 9], "def_end_pos": [325, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\nx\u271d\u00b3 : AssocList \u03b1 \u03b2\nx\u271d\u00b2 : x\u271d\u00b3 \u2208 (Buckets.mk (Array.size buckets.val * 2)).val.data\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx\u271d : x\u271d\u00b9 \u2208 AssocList.toList x\u271d\u00b3\n\u22a2 (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) < 0) x\u271d\u00b9.fst x\u271d\u00b9.snd", "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\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\n\u22a2 Buckets.WF (expand.go i source target)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\n\u22a2 Buckets.WF\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)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\n\u22a2 Buckets.WF\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)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nh\u271d : i < Array.size source\n\u22a2 Buckets.WF\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\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nh\u271d : \u00aci < Array.size source\n\u22a2 Buckets.WF target"}, {"tactic": "refine go (i+1) (fun _ hl => ?_) (fun i h => ?_) ?_", "annotated_tactic": ["refine go (i+1) (fun _ hl => ?_) (fun i h => ?_) ?_", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\n\u22a2 Buckets.WF\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)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nhl : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\n\u22a2 AssocList.All\n    (fun k x =>\n      USize.toNat (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)) = i)\n    (Array.set source { val := i\u271d, isLt := H } AssocList.nil)[i]\n\ncase refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\n\u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192\n        AssocList.All\n          (fun k x =>\n            USize.toNat\n                (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <\n              i + 1)\n          bucket"}, {"tactic": "match List.mem_or_eq_of_mem_set hl with\n| .inl hl => exact hs\u2081 _ hl\n| .inr e => exact e \u25b8 .nil", "annotated_tactic": ["match <a>List.mem_or_eq_of_mem_set</a> hl with\n        | .inl hl => exact hs\u2081 _ hl\n        | .inr e => exact e \u25b8 .nil", [{"full_name": "List.mem_or_eq_of_mem_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [965, 9], "def_end_pos": [965, 29]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nhl : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)", "state_after": "no goals"}, {"tactic": "exact hs\u2081 _ hl", "annotated_tactic": ["exact hs\u2081 _ hl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nhl\u271d : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data\nhl : x\u271d \u2208 source.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)", "state_after": "no goals"}, {"tactic": "exact e \u25b8 .nil", "annotated_tactic": ["exact e \u25b8 .nil", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nhl : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data\ne : x\u271d = AssocList.nil\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)", "state_after": "no goals"}, {"tactic": "simp [Array.getElem_eq_data_get, List.get_set]", "annotated_tactic": ["simp [<a>Array.getElem_eq_data_get</a>, <a>List.get_set</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]}, {"full_name": "List.get_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [961, 9], "def_end_pos": [961, 16]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\n\u22a2 AssocList.All\n    (fun k x =>\n      USize.toNat (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)) = i)\n    (Array.set source { val := i\u271d, isLt := H } AssocList.nil)[i]", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i)\n    (if i\u271d = i then AssocList.nil else List.get source.data { val := i, isLt := (_ : i < List.length source.data) })"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i)\n    (if i\u271d = i then AssocList.nil else List.get source.data { val := i, isLt := (_ : i < List.length source.data) })", "state_after": "case refine_2.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\nh\u271d : i\u271d = i\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i) AssocList.nil\n\ncase refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\nh\u271d : \u00aci\u271d = i\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i)\n    (List.get source.data { val := i, isLt := (_ : i < List.length source.data) })"}, {"tactic": "intro.", "annotated_tactic": ["intro.", []], "state_before": "case refine_2.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\nh\u271d : i\u271d = i\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i) AssocList.nil", "state_after": "no goals"}, {"tactic": "exact hs\u2082 _ (by simp_all)", "annotated_tactic": ["exact hs\u2082 _ (by simp_all)", []], "state_before": "case refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\nh\u271d : \u00aci\u271d = i\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i)\n    (List.get source.data { val := i, isLt := (_ : i < List.length source.data) })", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni\u271d : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket\nH : i\u271d < Array.size source\ni : Nat\nh : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)\nh\u271d : \u00aci\u271d = i\n\u22a2 i < Array.size source", "state_after": "no goals"}, {"tactic": "let rank (k : \u03b1) := ((hash k).toUSize % source.size).toNat", "annotated_tactic": ["let rank (k : \u03b1) := ((<a>hash</a> k).<a>toUSize</a> % source.size).<a>toNat</a>", [{"full_name": "Hashable.hash", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [3349, 3], "def_end_pos": [3349, 7]}, {"full_name": "UInt64.toUSize", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [3355, 8], "def_end_pos": [3355, 22]}, {"full_name": "USize.toNat", "def_path": "lake-packages/lean4/src/lean/Init/Data/UInt/Basic.lean", "def_pos": [294, 5], "def_end_pos": [294, 16]}]], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\n\u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192\n        AssocList.All\n          (fun k x =>\n            USize.toNat\n                (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <\n              i + 1)\n          bucket", "state_after": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\n\u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192\n        AssocList.All\n          (fun k x =>\n            USize.toNat\n                (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <\n              i + 1)\n          bucket"}, {"tactic": "have := expand_WF.foldl rank ?_ (hs\u2082 _ H) ht.1 (fun _ h\u2081 _ h\u2082 => ?_)", "annotated_tactic": ["have := <a>expand_WF.foldl</a> rank ?_ (hs\u2082 _ H) ht.1 (fun _ h\u2081 _ h\u2082 => ?_)", [{"full_name": "Std.HashMap.Imp.expand_WF.foldl", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [113, 9], "def_end_pos": [113, 24]}]], "state_before": "case refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\n\u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192\n        AssocList.All\n          (fun k x =>\n            USize.toNat\n                (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <\n              i + 1)\n          bucket", "state_after": "case refine_3.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nthis :\n  Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket\n\u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192\n        AssocList.All\n          (fun k x =>\n            USize.toNat\n                (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <\n              i + 1)\n          bucket\n\ncase refine_3.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\n\u22a2 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList source[i])\n\ncase refine_3.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nx\u271d\u00b9 : AssocList \u03b1 \u03b2\nh\u2081 : x\u271d\u00b9 \u2208 target.val.data\nx\u271d : \u03b1 \u00d7 \u03b2\nh\u2082 : x\u271d \u2208 AssocList.toList x\u271d\u00b9\n\u22a2 (fun k x =>\n      rank k \u2264 i \u2227\n        \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n          x \u2208 AssocList.toList source[i] \u2192 \u00ac(x.fst == k) = true)\n    x\u271d.fst x\u271d.snd"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_3.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nthis :\n  Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket\n\u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192\n        AssocList.All\n          (fun k x =>\n            USize.toNat\n                (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <\n              i + 1)\n          bucket", "state_after": "case refine_3.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nthis :\n  Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i + 1) bucket"}, {"tactic": "exact \u27e8this.1, fun _ h\u2081 _ h\u2082 => Nat.lt_succ_of_le (this.2 _ h\u2081 _ h\u2082)\u27e9", "annotated_tactic": ["exact \u27e8this.1, fun _ h\u2081 _ h\u2082 => <a>Nat.lt_succ_of_le</a> (this.2 _ h\u2081 _ h\u2082)\u27e9", [{"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": "case refine_3.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nthis :\n  Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i + 1) bucket", "state_after": "no goals"}, {"tactic": "exact hs\u2081 _ (Array.getElem_mem_data ..)", "annotated_tactic": ["exact hs\u2081 _ (<a>Array.getElem_mem_data</a> ..)", [{"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "case refine_3.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\n\u22a2 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList source[i])", "state_after": "no goals"}, {"tactic": "have := ht.2 _ h\u2081 _ h\u2082", "annotated_tactic": ["have := ht.2 _ h\u2081 _ h\u2082", []], "state_before": "case refine_3.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nx\u271d\u00b9 : AssocList \u03b1 \u03b2\nh\u2081 : x\u271d\u00b9 \u2208 target.val.data\nx\u271d : \u03b1 \u00d7 \u03b2\nh\u2082 : x\u271d \u2208 AssocList.toList x\u271d\u00b9\n\u22a2 (fun k x =>\n      rank k \u2264 i \u2227\n        \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n          x \u2208 AssocList.toList source[i] \u2192 \u00ac(x.fst == k) = true)\n    x\u271d.fst x\u271d.snd", "state_after": "case refine_3.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nx\u271d\u00b9 : AssocList \u03b1 \u03b2\nh\u2081 : x\u271d\u00b9 \u2208 target.val.data\nx\u271d : \u03b1 \u00d7 \u03b2\nh\u2082 : x\u271d \u2208 AssocList.toList x\u271d\u00b9\nthis : (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) x\u271d.fst x\u271d.snd\n\u22a2 (fun k x =>\n      rank k \u2264 i \u2227\n        \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n          x \u2208 AssocList.toList source[i] \u2192 \u00ac(x.fst == k) = true)\n    x\u271d.fst x\u271d.snd"}, {"tactic": "refine \u27e8Nat.le_of_lt this, fun _ h h' => Nat.ne_of_lt this ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>Nat.le_of_lt</a> this, fun _ h h' => <a>Nat.ne_of_lt</a> this ?_\u27e9", [{"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]}, {"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]}]], "state_before": "case refine_3.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nx\u271d\u00b9 : AssocList \u03b1 \u03b2\nh\u2081 : x\u271d\u00b9 \u2208 target.val.data\nx\u271d : \u03b1 \u00d7 \u03b2\nh\u2082 : x\u271d \u2208 AssocList.toList x\u271d\u00b9\nthis : (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) x\u271d.fst x\u271d.snd\n\u22a2 (fun k x =>\n      rank k \u2264 i \u2227\n        \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n          x \u2208 AssocList.toList source[i] \u2192 \u00ac(x.fst == k) = true)\n    x\u271d.fst x\u271d.snd", "state_after": "case refine_3.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nx\u271d\u00b2 : AssocList \u03b1 \u03b2\nh\u2081 : x\u271d\u00b2 \u2208 target.val.data\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nh\u2082 : x\u271d\u00b9 \u2208 AssocList.toList x\u271d\u00b2\nthis : (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) x\u271d\u00b9.fst x\u271d\u00b9.snd\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\nx\u271d : \u03b1 \u00d7 \u03b2\nh : x\u271d \u2208 AssocList.toList source[i]\nh' : (x\u271d.fst == x\u271d\u00b9.fst) = true\n\u22a2 USize.toNat (UInt64.toUSize (hash x\u271d\u00b9.fst) % Array.size source) = i"}, {"tactic": "exact LawfulHashable.hash_eq h' \u25b8 hs\u2082 _ H _ h", "annotated_tactic": ["exact <a>LawfulHashable.hash_eq</a> h' \u25b8 hs\u2082 _ H _ h", [{"full_name": "Std.HashMap.LawfulHashable.hash_eq", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [15, 3], "def_end_pos": [15, 10]}]], "state_before": "case refine_3.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH\u271d : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nH : i < Array.size source\nrank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source)\nx\u271d\u00b2 : AssocList \u03b1 \u03b2\nh\u2081 : x\u271d\u00b2 \u2208 target.val.data\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nh\u2082 : x\u271d\u00b9 \u2208 AssocList.toList x\u271d\u00b2\nthis : (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) x\u271d\u00b9.fst x\u271d\u00b9.snd\ninst\u271d\u00b9 : PartialEquivBEq \u03b1\ninst\u271d : LawfulHashable \u03b1\nx\u271d : \u03b1 \u00d7 \u03b2\nh : x\u271d \u2208 AssocList.toList source[i]\nh' : (x\u271d.fst == x\u271d\u00b9.fst) = true\n\u22a2 USize.toNat (UInt64.toUSize (hash x\u271d\u00b9.fst) % Array.size source) = i", "state_after": "no goals"}, {"tactic": "exact ht.1", "annotated_tactic": ["exact ht.1", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nsz : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\nH : Buckets.WF buckets\ni : Nat\nsource : Array (AssocList \u03b1 \u03b2)\nhs\u2081 :\n  \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket)\nhs\u2082 :\n  \u2200 (j : Nat) (h : j < Array.size source),\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j]\ntarget : Buckets \u03b1 \u03b2\nht :\n  Buckets.WF target \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 target.val.data \u2192\n        AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket\nh\u271d : \u00aci < Array.size source\n\u22a2 Buckets.WF target", "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_iff", "start": [121, 1], "end": [124, 6], "traced_tactics": [{"tactic": "rw [\u2190 sum_cond, aemeasurable_sum_measure_iff, Bool.forall_bool, and_comm]", "annotated_tactic": ["rw [\u2190 <a>sum_cond</a>, <a>aemeasurable_sum_measure_iff</a>, <a>Bool.forall_bool</a>, <a>and_comm</a>]", [{"full_name": "MeasureTheory.Measure.sum_cond", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2055, 9], "def_end_pos": [2055, 17]}, {"full_name": "aemeasurable_sum_measure_iff", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [115, 9], "def_end_pos": [115, 44]}, {"full_name": "Bool.forall_bool", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}, {"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\n\u22a2 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\n\u22a2 AEMeasurable f \u2227 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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 g : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\n\u22a2 AEMeasurable f \u2227 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f", "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_prod_smul", "start": [512, 1], "end": [521, 66], "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd"}, {"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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd"}, {"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\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd"}, {"tactic": "by_cases h : Integrable (fun z : \u03b1 \u00d7 \u03b2 => f z.1 \u2022 g z.2) (\u03bc.prod \u03bd)", "annotated_tactic": ["by_cases h : <a>Integrable</a> (fun z : \u03b1 \u00d7 \u03b2 => f z.1 \u2022 g z.2) (\u03bc.prod \u03bd)", [{"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\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : Integrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : \u00acIntegrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd"}, {"tactic": "have H : \u00acIntegrable f \u03bc \u2228 \u00acIntegrable g \u03bd := by\n  contrapose! h\n  exact h.1.prod_smul h.2", "annotated_tactic": ["have H : \u00ac<a>Integrable</a> f \u03bc \u2228 \u00ac<a>Integrable</a> g \u03bd := by\n    contrapose! h\n    exact h.1.<a>prod_smul</a> h.2", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Integrable.prod_smul", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [309, 9], "def_end_pos": [309, 29]}]], "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : \u00acIntegrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : \u00acIntegrable fun z => f z.1 \u2022 g z.2\nH : \u00acIntegrable f \u2228 \u00acIntegrable g\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd"}, {"tactic": "cases' H with H H <;> simp [integral_undef h, integral_undef H]", "annotated_tactic": ["cases' H with H H <;> simp [<a>integral_undef</a> h, <a>integral_undef</a> H]", [{"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\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : \u00acIntegrable fun z => f z.1 \u2022 g z.2\nH : \u00acIntegrable f \u2228 \u00acIntegrable g\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "state_after": "no goals"}, {"tactic": "rw [integral_prod _ h]", "annotated_tactic": ["rw [<a>integral_prod</a> _ h]", [{"full_name": "MeasureTheory.integral_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [455, 9], "def_end_pos": [455, 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\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : Integrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : Integrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y).1 \u2022 g (x, y).2 \u2202\u03bd \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd"}, {"tactic": "simp_rw [integral_smul, integral_smul_const]", "annotated_tactic": ["simp_rw [<a>integral_smul</a>, <a>integral_smul_const</a>]", [{"full_name": "MeasureTheory.integral_smul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [915, 9], "def_end_pos": [915, 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\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : Integrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y).1 \u2022 g (x, y).2 \u2202\u03bd \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd", "state_after": "no goals"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : \u00acIntegrable fun z => f z.1 \u2022 g z.2\n\u22a2 \u00acIntegrable f \u2228 \u00acIntegrable g", "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\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : Integrable f \u2227 Integrable g\n\u22a2 Integrable fun z => f z.1 \u2022 g z.2"}, {"tactic": "exact h.1.prod_smul h.2", "annotated_tactic": ["exact h.1.<a>prod_smul</a> h.2", [{"full_name": "MeasureTheory.Integrable.prod_smul", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [309, 9], "def_end_pos": [309, 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\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \u211d E'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhE : CompleteSpace E\nh : Integrable f \u2227 Integrable g\n\u22a2 Integrable fun z => f z.1 \u2022 g z.2", "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.subsingleton_iff_le_one", "start": [164, 1], "end": [168, 96], "traced_tactics": [{"tactic": "(match n with | 0 | 1 | n+2 => ?_) <;> try simp", "annotated_tactic": ["(match n with | 0 | 1 | n+2 => ?_) <;> try simp", []], "state_before": "n : Nat\n\u22a2 Subsingleton (Fin n) \u2194 n \u2264 1", "state_after": "case refine_1\nn : Nat\n\u22a2 Subsingleton (Fin 0)\n\ncase refine_2\nn : Nat\n\u22a2 Subsingleton (Fin 1)\n\ncase refine_3\nn\u271d n : Nat\n\u22a2 Subsingleton (Fin (n + 2)) \u2194 n + 2 \u2264 1"}, {"tactic": "match n with | 0 | 1 | n+2 => ?_", "annotated_tactic": ["match n with | 0 | 1 | n+2 => ?_", []], "state_before": "n : Nat\n\u22a2 Subsingleton (Fin n) \u2194 n \u2264 1", "state_after": "case refine_1\nn : Nat\n\u22a2 Subsingleton (Fin 0) \u2194 0 \u2264 1\n\ncase refine_2\nn : Nat\n\u22a2 Subsingleton (Fin 1) \u2194 1 \u2264 1\n\ncase refine_3\nn\u271d n : Nat\n\u22a2 Subsingleton (Fin (n + 2)) \u2194 n + 2 \u2264 1"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_2\nn : Nat\n\u22a2 Subsingleton (Fin 1) \u2194 1 \u2264 1", "state_after": "case refine_2\nn : Nat\n\u22a2 Subsingleton (Fin 1)"}, {"tactic": "exact \u27e8fun.\u27e9", "annotated_tactic": ["exact \u27e8fun.\u27e9", []], "state_before": "case refine_1\nn : Nat\n\u22a2 Subsingleton (Fin 0)", "state_after": "no goals"}, {"tactic": "exact \u27e8fun \u27e80, _\u27e9 \u27e80, _\u27e9 => rfl\u27e9", "annotated_tactic": ["exact \u27e8fun \u27e80, _\u27e9 \u27e80, _\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 refine_2\nn : Nat\n\u22a2 Subsingleton (Fin 1)", "state_after": "no goals"}, {"tactic": "exact iff_of_false (fun h => Fin.ne_of_lt zero_lt_one (h.elim ..)) (of_decide_eq_false rfl)", "annotated_tactic": ["exact <a>iff_of_false</a> (fun h => <a>Fin.ne_of_lt</a> <a>zero_lt_one</a> (h.elim ..)) (<a>of_decide_eq_false</a> <a>rfl</a>)", [{"full_name": "iff_of_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "Fin.ne_of_lt", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [87, 19], "def_end_pos": [87, 27]}, {"full_name": "Fin.zero_lt_one", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [111, 9], "def_end_pos": [111, 20]}, {"full_name": "of_decide_eq_false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [860, 9], "def_end_pos": [860, 27]}, {"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\nn\u271d n : Nat\n\u22a2 Subsingleton (Fin (n + 2)) \u2194 n + 2 \u2264 1", "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_restrict", "start": [600, 1], "end": [602, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Units.lean", "full_name": "Int.units_eq_one_or", "start": [30, 1], "end": [31, 61], "traced_tactics": [{"tactic": "simpa only [Units.ext_iff, units_natAbs] using natAbs_eq u", "annotated_tactic": ["simpa only [<a>Units.ext_iff</a>, <a>units_natAbs</a>] using <a>natAbs_eq</a> u", [{"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_natAbs", "def_path": "Mathlib/Data/Int/Units.lean", "def_pos": [23, 9], "def_end_pos": [23, 21]}, {"full_name": "Int.natAbs_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [162, 9], "def_end_pos": [162, 18]}]], "state_before": "u : \u2124\u02e3\n\u22a2 u = 1 \u2228 u = -1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.disjoint_union_right", "start": [674, 1], "end": [676, 92], "traced_tactics": [{"tactic": "rw [Disjoint.symm_iff, disjoint_union_left, Disjoint.symm_iff _ x, Disjoint.symm_iff _ x]", "annotated_tactic": ["rw [<a>Disjoint.symm_iff</a>, <a>disjoint_union_left</a>, <a>Disjoint.symm_iff</a> _ x, <a>Disjoint.symm_iff</a> _ x]", [{"full_name": "Finmap.Disjoint.symm_iff", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [660, 9], "def_end_pos": [660, 26]}, {"full_name": "Finmap.disjoint_union_left", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [670, 9], "def_end_pos": [670, 28]}, {"full_name": "Finmap.Disjoint.symm_iff", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [660, 9], "def_end_pos": [660, 26]}, {"full_name": "Finmap.Disjoint.symm_iff", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [660, 9], "def_end_pos": [660, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nx y z : Finmap \u03b2\n\u22a2 Disjoint x (y \u222a z) \u2194 Disjoint x y \u2227 Disjoint x z", "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_of_choice", "start": [37, 1], "end": [64, 80], "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\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\n\u22a2 p f", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\n\u22a2 p f"}, {"tactic": "induction' hs : univ.sigma f using Finset.strongInductionOn with s ihs generalizing f", "annotated_tactic": ["induction' hs : univ.sigma f using <a>Finset.strongInductionOn</a> with s ihs generalizing f", [{"full_name": "Finset.strongInductionOn", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [682, 5], "def_end_pos": [682, 22]}]], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\n\u22a2 p f", "state_after": "case intro.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs\u271d : Finset.sigma univ f\u271d = x\u271d\ns : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nihs : \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)), t \u2282 s \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhs : Finset.sigma univ f = s\n\u22a2 p f"}, {"tactic": "subst s", "annotated_tactic": ["subst s", []], "state_before": "case intro.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs\u271d : Finset.sigma univ f\u271d = x\u271d\ns : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nihs : \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)), t \u2282 s \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhs : Finset.sigma univ f = s\n\u22a2 p f", "state_after": "case intro.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\n\u22a2 p f"}, {"tactic": "cases' eq_empty_or_nonempty (univ.sigma f) with he hne", "annotated_tactic": ["cases' <a>eq_empty_or_nonempty</a> (univ.sigma f) with he hne", [{"full_name": "Finset.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 29]}]], "state_before": "case intro.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\n\u22a2 p f", "state_after": "case intro.a.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhe : Finset.sigma univ f = \u2205\n\u22a2 p f\n\ncase intro.a.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\n\u22a2 p f"}, {"tactic": "convert h0 using 1", "annotated_tactic": ["convert h0 using 1", []], "state_before": "case intro.a.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhe : Finset.sigma univ f = \u2205\n\u22a2 p f", "state_after": "case h.e'_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhe : Finset.sigma univ f = \u2205\n\u22a2 f = fun x => \u2205"}, {"tactic": "simpa [funext_iff] using he", "annotated_tactic": ["simpa [<a>funext_iff</a>] using he", [{"full_name": "Function.funext_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}]], "state_before": "case h.e'_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhe : Finset.sigma univ f = \u2205\n\u22a2 f = fun x => \u2205", "state_after": "no goals"}, {"tactic": "rcases sigma_nonempty.1 hne with \u27e8i, -, hi\u27e9", "annotated_tactic": ["rcases <a>sigma_nonempty</a>.1 hne with \u27e8i, -, hi\u27e9", [{"full_name": "Finset.sigma_nonempty", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [60, 9], "def_end_pos": [60, 23]}]], "state_before": "case intro.a.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\n\u22a2 p f", "state_after": "case intro.a.inr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\n\u22a2 p f"}, {"tactic": "rcases H_ex i (f i) hi with \u27e8x, x_mem, hr\u27e9", "annotated_tactic": ["rcases H_ex i (f i) hi with \u27e8x, x_mem, hr\u27e9", []], "state_before": "case intro.a.inr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\n\u22a2 p f", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\n\u22a2 p f"}, {"tactic": "set g := update f i ((f i).erase x) with hg", "annotated_tactic": ["set g := <a>update</a> f i ((f i).<a>erase</a> x) with hg", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Finset.erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1877, 5], "def_end_pos": [1877, 10]}]], "state_before": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\n\u22a2 p f", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\n\u22a2 p f"}, {"tactic": "have hx' : x \u2209 g i := by\n  rw [hg, update_same]\n  apply not_mem_erase", "annotated_tactic": ["have hx' : x \u2209 g i := by\n      rw [hg, <a>update_same</a>]\n      apply <a>not_mem_erase</a>", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"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 intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\n\u22a2 p f", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 p f"}, {"tactic": "rw [show f = update g i (insert x (g i)) by\n  rw [hg, update_idem, update_same, insert_erase x_mem, update_eq_self]] at hr ihs \u22a2", "annotated_tactic": ["rw [show f = <a>update</a> g i (<a>insert</a> x (g i)) by\n      rw [hg, <a>update_idem</a>, <a>update_same</a>, <a>insert_erase</a> x_mem, <a>update_eq_self</a>]] at hr ihs \u22a2", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}, {"full_name": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"full_name": "Function.update_idem", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [685, 9], "def_end_pos": [685, 20]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}, {"full_name": "Function.update_eq_self", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 23]}]], "state_before": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 p f", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (erase (update g i (insert x (g i)) i) x)\nhg : g = update f i (erase (f i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 p (update g i (insert x (g i)))"}, {"tactic": "clear hg", "annotated_tactic": ["clear hg", []], "state_before": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (erase (update g i (insert x (g i)) i) x)\nhg : g = update f i (erase (f i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 p (update g i (insert x (g i)))", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (erase (update g i (insert x (g i)) i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 p (update g i (insert x (g i)))"}, {"tactic": "rw [update_same, erase_insert hx'] at hr", "annotated_tactic": ["rw [<a>update_same</a>, <a>erase_insert</a> hx'] at hr", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.erase_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1941, 9], "def_end_pos": [1941, 21]}]], "state_before": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (erase (update g i (insert x (g i)) i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 p (update g i (insert x (g i)))", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 p (update g i (insert x (g i)))"}, {"tactic": "refine step _ _ _ hr (ihs (univ.sigma g) ?_ _ rfl)", "annotated_tactic": ["refine step _ _ _ hr (ihs (univ.sigma g) ?_ _ <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.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 p (update g i (insert x (g i)))", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 Finset.sigma univ g \u2282 Finset.sigma univ (update g i (insert x (g i)))"}, {"tactic": "rw [ssubset_iff_of_subset (sigma_mono (Subset.refl _) _)]", "annotated_tactic": ["rw [<a>ssubset_iff_of_subset</a> (<a>sigma_mono</a> (<a>Subset.refl</a> _) _)]", [{"full_name": "Finset.ssubset_iff_of_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [424, 9], "def_end_pos": [424, 30]}, {"full_name": "Finset.sigma_mono", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [69, 9], "def_end_pos": [69, 19]}, {"full_name": "Finset.Subset.refl", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}]], "state_before": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 Finset.sigma univ g \u2282 Finset.sigma univ (update g i (insert x (g i)))", "state_after": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 \u2203 x_1, x_1 \u2208 Finset.sigma univ (update g i (insert x (g i))) \u2227 \u00acx_1 \u2208 Finset.sigma univ g\n\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 \u2200 (i_1 : \u03b9), g i_1 \u2286 update g i (insert x (g i)) i_1"}, {"tactic": "exacts [\u27e8\u27e8i, x\u27e9, mem_sigma.2 \u27e8mem_univ _, by simp\u27e9, by simp [hx']\u27e9,\n  (@le_update_iff _ _ _ _ g g i _).2 \u27e8subset_insert _ _, fun _ _ \u21a6 le_rfl\u27e9]", "annotated_tactic": ["exacts [\u27e8\u27e8i, x\u27e9, <a>mem_sigma</a>.2 \u27e8<a>mem_univ</a> _, by simp\u27e9, by simp [hx']\u27e9,\n      (@<a>le_update_iff</a> _ _ _ _ g g i _).2 \u27e8<a>subset_insert</a> _ _, fun _ _ \u21a6 <a>le_rfl</a>\u27e9]", [{"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_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "le_update_iff", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [880, 9], "def_end_pos": [880, 22]}, {"full_name": "Finset.subset_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro.a.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 \u2203 x_1, x_1 \u2208 Finset.sigma univ (update g i (insert x (g i))) \u2227 \u00acx_1 \u2208 Finset.sigma univ g\n\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 \u2200 (i_1 : \u03b9), g i_1 \u2286 update g i (insert x (g i)) i_1", "state_after": "no goals"}, {"tactic": "rw [hg, update_same]", "annotated_tactic": ["rw [hg, <a>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": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\n\u22a2 \u00acx \u2208 g i", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\n\u22a2 \u00acx \u2208 erase (f i) x"}, {"tactic": "apply not_mem_erase", "annotated_tactic": ["apply <a>not_mem_erase</a>", [{"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1891, 9], "def_end_pos": [1891, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\nhr : r i x (erase (f i) x)\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nhg : g = update f i (erase (f i) x)\n\u22a2 \u00acx \u2208 erase (f i) x", "state_after": "no goals"}, {"tactic": "rw [hg, update_idem, update_same, insert_erase x_mem, update_eq_self]", "annotated_tactic": ["rw [hg, <a>update_idem</a>, <a>update_same</a>, <a>insert_erase</a> x_mem, <a>update_eq_self</a>]", [{"full_name": "Function.update_idem", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [685, 9], "def_end_pos": [685, 20]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}, {"full_name": "Function.update_eq_self", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (erase (update g i (insert x (g i)) i) x)\nhg : g = update f i (erase (f i) x)\nhx' : \u00acx \u2208 g i\n\u22a2 f = update g i (insert x (g i))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 { fst := i, snd := x }.snd \u2208 update g i (insert x (g i)) { fst := i, snd := x }.fst", "state_after": "no goals"}, {"tactic": "simp [hx']", "annotated_tactic": ["simp [hx']", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\nr : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop\nH_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x)\np : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop\nf\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh0 : p fun x => \u2205\nstep : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i)))\nval\u271d : Fintype \u03b9\nx\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i)\nhs : Finset.sigma univ f\u271d = x\u271d\nf : (i : \u03b9) \u2192 Finset (\u03b1 i)\nhne : Finset.Nonempty (Finset.sigma univ f)\ni : \u03b9\nhi : Finset.Nonempty (f i)\nx : \u03b1 i\nx_mem : x \u2208 f i\ng : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x)\nihs :\n  \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),\n    t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f\nhr : r i x (g i)\nhx' : \u00acx \u2208 g i\n\u22a2 \u00ac{ fst := i, snd := x } \u2208 Finset.sigma univ g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.integrable_compProd_iff", "start": [124, 1], "end": [129, 59], "traced_tactics": [{"tactic": "simp only [Integrable, hasFiniteIntegral_compProd_iff' hf, hf.norm.integral_kernel_compProd,\n  hf, hf.compProd_mk_left, eventually_and, true_and_iff]", "annotated_tactic": ["simp only [<a>Integrable</a>, <a>hasFiniteIntegral_compProd_iff'</a> hf, hf.norm.integral_kernel_compProd,\n    hf, hf.compProd_mk_left, <a>eventually_and</a>, <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": "ProbabilityTheory.hasFiniteIntegral_compProd_iff'", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [107, 9], "def_end_pos": [107, 40]}, {"full_name": "Filter.eventually_and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1151, 9], "def_end_pos": [1151, 23]}, {"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\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\nhf : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 Integrable f \u2194 (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, Integrable fun y => f (x, y)) \u2227 Integrable fun x => \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/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite", "start": [314, 1], "end": [323, 76], "traced_tactics": [{"tactic": "apply ae_of_forall_measure_lt_top_ae_restrict", "annotated_tactic": ["apply <a>ae_of_forall_measure_lt_top_ae_restrict</a>", [{"full_name": "MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3798, 9], "def_end_pos": [3798, 48]}]], "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\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f", "state_after": "case h\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\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, OfNat.ofNat 0 x \u2264 f x"}, {"tactic": "intro t t_meas t_lt_top", "annotated_tactic": ["intro t t_meas t_lt_top", []], "state_before": "case h\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\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, OfNat.ofNat 0 x \u2264 f x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, OfNat.ofNat 0 x \u2264 f x"}, {"tactic": "apply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t t_meas t_lt_top)", "annotated_tactic": ["apply <a>ae_nonneg_restrict_of_forall_set_integral_nonneg_inter</a> (hf_int_finite t t_meas t_lt_top)", [{"full_name": "MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [304, 9], "def_end_pos": [304, 63]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, OfNat.ofNat 0 x \u2264 f x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc"}, {"tactic": "intro s s_meas _", "annotated_tactic": ["intro s s_meas _", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc", "state_after": "case h\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\ns : Set \u03b1\ns_meas : MeasurableSet s\na\u271d : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc"}, {"tactic": "exact\n  hf_zero _ (s_meas.inter t_meas)\n    (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)", "annotated_tactic": ["exact\n    hf_zero _ (s_meas.inter t_meas)\n      (<a>lt_of_le_of_lt</a> (<a>measure_mono</a> (<a>Set.inter_subset_right</a> _ _)) t_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": "case h\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\ns : Set \u03b1\ns_meas : MeasurableSet s\na\u271d : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.eqvGen_mono", "start": [263, 1], "end": [265, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.mk_preimage_prod_left_fn_eq_if", "start": [281, 1], "end": [283, 76], "traced_tactics": [{"tactic": "rw [\u2190 mk_preimage_prod_left_eq_if, prod_preimage_left, preimage_preimage]", "annotated_tactic": ["rw [\u2190 <a>mk_preimage_prod_left_eq_if</a>, <a>prod_preimage_left</a>, <a>preimage_preimage</a>]", [{"full_name": "Set.mk_preimage_prod_left_eq_if", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [273, 9], "def_end_pos": [273, 36]}, {"full_name": "Set.prod_preimage_left", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [229, 9], "def_end_pos": [229, 27]}, {"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [176, 9], "def_end_pos": [176, 26]}]], "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\ninst\u271d : DecidablePred fun x => x \u2208 t\nf : \u03b3 \u2192 \u03b1\n\u22a2 (fun a => (f a, b)) \u207b\u00b9' s \u00d7\u02e2 t = if b \u2208 t then f \u207b\u00b9' s else \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_left", "start": [1329, 1], "end": [1333, 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\nc : \u211d\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) 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\nc : \u211d\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) 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\nc : \u211d\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) f = c \u2022 setToFun \u03bc T hT f"}, {"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left hT c]", "annotated_tactic": ["simp_rw [<a>setToFun_eq</a> _ hf, <a>L1.setToL1_smul_left</a> hT 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.L1.setToL1_smul_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1105, 9], "def_end_pos": [1105, 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\nc : \u211d\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) 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\nc : \u211d\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) f = c \u2022 setToFun \u03bc T hT f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.null_of_totalVariation_zero", "start": [506, 1], "end": [513, 21], "traced_tactics": [{"tactic": "rw [totalVariation, Measure.coe_add, Pi.add_apply, add_eq_zero_iff] at hs", "annotated_tactic": ["rw [<a>totalVariation</a>, <a>Measure.coe_add</a>, <a>Pi.add_apply</a>, <a>add_eq_zero_iff</a>] at hs", [{"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.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": "add_eq_zero_iff", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(totalVariation s) i = 0\n\u22a2 \u2191s i = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 \u2191s i = 0"}, {"tactic": "rw [\u2190 toSignedMeasure_toJordanDecomposition s, toSignedMeasure, VectorMeasure.coe_sub,\n  Pi.sub_apply, Measure.toSignedMeasure_apply, Measure.toSignedMeasure_apply]", "annotated_tactic": ["rw [\u2190 <a>toSignedMeasure_toJordanDecomposition</a> s, <a>toSignedMeasure</a>, <a>VectorMeasure.coe_sub</a>,\n    <a>Pi.sub_apply</a>, <a>Measure.toSignedMeasure_apply</a>, <a>Measure.toSignedMeasure_apply</a>]", [{"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.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.VectorMeasure.coe_sub", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [367, 9], "def_end_pos": [367, 16]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [409, 3], "def_end_pos": [409, 8]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [409, 3], "def_end_pos": [409, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 \u2191s i = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0"}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : \u00acMeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0"}, {"tactic": "rw [if_pos hi, if_pos hi]", "annotated_tactic": ["rw [<a>if_pos</a> hi, <a>if_pos</a> hi]", [{"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_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\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = 0"}, {"tactic": "simp [hs.1, hs.2]", "annotated_tactic": ["simp [hs.1, hs.2]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = 0", "state_after": "no goals"}, {"tactic": "simp [if_neg hi]", "annotated_tactic": ["simp [<a>if_neg</a> hi]", [{"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\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : \u00acMeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0", "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_smul", "start": [379, 1], "end": [387, 94], "traced_tactics": [{"tactic": "simp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_smul, testAgainstNN_coe_eq,\n  ENNReal.coe_smul]", "annotated_tactic": ["simp only [\u2190 <a>ENNReal.coe_eq_coe</a>, <a>BoundedContinuousFunction.coe_smul</a>, <a>testAgainstNN_coe_eq</a>,\n    <a>ENNReal.coe_smul</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_smul", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 17]}, {"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_coe_eq", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [326, 9], "def_end_pos": [326, 29]}, {"full_name": "ENNReal.coe_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [539, 9], "def_end_pos": [539, 17]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc (c \u2022 f) = c \u2022 testAgainstNN \u03bc f", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc"}, {"tactic": "simp_rw [\u2190 smul_one_smul \u211d\u22650\u221e c (f _ : \u211d\u22650\u221e), \u2190 smul_one_smul \u211d\u22650\u221e c (lintegral _ _ : \u211d\u22650\u221e),\n  smul_eq_mul]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_one_smul</a> \u211d\u22650\u221e c (f _ : \u211d\u22650\u221e), \u2190 <a>smul_one_smul</a> \u211d\u22650\u221e c (<a>lintegral</a> _ _ : \u211d\u22650\u221e),\n    <a>smul_eq_mul</a>]", [{"full_name": "smul_one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 22]}, {"full_name": "smul_one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 22]}, {"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [60, 17], "def_end_pos": [60, 26]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 1 * \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 1 * \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc"}, {"tactic": "exact\n  @lintegral_const_mul _ _ (\u03bc : Measure \u03a9) (c \u2022 (1 : \u211d\u22650\u221e)) _ f.measurable_coe_ennreal_comp", "annotated_tactic": ["exact\n    @<a>lintegral_const_mul</a> _ _ (\u03bc : <a>Measure</a> \u03a9) (c \u2022 (1 : \u211d\u22650\u221e)) _ f.measurable_coe_ennreal_comp", [{"full_name": "MeasureTheory.lintegral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [671, 9], "def_end_pos": [671, 28]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 1 * \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 1 * \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\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.get?_of_valid", "start": [241, 1], "end": [242, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.cast_succ", "start": [1327, 1], "end": [1328, 37], "traced_tactics": [{"tactic": "rw [\u2190 add_one, cast_add, cast_one]", "annotated_tactic": ["rw [\u2190 <a>add_one</a>, <a>cast_add</a>, <a>cast_one</a>]", [{"full_name": "ZNum.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 16]}, {"full_name": "ZNum.cast_add", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 17]}, {"full_name": "ZNum.cast_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1033, 9], "def_end_pos": [1033, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(succ n) = \u2191n + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "full_name": "volume_regionBetween_eq_integral", "start": [37, 1], "end": [41, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.indepFun_prod_range_succ", "start": [624, 1], "end": [628, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primcodable.mem_range_encode", "start": [1200, 1], "end": [1208, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.continuous_set_integral", "start": [968, 1], "end": [977, 76], "traced_tactics": [{"tactic": "haveI : Fact ((1 : \u211d\u22650\u221e) \u2264 1) := \u27e8le_rfl\u27e9", "annotated_tactic": ["haveI : <a>Fact</a> ((1 : \u211d\u22650\u221e) \u2264 1) := \u27e8<a>le_rfl</a>\u27e9", [{"full_name": "Fact", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 11]}, {"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\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "have h_comp :\n  (fun f : \u03b1 \u2192\u2081[\u03bc] E => \u222b x in s, f x \u2202\u03bc) =\n    integral (\u03bc.restrict s) \u2218 fun f => LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s f := by\n  ext1 f\n  rw [Function.comp_apply, integral_congr_ae (LpToLpRestrictCLM_coeFn \u211d s f)]", "annotated_tactic": ["have h_comp :\n    (fun f : \u03b1 \u2192\u2081[\u03bc] E => \u222b x in s, f x \u2202\u03bc) =\n      <a>integral</a> (\u03bc.restrict s) \u2218 fun f => <a>LpToLpRestrictCLM</a> \u03b1 E \u211d \u03bc 1 s f := by\n    ext1 f\n    rw [<a>Function.comp_apply</a>, <a>integral_congr_ae</a> (<a>LpToLpRestrictCLM_coeFn</a> \u211d s f)]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.LpToLpRestrictCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [946, 5], "def_end_pos": [946, 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": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.LpToLpRestrictCLM_coeFn", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [959, 9], "def_end_pos": [959, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\nh_comp :\n  (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =\n    integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc"}, {"tactic": "rw [h_comp]", "annotated_tactic": ["rw [h_comp]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\nh_comp :\n  (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =\n    integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)\n\u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\nh_comp :\n  (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =\n    integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)\n\u22a2 Continuous (integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f))"}, {"tactic": "exact continuous_integral.comp (LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s).continuous", "annotated_tactic": ["exact continuous_integral.comp (<a>LpToLpRestrictCLM</a> \u03b1 E \u211d \u03bc 1 s).<a>continuous</a>", [{"full_name": "MeasureTheory.LpToLpRestrictCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [946, 5], "def_end_pos": [946, 22]}, {"full_name": "ContinuousLinearMap.continuous", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [448, 19], "def_end_pos": [448, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\nh_comp :\n  (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =\n    integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)\n\u22a2 Continuous (integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f))", "state_after": "no goals"}, {"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\n\u22a2 (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =\n    integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = (integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)) f"}, {"tactic": "rw [Function.comp_apply, integral_congr_ae (LpToLpRestrictCLM_coeFn \u211d s f)]", "annotated_tactic": ["rw [<a>Function.comp_apply</a>, <a>integral_congr_ae</a> (<a>LpToLpRestrictCLM_coeFn</a> \u211d s f)]", [{"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.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.LpToLpRestrictCLM_coeFn", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [959, 9], "def_end_pos": [959, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : NormedSpace \u211d E\ns : Set \u03b1\nthis : Fact (1 \u2264 1)\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = (integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.smn", "start": [693, 1], "end": [695, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_hasFDerivAt", "start": [717, 1], "end": [722, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.sym2_eq_empty", "start": [72, 1], "end": [73, 66], "traced_tactics": [{"tactic": "rw [Finset.sym2, image_eq_empty, product_eq_empty, or_self_iff]", "annotated_tactic": ["rw [<a>Finset.sym2</a>, <a>image_eq_empty</a>, <a>product_eq_empty</a>, <a>or_self_iff</a>]", [{"full_name": "Finset.sym2", "def_path": "Mathlib/Data/Finset/Sym.lean", "def_pos": [51, 15], "def_end_pos": [51, 19]}, {"full_name": "Finset.image_eq_empty", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [543, 9], "def_end_pos": [543, 23]}, {"full_name": "Finset.product_eq_empty", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [214, 9], "def_end_pos": [214, 25]}, {"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 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Finset.sym2 s = \u2205 \u2194 s = \u2205", "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.aemeasurable", "start": [604, 11], "end": [606, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval_pow", "start": [1184, 1], "end": [1185, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top", "start": [143, 1], "end": [146, 71], "traced_tactics": [{"tactic": "rw [lintegral_rpow_nnnorm_eq_rpow_snorm' hq0_lt]", "annotated_tactic": ["rw [<a>lintegral_rpow_nnnorm_eq_rpow_snorm'</a> hq0_lt]", [{"full_name": "MeasureTheory.lintegral_rpow_nnnorm_eq_rpow_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [125, 9], "def_end_pos": [125, 45]}]], "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\nhq0_lt : 0 < q\nhfq : snorm' f q \u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc < \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\nhq0_lt : 0 < q\nhfq : snorm' f q \u03bc < \u22a4\n\u22a2 snorm' (fun a => f a) q \u03bc ^ q < \u22a4"}, {"tactic": "exact ENNReal.rpow_lt_top_of_nonneg (le_of_lt hq0_lt) (ne_of_lt hfq)", "annotated_tactic": ["exact <a>ENNReal.rpow_lt_top_of_nonneg</a> (<a>le_of_lt</a> hq0_lt) (<a>ne_of_lt</a> hfq)", [{"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": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"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\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 : 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"def_end_pos": [768, 42]}]], "state_before": "a b : Int\nh : b \u2223 a\n\u22a2 div a b = a / b", "state_after": "no goals"}, {"tactic": "simp [b0]", "annotated_tactic": ["simp [b0]", []], "state_before": "a b : Int\nh : b \u2223 a\nb0 : b = 0\n\u22a2 div a b = a / b", "state_after": "no goals"}, {"tactic": "rw [Int.div_eq_iff_eq_mul_left b0 h, \u2190 Int.ediv_eq_iff_eq_mul_left b0 h]", "annotated_tactic": ["rw [<a>Int.div_eq_iff_eq_mul_left</a> b0 h, \u2190 <a>Int.ediv_eq_iff_eq_mul_left</a> b0 h]", [{"full_name": "Int.div_eq_iff_eq_mul_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [764, 19], "def_end_pos": [764, 41]}, {"full_name": "Int.ediv_eq_iff_eq_mul_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [768, 19], "def_end_pos": [768, 42]}]], "state_before": "a b : Int\nh : b \u2223 a\nb0 : \u00acb = 0\n\u22a2 div a b = 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.measurePreserving_prod_inv_mul_swap", "start": [137, 1], "end": [139, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.range_diff_image", "start": [1136, 1], "end": [1138, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mem_bind_iff", "start": [494, 1], "end": [497, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrableOn.continuousOn_smul", "start": [593, 1], "end": [598, 88], "traced_tactics": [{"tactic": "rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2", "annotated_tactic": ["rw [<a>MeasureTheory.locallyIntegrableOn_iff</a> (<a>Or.inr</a> hs)] at hf \u22a2", [{"full_name": "MeasureTheory.locallyIntegrableOn_iff", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"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": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 E\ng : X \u2192 \ud835\udd5c\ns : Set X\nhs : IsOpen s\nhf : LocallyIntegrableOn f s\nhg : ContinuousOn g s\n\u22a2 LocallyIntegrableOn (fun x => g x \u2022 f x) s", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c 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hfmeas fun \u03b5 h\u03b5 => _", "annotated_tactic": ["refine' <a>uniformIntegrable_of'</a> hp hp' hfmeas fun \u03b5 h\u03b5 => _", [{"full_name": "MeasureTheory.uniformIntegrable_of'", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [794, 9], "def_end_pos": [794, 30]}]], "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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"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": "\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\ncase pos\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8C, hC\u2081, hC\u2082\u27e9 := h\u2112p.snorm_indicator_norm_ge_pos_le \u03bc (hfmeas _) h\u03b5", "annotated_tactic": ["obtain \u27e8C, hC\u2081, hC\u2082\u27e9 := h\u2112p.snorm_indicator_norm_ge_pos_le \u03bc (hfmeas _) h\u03b5", []], "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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8\u27e8C, hC\u2081.le\u27e9, fun i => le_trans (le_of_eq _) hC\u2082\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8C, hC\u2081.le\u27e9, fun i => <a>le_trans</a> (<a>le_of_eq</a> _) hC\u2082\u27e9", [{"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]}]], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 snorm (Set.indicator {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc =\n    snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc"}, {"tactic": "have : {x | (\u27e8C, hC\u2081.le\u27e9 : \u211d\u22650) \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} := by\n  ext x\n  simp_rw [\u2190 norm_toNNReal]\n  exact Real.le_toNNReal_iff_coe_le (norm_nonneg _)", "annotated_tactic": ["have : {x | (\u27e8C, hC\u2081.le\u27e9 : \u211d\u22650) \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} := by\n    ext x\n    simp_rw [\u2190 <a>norm_toNNReal</a>]\n    exact <a>Real.le_toNNReal_iff_coe_le</a> (<a>norm_nonneg</a> _)", [{"full_name": "norm_toNNReal", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [917, 15], "def_end_pos": [917, 28]}, {"full_name": "Real.le_toNNReal_iff_coe_le", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [689, 9], "def_end_pos": [689, 31]}, {"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.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 snorm (Set.indicator {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc =\n    snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\n\u22a2 snorm (Set.indicator {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc =\n    snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc"}, {"tactic": "rw [this, \u2190 snorm_norm, \u2190 snorm_norm (Set.indicator _ _)]", "annotated_tactic": ["rw [this, \u2190 <a>snorm_norm</a>, \u2190 <a>snorm_norm</a> (<a>Set.indicator</a> _ _)]", [{"full_name": "MeasureTheory.snorm_norm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [501, 9], "def_end_pos": [501, 19]}, {"full_name": "MeasureTheory.snorm_norm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [501, 9], "def_end_pos": [501, 19]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\n\u22a2 snorm (Set.indicator {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc =\n    snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\n\u22a2 snorm (fun x => \u2016Set.indicator {x | C \u2264 \u2016f i x\u2016} (f i) x\u2016) p \u03bc =\n    snorm (fun x => \u2016Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j) x\u2016) p \u03bc"}, {"tactic": "simp_rw [norm_indicator_eq_indicator_norm, coe_nnnorm]", "annotated_tactic": ["simp_rw [<a>norm_indicator_eq_indicator_norm</a>, <a>coe_nnnorm</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": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}]], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\n\u22a2 snorm (fun x => \u2016Set.indicator {x | C \u2264 \u2016f i x\u2016} (f i) x\u2016) p \u03bc =\n    snorm (fun x => \u2016Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j) x\u2016) p \u03bc", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc"}, {"tactic": "let F : E \u2192 \u211d := (fun x : E => if (\u27e8C, hC\u2081.le\u27e9 : \u211d\u22650) \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0)", "annotated_tactic": ["let F : E \u2192 \u211d := (fun x : E => if (\u27e8C, hC\u2081.le\u27e9 : \u211d\u22650) \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0)", []], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc"}, {"tactic": "have F_meas : Measurable F := by\n  apply measurable_norm.indicator (measurableSet_le measurable_const measurable_nnnorm)", "annotated_tactic": ["have F_meas : <a>Measurable</a> F := by\n    apply measurable_norm.indicator (<a>measurableSet_le</a> <a>measurable_const</a> <a>measurable_nnnorm</a>)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"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": [2256, 9], "def_end_pos": [2256, 26]}]], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc"}, {"tactic": "have : \u2200 k, (fun x \u21a6 Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a \u21a6 \u2016f k a\u2016) x) = F \u2218 f k := by\n  intro k\n  ext x\n  simp only [Set.indicator, Set.mem_setOf_eq]; norm_cast", "annotated_tactic": ["have : \u2200 k, (fun x \u21a6 <a>Set.indicator</a> {x | C \u2264 \u2016f k x\u2016} (fun a \u21a6 \u2016f k a\u2016) x) = F \u2218 f k := by\n    intro k\n    ext x\n    simp only [<a>Set.indicator</a>, <a>Set.mem_setOf_eq</a>]; norm_cast", [{"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": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc", "state_after": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis\u271d : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nthis : \u2200 (k : \u03b9), (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc"}, {"tactic": "rw [this, this, \u2190 snorm_map_measure F_meas.aestronglyMeasurable (hf i).aemeasurable_fst,\n  (hf i).map_eq, snorm_map_measure F_meas.aestronglyMeasurable (hf j).aemeasurable_fst]", "annotated_tactic": ["rw [this, this, \u2190 <a>snorm_map_measure</a> F_meas.aestronglyMeasurable (hf i).<a>aemeasurable_fst</a>,\n    (hf i).<a>map_eq</a>, <a>snorm_map_measure</a> F_meas.aestronglyMeasurable (hf j).<a>aemeasurable_fst</a>]", [{"full_name": "MeasureTheory.snorm_map_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [920, 9], "def_end_pos": [920, 26]}, {"full_name": "ProbabilityTheory.IdentDistrib.aemeasurable_fst", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [71, 3], "def_end_pos": [71, 19]}, {"full_name": "ProbabilityTheory.IdentDistrib.map_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [73, 3], "def_end_pos": [73, 9]}, {"full_name": "MeasureTheory.snorm_map_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [920, 9], "def_end_pos": [920, 26]}, {"full_name": "ProbabilityTheory.IdentDistrib.aemeasurable_fst", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [71, 3], "def_end_pos": [71, 19]}]], "state_before": "case pos.intro.intro\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis\u271d : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nthis : \u2200 (k : \u03b9), (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k\n\u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =\n    snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc", "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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\n\u22a2 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 x \u2208 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} \u2194 x \u2208 {x | C \u2264 \u2016f i x\u2016}"}, {"tactic": "simp_rw [\u2190 norm_toNNReal]", "annotated_tactic": ["simp_rw [\u2190 <a>norm_toNNReal</a>]", [{"full_name": "norm_toNNReal", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [917, 15], "def_end_pos": [917, 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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 x \u2208 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} \u2194 x \u2208 {x | C \u2264 \u2016f i x\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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 x \u2208 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 Real.toNNReal \u2016f i x\u2016} \u2194 x \u2208 {x | C \u2264 \u2016f i x\u2016}"}, {"tactic": "exact Real.le_toNNReal_iff_coe_le (norm_nonneg _)", "annotated_tactic": ["exact <a>Real.le_toNNReal_iff_coe_le</a> (<a>norm_nonneg</a> _)", [{"full_name": "Real.le_toNNReal_iff_coe_le", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [689, 9], "def_end_pos": [689, 31]}, {"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\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nx : \u03b1\n\u22a2 x \u2208 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 Real.toNNReal \u2016f i x\u2016} \u2194 x \u2208 {x | C \u2264 \u2016f i x\u2016}", "state_after": "no goals"}, {"tactic": "apply measurable_norm.indicator (measurableSet_le measurable_const measurable_nnnorm)", "annotated_tactic": ["apply measurable_norm.indicator (<a>measurableSet_le</a> <a>measurable_const</a> <a>measurable_nnnorm</a>)", [{"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": [2256, 9], "def_end_pos": [2256, 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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\n\u22a2 Measurable F", "state_after": "no goals"}, {"tactic": "intro k", "annotated_tactic": ["intro k", []], "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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\n\u22a2 \u2200 (k : \u03b9), (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k", "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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nk : \u03b9\n\u22a2 (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nk : \u03b9\n\u22a2 (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k", "state_after": "case h\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nk : \u03b9\nx : \u03b1\n\u22a2 Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x = (F \u2218 f k) x"}, {"tactic": "simp only [Set.indicator, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>Set.indicator</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.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]}]], "state_before": "case h\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nk : \u03b9\nx : \u03b1\n\u22a2 Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x = (F \u2218 f k) 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\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nk : \u03b9\nx : \u03b1\n\u22a2 (if C \u2264 \u2016f k x\u2016 then \u2016f k x\u2016 else 0) =\n    ((fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0) \u2218 f k) x"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h\n\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\nhfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\nC : \u211d\nhC\u2081 : 0 < C\nhC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : \u03b9\nthis : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016}\nF : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0\nF_meas : Measurable F\nk : \u03b9\nx : \u03b1\n\u22a2 (if C \u2264 \u2016f k x\u2016 then \u2016f k x\u2016 else 0) =\n    ((fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0) \u2218 f k) x", "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_Ioo_ae_eq_Icc", "start": [515, 1], "end": [517, 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 => Ioo (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 => Ioo (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i)"}, {"tactic": "exact pi_Ioo_ae_eq_pi_Icc", "annotated_tactic": ["exact <a>pi_Ioo_ae_eq_pi_Icc</a>", [{"full_name": "MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [505, 9], "def_end_pos": [505, 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 => Ioo (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/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_sdiff_left'", "start": [2334, 1], "end": [2335, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one", "start": [96, 1], "end": [101, 19], "traced_tactics": [{"tactic": "simp_rw [funMulInvSnorm_rpow hp0_lt]", "annotated_tactic": ["simp_rw [<a>funMulInvSnorm_rpow</a> hp0_lt]", [{"full_name": "ENNReal.funMulInvSnorm_rpow", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [88, 9], "def_end_pos": [88, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (c : \u03b1), funMulInvSnorm f p \u03bc c ^ p \u2202\u03bc = 1", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (c : \u03b1), f c ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2202\u03bc = 1"}, {"tactic": "rw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top]", "annotated_tactic": ["rw [<a>lintegral_mul_const'</a>, <a>ENNReal.mul_inv_cancel</a> hf_nonzero hf_top]", [{"full_name": "MeasureTheory.lintegral_mul_const'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [737, 9], "def_end_pos": [737, 29]}, {"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (c : \u03b1), f c ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2202\u03bc = 1", "state_after": "case hr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "rwa [inv_ne_top]", "annotated_tactic": ["rwa [<a>inv_ne_top</a>]", [{"full_name": "ENNReal.inv_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 19]}]], "state_before": "case hr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2260 \u22a4", "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_uIoc", "start": [251, 1], "end": [252, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.liftpPreservation_iff_uniform", "start": [723, 1], "end": [724, 78], "traced_tactics": [{"tactic": "rw [\u2190 suppPreservation_iff_liftpPreservation, suppPreservation_iff_uniform]", "annotated_tactic": ["rw [\u2190 <a>suppPreservation_iff_liftpPreservation</a>, <a>suppPreservation_iff_uniform</a>]", [{"full_name": "QPF.suppPreservation_iff_liftpPreservation", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [709, 9], "def_end_pos": [709, 47]}, {"full_name": "QPF.suppPreservation_iff_uniform", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 37]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u22a2 LiftpPreservation \u2194 IsUniform", "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.next", "start": [565, 1], "end": [569, 60], "traced_tactics": [{"tactic": "cases h.out'", "annotated_tactic": ["cases h.out'", []], "state_before": "l : List Char\nc : Char\nr : List Char\nit : Iterator\nh : ValidFor l (c :: r) it\n\u22a2 ValidFor (c :: l) r (Iterator.next it)", "state_after": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor (c :: l) r\n    (Iterator.next { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } })"}, {"tactic": "simp only [Iterator.next, next_of_valid l.reverse c r]", "annotated_tactic": ["simp only [<a>Iterator.next</a>, <a>next_of_valid</a> l.reverse c r]", [{"full_name": "String.Iterator.next", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [323, 5], "def_end_pos": [323, 9]}, {"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": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor (c :: l) r\n    (Iterator.next { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } })", "state_after": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor (c :: l) r\n    { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) + csize c } }"}, {"tactic": "rw [\u2190 List.reverseAux_eq, utf8Len_reverse]", "annotated_tactic": ["rw [\u2190 <a>List.reverseAux_eq</a>, <a>utf8Len_reverse</a>]", [{"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}, {"full_name": "String.utf8Len_reverse", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [71, 17], "def_end_pos": [71, 32]}]], "state_before": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor (c :: l) r\n    { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) + csize c } }", "state_after": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor (c :: l) r { s := { data := List.reverseAux l (c :: r) }, i := { byteIdx := utf8Len l + csize c } }"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n\u22a2 ValidFor (c :: l) r { s := { data := List.reverseAux l (c :: r) }, i := { byteIdx := utf8Len l + csize c } }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.cprank_upper_bound", "start": [406, 1], "end": [410, 30], "traced_tactics": []}, {"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": [929, 1], "end": [932, 54], "traced_tactics": [{"tactic": "simpa only [\u2190 integrableOn_Ioi_comp_rpow_iff f hp, mul_smul] using\n  (integrable_smul_iff (abs_pos.mpr hp).ne' _).symm", "annotated_tactic": ["simpa only [\u2190 <a>integrableOn_Ioi_comp_rpow_iff</a> f hp, <a>mul_smul</a>] using\n    (<a>integrable_smul_iff</a> (abs_pos.mpr hp).<a>ne'</a> _).<a>symm</a>", [{"full_name": "MeasureTheory.integrableOn_Ioi_comp_rpow_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [902, 9], "def_end_pos": [902, 39]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "MeasureTheory.integrable_smul_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 28]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"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": "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 => x ^ (p - 1) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 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_union_eq", "start": [113, 1], "end": [116, 82], "traced_tactics": [{"tactic": "classical\nhave e := (Equiv.Set.union (by rwa [subset_empty_iff, \u2190disjoint_iff_inter_eq_empty])).symm\nsimp [encard, \u2190PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv]", "annotated_tactic": ["classical\n  have e := (<a>Equiv.Set.union</a> (by rwa [<a>subset_empty_iff</a>, \u2190<a>disjoint_iff_inter_eq_empty</a>])).<a>symm</a>\n  simp [<a>encard</a>, \u2190<a>PartENat.card_congr</a> e, <a>PartENat.card_sum</a>, <a>PartENat.withTopEquiv</a>]", [{"full_name": "Equiv.Set.union", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [234, 15], "def_end_pos": [234, 20]}, {"full_name": "Set.subset_empty_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [578, 9], "def_end_pos": [578, 25]}, {"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": "Equiv.symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [157, 15], "def_end_pos": [157, 19]}, {"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [66, 19], "def_end_pos": [66, 25]}, {"full_name": "PartENat.card_congr", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [162, 9], "def_end_pos": [162, 19]}, {"full_name": "PartENat.card_sum", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [158, 9], "def_end_pos": [158, 17]}, {"full_name": "PartENat.withTopEquiv", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [675, 19], "def_end_pos": [675, 31]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : Disjoint s t\n\u22a2 encard (s \u222a t) = encard s + encard t", "state_after": "no goals"}, {"tactic": "have e := (Equiv.Set.union (by rwa [subset_empty_iff, \u2190disjoint_iff_inter_eq_empty])).symm", "annotated_tactic": ["have e := (<a>Equiv.Set.union</a> (by rwa [<a>subset_empty_iff</a>, \u2190<a>disjoint_iff_inter_eq_empty</a>])).<a>symm</a>", [{"full_name": "Equiv.Set.union", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [234, 15], "def_end_pos": [234, 20]}, {"full_name": "Set.subset_empty_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [578, 9], "def_end_pos": [578, 25]}, {"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": "Equiv.symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [157, 15], "def_end_pos": [157, 19]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : Disjoint s t\n\u22a2 encard (s \u222a t) = encard s + encard t", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : Disjoint s t\ne : \u2191s \u2295 \u2191t \u2243 \u2191(s \u222a t)\n\u22a2 encard (s \u222a t) = encard s + encard t"}, {"tactic": "simp [encard, \u2190PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv]", "annotated_tactic": ["simp [<a>encard</a>, \u2190<a>PartENat.card_congr</a> e, <a>PartENat.card_sum</a>, <a>PartENat.withTopEquiv</a>]", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [66, 19], "def_end_pos": [66, 25]}, {"full_name": "PartENat.card_congr", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [162, 9], "def_end_pos": [162, 19]}, {"full_name": "PartENat.card_sum", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [158, 9], "def_end_pos": [158, 17]}, {"full_name": "PartENat.withTopEquiv", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [675, 19], "def_end_pos": [675, 31]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : Disjoint s t\ne : \u2191s \u2295 \u2191t \u2243 \u2191(s \u222a t)\n\u22a2 encard (s \u222a t) = encard s + encard t", "state_after": "no goals"}, {"tactic": "rwa [subset_empty_iff, \u2190disjoint_iff_inter_eq_empty]", "annotated_tactic": ["rwa [<a>subset_empty_iff</a>, \u2190<a>disjoint_iff_inter_eq_empty</a>]", [{"full_name": "Set.subset_empty_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [578, 9], "def_end_pos": [578, 25]}, {"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\ns t : Set \u03b1\nh : Disjoint s t\n\u22a2 ?m.6374 \u2229 ?m.6375 \u2286 \u2205", "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_union_of_top_of_nonempty_inter", "start": [735, 1], "end": [765, 71], "traced_tactics": [{"tactic": "refine' le_antisymm (OuterMeasure.union _ _ _) (le_iInf fun f => le_iInf fun hf => _)", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>OuterMeasure.union</a> _ _ _) (<a>le_iInf</a> fun f => <a>le_iInf</a> fun hf => _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.OuterMeasure.union", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [150, 19], "def_end_pos": [150, 24]}, {"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\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) (s \u222a t) =\n    \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "set \u03bc := OuterMeasure.ofFunction m m_empty", "annotated_tactic": ["set \u03bc := <a>OuterMeasure.ofFunction</a> m m_empty", [{"full_name": "MeasureTheory.OuterMeasure.ofFunction", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [654, 15], "def_end_pos": [654, 25]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "rcases Classical.em (\u2203 i, (s \u2229 f i).Nonempty \u2227 (t \u2229 f i).Nonempty) with (\u27e8i, hs, ht\u27e9 | he)", "annotated_tactic": ["rcases <a>Classical.em</a> (\u2203 i, (s \u2229 f i).<a>Nonempty</a> \u2227 (t \u2229 f i).<a>Nonempty</a>) with (\u27e8i, hs, ht\u27e9 | he)", [{"full_name": "Classical.em", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [26, 9], "def_end_pos": [26, 11]}, {"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\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "case inl.intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\ni : \u2115\nhs : Set.Nonempty (s \u2229 f i)\nht : Set.Nonempty (t \u2229 f i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "set I := fun s => { i : \u2115 | (s \u2229 f i).Nonempty }", "annotated_tactic": ["set I := fun s => { i : \u2115 | (s \u2229 f i).<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 inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "case inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "have hd : Disjoint (I s) (I t) := disjoint_iff_inf_le.mpr fun i hi => he \u27e8i, hi\u27e9", "annotated_tactic": ["have hd : <a>Disjoint</a> (I s) (I t) := disjoint_iff_inf_le.mpr fun i hi => he \u27e8i, hi\u27e9", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "case inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\nhd : Disjoint (I s) (I t)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "have hI : \u2200 (u) (_ : u \u2286 s \u222a t), \u03bc u \u2264 \u2211' i : I u, \u03bc (f i) := fun u hu =>\n  calc\n    \u03bc u \u2264 \u03bc (\u22c3 i : I u, f i) :=\n      \u03bc.mono fun x hx =>\n        let \u27e8i, hi\u27e9 := mem_iUnion.1 (hf (hu hx))\n        mem_iUnion.2 \u27e8\u27e8i, \u27e8x, hx, hi\u27e9\u27e9, hi\u27e9\n    _ \u2264 \u2211' i : I u, \u03bc (f i) := \u03bc.iUnion _", "annotated_tactic": ["have hI : \u2200 (u) (_ : u \u2286 s \u222a t), \u03bc u \u2264 \u2211' i : I u, \u03bc (f i) := fun u hu =>\n    calc\n      \u03bc u \u2264 \u03bc (\u22c3 i : I u, f i) :=\n        \u03bc.mono fun x hx =>\n          let \u27e8i, hi\u27e9 := <a>mem_iUnion</a>.1 (hf (hu hx))\n          <a>mem_iUnion</a>.2 \u27e8\u27e8i, \u27e8x, hx, hi\u27e9\u27e9, hi\u27e9\n      _ \u2264 \u2211' i : I u, \u03bc (f i) := \u03bc.iUnion _", [{"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_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\nhd : Disjoint (I s) (I t)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "case inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\nhd : Disjoint (I s) (I t)\nhI : \u2200 (u : Set \u03b1), u \u2286 s \u222a t \u2192 \u2191\u03bc u \u2264 \u2211' (i : \u2191(I u)), \u2191\u03bc (f \u2191i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)"}, {"tactic": "calc\n  \u03bc s + \u03bc t \u2264 (\u2211' i : I s, \u03bc (f i)) + \u2211' i : I t, \u03bc (f i) :=\n    add_le_add (hI _ <| subset_union_left _ _) (hI _ <| subset_union_right _ _)\n  _ = \u2211' i : \u2191(I s \u222a I t), \u03bc (f i) :=\n    (tsum_union_disjoint (f := fun i => \u03bc (f i)) hd ENNReal.summable ENNReal.summable).symm\n  _ \u2264 \u2211' i, \u03bc (f i) :=\n    (tsum_le_tsum_of_inj (\u2191) Subtype.coe_injective (fun _ _ => zero_le _) (fun _ => le_rfl)\n      ENNReal.summable ENNReal.summable)\n  _ \u2264 \u2211' i, m (f i) := ENNReal.tsum_le_tsum fun i => ofFunction_le _", "annotated_tactic": ["calc\n    \u03bc s + \u03bc t \u2264 (\u2211' i : I s, \u03bc (f i)) + \u2211' i : I t, \u03bc (f i) :=\n      <a>add_le_add</a> (hI _ <| <a>subset_union_left</a> _ _) (hI _ <| <a>subset_union_right</a> _ _)\n    _ = \u2211' i : \u2191(I s \u222a I t), \u03bc (f i) :=\n      (<a>tsum_union_disjoint</a> (f := fun i => \u03bc (f i)) hd <a>ENNReal.summable</a> <a>ENNReal.summable</a>).<a>symm</a>\n    _ \u2264 \u2211' i, \u03bc (f i) :=\n      (<a>tsum_le_tsum_of_inj</a> (\u2191) <a>Subtype.coe_injective</a> (fun _ _ => <a>zero_le</a> _) (fun _ => <a>le_rfl</a>)\n        <a>ENNReal.summable</a> <a>ENNReal.summable</a>)\n    _ \u2264 \u2211' i, m (f i) := <a>ENNReal.tsum_le_tsum</a> fun i => <a>ofFunction_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": "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": "tsum_union_disjoint", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [782, 9], "def_end_pos": [782, 28]}, {"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": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "tsum_le_tsum_of_inj", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 28]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 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": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "MeasureTheory.OuterMeasure.ofFunction_le", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [698, 9], "def_end_pos": [698, 22]}]], "state_before": "case inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\nhd : Disjoint (I s) (I t)\nhI : \u2200 (u : Set \u03b1), u \u2286 s \u222a t \u2192 \u2191\u03bc u \u2264 \u2211' (i : \u2191(I u)), \u2191\u03bc (f \u2191i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "no goals"}, {"tactic": "calc\n  \u03bc s + \u03bc t \u2264 \u221e := le_top\n  _ = m (f i) := (h (f i) hs ht).symm\n  _ \u2264 \u2211' i, m (f i) := ENNReal.le_tsum i", "annotated_tactic": ["calc\n      \u03bc s + \u03bc t \u2264 \u221e := <a>le_top</a>\n      _ = m (f i) := (h (f i) hs ht).<a>symm</a>\n      _ \u2264 \u2211' i, m (f i) := <a>ENNReal.le_tsum</a> i", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"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": "ENNReal.le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [857, 19], "def_end_pos": [857, 26]}]], "state_before": "case inl.intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 i, f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\ni : \u2115\nhs : Set.Nonempty (s \u2229 f i)\nht : Set.Nonempty (t \u2229 f i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.card_Icc_finset", "start": [101, 1], "end": [106, 78], "traced_tactics": [{"tactic": "rw [\u2190 card_sdiff h, \u2190 card_powerset, Icc_eq_image_powerset h, Finset.card_image_iff]", "annotated_tactic": ["rw [\u2190 <a>card_sdiff</a> h, \u2190 <a>card_powerset</a>, <a>Icc_eq_image_powerset</a> h, <a>Finset.card_image_iff</a>]", [{"full_name": "Finset.card_sdiff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [440, 9], "def_end_pos": [440, 19]}, {"full_name": "Finset.card_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}, {"full_name": "Finset.Icc_eq_image_powerset", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [80, 9], "def_end_pos": [80, 30]}, {"full_name": "Finset.card_image_iff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [246, 9], "def_end_pos": [246, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 card (Icc s t) = 2 ^ (card t - card s)", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 Set.InjOn ((fun x x_1 => x \u222a x_1) s) \u2191(powerset (t \\ s))"}, {"tactic": "rintro u hu v hv (huv : s \u2294 u = s \u2294 v)", "annotated_tactic": ["rintro u hu v hv (huv : s \u2294 u = s \u2294 v)", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 Set.InjOn ((fun x x_1 => x \u222a x_1) s) \u2191(powerset (t \\ s))", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhu : u \u2208 \u2191(powerset (t \\ s))\nv : Finset \u03b1\nhv : v \u2208 \u2191(powerset (t \\ s))\nhuv : s \u2294 u = s \u2294 v\n\u22a2 u = v"}, {"tactic": "rw [mem_coe, mem_powerset] at hu hv", "annotated_tactic": ["rw [<a>mem_coe</a>, <a>mem_powerset</a>] at hu hv", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finset.mem_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhu : u \u2208 \u2191(powerset (t \\ s))\nv : Finset \u03b1\nhv : v \u2208 \u2191(powerset (t \\ s))\nhuv : s \u2294 u = s \u2294 v\n\u22a2 u = v", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhu : u \u2286 t \\ s\nv : Finset \u03b1\nhv : v \u2286 t \\ s\nhuv : s \u2294 u = s \u2294 v\n\u22a2 u = v"}, {"tactic": "rw [\u2190 (disjoint_sdiff.mono_right hu : Disjoint s u).sup_sdiff_cancel_left, \u2190\n  (disjoint_sdiff.mono_right hv : Disjoint s v).sup_sdiff_cancel_left, huv]", "annotated_tactic": ["rw [\u2190 (disjoint_sdiff.mono_right hu : <a>Disjoint</a> s u).<a>sup_sdiff_cancel_left</a>, \u2190\n    (disjoint_sdiff.mono_right hv : <a>Disjoint</a> s v).<a>sup_sdiff_cancel_left</a>, huv]", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Disjoint.sup_sdiff_cancel_left", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 39]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Disjoint.sup_sdiff_cancel_left", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhu : u \u2286 t \\ s\nv : Finset \u03b1\nhv : v \u2286 t \\ s\nhuv : s \u2294 u = s \u2294 v\n\u22a2 u = v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded", "start": [590, 1], "end": [599, 27], "traced_tactics": [{"tactic": "refine (aecover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I\n  (fun i => (hfi i).restrict measurableSet_Ioc) (h.mono fun i hi \u21a6 ?_)", "annotated_tactic": ["refine (<a>aecover_Ioc_of_Ioc</a> ha hb).<a>integrable_of_integral_norm_bounded</a> I\n    (fun i => (hfi i).<a>restrict</a> <a>measurableSet_Ioc</a>) (h.mono fun i hi \u21a6 ?_)", [{"full_name": "MeasureTheory.aecover_Ioc_of_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [224, 9], "def_end_pos": [224, 27]}, {"full_name": "MeasureTheory.AECover.integrable_of_integral_norm_bounded", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [436, 9], "def_end_pos": [436, 52]}, {"full_name": "MeasureTheory.IntegrableOn.restrict", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [167, 9], "def_end_pos": [167, 30]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 IntegrableOn f (Ioc a\u2080 b\u2080)", "state_after": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2202Measure.restrict volume (Ioc a\u2080 b\u2080) \u2264 I"}, {"tactic": "rw [Measure.restrict_restrict measurableSet_Ioc]", "annotated_tactic": ["rw [<a>Measure.restrict_restrict</a> <a>measurableSet_Ioc</a>]", [{"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2202Measure.restrict volume (Ioc a\u2080 b\u2080) \u2264 I", "state_after": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u222b (x : \u211d) in Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080, \u2016f x\u2016 \u2264 I"}, {"tactic": "refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi <;> apply ae_of_all", "annotated_tactic": ["refine' <a>le_trans</a> (<a>set_integral_mono_set</a> (hfi i).<a>norm</a> _ _) hi <;> apply <a>ae_of_all</a>", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.set_integral_mono_set", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [736, 9], "def_end_pos": [736, 30]}, {"full_name": "MeasureTheory.Integrable.norm", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [706, 9], "def_end_pos": [706, 24]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u222b (x : \u211d) in Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080, \u2016f x\u2016 \u2264 I", "state_after": "case refine'_1.a\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u2200 (a : \u211d), OfNat.ofNat 0 a \u2264 (fun x => \u2016f x\u2016) a\n\ncase refine'_2.a\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u2200 (a_1 : \u211d), (Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080) a_1 \u2264 Ioc (a i) (b i) a_1"}, {"tactic": "simp only [Pi.zero_apply, norm_nonneg, forall_const]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>norm_nonneg</a>, <a>forall_const</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_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}]], "state_before": "case refine'_1.a\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u2200 (a : \u211d), OfNat.ofNat 0 a \u2264 (fun x => \u2016f x\u2016) a", "state_after": "no goals"}, {"tactic": "intro c hc", "annotated_tactic": ["intro c hc", []], "state_before": "case refine'_2.a\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\n\u22a2 \u2200 (a_1 : \u211d), (Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080) a_1 \u2264 Ioc (a i) (b i) a_1", "state_after": "case refine'_2.a\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\nc : \u211d\nhc : (Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080) c\n\u22a2 Ioc (a i) (b i) c"}, {"tactic": "exact hc.1", "annotated_tactic": ["exact hc.1", []], "state_before": "case refine'_2.a\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : NeBot l\ninst\u271d\u00b9 : IsCountablyGenerated l\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI a\u2080 b\u2080 : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i))\nha : Tendsto a l (\ud835\udcdd a\u2080)\nhb : Tendsto b l (\ud835\udcdd b\u2080)\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\ni : \u03b9\nhi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I\nc : \u211d\nhc : (Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080) c\n\u22a2 Ioc (a i) (b i) c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.hasSum_of_disjoint_iUnion", "start": [145, 1], "end": [177, 78], "traced_tactics": [{"tactic": "cases nonempty_encodable \u03b2", "annotated_tactic": ["cases <a>nonempty_encodable</a> \u03b2", [{"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\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))"}, {"tactic": "set g := fun i : \u2115 => \u22c3 (b : \u03b2) (_ : b \u2208 Encodable.decode\u2082 \u03b2 i), f b with hg", "annotated_tactic": ["set g := fun i : \u2115 => \u22c3 (b : \u03b2) (_ : b \u2208 <a>Encodable.decode\u2082</a> \u03b2 i), f b with hg", [{"full_name": "Encodable.decode\u2082", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [188, 5], "def_end_pos": [188, 12]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))"}, {"tactic": "have hg\u2081 : \u2200 i, MeasurableSet (g i) :=\n  fun _ => MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => hf\u2081 b", "annotated_tactic": ["have hg\u2081 : \u2200 i, <a>MeasurableSet</a> (g i) :=\n    fun _ => <a>MeasurableSet.iUnion</a> fun b => <a>MeasurableSet.iUnion</a> fun _ => hf\u2081 b", [{"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": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))"}, {"tactic": "have hg\u2082 : Pairwise (Disjoint on g) := Encodable.iUnion_decode\u2082_disjoint_on hf\u2082", "annotated_tactic": ["have hg\u2082 : <a>Pairwise</a> (<a>Disjoint</a> on g) := <a>Encodable.iUnion_decode\u2082_disjoint_on</a> hf\u2082", [{"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": "Encodable.iUnion_decode\u2082_disjoint_on", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [52, 9], "def_end_pos": [52, 35]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))"}, {"tactic": "have := v.of_disjoint_iUnion_nat hg\u2081 hg\u2082", "annotated_tactic": ["have := v.of_disjoint_iUnion_nat hg\u2081 hg\u2082", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 i, g i) = \u2211' (i : \u2115), \u2191v (g i)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))"}, {"tactic": "rw [hg, Encodable.iUnion_decode\u2082] at this", "annotated_tactic": ["rw [hg, <a>Encodable.iUnion_decode\u2082</a>] at this", [{"full_name": "Encodable.iUnion_decode\u2082", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [35, 9], "def_end_pos": [35, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 i, g i) = \u2211' (i : \u2115), \u2191v (g i)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))"}, {"tactic": "rw [Summable.hasSum_iff, this, \u2190 tsum_iUnion_decode\u2082]", "annotated_tactic": ["rw [<a>Summable.hasSum_iff</a>, this, \u2190 <a>tsum_iUnion_decode\u2082</a>]", [{"full_name": "Summable.hasSum_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [571, 9], "def_end_pos": [571, 28]}, {"full_name": "tsum_iUnion_decode\u2082", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 28]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i))", "state_after": "case intro.m0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 \u2191v \u2205 = 0\n\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable fun i => \u2191v (f i)"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\n\u22a2 (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 \u2191v (f x) = \u2191v (g (Encodable.encode x))"}, {"tactic": "rw [hg]", "annotated_tactic": ["rw [hg]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 \u2191v (f x) = \u2191v (g (Encodable.encode x))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 \u2191v (f x) = \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) (Encodable.encode x))"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 \u2191v (f x) = \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) (Encodable.encode x))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 \u2191v (f x) = \u2191v (\u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 \u2191v (f x) = \u2191v (\u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b)", "state_after": "case h.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 f x = \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b"}, {"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "case h.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\n\u22a2 f x = \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b", "state_after": "case h.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 y \u2208 f x \u2194 y \u2208 \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b"}, {"tactic": "simp only [exists_prop, Set.mem_iUnion, Option.mem_def]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>Set.mem_iUnion</a>, <a>Option.mem_def</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": "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 h.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 y \u2208 f x \u2194 y \u2208 \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b", "state_after": "case h.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 y \u2208 f x \u2194 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 y \u2208 f x \u2194 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i", "state_after": "case h.e_a.h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 y \u2208 f x \u2192 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i\n\ncase h.e_a.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 (\u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i) \u2192 y \u2208 f x"}, {"tactic": "intro hy", "annotated_tactic": ["intro hy", []], "state_before": "case h.e_a.h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 y \u2208 f x \u2192 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i", "state_after": "case h.e_a.h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\nhy : y \u2208 f x\n\u22a2 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i"}, {"tactic": "refine' \u27e8x, (Encodable.decode\u2082_is_partial_inv _ _).2 rfl, hy\u27e9", "annotated_tactic": ["refine' \u27e8x, (<a>Encodable.decode\u2082_is_partial_inv</a> _ _).2 <a>rfl</a>, hy\u27e9", [{"full_name": "Encodable.decode\u2082_is_partial_inv", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 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 h.e_a.h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\nhy : y \u2208 f x\n\u22a2 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i", "state_after": "no goals"}, {"tactic": "rintro \u27e8b, hb\u2081, hb\u2082\u27e9", "annotated_tactic": ["rintro \u27e8b, hb\u2081, hb\u2082\u27e9", []], "state_before": "case h.e_a.h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\n\u22a2 (\u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i) \u2192 y \u2208 f x", "state_after": "case h.e_a.h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\nb : \u03b2\nhb\u2081 : Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some b\nhb\u2082 : y \u2208 f b\n\u22a2 y \u2208 f x"}, {"tactic": "rw [Encodable.decode\u2082_is_partial_inv _ _] at hb\u2081", "annotated_tactic": ["rw [<a>Encodable.decode\u2082_is_partial_inv</a> _ _] at hb\u2081", [{"full_name": "Encodable.decode\u2082_is_partial_inv", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 31]}]], "state_before": "case h.e_a.h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\nb : \u03b2\nhb\u2081 : Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some b\nhb\u2082 : y \u2208 f b\n\u22a2 y \u2208 f x", "state_after": "case h.e_a.h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\nb : \u03b2\nhb\u2081 : Encodable.encode b = Encodable.encode x\nhb\u2082 : y \u2208 f b\n\u22a2 y \u2208 f x"}, {"tactic": "rwa [\u2190 Encodable.encode_injective hb\u2081]", "annotated_tactic": ["rwa [\u2190 <a>Encodable.encode_injective</a> hb\u2081]", [{"full_name": "Encodable.encode_injective", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}]], "state_before": "case h.e_a.h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nx : \u03b2\ny : \u03b1\nb : \u03b2\nhb\u2081 : Encodable.encode b = Encodable.encode x\nhb\u2082 : y \u2208 f b\n\u22a2 y \u2208 f x", "state_after": "no goals"}, {"tactic": "exact v.empty", "annotated_tactic": ["exact v.empty", []], "state_before": "case intro.m0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 \u2191v \u2205 = 0", "state_after": "no goals"}, {"tactic": "rw [hg\u2083]", "annotated_tactic": ["rw [hg\u2083]", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable fun i => \u2191v (f i)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable fun i => \u2191v (g (Encodable.encode i))"}, {"tactic": "change Summable ((fun i => v (g i)) \u2218 Encodable.encode)", "annotated_tactic": ["change <a>Summable</a> ((fun i => v (g i)) \u2218 <a>Encodable.encode</a>)", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"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\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable fun i => \u2191v (g (Encodable.encode i))", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable ((fun i => \u2191v (g i)) \u2218 Encodable.encode)"}, {"tactic": "rw [Function.Injective.summable_iff Encodable.encode_injective]", "annotated_tactic": ["rw [<a>Function.Injective.summable_iff</a> <a>Encodable.encode_injective</a>]", [{"full_name": "Function.Injective.summable_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [138, 9], "def_end_pos": [138, 40]}, {"full_name": "Encodable.encode_injective", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 25]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable ((fun i => \u2191v (g i)) \u2218 Encodable.encode)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable fun i => \u2191v (g i)\n\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 \u2200 (x : \u2115), \u00acx \u2208 range Encodable.encode \u2192 \u2191v (g x) = 0"}, {"tactic": "exact (v.m_iUnion hg\u2081 hg\u2082).summable", "annotated_tactic": ["exact (v.m_iUnion hg\u2081 hg\u2082).<a>summable</a>", [{"full_name": "HasSum.summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 24]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 Summable fun i => \u2191v (g i)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\n\u22a2 \u2200 (x : \u2115), \u00acx \u2208 range Encodable.encode \u2192 \u2191v (g x) = 0", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u00acx \u2208 range Encodable.encode\n\u22a2 \u2191v (g x) = 0"}, {"tactic": "convert v.empty", "annotated_tactic": ["convert v.empty", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u00acx \u2208 range Encodable.encode\n\u22a2 \u2191v (g x) = 0", "state_after": "case h.e'_2.h.e'_7\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u00acx \u2208 range Encodable.encode\n\u22a2 g x = \u2205"}, {"tactic": "simp only [Set.iUnion_eq_empty, Option.mem_def, not_exists, Set.mem_range] at hx \u22a2", "annotated_tactic": ["simp only [<a>Set.iUnion_eq_empty</a>, <a>Option.mem_def</a>, <a>not_exists</a>, <a>Set.mem_range</a>] at hx \u22a2", [{"full_name": "Set.iUnion_eq_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [824, 9], "def_end_pos": [824, 24]}, {"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": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"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.e'_2.h.e'_7\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u00acx \u2208 range Encodable.encode\n\u22a2 g x = \u2205", "state_after": "case h.e'_2.h.e'_7\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u2200 (x_1 : \u03b2), \u00acEncodable.encode x_1 = x\n\u22a2 \u2200 (i : \u03b2), Encodable.decode\u2082 \u03b2 x = Option.some i \u2192 f i = \u2205"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "case h.e'_2.h.e'_7\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u2200 (x_1 : \u03b2), \u00acEncodable.encode x_1 = x\n\u22a2 \u2200 (i : \u03b2), Encodable.decode\u2082 \u03b2 x = Option.some i \u2192 f i = \u2205", "state_after": "case h.e'_2.h.e'_7\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u2200 (x_1 : \u03b2), \u00acEncodable.encode x_1 = x\ni : \u03b2\nhi : Encodable.decode\u2082 \u03b2 x = Option.some i\n\u22a2 f i = \u2205"}, {"tactic": "exact False.elim ((hx i) ((Encodable.decode\u2082_is_partial_inv _ _).1 hi))", "annotated_tactic": ["exact <a>False.elim</a> ((hx i) ((<a>Encodable.decode\u2082_is_partial_inv</a> _ _).1 hi))", [{"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": "Encodable.decode\u2082_is_partial_inv", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 31]}]], "state_before": "case h.e'_2.h.e'_7\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : T2Space M\nv : VectorMeasure \u03b1 M\nf\u271d : \u2115 \u2192 Set \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\nval\u271d : Encodable \u03b2\ng : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b\nhg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i)\nhg\u2082 : Pairwise (Disjoint on g)\nthis : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i)\nhg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i))\nx : \u2115\nhx : \u2200 (x_1 : \u03b2), \u00acEncodable.encode x_1 = x\ni : \u03b2\nhi : Encodable.decode\u2082 \u03b2 x = Option.some i\n\u22a2 f i = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.support_sum", "start": [570, 1], "end": [572, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.lintegral_eq_of_measure_preimage", "start": [1116, 1], "end": [1122, 41], "traced_tactics": [{"tactic": "simp only [lintegral, \u2190 H]", "annotated_tactic": ["simp only [<a>lintegral</a>, \u2190 H]", [{"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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 lintegral f \u03bc = lintegral g \u03bd", "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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 \u2211 x in SimpleFunc.range f, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x}) = \u2211 x in SimpleFunc.range g, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x})"}, {"tactic": "apply lintegral_eq_of_subset", "annotated_tactic": ["apply <a>lintegral_eq_of_subset</a>", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_eq_of_subset", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [954, 9], "def_end_pos": [954, 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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 \u2211 x in SimpleFunc.range f, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x}) = \u2211 x in SimpleFunc.range g, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x})", "state_after": "case hs\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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 SimpleFunc.range g"}, {"tactic": "simp only [H]", "annotated_tactic": ["simp only [H]", []], "state_before": "case hs\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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 SimpleFunc.range g", "state_after": "case hs\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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 SimpleFunc.range g"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case hs\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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\n\u22a2 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 SimpleFunc.range g", "state_after": "case hs\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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\nx\u271d : \u03b1\na\u271d\u00b9 : \u2191f x\u271d \u2260 0\na\u271d : \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {\u2191f x\u271d}) \u2260 0\n\u22a2 \u2191f x\u271d \u2208 SimpleFunc.range g"}, {"tactic": "exact mem_range_of_measure_ne_zero \u2039_\u203a", "annotated_tactic": ["exact <a>mem_range_of_measure_ne_zero</a> \u2039_\u203a", [{"full_name": "MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [121, 9], "def_end_pos": [121, 37]}]], "state_before": "case hs\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\u271d : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ng : \u03b2 \u2192\u209b \u211d\u22650\u221e\n\u03bd : Measure \u03b2\nH : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y})\nx\u271d : \u03b1\na\u271d\u00b9 : \u2191f x\u271d \u2260 0\na\u271d : \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {\u2191f x\u271d}) \u2260 0\n\u22a2 \u2191f x\u271d \u2208 SimpleFunc.range g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/MulAntidiagonal.lean", "full_name": "Set.IsPwo.mul", "start": [25, 1], "end": [27, 72], "traced_tactics": [{"tactic": "rw [\u2190 image_mul_prod]", "annotated_tactic": ["rw [\u2190 <a>image_mul_prod</a>]", [{"full_name": "Set.image_mul_prod", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [349, 9], "def_end_pos": [349, 23]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : OrderedCancelCommMonoid \u03b1\nhs : IsPwo s\nht : IsPwo t\n\u22a2 IsPwo (s * t)", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : OrderedCancelCommMonoid \u03b1\nhs : IsPwo s\nht : IsPwo t\n\u22a2 IsPwo ((fun x => x.1 * x.2) '' s \u00d7\u02e2 t)"}, {"tactic": "exact (hs.prod ht).image_of_monotone (monotone_fst.mul' monotone_snd)", "annotated_tactic": ["exact (hs.prod ht).<a>image_of_monotone</a> (monotone_fst.mul' <a>monotone_snd</a>)", [{"full_name": "Set.IsPwo.image_of_monotone", "def_path": "Mathlib/Order/WellFoundedSet.lean", "def_pos": [460, 9], "def_end_pos": [460, 32]}, {"full_name": "monotone_snd", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 21]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : OrderedCancelCommMonoid \u03b1\nhs : IsPwo s\nht : IsPwo t\n\u22a2 IsPwo ((fun x => x.1 * x.2) '' s \u00d7\u02e2 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Multiset.toFinset_nonempty", "start": [3249, 1], "end": [3250, 70], "traced_tactics": [{"tactic": "simp only [toFinset_eq_empty, Ne.def, Finset.nonempty_iff_ne_empty]", "annotated_tactic": ["simp only [<a>toFinset_eq_empty</a>, <a>Ne.def</a>, <a>Finset.nonempty_iff_ne_empty</a>]", [{"full_name": "Multiset.toFinset_eq_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3244, 9], "def_end_pos": [3244, 26]}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 Finset.Nonempty (toFinset s) \u2194 s \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr'", "start": [361, 1], "end": [377, 67], "traced_tactics": [{"tactic": "have h_pair : Integrable (f.pair g) \u03bc := integrable_pair hf hg", "annotated_tactic": ["have h_pair : <a>Integrable</a> (f.pair g) \u03bc := <a>integrable_pair</a> hf hg", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.SimpleFunc.integrable_pair", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [379, 9], "def_end_pos": [379, 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\n\u22a2 setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\n\u22a2 setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f g))"}, {"tactic": "rw [map_setToSimpleFunc T h_add h_pair Prod.fst_zero]", "annotated_tactic": ["rw [<a>map_setToSimpleFunc</a> T h_add h_pair <a>Prod.fst_zero</a>]", [{"full_name": "MeasureTheory.SimpleFunc.map_setToSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [317, 9], "def_end_pos": [317, 28]}, {"full_name": "Prod.fst_zero", "def_path": "Mathlib/Algebra/Group/Prod.lean", "def_pos": [93, 3], "def_end_pos": [93, 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\n\u22a2 setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\n\u22a2 \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.1 = setToSimpleFunc T (map Prod.snd (pair f g))"}, {"tactic": "rw [map_setToSimpleFunc T h_add h_pair Prod.snd_zero]", "annotated_tactic": ["rw [<a>map_setToSimpleFunc</a> T h_add h_pair <a>Prod.snd_zero</a>]", [{"full_name": "MeasureTheory.SimpleFunc.map_setToSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [317, 9], "def_end_pos": [317, 28]}, {"full_name": "Prod.snd_zero", "def_path": "Mathlib/Algebra/Group/Prod.lean", "def_pos": [99, 3], "def_end_pos": [99, 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\n\u22a2 \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.1 = setToSimpleFunc T (map Prod.snd (pair f 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\n\u22a2 \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.1 =\n    \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.2"}, {"tactic": "refine' Finset.sum_congr rfl fun p hp => _", "annotated_tactic": ["refine' <a>Finset.sum_congr</a> <a>rfl</a> fun p hp => _", [{"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\nE : Type u_2\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\n\u22a2 \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.1 =\n    \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.2", "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\u271d : \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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\np : E \u00d7 E\nhp : p \u2208 SimpleFunc.range (pair f g)\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {p})) p.1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {p})) p.2"}, {"tactic": "rcases mem_range.1 hp with \u27e8a, rfl\u27e9", "annotated_tactic": ["rcases <a>mem_range</a>.1 hp with \u27e8a, rfl\u27e9", [{"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\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np\u271d : \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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\np : E \u00d7 E\nhp : p \u2208 SimpleFunc.range (pair f g)\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {p})) p.1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {p})) p.2", "state_after": "case 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2"}, {"tactic": "by_cases eq : f a = g a", "annotated_tactic": ["by_cases eq : f a = g a", []], "state_before": "case 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2", "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u2191f a = \u2191g a\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2"}, {"tactic": "dsimp only [pair_apply]", "annotated_tactic": ["dsimp only [<a>pair_apply</a>]", [{"full_name": "MeasureTheory.SimpleFunc.pair_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [402, 9], "def_end_pos": [402, 19]}]], "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u2191f a = \u2191g a\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2", "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u2191f a = \u2191g a\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)})) (\u2191f a) = \u2191(T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)})) (\u2191g a)"}, {"tactic": "rw [eq]", "annotated_tactic": ["rw [eq]", []], "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u2191f a = \u2191g a\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)})) (\u2191f a) = \u2191(T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)})) (\u2191g a)", "state_after": "no goals"}, {"tactic": "have : T (pair f g \u207b\u00b9' {(f a, g a)}) = 0 := by\n  have h_eq : T ((\u21d1(f.pair g)) \u207b\u00b9' {(f a, g a)}) = T (f \u207b\u00b9' {f a} \u2229 g \u207b\u00b9' {g a}) := by\n    congr; rw [pair_preimage_singleton f g]\n  rw [h_eq]\n  exact h (f a) (g a) eq", "annotated_tactic": ["have : T (<a>pair</a> f g \u207b\u00b9' {(f a, g a)}) = 0 := by\n        have h_eq : T ((\u21d1(f.pair g)) \u207b\u00b9' {(f a, g a)}) = T (f \u207b\u00b9' {f a} \u2229 g \u207b\u00b9' {g a}) := by\n          congr; rw [<a>pair_preimage_singleton</a> f g]\n        rw [h_eq]\n        exact h (f a) (g a) eq", [{"full_name": "MeasureTheory.SimpleFunc.pair", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [397, 5], "def_end_pos": [397, 9]}, {"full_name": "MeasureTheory.SimpleFunc.pair_preimage_singleton", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [412, 9], "def_end_pos": [412, 32]}]], "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2", "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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\nthis : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 0\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2"}, {"tactic": "simp only [this, ContinuousLinearMap.zero_apply, pair_apply]", "annotated_tactic": ["simp only [this, <a>ContinuousLinearMap.zero_apply</a>, <a>pair_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.pair_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [402, 9], "def_end_pos": [402, 19]}]], "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\nthis : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 0\n\u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2", "state_after": "no goals"}, {"tactic": "have h_eq : T ((\u21d1(f.pair g)) \u207b\u00b9' {(f a, g a)}) = T (f \u207b\u00b9' {f a} \u2229 g \u207b\u00b9' {g a}) := by\n  congr; rw [pair_preimage_singleton f g]", "annotated_tactic": ["have h_eq : T ((\u21d1(f.pair g)) \u207b\u00b9' {(f a, g a)}) = T (f \u207b\u00b9' {f a} \u2229 g \u207b\u00b9' {g a}) := by\n          congr; rw [<a>pair_preimage_singleton</a> f g]", [{"full_name": "MeasureTheory.SimpleFunc.pair_preimage_singleton", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [412, 9], "def_end_pos": [412, 32]}]], "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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\n\u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\nh_eq : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a})\n\u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 0"}, {"tactic": "rw [h_eq]", "annotated_tactic": ["rw [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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\nh_eq : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a})\n\u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 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\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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\nh_eq : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a})\n\u22a2 T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}) = 0"}, {"tactic": "exact h (f a) (g a) eq", "annotated_tactic": ["exact h (f a) (g a) 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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\nh_eq : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a})\n\u22a2 T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}) = 0", "state_after": "no goals"}, {"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\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\n\u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a})", "state_after": "case 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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\n\u22a2 \u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)} = \u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}"}, {"tactic": "rw [pair_preimage_singleton f g]", "annotated_tactic": ["rw [<a>pair_preimage_singleton</a> f g]", [{"full_name": "MeasureTheory.SimpleFunc.pair_preimage_singleton", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [412, 9], "def_end_pos": [412, 32]}]], "state_before": "case 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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nT : Set \u03b1 \u2192 E \u2192L[\u211d] F\nh_add : FinMeasAdditive \u03bc T\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nhg : Integrable \u2191g\nh : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0\nh_pair : Integrable \u2191(pair f g)\na : \u03b1\nhp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g)\neq : \u00ac\u2191f a = \u2191g a\n\u22a2 \u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)} = \u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g 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.dest_corec\u2081", "start": [548, 1], "end": [553, 49], "traced_tactics": [{"tactic": "rw [Cofix.corec\u2081, Cofix.dest_corec', \u2190 h]", "annotated_tactic": ["rw [<a>Cofix.corec\u2081</a>, <a>Cofix.dest_corec'</a>, \u2190 h]", [{"full_name": "MvQPF.Cofix.corec\u2081", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [179, 5], "def_end_pos": [179, 17]}, {"full_name": "MvQPF.Cofix.dest_corec'", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "def_pos": [529, 9], "def_end_pos": [529, 26]}]], "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 : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)\nx : \u03b2\nh :\n  \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x\n\u22a2 dest (corec\u2081 g x) = g _root_.id (corec\u2081 g) x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)\nx : \u03b2\nh :\n  \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x\n\u22a2 g (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inl)\n      (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inr) x =\n    g _root_.id (corec\u2081 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 : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)\nx : \u03b2\nh :\n  \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x\n\u22a2 g (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inl)\n      (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inr) x =\n    g _root_.id (corec\u2081 g) x", "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_real_prod", "start": [48, 1], "end": [49, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.boundedSMul", "start": [518, 11], "end": [519, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.ToPartrec.stepNormal_eval", "start": [718, 1], "end": [719, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral", "start": [1760, 1], "end": [1770, 68], "traced_tactics": [{"tactic": "simp_rw [lintegral_eq_nnreal, lt_iSup_iff] at hL", "annotated_tactic": ["simp_rw [<a>lintegral_eq_nnreal</a>, <a>lt_iSup_iff</a>] at hL", [{"full_name": "MeasureTheory.lintegral_eq_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}, {"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g 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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\nhL : \u2203 i i_1, L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "rcases hL with \u27e8g\u2080, hg\u2080, g\u2080L\u27e9", "annotated_tactic": ["rcases hL with \u27e8g\u2080, hg\u2080, g\u2080L\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\nhL : \u2203 i i_1, L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "have h'L : L < \u222b\u207b x, g\u2080 x \u2202\u03bc := by\n  convert g\u2080L\n  rw [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_map]\n  simp only [Function.comp_apply]", "annotated_tactic": ["have h'L : L < \u222b\u207b x, g\u2080 x \u2202\u03bc := by\n    convert g\u2080L\n    rw [\u2190 <a>SimpleFunc.lintegral_eq_lintegral</a>, <a>coe_map</a>]\n    simp only [<a>Function.comp_apply</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_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]}]], "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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\nh'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "rcases SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral h'L with \u27e8g, hg, gL, gtop\u27e9", "annotated_tactic": ["rcases <a>SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral</a> h'L with \u27e8g, hg, gL, gtop\u27e9", [{"full_name": "MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1705, 9], "def_end_pos": [1705, 66]}]], "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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\nh'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "case 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\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\nh'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\nhg : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191g\u2080 x\ngL : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngtop : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc"}, {"tactic": "exact \u27e8g, fun x => (hg x).trans (coe_le_coe.1 (hg\u2080 x)), gL, gtop\u27e9", "annotated_tactic": ["exact \u27e8g, fun x => (hg x).<a>trans</a> (<a>coe_le_coe</a>.1 (hg\u2080 x)), gL, gtop\u27e9", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"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.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\nh'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\nhg : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191g\u2080 x\ngL : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngtop : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "convert g\u2080L", "annotated_tactic": ["convert g\u2080L", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc", "state_after": "case h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc"}, {"tactic": "rw [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_map]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.lintegral_eq_lintegral</a>, <a>coe_map</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_map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [303, 9], "def_end_pos": [303, 16]}]], "state_before": "case h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc", "state_after": "case h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = \u222b\u207b (a : \u03b1), (ENNReal.some \u2218 \u2191g\u2080) 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 h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = \u222b\u207b (a : \u03b1), (ENNReal.some \u2218 \u2191g\u2080) a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.continuous_condexpIndL1", "start": [244, 1], "end": [245, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.univ_mul_univ", "start": [944, 1], "end": [945, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.mul_inv_of_unit", "start": [767, 1], "end": [769, 31], "traced_tactics": [{"tactic": "rcases h with \u27e8u, rfl\u27e9", "annotated_tactic": ["rcases h with \u27e8u, rfl\u27e9", []], "state_before": "n : \u2115\na : ZMod n\nh : IsUnit a\n\u22a2 a * a\u207b\u00b9 = 1", "state_after": "case intro\nn : \u2115\nu : (ZMod n)\u02e3\n\u22a2 \u2191u * (\u2191u)\u207b\u00b9 = 1"}, {"tactic": "rw [inv_coe_unit, u.mul_inv]", "annotated_tactic": ["rw [<a>inv_coe_unit</a>, u.mul_inv]", [{"full_name": "ZMod.inv_coe_unit", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [756, 9], "def_end_pos": [756, 21]}]], "state_before": "case intro\nn : \u2115\nu : (ZMod n)\u02e3\n\u22a2 \u2191u * (\u2191u)\u207b\u00b9 = 1", "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.getLast?_eq_get?", "start": [718, 1], "end": [720, 81], "traced_tactics": [{"tactic": "rw [getLast?_eq_getLast (a::l) fun., getLast_eq_get, get?_eq_get]", "annotated_tactic": ["rw [<a>getLast?_eq_getLast</a> (a::l) fun., <a>getLast_eq_get</a>, <a>get?_eq_get</a>]", [{"full_name": "List.getLast?_eq_getLast", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [556, 9], "def_end_pos": [556, 28]}, {"full_name": "List.getLast_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [712, 9], "def_end_pos": [712, 23]}, {"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\na : \u03b1\nl : List \u03b1\n\u22a2 getLast? (a :: l) = get? (a :: l) (length (a :: l) - 1)", "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", "start": [1023, 1], "end": [1026, 8], "traced_tactics": [{"tactic": "rw [ncard_le_one hs]", "annotated_tactic": ["rw [<a>ncard_le_one</a> hs]", [{"full_name": "Set.ncard_le_one", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [1018, 9], "def_end_pos": [1018, 21]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard 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\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b) \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b) \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Nat.Primrec'.of_prim", "start": [1522, 1], "end": [1540, 93], "traced_tactics": [{"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": "n : \u2115\nf : Vector \u2115 n \u2192 \u2115\nthis : \u2200 (f : \u2115 \u2192 \u2115), Nat.Primrec f \u2192 Primrec' fun v => f (Vector.head v)\nhf : Primrec f\ni : Vector \u2115 n\n\u22a2 Nat.pred ((fun m => encode (Option.map f (decode m))) (encode i)) = f i", "state_after": "no goals"}, {"tactic": "induction hf", "annotated_tactic": ["induction hf", []], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\n\u22a2 Primrec' fun v => f (Vector.head v)", "state_after": "case zero\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)\n\ncase succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case zero => exact const 0", "annotated_tactic": ["case zero => exact <a>const</a> 0", [{"full_name": "Nat.Primrec'.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 14]}]], "state_before": "case zero\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)\n\ncase succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case succ => exact succ", "annotated_tactic": ["case succ => exact <a>succ</a>", [{"full_name": "Nat.Primrec'.succ", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1363, 5], "def_end_pos": [1363, 9]}]], "state_before": "case succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case left => exact unpair\u2081 head", "annotated_tactic": ["case left => exact <a>unpair\u2081</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case right => exact unpair\u2082 head", "annotated_tactic": ["case right => exact <a>unpair\u2082</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case pair f g _ _ hf hg => exact natPair.comp\u2082 _ hf hg", "annotated_tactic": ["case pair f g _ _ hf hg => exact natPair.comp\u2082 _ hf hg", []], "state_before": "case pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case comp f g _ _ hf hg => exact hf.comp\u2081 _ hg", "annotated_tactic": ["case comp f g _ _ hf hg => exact hf.comp\u2081 _ hg", []], "state_before": "case comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case prec f g _ _ hf hg =>\n  simpa using\n    prec' (unpair\u2082 head) (hf.comp\u2081 _ (unpair\u2081 head))\n      (hg.comp\u2081 _ <|\n        natPair.comp\u2082 _ (unpair\u2081 <| tail <| tail head) (natPair.comp\u2082 _ head (tail head)))", "annotated_tactic": ["case prec f g _ _ hf hg =>\n    simpa using\n      <a>prec'</a> (<a>unpair\u2082</a> <a>head</a>) (hf.comp\u2081 _ (<a>unpair\u2081</a> <a>head</a>))\n        (hg.comp\u2081 _ <|\n          natPair.comp\u2082 _ (<a>unpair\u2081</a> <| <a>tail</a> <| <a>tail</a> <a>head</a>) (natPair.comp\u2082 _ <a>head</a> (<a>tail</a> <a>head</a>)))", [{"full_name": "Nat.Primrec'.prec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 14]}, {"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"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]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"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 prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact const 0", "annotated_tactic": ["exact <a>const</a> 0", [{"full_name": "Nat.Primrec'.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 14]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact succ", "annotated_tactic": ["exact <a>succ</a>", [{"full_name": "Nat.Primrec'.succ", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1363, 5], "def_end_pos": [1363, 9]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact unpair\u2081 head", "annotated_tactic": ["exact <a>unpair\u2081</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact unpair\u2082 head", "annotated_tactic": ["exact <a>unpair\u2082</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact natPair.comp\u2082 _ hf hg", "annotated_tactic": ["exact natPair.comp\u2082 _ hf hg", []], "state_before": "n : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf\u271d f g : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\na\u271d : Nat.Primrec g\nhf : Primrec' fun v => f (Vector.head v)\nhg : Primrec' fun v => g (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f n) (g n)) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact hf.comp\u2081 _ hg", "annotated_tactic": ["exact hf.comp\u2081 _ hg", []], "state_before": "n : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf\u271d f g : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\na\u271d : Nat.Primrec g\nhf : Primrec' fun v => f (Vector.head v)\nhg : Primrec' fun v => g (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f (g n)) (Vector.head v)", "state_after": "no goals"}, {"tactic": "simpa using\n  prec' (unpair\u2082 head) (hf.comp\u2081 _ (unpair\u2081 head))\n    (hg.comp\u2081 _ <|\n      natPair.comp\u2082 _ (unpair\u2081 <| tail <| tail head) (natPair.comp\u2082 _ head (tail head)))", "annotated_tactic": ["simpa using\n      <a>prec'</a> (<a>unpair\u2082</a> <a>head</a>) (hf.comp\u2081 _ (<a>unpair\u2081</a> <a>head</a>))\n        (hg.comp\u2081 _ <|\n          natPair.comp\u2082 _ (<a>unpair\u2081</a> <| <a>tail</a> <| <a>tail</a> <a>head</a>) (natPair.comp\u2082 _ <a>head</a> (<a>tail</a> <a>head</a>)))", [{"full_name": "Nat.Primrec'.prec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 14]}, {"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"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]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"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": "n : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf\u271d f g : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\na\u271d : Nat.Primrec g\nhf : Primrec' fun v => f (Vector.head v)\nhg : Primrec' fun v => g (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f z) (fun y IH => g (pair z (pair y IH))) n) (Vector.head v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.lookup_union_left", "start": [583, 1], "end": [584, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le_sub_one_of_lt", "start": [1267, 1], "end": [1267, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.essSup_trim", "start": [1009, 1], "end": [1012, 26], "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\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 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 essSup f (Measure.trim \u03bd hm) = essSup f \u03bd", "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\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 limsup f (Measure.ae (Measure.trim \u03bd hm)) = limsup f (Measure.ae \u03bd)"}, {"tactic": "exact limsup_trim hm hf", "annotated_tactic": ["exact <a>limsup_trim</a> hm hf", [{"full_name": "MeasureTheory.limsup_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [996, 9], "def_end_pos": [996, 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\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 limsup f (Measure.ae (Measure.trim \u03bd hm)) = limsup f (Measure.ae \u03bd)", "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.subset_append_of_subset_right", "start": [303, 1], "end": [304, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "full_name": "Set.pairwise_eq_iff_exists_eq", "start": [132, 1], "end": [134, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Multiset.map_finset_sup", "start": [1829, 1], "end": [1831, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_preserving_piFinSuccAboveEquiv", "start": [807, 1], "end": [810, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_sub", "start": [422, 1], "end": [426, 32], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, lpMeasSubgroupToLpTrim_add,\n  lpMeasSubgroupToLpTrim_neg]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>sub_eq_add_neg</a>, <a>lpMeasSubgroupToLpTrim_add</a>,\n    <a>lpMeasSubgroupToLpTrim_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": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_add", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [392, 9], "def_end_pos": [392, 35]}, {"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_neg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [409, 9], "def_end_pos": [409, 35]}]], "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": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProdFun_eq_tsum", "start": [151, 1], "end": [154, 75], "traced_tactics": [{"tactic": "simp_rw [compProdFun_tsum_left \u03ba \u03b7 a s, compProdFun_tsum_right _ \u03b7 a hs]", "annotated_tactic": ["simp_rw [<a>compProdFun_tsum_left</a> \u03ba \u03b7 a s, <a>compProdFun_tsum_right</a> _ \u03b7 a hs]", [{"full_name": "ProbabilityTheory.kernel.compProdFun_tsum_left", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [146, 9], "def_end_pos": [146, 30]}, {"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": "\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 = \u2211' (n : \u2115) (m : \u2115), compProdFun (seq \u03ba n) (seq \u03b7 m) a s", "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_eq_addHaar_image", "start": [1104, 1], "end": [1108, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integrableOn_iUnion_of_summable_integral_norm", "start": [819, 1], "end": [832, 32], "traced_tactics": [{"tactic": "refine' \u27e8AEStronglyMeasurable.iUnion fun i => (hi i).1, (lintegral_iUnion_le _ _).trans_lt _\u27e9", "annotated_tactic": ["refine' \u27e8<a>AEStronglyMeasurable.iUnion</a> fun i => (hi i).1, (<a>lintegral_iUnion_le</a> _ _).<a>trans_lt</a> _\u27e9", [{"full_name": "MeasureTheory.AEStronglyMeasurable.iUnion", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1738, 19], "def_end_pos": [1738, 25]}, {"full_name": "MeasureTheory.lintegral_iUnion_le", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1237, 9], "def_end_pos": [1237, 28]}, {"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\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\n\u22a2 IntegrableOn f (iUnion 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\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\n\u22a2 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "have B := fun b : \u03b2 => lintegral_coe_eq_integral (fun a : \u03b1 => \u2016f a\u2016\u208a) (hi b).norm", "annotated_tactic": ["have B := fun b : \u03b2 => <a>lintegral_coe_eq_integral</a> (fun a : \u03b1 => \u2016f a\u2016\u208a) (hi b).<a>norm</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": "MeasureTheory.Integrable.norm", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [706, 9], "def_end_pos": [706, 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\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\n\u22a2 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "rw [tsum_congr B]", "annotated_tactic": ["rw [<a>tsum_congr</a> B]", [{"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4"}, {"tactic": "have S' :\n  Summable fun b : \u03b2 =>\n    (\u27e8\u222b a : \u03b1 in s b, \u2016f a\u2016\u208a \u2202\u03bc, set_integral_nonneg (hs b) fun a _ => NNReal.coe_nonneg _\u27e9 :\n      NNReal) :=\n  by rw [\u2190 NNReal.summable_coe]; exact h", "annotated_tactic": ["have S' :\n    <a>Summable</a> fun b : \u03b2 =>\n      (\u27e8\u222b a : \u03b1 in s b, \u2016f a\u2016\u208a \u2202\u03bc, <a>set_integral_nonneg</a> (hs b) fun a _ => <a>NNReal.coe_nonneg</a> _\u27e9 :\n        <a>NNReal</a>) :=\n    by rw [\u2190 <a>NNReal.summable_coe</a>]; exact h", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "MeasureTheory.set_integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [766, 9], "def_end_pos": [766, 28]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}, {"full_name": "NNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [59, 5], "def_end_pos": [59, 11]}, {"full_name": "NNReal.summable_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\nS' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4"}, {"tactic": "have S'' := ENNReal.tsum_coe_eq S'.hasSum", "annotated_tactic": ["have S'' := <a>ENNReal.tsum_coe_eq</a> S'.hasSum", [{"full_name": "ENNReal.tsum_coe_eq", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [764, 19], "def_end_pos": [764, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\nS' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\nS' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }\nS'' :\n  \u2211' (a : \u03b2), \u2191{ val := \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } =\n    \u2191(\u2211' (b : \u03b2), { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) })\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4"}, {"tactic": "simp_rw [ENNReal.coe_nnreal_eq, NNReal.coe_mk, coe_nnnorm] at S''", "annotated_tactic": ["simp_rw [<a>ENNReal.coe_nnreal_eq</a>, <a>NNReal.coe_mk</a>, <a>coe_nnnorm</a>] at S''", [{"full_name": "ENNReal.coe_nnreal_eq", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [203, 9], "def_end_pos": [203, 22]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [137, 28], "def_end_pos": [137, 34]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\nS' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }\nS'' :\n  \u2211' (a : \u03b2), \u2191{ val := \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } =\n    \u2191(\u2211' (b : \u03b2), { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) })\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\nS' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }\nS'' :\n  \u2211' (a : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s a, \u2016f a\u2016 \u2202\u03bc) =\n    ENNReal.ofReal\n      \u2191(\u2211' (b : \u03b2),\n          { val := \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc, property := (_ : (fun r => 0 \u2264 r) (\u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc)) })\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4"}, {"tactic": "convert ENNReal.ofReal_lt_top", "annotated_tactic": ["convert <a>ENNReal.ofReal_lt_top</a>", [{"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\nS' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }\nS'' :\n  \u2211' (a : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s a, \u2016f a\u2016 \u2202\u03bc) =\n    ENNReal.ofReal\n      \u2191(\u2211' (b : \u03b2),\n          { val := \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc, property := (_ : (fun r => 0 \u2264 r) (\u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc)) })\n\u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190 NNReal.summable_coe]", "annotated_tactic": ["rw [\u2190 <a>NNReal.summable_coe</a>]", [{"full_name": "NNReal.summable_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \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\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 Summable fun a => \u2191{ val := \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 E\ns : \u03b2 \u2192 Set \u03b1\nhs : \u2200 (b : \u03b2), MeasurableSet (s b)\nhi : \u2200 (b : \u03b2), IntegrableOn f (s b)\nh : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc\nB : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc)\n\u22a2 Summable fun a => \u2191{ val := \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }", "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_subset_restrict_nonpos", "start": [266, 1], "end": [329, 34], "traced_tactics": [{"tactic": "have hi\u2081 : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h", "annotated_tactic": ["have hi\u2081 : <a>MeasurableSet</a> i := <a>by_contradiction</a> fun h => <a>ne_of_lt</a> hi <| s.not_measurable h", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "by_contradiction", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 25]}, {"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\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 : \u2191s i < 0\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "by_cases s \u2264[i] 0", "annotated_tactic": ["by_cases s \u2264[i] 0", []], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : restrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0\n\ncase neg\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "by_cases hn : \u2200 n : \u2115, \u00acs \u2264[i \\ \u22c3 l < n, restrictNonposSeq s i l] 0", "annotated_tactic": ["by_cases hn : \u2200 n : \u2115, \u00acs \u2264[i \\ \u22c3 l < n, <a>restrictNonposSeq</a> s i l] 0", [{"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 neg\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0\n\ncase neg\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0\n\ncase neg\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case neg\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0\n\ncase pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "set A := i \\ \u22c3 l, restrictNonposSeq s i l with hA", "annotated_tactic": ["set A := i \\ \u22c3 l, <a>restrictNonposSeq</a> s i l with hA", [{"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 pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "set bdd : \u2115 \u2192 \u2115 := fun n => findExistsOneDivLT s (i \\ \u22c3 k \u2264 n, restrictNonposSeq s i k)", "annotated_tactic": ["set bdd : \u2115 \u2192 \u2115 := fun n => <a>findExistsOneDivLT</a> s (i \\ \u22c3 k \u2264 n, <a>restrictNonposSeq</a> s i k)", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [107, 13], "def_end_pos": [107, 31]}, {"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 pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have h\u2081 : s i = s A + \u2211' l, s (restrictNonposSeq s i l) := by\n  rw [hA, \u2190 s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]\n  exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _\n  exacts [hi\u2081, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>\n    restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint]", "annotated_tactic": ["have h\u2081 : s i = s A + \u2211' l, s (<a>restrictNonposSeq</a> s i l) := by\n    rw [hA, \u2190 s.of_disjoint_iUnion_nat, <a>add_comm</a>, <a>of_add_of_diff</a>]\n    exact <a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</a> _\n    exacts [hi\u2081, <a>Set.iUnion_subset</a> fun _ => <a>restrictNonposSeq_subset</a> _, fun _ =>\n      <a>restrictNonposSeq_measurableSet</a> _, <a>restrictNonposSeq_disjoint</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]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.VectorMeasure.of_add_of_diff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [191, 9], "def_end_pos": [191, 23]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_subset", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [176, 17], "def_end_pos": [176, 41]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [217, 17], "def_end_pos": [217, 43]}]], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have h\u2082 : s A \u2264 s i := by\n  rw [h\u2081]\n  apply le_add_of_nonneg_right\n  exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))", "annotated_tactic": ["have h\u2082 : s A \u2264 s i := by\n    rw [h\u2081]\n    apply <a>le_add_of_nonneg_right</a>\n    exact <a>tsum_nonneg</a> fun n => <a>le_of_lt</a> (<a>measure_of_restrictNonposSeq</a> h _ (hn n))", [{"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}, {"full_name": "tsum_nonneg", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [185, 17], "def_end_pos": [185, 45]}]], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have h\u2083 : Tendsto (fun n => (bdd n : \u211d) + 1) atTop atTop := by\n  simp only [one_div] at h\u2083'\n  exact Summable.tendsto_atTop_of_pos h\u2083' fun n => Nat.cast_add_one_pos (bdd n)", "annotated_tactic": ["have h\u2083 : <a>Tendsto</a> (fun n => (bdd n : \u211d) + 1) <a>atTop</a> <a>atTop</a> := by\n    simp only [<a>one_div</a>] at h\u2083'\n    exact <a>Summable.tendsto_atTop_of_pos</a> h\u2083' fun n => <a>Nat.cast_add_one_pos</a> (bdd n)", [{"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": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Summable.tendsto_atTop_of_pos", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [282, 9], "def_end_pos": [282, 38]}, {"full_name": "Nat.cast_add_one_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [62, 9], "def_end_pos": [62, 25]}]], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have h\u2084 : Tendsto (fun n => (bdd n : \u211d)) atTop atTop := by\n  convert atTop.tendsto_atTop_add_const_right (-1) h\u2083; simp", "annotated_tactic": ["have h\u2084 : <a>Tendsto</a> (fun n => (bdd n : \u211d)) <a>atTop</a> <a>atTop</a> := by\n    convert atTop.tendsto_atTop_add_const_right (-1) h\u2083; simp", [{"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]}]], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have A_meas : MeasurableSet A :=\n  hi\u2081.diff (MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _)", "annotated_tactic": ["have A_meas : <a>MeasurableSet</a> A :=\n    hi\u2081.diff (<a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</a> _)", [{"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": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}]], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "refine' \u27e8A, A_meas, Set.diff_subset _ _, _, h\u2082.trans_lt hi\u27e9", "annotated_tactic": ["refine' \u27e8A, A_meas, <a>Set.diff_subset</a> _ _, _, h\u2082.trans_lt hi\u27e9", [{"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\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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\n\u22a2 restrict s A \u2264 restrict 0 A"}, {"tactic": "by_contra hnn", "annotated_tactic": ["by_contra hnn", []], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\n\u22a2 restrict s A \u2264 restrict 0 A", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nhnn : \u00acrestrict s A \u2264 restrict 0 A\n\u22a2 False"}, {"tactic": "rw [restrict_le_restrict_iff _ _ A_meas] at hnn", "annotated_tactic": ["rw [<a>restrict_le_restrict_iff</a> _ _ A_meas] at hnn", [{"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 pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nhnn : \u00acrestrict s A \u2264 restrict 0 A\n\u22a2 False", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nhnn : \u00ac\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 A \u2192 \u2191s j \u2264 \u21910 j\n\u22a2 False"}, {"tactic": "push_neg at hnn", "annotated_tactic": ["push_neg at hnn", []], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nhnn : \u00ac\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 A \u2192 \u2191s j \u2264 \u21910 j\n\u22a2 False", "state_after": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nhnn : Exists fun \u2983j\u2984 => MeasurableSet j \u2227 j \u2286 A \u2227 \u21910 j < \u2191s j\n\u22a2 False"}, {"tactic": "obtain \u27e8E, hE\u2081, hE\u2082, hE\u2083\u27e9 := hnn", "annotated_tactic": ["obtain \u27e8E, hE\u2081, hE\u2082, hE\u2083\u27e9 := hnn", []], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nhnn : Exists fun \u2983j\u2984 => MeasurableSet j \u2227 j \u2286 A \u2227 \u21910 j < \u2191s j\n\u22a2 False", "state_after": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\n\u22a2 False"}, {"tactic": "obtain \u27e8k, hk\u2081, hk\u2082\u27e9 := this", "annotated_tactic": ["obtain \u27e8k, hk\u2081, hk\u2082\u27e9 := this", []], "state_before": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nthis : \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E\n\u22a2 False", "state_after": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\n\u22a2 False"}, {"tactic": "have hA' : A \u2286 i \\ \u22c3 l \u2264 k, restrictNonposSeq s i l := by\n  apply Set.diff_subset_diff_right\n  intro x; simp only [Set.mem_iUnion]\n  rintro \u27e8n, _, hn\u2082\u27e9\n  exact \u27e8n, hn\u2082\u27e9", "annotated_tactic": ["have hA' : A \u2286 i \\ \u22c3 l \u2264 k, <a>restrictNonposSeq</a> s i l := by\n    apply <a>Set.diff_subset_diff_right</a>\n    intro x; simp only [<a>Set.mem_iUnion</a>]\n    rintro \u27e8n, _, hn\u2082\u27e9\n    exact \u27e8n, hn\u2082\u27e9", [{"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]}, {"full_name": "Set.diff_subset_diff_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1912, 9], "def_end_pos": [1912, 31]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\n\u22a2 False", "state_after": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 False"}, {"tactic": "refine'\n  findExistsOneDivLT_min (hn' k) (Nat.sub_lt hk\u2081 Nat.zero_lt_one)\n    \u27e8E, Set.Subset.trans hE\u2082 hA', hE\u2081, _\u27e9", "annotated_tactic": ["refine'\n    <a>findExistsOneDivLT_min</a> (hn' k) (<a>Nat.sub_lt</a> hk\u2081 <a>Nat.zero_lt_one</a>)\n      \u27e8E, <a>Set.Subset.trans</a> hE\u2082 hA', hE\u2081, _\u27e9", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT_min", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [115, 17], "def_end_pos": [115, 39]}, {"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": "Nat.zero_lt_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [401, 19], "def_end_pos": [401, 30]}, {"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}]], "state_before": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 False", "state_after": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 1 /\n      (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n              (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -\n            1) +\n        1) <\n    \u2191s E"}, {"tactic": "convert hk\u2082", "annotated_tactic": ["convert hk\u2082", []], "state_before": "case pos.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 1 /\n      (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n              (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -\n            1) +\n        1) <\n    \u2191s E", "state_after": "case h.e'_3.h.e'_6\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 \u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -\n          1) +\n      1 =\n    \u2191(bdd k)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h.e'_3.h.e'_6\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 \u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -\n          1) +\n      1 =\n    \u2191(bdd k)", "state_after": "case h.e'_3.h.e'_6\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 MeasureTheory.SignedMeasure.findExistsOneDivLT s\n          (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -\n        1 +\n      1 =\n    bdd k"}, {"tactic": "exact tsub_add_cancel_of_le hk\u2081", "annotated_tactic": ["exact <a>tsub_add_cancel_of_le</a> hk\u2081", [{"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.e'_3.h.e'_6\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nhA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l\n\u22a2 MeasureTheory.SignedMeasure.findExistsOneDivLT s\n          (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -\n        1 +\n      1 =\n    bdd k", "state_after": "no goals"}, {"tactic": "exact \u27e8i, hi\u2081, Set.Subset.refl _, h, hi\u27e9", "annotated_tactic": ["exact \u27e8i, hi\u2081, <a>Set.Subset.refl</a> _, h, hi\u27e9", [{"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "case pos\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : restrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "no goals"}, {"tactic": "exact exists_subset_restrict_nonpos' hi\u2081 hi hn", "annotated_tactic": ["exact <a>exists_subset_restrict_nonpos'</a> hi\u2081 hi hn", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos'", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [224, 17], "def_end_pos": [224, 47]}]], "state_before": "case neg\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\n\u22a2 \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nn : \u2115\n\u22a2 \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "ext l", "annotated_tactic": ["ext l", []], "state_before": "case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nn : \u2115\n\u22a2 (fun l => \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) = fun l =>\n    \u22c3 (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nn l : \u2115\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194\n    x\u271d \u2208 \u22c3 (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l"}, {"tactic": "simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>Set.mem_iUnion</a>, <a>and_congr_left_iff</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": "and_congr_left_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [222, 17], "def_end_pos": [222, 35]}]], "state_before": "case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nn l : \u2115\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194\n    x\u271d \u2208 \u22c3 (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nn l : \u2115\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192 (l \u2264 n \u2194 l < n + 1)"}, {"tactic": "exact fun _ => Nat.lt_succ_iff.symm", "annotated_tactic": ["exact fun _ => Nat.lt_succ_iff.symm", []], "state_before": "case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nn l : \u2115\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192 (l \u2264 n \u2194 l < n + 1)", "state_after": "no goals"}, {"tactic": "rw [hA, \u2190 s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]", "annotated_tactic": ["rw [hA, \u2190 s.of_disjoint_iUnion_nat, <a>add_comm</a>, <a>of_add_of_diff</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.VectorMeasure.of_add_of_diff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [191, 9], "def_end_pos": [191, 23]}]], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case hA\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 MeasurableSet (\u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase hB\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 MeasurableSet i\n\ncase h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i\n\ncase hf\u2081\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase hf\u2082\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</a> _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}]], "state_before": "case hA\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 MeasurableSet (\u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase hB\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 MeasurableSet i\n\ncase h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i\n\ncase hf\u2081\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase hf\u2082\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case hB\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 MeasurableSet i\n\ncase h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i\n\ncase hf\u2081\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase hf\u2082\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "exacts [hi\u2081, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>\n  restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint]", "annotated_tactic": ["exacts [hi\u2081, <a>Set.iUnion_subset</a> fun _ => <a>restrictNonposSeq_subset</a> _, fun _ =>\n      <a>restrictNonposSeq_measurableSet</a> _, <a>restrictNonposSeq_disjoint</a>]", [{"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_subset", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [176, 17], "def_end_pos": [176, 41]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [217, 17], "def_end_pos": [217, 43]}]], "state_before": "case hB\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 MeasurableSet i\n\ncase h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i\n\ncase hf\u2081\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase hf\u2082\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "no goals"}, {"tactic": "rw [h\u2081]", "annotated_tactic": ["rw [h\u2081]", []], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s A \u2264 \u2191s 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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s A \u2264 \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "apply le_add_of_nonneg_right", "annotated_tactic": ["apply <a>le_add_of_nonneg_right</a>", [{"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [391, 15], "def_end_pos": [391, 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s A \u2264 \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 0 \u2264 \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))", "annotated_tactic": ["exact <a>tsum_nonneg</a> fun n => <a>le_of_lt</a> (<a>measure_of_restrictNonposSeq</a> h _ (hn n))", [{"full_name": "tsum_nonneg", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [185, 17], "def_end_pos": [185, 45]}]], "state_before": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 0 \u2264 \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "no goals"}, {"tactic": "have : Summable fun l => s (restrictNonposSeq s i l) :=\n  HasSum.summable\n    (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint)", "annotated_tactic": ["have : <a>Summable</a> fun l => s (<a>restrictNonposSeq</a> s i l) :=\n      <a>HasSum.summable</a>\n        (s.m_iUnion (fun _ => <a>restrictNonposSeq_measurableSet</a> _) <a>restrictNonposSeq_disjoint</a>)", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"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]}, {"full_name": "HasSum.summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 24]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [217, 17], "def_end_pos": [217, 43]}]], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\n\u22a2 Summable fun n => 1 / (\u2191(bdd n) + 1)", "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nthis : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Summable fun n => 1 / (\u2191(bdd n) + 1)"}, {"tactic": "refine'\n  summable_of_nonneg_of_le (fun n => _) (fun n => _)\n    (Summable.comp_injective this Nat.succ_injective)", "annotated_tactic": ["refine'\n      <a>summable_of_nonneg_of_le</a> (fun n => _) (fun n => _)\n        (<a>Summable.comp_injective</a> this <a>Nat.succ_injective</a>)", [{"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": "Summable.comp_injective", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [1182, 9], "def_end_pos": [1182, 32]}, {"full_name": "Nat.succ_injective", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 23]}]], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nthis : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Summable fun n => 1 / (\u2191(bdd n) + 1)", "state_after": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nthis : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nn : \u2115\n\u22a2 0 \u2264 1 / (\u2191(bdd n) + 1)\n\ncase 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nthis : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nn : \u2115\n\u22a2 1 / (\u2191(bdd n) + 1) \u2264 ((fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)) \u2218 Nat.succ) n"}, {"tactic": "exact le_of_lt Nat.one_div_pos_of_nat", "annotated_tactic": ["exact <a>le_of_lt</a> <a>Nat.one_div_pos_of_nat</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.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 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nthis : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nn : \u2115\n\u22a2 0 \u2264 1 / (\u2191(bdd n) + 1)", "state_after": "no goals"}, {"tactic": "exact le_of_lt (restrictNonposSeq_lt n (hn' n))", "annotated_tactic": ["exact <a>le_of_lt</a> (<a>restrictNonposSeq_lt</a> n (hn' n))", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_lt", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [179, 17], "def_end_pos": [179, 37]}]], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nthis : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nn : \u2115\n\u22a2 1 / (\u2191(bdd n) + 1) \u2264 ((fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)) \u2218 Nat.succ) n", "state_after": "no goals"}, {"tactic": "simp only [one_div] at h\u2083'", "annotated_tactic": ["simp only [<a>one_div</a>] at h\u2083'", [{"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\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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\n\u22a2 Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop", "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' :\n  Summable fun n =>\n    (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +\n        1)\u207b\u00b9\n\u22a2 Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop"}, {"tactic": "exact Summable.tendsto_atTop_of_pos h\u2083' fun n => Nat.cast_add_one_pos (bdd n)", "annotated_tactic": ["exact <a>Summable.tendsto_atTop_of_pos</a> h\u2083' fun n => <a>Nat.cast_add_one_pos</a> (bdd n)", [{"full_name": "Summable.tendsto_atTop_of_pos", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [282, 9], "def_end_pos": [282, 38]}, {"full_name": "Nat.cast_add_one_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [62, 9], "def_end_pos": [62, 25]}]], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' :\n  Summable fun n =>\n    (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n            (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +\n        1)\u207b\u00b9\n\u22a2 Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop", "state_after": "no goals"}, {"tactic": "convert atTop.tendsto_atTop_add_const_right (-1) h\u2083", "annotated_tactic": ["convert atTop.tendsto_atTop_add_const_right (-1) h\u2083", []], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\n\u22a2 Tendsto (fun n => \u2191(bdd n)) atTop atTop", "state_after": "case h.e'_3.h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nx\u271d : \u2115\n\u22a2 \u2191(bdd x\u271d) = \u2191(bdd x\u271d) + 1 + -1"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3.h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nx\u271d : \u2115\n\u22a2 \u2191(bdd x\u271d) = \u2191(bdd x\u271d) + 1 + -1", "state_after": "no goals"}, {"tactic": "rw [tendsto_atTop_atTop] at h\u2084", "annotated_tactic": ["rw [<a>tendsto_atTop_atTop</a>] at h\u2084", [{"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\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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\n\u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E", "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\n\u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E"}, {"tactic": "obtain \u27e8k, hk\u27e9 := h\u2084 (max (1 / s E + 1) 1)", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := h\u2084 (<a>max</a> (1 / s E + 1) 1)", [{"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\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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\n\u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E", "state_after": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E"}, {"tactic": "refine' \u27e8k, _, _\u27e9", "annotated_tactic": ["refine' \u27e8k, _, _\u27e9", []], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E", "state_after": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 1 \u2264 bdd k\n\ncase 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 1 / \u2191(bdd k) < \u2191s E"}, {"tactic": "have hle := le_of_max_le_right (hk k le_rfl)", "annotated_tactic": ["have hle := <a>le_of_max_le_right</a> (hk k <a>le_rfl</a>)", [{"full_name": "le_of_max_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [290, 9], "def_end_pos": [290, 27]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 1 \u2264 bdd k", "state_after": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\nhle : 1 \u2264 \u2191(bdd k)\n\u22a2 1 \u2264 bdd k"}, {"tactic": "norm_cast at hle", "annotated_tactic": ["norm_cast at hle", []], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\nhle : 1 \u2264 \u2191(bdd k)\n\u22a2 1 \u2264 bdd k", "state_after": "no goals"}, {"tactic": "have : 1 / s E < bdd k := by\n  linarith only [le_of_max_le_left (hk k le_rfl)]", "annotated_tactic": ["have : 1 / s E < bdd k := by\n        linarith only [<a>le_of_max_le_left</a> (hk k <a>le_rfl</a>)]", [{"full_name": "le_of_max_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [286, 9], "def_end_pos": [286, 26]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 1 / \u2191(bdd k) < \u2191s E", "state_after": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\nthis : 1 / \u2191s E < \u2191(bdd k)\n\u22a2 1 / \u2191(bdd k) < \u2191s E"}, {"tactic": "rw [one_div] at this \u22a2", "annotated_tactic": ["rw [<a>one_div</a>] at this \u22a2", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\nthis : 1 / \u2191s E < \u2191(bdd k)\n\u22a2 1 / \u2191(bdd k) < \u2191s E", "state_after": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\nthis : (\u2191s E)\u207b\u00b9 < \u2191(bdd k)\n\u22a2 (\u2191(bdd k))\u207b\u00b9 < \u2191s E"}, {"tactic": "rwa [inv_lt (lt_trans (inv_pos.2 hE\u2083) this) hE\u2083]", "annotated_tactic": ["rwa [<a>inv_lt</a> (<a>lt_trans</a> (<a>inv_pos</a>.2 hE\u2083) this) hE\u2083]", [{"full_name": "inv_lt", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [285, 9], "def_end_pos": [285, 15]}, {"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "case 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\nthis : (\u2191s E)\u207b\u00b9 < \u2191(bdd k)\n\u22a2 (\u2191(bdd k))\u207b\u00b9 < \u2191s E", "state_after": "no goals"}, {"tactic": "linarith only [le_of_max_le_left (hk k le_rfl)]", "annotated_tactic": ["linarith only [<a>le_of_max_le_left</a> (hk k <a>le_rfl</a>)]", [{"full_name": "le_of_max_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [286, 9], "def_end_pos": [286, 26]}, {"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\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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a)\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a)\n\u22a2 1 / \u2191s E < \u2191(bdd k)", "state_after": "no goals"}, {"tactic": "apply Set.diff_subset_diff_right", "annotated_tactic": ["apply <a>Set.diff_subset_diff_right</a>", [{"full_name": "Set.diff_subset_diff_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1912, 9], "def_end_pos": [1912, 31]}]], "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 : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\n\u22a2 A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\n\u22a2 \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2286\n    \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\n\u22a2 \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2286\n    \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nx : \u03b1\n\u22a2 x \u2208 \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192\n    x \u2208 \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l"}, {"tactic": "simp only [Set.mem_iUnion]", "annotated_tactic": ["simp only [<a>Set.mem_iUnion</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nx : \u03b1\n\u22a2 x \u2208 \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192\n    x \u2208 \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nx : \u03b1\n\u22a2 (\u2203 i_1 i_2, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) \u2192\n    \u2203 i_1, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1"}, {"tactic": "rintro \u27e8n, _, hn\u2082\u27e9", "annotated_tactic": ["rintro \u27e8n, _, hn\u2082\u27e9", []], "state_before": "case h\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\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nx : \u03b1\n\u22a2 (\u2203 i_1 i_2, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) \u2192\n    \u2203 i_1, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1", "state_after": "case h.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nx : \u03b1\nn : \u2115\nw\u271d : n \u2264 k\nhn\u2082 : x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i n\n\u22a2 \u2203 i_1, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1"}, {"tactic": "exact \u27e8n, hn\u2082\u27e9", "annotated_tactic": ["exact \u27e8n, hn\u2082\u27e9", []], "state_before": "case h.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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi : \u2191s i < 0\nhi\u2081 : MeasurableSet i\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nA : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nhA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l\nbdd : \u2115 \u2192 \u2115 :=\n  fun n =>\n    MeasureTheory.SignedMeasure.findExistsOneDivLT s\n      (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)\nhn' :\n  \u2200 (n : \u2115),\n    \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n        restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2082 : \u2191s A \u2264 \u2191s i\nh\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1)\nh\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop\nh\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop\nA_meas : MeasurableSet A\nE : Set \u03b1\nhE\u2081 : MeasurableSet E\nhE\u2082 : E \u2286 A\nhE\u2083 : \u21910 E < \u2191s E\nk : \u2115\nhk\u2081 : 1 \u2264 bdd k\nhk\u2082 : 1 / \u2191(bdd k) < \u2191s E\nx : \u03b1\nn : \u2115\nw\u271d : n \u2264 k\nhn\u2082 : x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i n\n\u22a2 \u2203 i_1, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.finSuccEquiv_coeff_coeff", "start": [365, 1], "end": [386, 38], "traced_tactics": [{"tactic": "induction' f using MvPolynomial.induction_on' with j r p q hp hq generalizing i m", "annotated_tactic": ["induction' f using <a>MvPolynomial.induction_on'</a> with j r p q hp hq generalizing i m", [{"full_name": "MvPolynomial.induction_on'", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [418, 9], "def_end_pos": [418, 22]}]], "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\nm : Fin n \u2192\u2080 \u2115\nf : MvPolynomial (Fin (n + 1)) R\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) = coeff (cons i m) f", "state_after": "case h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (\u2191(monomial j) r)) i) = coeff (cons i m) (\u2191(monomial j) r)\n\ncase h2\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\np q : MvPolynomial (Fin (n + 1)) R\nhp : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (\u2191(monomial j) r)) i) = coeff (cons i m) (\u2191(monomial j) r)\n\ncase h2\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\np q : MvPolynomial (Fin (n + 1)) R\nhp : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)", "state_after": "case h2\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\np q : MvPolynomial (Fin (n + 1)) R\nhp : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)\n\ncase h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (\u2191(monomial j) r)) i) = coeff (cons i m) (\u2191(monomial j) r)"}, {"tactic": "simp only [finSuccEquiv_apply, coe_eval\u2082Hom, eval\u2082_monomial, RingHom.coe_comp, prod_pow,\n  Polynomial.coeff_C_mul, coeff_C_mul, coeff_monomial, Fin.prod_univ_succ, Fin.cases_zero,\n  Fin.cases_succ, \u2190 map_prod, \u2190 RingHom.map_pow, Function.comp_apply]", "annotated_tactic": ["simp only [<a>finSuccEquiv_apply</a>, <a>coe_eval\u2082Hom</a>, <a>eval\u2082_monomial</a>, <a>RingHom.coe_comp</a>, <a>prod_pow</a>,\n    <a>Polynomial.coeff_C_mul</a>, <a>coeff_C_mul</a>, <a>coeff_monomial</a>, <a>Fin.prod_univ_succ</a>, <a>Fin.cases_zero</a>,\n    <a>Fin.cases_succ</a>, \u2190 <a>map_prod</a>, \u2190 <a>RingHom.map_pow</a>, <a>Function.comp_apply</a>]", [{"full_name": "MvPolynomial.finSuccEquiv_apply", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [332, 9], "def_end_pos": [332, 27]}, {"full_name": "MvPolynomial.coe_eval\u2082Hom", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1051, 9], "def_end_pos": [1051, 21]}, {"full_name": "MvPolynomial.eval\u2082_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [983, 9], "def_end_pos": [983, 23]}, {"full_name": "RingHom.coe_comp", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [668, 9], "def_end_pos": [668, 17]}, {"full_name": "Finsupp.prod_pow", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Polynomial.coeff_C_mul", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [161, 9], "def_end_pos": [161, 20]}, {"full_name": "MvPolynomial.coeff_C_mul", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [702, 9], "def_end_pos": [702, 20]}, {"full_name": "MvPolynomial.coeff_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 23]}, {"full_name": "Fin.prod_univ_succ", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}, {"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": "Fin.cases_succ", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [620, 17], "def_end_pos": [620, 27]}, {"full_name": "map_prod", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [205, 9], "def_end_pos": [205, 17]}, {"full_name": "RingHom.map_pow", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [153, 19], "def_end_pos": [153, 34]}, {"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 h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (\u2191(monomial j) r)) i) = coeff (cons i m) (\u2191(monomial j) r)", "state_after": "case h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 r * coeff m (Polynomial.coeff (Polynomial.X ^ \u2191j 0 * \u2191Polynomial.C (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x))) i) =\n    if j = cons i m then r else 0"}, {"tactic": "rw [\u2190 mul_boole, mul_comm (Polynomial.X ^ j 0), Polynomial.coeff_C_mul_X_pow]", "annotated_tactic": ["rw [\u2190 <a>mul_boole</a>, <a>mul_comm</a> (<a>Polynomial.X</a> ^ j 0), <a>Polynomial.coeff_C_mul_X_pow</a>]", [{"full_name": "mul_boole", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [217, 9], "def_end_pos": [217, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "Polynomial.coeff_C_mul_X_pow", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [149, 9], "def_end_pos": [149, 26]}]], "state_before": "case h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 r * coeff m (Polynomial.coeff (Polynomial.X ^ \u2191j 0 * \u2191Polynomial.C (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x))) i) =\n    if j = cons i m then r else 0", "state_after": "case h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 r * coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = r * if j = cons i m then 1 else 0"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h1\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 r * coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = r * if j = cons i m then 1 else 0", "state_after": "case h1.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 : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = if j = cons i m then 1 else 0"}, {"tactic": "obtain rfl | hjmi := eq_or_ne j (m.cons i)", "annotated_tactic": ["obtain rfl | hjmi := <a>eq_or_ne</a> j (m.cons i)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case h1.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 : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = if j = cons i m then 1 else 0", "state_after": "case h1.e_a.inl\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (if i = \u2191(cons i m) 0 then \u220f x : Fin n, X x ^ \u2191(cons i m) (Fin.succ x) else 0) =\n    if cons i m = cons i m then 1 else 0\n\ncase h1.e_a.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nhjmi : j \u2260 cons i m\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = if j = cons i m then 1 else 0"}, {"tactic": "simp only [(finSuccEquiv R n).map_add, Polynomial.coeff_add, coeff_add, hp, hq]", "annotated_tactic": ["simp only [(<a>finSuccEquiv</a> R n).<a>map_add</a>, <a>Polynomial.coeff_add</a>, <a>coeff_add</a>, hp, hq]", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "AlgEquiv.map_add", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [214, 19], "def_end_pos": [214, 26]}, {"full_name": "Polynomial.coeff_add", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "MvPolynomial.coeff_add", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [614, 9], "def_end_pos": [614, 18]}]], "state_before": "case h2\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\np q : MvPolynomial (Fin (n + 1)) R\nhp : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)", "state_after": "no goals"}, {"tactic": "simpa only [cons_zero, cons_succ, if_pos rfl, monomial_eq, C_1, one_mul, prod_pow] using\n  coeff_monomial m m (1 : R)", "annotated_tactic": ["simpa only [<a>cons_zero</a>, <a>cons_succ</a>, <a>if_pos</a> <a>rfl</a>, <a>monomial_eq</a>, <a>C_1</a>, <a>one_mul</a>, <a>prod_pow</a>] using\n      <a>coeff_monomial</a> m m (1 : R)", [{"full_name": "Finsupp.cons_zero", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [44, 9], "def_end_pos": [44, 18]}, {"full_name": "Finsupp.cons_succ", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"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]}, {"full_name": "MvPolynomial.monomial_eq", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 20]}, {"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]}, {"full_name": "Finsupp.prod_pow", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "MvPolynomial.coeff_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 23]}]], "state_before": "case h1.e_a.inl\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\n\u22a2 coeff m (if i = \u2191(cons i m) 0 then \u220f x : Fin n, X x ^ \u2191(cons i m) (Fin.succ x) else 0) =\n    if cons i m = cons i m then 1 else 0", "state_after": "no goals"}, {"tactic": "simp only [hjmi, if_false]", "annotated_tactic": ["simp only [hjmi, <a>if_false</a>]", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case h1.e_a.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nhjmi : j \u2260 cons i m\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = if j = cons i m then 1 else 0", "state_after": "case h1.e_a.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nhjmi : j \u2260 cons i m\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0"}, {"tactic": "obtain hij | rfl := ne_or_eq i (j 0)", "annotated_tactic": ["obtain hij | rfl := <a>ne_or_eq</a> i (j 0)", [{"full_name": "ne_or_eq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "case h1.e_a.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nhjmi : j \u2260 cons i m\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0", "state_after": "case h1.e_a.inr.inl\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nhjmi : j \u2260 cons i m\nhij : i \u2260 \u2191j 0\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0\n\ncase h1.e_a.inr.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\n\u22a2 coeff m (if \u2191j 0 = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0"}, {"tactic": "simp only [eq_self_iff_true, if_true]", "annotated_tactic": ["simp only [<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": "case h1.e_a.inr.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\n\u22a2 coeff m (if \u2191j 0 = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0", "state_after": "case h1.e_a.inr.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\n\u22a2 coeff m (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x)) = 0"}, {"tactic": "have hmj : m \u2260 j.tail := by\n  rintro rfl\n  rw [cons_tail] at hjmi\n  contradiction", "annotated_tactic": ["have hmj : m \u2260 j.tail := by\n      rintro rfl\n      rw [<a>cons_tail</a>] at hjmi\n      contradiction", [{"full_name": "Finsupp.cons_tail", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [60, 9], "def_end_pos": [60, 18]}]], "state_before": "case h1.e_a.inr.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\n\u22a2 coeff m (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x)) = 0", "state_after": "case h1.e_a.inr.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\nhmj : m \u2260 tail j\n\u22a2 coeff m (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x)) = 0"}, {"tactic": "simpa only [monomial_eq, C_1, one_mul, prod_pow, Finsupp.tail_apply, if_neg hmj.symm] using\n  coeff_monomial m j.tail (1 : R)", "annotated_tactic": ["simpa only [<a>monomial_eq</a>, <a>C_1</a>, <a>one_mul</a>, <a>prod_pow</a>, <a>Finsupp.tail_apply</a>, <a>if_neg</a> hmj.symm] using\n      <a>coeff_monomial</a> m j.tail (1 : R)", [{"full_name": "MvPolynomial.monomial_eq", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 20]}, {"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]}, {"full_name": "Finsupp.prod_pow", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Finsupp.tail_apply", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [39, 9], "def_end_pos": [39, 19]}, {"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": "MvPolynomial.coeff_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 23]}]], "state_before": "case h1.e_a.inr.inr\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\nhmj : m \u2260 tail j\n\u22a2 coeff m (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x)) = 0", "state_after": "no goals"}, {"tactic": "simp only [hij, if_false, coeff_zero]", "annotated_tactic": ["simp only [hij, <a>if_false</a>, <a>coeff_zero</a>]", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "MvPolynomial.coeff_zero", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [625, 9], "def_end_pos": [625, 19]}]], "state_before": "case h1.e_a.inr.inl\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\nm\u271d : Fin n \u2192\u2080 \u2115\ni\u271d : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nhjmi : j \u2260 cons i m\nhij : i \u2260 \u2191j 0\n\u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro 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\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nm\u271d : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nm : Fin n \u2192\u2080 \u2115\nhjmi : j \u2260 cons (\u2191j 0) m\n\u22a2 m \u2260 tail j", "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\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nhjmi : j \u2260 cons (\u2191j 0) (tail j)\n\u22a2 False"}, {"tactic": "rw [cons_tail] at hjmi", "annotated_tactic": ["rw [<a>cons_tail</a>] at hjmi", [{"full_name": "Finsupp.cons_tail", "def_path": "Mathlib/Data/Finsupp/Fin.lean", "def_pos": [60, 9], "def_end_pos": [60, 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\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nhjmi : j \u2260 cons (\u2191j 0) (tail j)\n\u22a2 False", "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\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nhjmi : j \u2260 j\n\u22a2 False"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "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\nm : Fin n \u2192\u2080 \u2115\ni : \u2115\nj : Fin (n + 1) \u2192\u2080 \u2115\nr : R\nhjmi : j \u2260 j\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.measure_Iic", "start": [423, 1], "end": [427, 56], "traced_tactics": [{"tactic": "refine' tendsto_nhds_unique (tendsto_measure_Ioc_atBot _ _) _", "annotated_tactic": ["refine' <a>tendsto_nhds_unique</a> (<a>tendsto_measure_Ioc_atBot</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_measure_Ioc_atBot", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2768, 9], "def_end_pos": [2768, 34]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atBot (\ud835\udcdd l)\nx : \u211d\n\u22a2 \u2191\u2191(StieltjesFunction.measure f) (Iic x) = ofReal (\u2191f x - l)", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atBot (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => \u2191\u2191(StieltjesFunction.measure f) (Ioc x_1 x)) atBot (\ud835\udcdd (ofReal (\u2191f x - l)))"}, {"tactic": "simp_rw [measure_Ioc]", "annotated_tactic": ["simp_rw [<a>measure_Ioc</a>]", [{"full_name": "StieltjesFunction.measure_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [352, 9], "def_end_pos": [352, 20]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atBot (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => \u2191\u2191(StieltjesFunction.measure f) (Ioc x_1 x)) atBot (\ud835\udcdd (ofReal (\u2191f x - l)))", "state_after": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atBot (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => ofReal (\u2191f x - \u2191f x_1)) atBot (\ud835\udcdd (ofReal (\u2191f x - l)))"}, {"tactic": "exact ENNReal.tendsto_ofReal (Tendsto.const_sub _ hf)", "annotated_tactic": ["exact <a>ENNReal.tendsto_ofReal</a> (<a>Tendsto.const_sub</a> _ hf)", [{"full_name": "ENNReal.tendsto_ofReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [108, 9], "def_end_pos": [108, 23]}, {"full_name": "Filter.Tendsto.const_sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1088, 15], "def_end_pos": [1088, 24]}]], "state_before": "f : StieltjesFunction\nl : \u211d\nhf : Tendsto (\u2191f) atBot (\ud835\udcdd l)\nx : \u211d\n\u22a2 Tendsto (fun x_1 => ofReal (\u2191f x - \u2191f x_1)) atBot (\ud835\udcdd (ofReal (\u2191f x - l)))", "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'_isMetric", "start": [323, 1], "end": [333, 52], "traced_tactics": [{"tactic": "rintro s t \u27e8r, r0, hr\u27e9", "annotated_tactic": ["rintro s t \u27e8r, r0, hr\u27e9", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\n\u22a2 IsMetric (mkMetric' m)", "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\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 \u2191(mkMetric' m) (s \u222a t) = \u2191(mkMetric' m) s + \u2191(mkMetric' m) t"}, {"tactic": "refine' tendsto_nhds_unique_of_eventuallyEq\n  (mkMetric'.tendsto_pre _ _) ((mkMetric'.tendsto_pre _ _).add (mkMetric'.tendsto_pre _ _)) _", "annotated_tactic": ["refine' <a>tendsto_nhds_unique_of_eventuallyEq</a>\n    (<a>mkMetric'.tendsto_pre</a> _ _) ((<a>mkMetric'.tendsto_pre</a> _ _).<a>add</a> (<a>mkMetric'.tendsto_pre</a> _ _)) _", [{"full_name": "tendsto_nhds_unique_of_eventuallyEq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1004, 9], "def_end_pos": [1004, 44]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [288, 9], "def_end_pos": [288, 20]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [288, 9], "def_end_pos": [288, 20]}, {"full_name": "Filter.Tendsto.add", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [118, 3], "def_end_pos": [118, 14]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [288, 9], "def_end_pos": [288, 20]}]], "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\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 \u2191(mkMetric' m) (s \u222a t) = \u2191(mkMetric' m) s + \u2191(mkMetric' m) t", "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\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t"}, {"tactic": "rw [\u2190 pos_iff_ne_zero] at r0", "annotated_tactic": ["rw [\u2190 <a>pos_iff_ne_zero</a>] at r0", [{"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 intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t", "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\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t"}, {"tactic": "filter_upwards [Ioo_mem_nhdsWithin_Ioi \u27e8le_rfl, r0\u27e9]", "annotated_tactic": ["filter_upwards [<a>Ioo_mem_nhdsWithin_Ioi</a> \u27e8<a>le_rfl</a>, r0\u27e9]", [{"full_name": "Ioo_mem_nhdsWithin_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [427, 9], "def_end_pos": [427, 31]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "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\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t", "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 : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 \u2200 (a : \u211d\u22650\u221e), a \u2208 Ioo 0 r \u2192 \u2191(mkMetric'.pre m a) (s \u222a t) = \u2191(mkMetric'.pre m a) s + \u2191(mkMetric'.pre m a) t"}, {"tactic": "rintro \u03b5 \u27e8_, \u03b5r\u27e9", "annotated_tactic": ["rintro \u03b5 \u27e8_, \u03b5r\u27e9", []], "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 : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 \u2200 (a : \u211d\u22650\u221e), a \u2208 Ioo 0 r \u2192 \u2191(mkMetric'.pre m a) (s \u222a t) = \u2191(mkMetric'.pre m a) s + \u2191(mkMetric'.pre m a) t", "state_after": "case h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\n\u22a2 \u2191(mkMetric'.pre m \u03b5) (s \u222a t) = \u2191(mkMetric'.pre m \u03b5) s + \u2191(mkMetric'.pre m \u03b5) t"}, {"tactic": "refine' boundedBy_union_of_top_of_nonempty_inter _", "annotated_tactic": ["refine' <a>boundedBy_union_of_top_of_nonempty_inter</a> _", [{"full_name": "MeasureTheory.OuterMeasure.boundedBy_union_of_top_of_nonempty_inter", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [916, 9], "def_end_pos": [916, 49]}]], "state_before": "case h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\n\u22a2 \u2191(mkMetric'.pre m \u03b5) (s \u222a t) = \u2191(mkMetric'.pre m \u03b5) s + \u2191(mkMetric'.pre m \u03b5) t", "state_after": "case h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\n\u22a2 \u2200 (u : Set X), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 extend (fun s x => m s) u = \u22a4"}, {"tactic": "rintro u \u27e8x, hxs, hxu\u27e9 \u27e8y, hyt, hyu\u27e9", "annotated_tactic": ["rintro u \u27e8x, hxs, hxu\u27e9 \u27e8y, hyt, hyu\u27e9", []], "state_before": "case h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\n\u22a2 \u2200 (u : Set X), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 extend (fun s x => m s) u = \u22a4", "state_after": "case h.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\nu : Set X\nx : X\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : X\nhyt : y \u2208 t\nhyu : y \u2208 u\n\u22a2 extend (fun s x => m s) u = \u22a4"}, {"tactic": "have : \u03b5 < diam u := \u03b5r.trans_le ((hr x hxs y hyt).trans <| edist_le_diam_of_mem hxu hyu)", "annotated_tactic": ["have : \u03b5 < <a>diam</a> u := \u03b5r.trans_le ((hr x hxs y hyt).<a>trans</a> <| <a>edist_le_diam_of_mem</a> hxu hyu)", [{"full_name": "EMetric.diam", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [881, 19], "def_end_pos": [881, 23]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "EMetric.edist_le_diam_of_mem", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [901, 9], "def_end_pos": [901, 29]}]], "state_before": "case h.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\nu : Set X\nx : X\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : X\nhyt : y \u2208 t\nhyu : y \u2208 u\n\u22a2 extend (fun s x => m s) u = \u22a4", "state_after": "case h.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\nu : Set X\nx : X\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : X\nhyt : y \u2208 t\nhyu : y \u2208 u\nthis : \u03b5 < diam u\n\u22a2 extend (fun s x => m s) u = \u22a4"}, {"tactic": "exact iInf_eq_top.2 fun h => (this.not_le h).elim", "annotated_tactic": ["exact <a>iInf_eq_top</a>.2 fun h => (this.not_le h).<a>elim</a>", [{"full_name": "iInf_eq_top", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 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": "case h.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\nu : Set X\nx : X\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : X\nhyt : y \u2208 t\nhyu : y \u2208 u\nthis : \u03b5 < diam u\n\u22a2 extend (fun s x => m s) u = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.Countable.insert", "start": [231, 11], "end": [232, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.tendsto_sum_indicator_atTop_iff'", "start": [357, 1], "end": [370, 19], "traced_tactics": [{"tactic": "have := tendsto_sum_indicator_atTop_iff (eventually_of_forall fun \u03c9 n => ?_) (adapted_process hs)\n  (integrable_process \u03bc hs) (eventually_of_forall <| process_difference_le s)", "annotated_tactic": ["have := <a>tendsto_sum_indicator_atTop_iff</a> (<a>eventually_of_forall</a> fun \u03c9 n => ?_) (<a>adapted_process</a> hs)\n    (<a>integrable_process</a> \u03bc hs) (<a>eventually_of_forall</a> <| <a>process_difference_le</a> s)", [{"full_name": "MeasureTheory.tendsto_sum_indicator_atTop_iff", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [326, 9], "def_end_pos": [326, 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.BorelCantelli.adapted_process", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [284, 9], "def_end_pos": [284, 24]}, {"full_name": "MeasureTheory.BorelCantelli.integrable_process", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [314, 9], "def_end_pos": [314, 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": "MeasureTheory.BorelCantelli.process_difference_le", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [305, 9], "def_end_pos": [305, 30]}]], "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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop", "state_after": "case refine_2\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 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194\n      Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n\ncase refine_1\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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine_2\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 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194\n      Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n\ncase refine_1\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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9", "state_after": "case refine_1\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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9\n\ncase refine_2\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 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194\n      Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop"}, {"tactic": "simp_rw [process, predictablePart_process_ae_eq] at this", "annotated_tactic": ["simp_rw [<a>process</a>, <a>predictablePart_process_ae_eq</a>] at this", [{"full_name": "MeasureTheory.BorelCantelli.process", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [275, 19], "def_end_pos": [275, 26]}, {"full_name": "MeasureTheory.BorelCantelli.predictablePart_process_ae_eq", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [297, 9], "def_end_pos": [297, 38]}]], "state_before": "case refine_2\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 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194\n      Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop", "state_after": "case refine_2\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 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => Finset.sum (Finset.range n) (fun k => Set.indicator (s (k + 1)) 1) \u03c9) atTop atTop \u2194\n      Tendsto (fun n => Finset.sum (Finset.range n) (fun k => \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "state_before": "case refine_2\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 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => Finset.sum (Finset.range n) (fun k => Set.indicator (s (k + 1)) 1) \u03c9) atTop atTop \u2194\n      Tendsto (fun n => Finset.sum (Finset.range n) (fun k => \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop", "state_after": "no goals"}, {"tactic": "rw [process, process, \u2190 sub_nonneg, Finset.sum_apply, Finset.sum_apply,\n  Finset.sum_range_succ_sub_sum]", "annotated_tactic": ["rw [<a>process</a>, <a>process</a>, \u2190 <a>sub_nonneg</a>, <a>Finset.sum_apply</a>, <a>Finset.sum_apply</a>,\n      <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": "MeasureTheory.BorelCantelli.process", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [275, 19], "def_end_pos": [275, 26]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"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_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"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": "case refine_1\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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9", "state_after": "case refine_1\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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 0 \u2264 Set.indicator (s (n + 1)) 1 \u03c9"}, {"tactic": "exact Set.indicator_nonneg (fun _ _ => zero_le_one) _", "annotated_tactic": ["exact <a>Set.indicator_nonneg</a> (fun _ _ => <a>zero_le_one</a>) _", [{"full_name": "Set.indicator_nonneg", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [823, 15], "def_end_pos": [823, 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 refine_1\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\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 0 \u2264 Set.indicator (s (n + 1)) 1 \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.suppPreservation_iff_liftpPreservation", "start": [709, 1], "end": [720, 24], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u22a2 SuppPreservation \u2194 LiftpPreservation", "state_after": "case mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u22a2 LiftpPreservation\n\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF 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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u22a2 LiftpPreservation", "state_after": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }", "state_after": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : SuppPreservation\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }"}, {"tactic": "rw [suppPreservation_iff_uniform] at h'", "annotated_tactic": ["rw [<a>suppPreservation_iff_uniform</a>] at h'", [{"full_name": "QPF.suppPreservation_iff_uniform", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 37]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : SuppPreservation\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }", "state_after": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \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": "QPF.SuppPreservation", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [667, 5], "def_end_pos": [667, 21]}, {"full_name": "Functor.supp", "def_path": "Mathlib/Control/Functor.lean", "def_pos": [292, 5], "def_end_pos": [292, 9]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }", "state_after": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh :\n  \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }"}, {"tactic": "rw [liftp_iff_of_isUniform h', supp_eq_of_isUniform h', PFunctor.liftp_iff']", "annotated_tactic": ["rw [<a>liftp_iff_of_isUniform</a> h', <a>supp_eq_of_isUniform</a> h', <a>PFunctor.liftp_iff'</a>]", [{"full_name": "QPF.liftp_iff_of_isUniform", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [680, 9], "def_end_pos": [680, 31]}, {"full_name": "QPF.supp_eq_of_isUniform", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [671, 9], "def_end_pos": [671, 29]}, {"full_name": "PFunctor.liftp_iff'", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [202, 9], "def_end_pos": [202, 19]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh :\n  \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }", "state_after": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh :\n  \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1), u \u2208 f '' univ \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i)"}, {"tactic": "simp only [image_univ, mem_range, exists_imp]", "annotated_tactic": ["simp only [<a>image_univ</a>, <a>mem_range</a>, <a>exists_imp</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.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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh :\n  \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1), u \u2208 f '' univ \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i)", "state_after": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh :\n  \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i)"}, {"tactic": "constructor <;> intros <;> subst_vars <;> solve_by_elim", "annotated_tactic": ["constructor <;> intros <;> subst_vars <;> solve_by_elim", []], "state_before": "case mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh :\n  \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i)", "state_after": "no goals"}, {"tactic": "rintro \u03b1 \u27e8a, f\u27e9", "annotated_tactic": ["rintro \u03b1 \u27e8a, f\u27e9", []], "state_before": "case mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nh : LiftpPreservation\n\u22a2 SuppPreservation", "state_after": "case mpr.mk\nF : 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\u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)"}, {"tactic": "set f : (n : \u2115) \u2192 (g : (m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k : Iio n => g k.1 k.2)", "annotated_tactic": ["set f : (n : \u2115) \u2192 (g : (m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (<a>range</a> fun k : <a>Iio</a> n => g k.1 k.2)", [{"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]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\n\u22a2 \u2203 u, \u2200 (n : 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"full_name": "Fin.coe_clamp", "start": [708, 9], "end": [708, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasureTheory.Measure.ext_of_Ioc_finite", "start": [754, 1], "end": [760, 14], "traced_tactics": [{"tactic": "refine' @ext_of_Ico_finite \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a \u03bc \u03bd _ h\u03bc\u03bd fun a b hab => _", "annotated_tactic": ["refine' @<a>ext_of_Ico_finite</a> \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a \u03bc \u03bd _ h\u03bc\u03bd fun a b hab => _", [{"full_name": "MeasureTheory.Measure.ext_of_Ico_finite", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [739, 9], "def_end_pos": [739, 26]}]], "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 (Ioc a b) = \u2191\u2191\u03bd (Ioc 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\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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)"}, {"tactic": "erw [dual_Ico (\u03b1 := \u03b1)]", "annotated_tactic": ["erw [<a>dual_Ico</a> (\u03b1 := \u03b1)]", [{"full_name": "Set.dual_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [256, 9], "def_end_pos": [256, 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\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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b)", "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\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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) = \u2191\u2191\u03bd (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a)"}, {"tactic": "exact h hab", "annotated_tactic": ["exact h hab", []], "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\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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) = \u2191\u2191\u03bd (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "ProbabilityTheory.condDistrib_ae_eq_condexp", "start": [211, 1], "end": [222, 82], "traced_tactics": [{"tactic": "refine' ae_eq_condexp_of_forall_set_integral_eq hX.comap_le _ _ _ _", "annotated_tactic": ["refine' <a>ae_eq_condexp_of_forall_set_integral_eq</a> hX.comap_le _ _ _ _", [{"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]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) =\u1d50[\u03bc]\n    \u03bc[indicator (Y \u207b\u00b9' s) fun \u03c9 => 1|MeasurableSpace.comap X m\u03b2]", "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\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 Integrable (indicator (Y \u207b\u00b9' s) fun \u03c9 => 1)\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\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 < \u22a4 \u2192 IntegrableOn (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) s_1\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\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 < \u22a4 \u2192\n        \u222b (x : \u03b1) in s_1, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc =\n          \u222b (x : \u03b1) in s_1, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc\n\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) \u03bc"}, {"tactic": "exact (integrable_const _).indicator (hY hs)", "annotated_tactic": ["exact (<a>integrable_const</a> _).<a>indicator</a> (hY hs)", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}, {"full_name": "MeasureTheory.Integrable.indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}]], "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\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 Integrable (indicator (Y \u207b\u00b9' s) fun \u03c9 => 1)", "state_after": "no goals"}, {"tactic": "exact fun t _ _ => (integrable_toReal_condDistrib hX.aemeasurable hs).integrableOn", "annotated_tactic": ["exact fun t _ _ => (<a>integrable_toReal_condDistrib</a> hX.aemeasurable hs).<a>integrableOn</a>", [{"full_name": "ProbabilityTheory.integrable_toReal_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [127, 9], "def_end_pos": [127, 38]}, {"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'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 < \u22a4 \u2192 IntegrableOn (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) s_1", "state_after": "no goals"}, {"tactic": "intro t ht _", "annotated_tactic": ["intro t ht _", []], "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\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 < \u22a4 \u2192\n        \u222b (x : \u03b1) in s_1, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc =\n          \u222b (x : \u03b1) in s_1, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc =\n    \u222b (x : \u03b1) in t, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc"}, {"tactic": "rw [integral_toReal ((measurable_condDistrib hs).mono hX.comap_le le_rfl).aemeasurable\n  (eventually_of_forall fun \u03c9 => measure_lt_top (condDistrib Y X \u03bc (X \u03c9)) _),\n  integral_indicator_const _ (hY hs), Measure.restrict_apply (hY hs), smul_eq_mul, mul_one,\n  inter_comm, set_lintegral_condDistrib_of_measurableSet hX hY.aemeasurable hs ht]", "annotated_tactic": ["rw [<a>integral_toReal</a> ((<a>measurable_condDistrib</a> hs).<a>mono</a> hX.comap_le <a>le_rfl</a>).<a>aemeasurable</a>\n      (<a>eventually_of_forall</a> fun \u03c9 => <a>measure_lt_top</a> (<a>condDistrib</a> Y X \u03bc (X \u03c9)) _),\n      <a>integral_indicator_const</a> _ (hY hs), <a>Measure.restrict_apply</a> (hY hs), <a>smul_eq_mul</a>, <a>mul_one</a>,\n      <a>inter_comm</a>, <a>set_lintegral_condDistrib_of_measurableSet</a> hX hY.aemeasurable hs ht]", [{"full_name": "MeasureTheory.integral_toReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 24]}, {"full_name": "ProbabilityTheory.measurable_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [74, 9], "def_end_pos": [74, 31]}, {"full_name": "Measurable.mono", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 24]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 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": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "ProbabilityTheory.condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [62, 31], "def_end_pos": [62, 42]}, {"full_name": "MeasureTheory.integral_indicator_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [479, 9], "def_end_pos": [479, 33]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 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": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [202, 9], "def_end_pos": [202, 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\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\na\u271d : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc =\n    \u222b (x : \u03b1) in t, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' (Measurable.stronglyMeasurable _).aeStronglyMeasurable'", "annotated_tactic": ["refine' (<a>Measurable.stronglyMeasurable</a> _).<a>aeStronglyMeasurable'</a>", [{"full_name": "Measurable.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 45]}, {"full_name": "MeasureTheory.StronglyMeasurable.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [138, 9], "def_end_pos": [138, 49]}]], "state_before": "case refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) \u03bc", "state_after": "case refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 Measurable fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)"}, {"tactic": "exact @Measurable.ennreal_toReal _ (m\u03b2.comap X) _ (measurable_condDistrib hs)", "annotated_tactic": ["exact @<a>Measurable.ennreal_toReal</a> _ (m\u03b2.comap X) _ (<a>measurable_condDistrib</a> hs)", [{"full_name": "Measurable.ennreal_toReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2123, 9], "def_end_pos": [2123, 34]}, {"full_name": "ProbabilityTheory.measurable_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [74, 9], "def_end_pos": [74, 31]}]], "state_before": "case refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : Measurable Y\nhs : MeasurableSet s\n\u22a2 Measurable fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "integrableOn_Icc_iff_integrableOn_Ico", "start": [703, 1], "end": [705, 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/Num/Lemmas.lean", "full_name": "Num.castNum_and", "start": [933, 1], "end": [934, 85], "traced_tactics": [{"tactic": "apply castNum_eq_bitwise PosNum.land <;> intros <;> (try cases_type* Bool) <;> rfl", "annotated_tactic": ["apply <a>castNum_eq_bitwise</a> <a>PosNum.land</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.land", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [42, 5], "def_end_pos": [42, 9]}, {"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(m &&& n) = \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.land (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && b\u271d) (PosNum.land m\u271d n\u271d)", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false && false) (PosNum.land m\u271d n\u271d)\n\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (false && true) (PosNum.land m\u271d n\u271d)\n\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (true && false) (PosNum.land m\u271d n\u271d)\n\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (true && true) (PosNum.land m\u271d n\u271d)"}, {"tactic": "cases_type* Bool", "annotated_tactic": ["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.land (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && b\u271d) (PosNum.land m\u271d n\u271d)", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false && false) (PosNum.land m\u271d n\u271d)\n\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (false && true) (PosNum.land m\u271d n\u271d)\n\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (true && false) (PosNum.land m\u271d n\u271d)\n\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (true && true) (PosNum.land m\u271d n\u271d)"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_congr", "start": [2774, 1], "end": [2775, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_comp_mul_deriv", "start": [1499, 1], "end": [1502, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.setLaverage_lt_top", "start": [138, 1], "end": [139, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.mkOfClosure_sets", "start": [418, 1], "end": [420, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "full_name": "Sum.liftRel_swap_iff", "start": [188, 9], "end": [189, 78], "traced_tactics": [{"tactic": "rw [\u2190 swap_swap x, \u2190 swap_swap y]", "annotated_tactic": ["rw [\u2190 <a>swap_swap</a> x, \u2190 <a>swap_swap</a> y]", [{"full_name": "Sum.swap_swap", "def_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "def_pos": [155, 17], "def_end_pos": [155, 26]}, {"full_name": "Sum.swap_swap", "def_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "def_pos": [155, 17], "def_end_pos": [155, 26]}]], "state_before": "\u03b2\u271d\u00b9 : Type u_1\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d\u00b9 \u2192 \u03b2\u271d \u2192 Prop\n\u03b1\u271d\u00b9 : Type u_3\n\u03b1\u271d : Type u_4\nr : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d \u2192 Prop\nx : \u03b1\u271d\u00b9 \u2295 \u03b2\u271d\u00b9\ny : \u03b1\u271d \u2295 \u03b2\u271d\nh : LiftRel s r (swap x) (swap y)\n\u22a2 LiftRel r s x y", "state_after": "\u03b2\u271d\u00b9 : Type u_1\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d\u00b9 \u2192 \u03b2\u271d \u2192 Prop\n\u03b1\u271d\u00b9 : Type u_3\n\u03b1\u271d : Type u_4\nr : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d \u2192 Prop\nx : \u03b1\u271d\u00b9 \u2295 \u03b2\u271d\u00b9\ny : \u03b1\u271d \u2295 \u03b2\u271d\nh : LiftRel s r (swap x) (swap y)\n\u22a2 LiftRel r s (swap (swap x)) (swap (swap y))"}, {"tactic": "exact h.swap", "annotated_tactic": ["exact h.swap", []], "state_before": "\u03b2\u271d\u00b9 : Type u_1\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d\u00b9 \u2192 \u03b2\u271d \u2192 Prop\n\u03b1\u271d\u00b9 : Type u_3\n\u03b1\u271d : Type u_4\nr : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d \u2192 Prop\nx : \u03b1\u271d\u00b9 \u2295 \u03b2\u271d\u00b9\ny : \u03b1\u271d \u2295 \u03b2\u271d\nh : LiftRel s r (swap x) (swap y)\n\u22a2 LiftRel r s (swap (swap x)) (swap (swap y))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.Martingale.ae_not_tendsto_atTop_atBot", "start": [264, 1], "end": [268, 89], "traced_tactics": [{"tactic": "filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n  unbounded_of_tendsto_atBot htop (h\u03c9.1 <| bddAbove_range_of_tendsto_atTop_atBot htop)", "annotated_tactic": ["filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n    <a>unbounded_of_tendsto_atBot</a> htop (h\u03c9.1 <| <a>bddAbove_range_of_tendsto_atTop_atBot</a> htop)", [{"full_name": "Filter.unbounded_of_tendsto_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1759, 9], "def_end_pos": [1759, 35]}, {"full_name": "Filter.bddAbove_range_of_tendsto_atTop_atBot", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [286, 9], "def_end_pos": [286, 46]}]], "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 : Martingale 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, \u00acTendsto (fun n => f n \u03c9) atTop atBot", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.norm_Integral_le_one", "start": [727, 1], "end": [728, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/AssocList.lean", "full_name": "Std.AssocList.mapVal_toList", "start": [94, 9], "end": [96, 27], "traced_tactics": [{"tactic": "induction l <;> simp [*]", "annotated_tactic": ["induction l <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nl : AssocList \u03b1 \u03b2\n\u22a2 toList (mapVal f l) =\n    List.map\n      (fun x =>\n        match x with\n        | (a, b) => (a, f a b))\n      (toList l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.preimage_id", "start": [198, 1], "end": [199, 41], "traced_tactics": [{"tactic": "simp only [preimage, inv_id, image_id]", "annotated_tactic": ["simp only [<a>preimage</a>, <a>inv_id</a>, <a>image_id</a>]", [{"full_name": "Rel.preimage", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [174, 5], "def_end_pos": [174, 13]}, {"full_name": "Rel.inv_id", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [119, 9], "def_end_pos": [119, 15]}, {"full_name": "Rel.image_id", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [157, 9], "def_end_pos": [157, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 preimage Eq s = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.setIntegral_setAverage_sub", "start": [428, 1], "end": [431, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/WithDensity.lean", "full_name": "ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded", "start": [139, 1], "end": [151, 19], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\n\u22a2 IsFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : Measurable (Function.uncurry f)\n\u22a2 IsFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsFiniteKernel (withDensity \u03ba f)"}, {"tactic": "exact \u27e8\u27e8B * IsFiniteKernel.bound \u03ba, ENNReal.mul_lt_top hB_top (IsFiniteKernel.bound_ne_top \u03ba),\n  fun a => by\n    rw [withDensity_apply' \u03ba hf a MeasurableSet.univ]\n    calc\n      \u222b\u207b b in Set.univ, f a b \u2202\u03ba a \u2264 \u222b\u207b _ in Set.univ, B \u2202\u03ba a := lintegral_mono (hf_B a)\n      _ = B * \u03ba a Set.univ := by\n        simp only [Measure.restrict_univ, MeasureTheory.lintegral_const]\n      _ \u2264 B * IsFiniteKernel.bound \u03ba := mul_le_mul_left' (measure_le_bound \u03ba a Set.univ) _\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\u27e8B * <a>IsFiniteKernel.bound</a> \u03ba, <a>ENNReal.mul_lt_top</a> hB_top (<a>IsFiniteKernel.bound_ne_top</a> \u03ba),\n      fun a => by\n        rw [<a>withDensity_apply'</a> \u03ba hf a <a>MeasurableSet.univ</a>]\n        calc\n          \u222b\u207b b in <a>Set.univ</a>, f a b \u2202\u03ba a \u2264 \u222b\u207b _ in <a>Set.univ</a>, B \u2202\u03ba a := <a>lintegral_mono</a> (hf_B a)\n          _ = B * \u03ba a <a>Set.univ</a> := by\n            simp only [<a>Measure.restrict_univ</a>, <a>MeasureTheory.lintegral_const</a>]\n          _ \u2264 B * <a>IsFiniteKernel.bound</a> \u03ba := <a>mul_le_mul_left'</a> (<a>measure_le_bound</a> \u03ba a <a>Set.univ</a>) _\u27e9\u27e9", [{"full_name": "ProbabilityTheory.IsFiniteKernel.bound", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [128, 19], "def_end_pos": [128, 39]}, {"full_name": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}, {"full_name": "ProbabilityTheory.IsFiniteKernel.bound_ne_top", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [137, 9], "def_end_pos": [137, 36]}, {"full_name": "ProbabilityTheory.kernel.withDensity_apply'", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ProbabilityTheory.IsFiniteKernel.bound", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [128, 19], "def_end_pos": [128, 39]}, {"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": "ProbabilityTheory.kernel.measure_le_bound", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [142, 9], "def_end_pos": [142, 32]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : Measurable (Function.uncurry f)\n\u22a2 IsFiniteKernel (withDensity \u03ba f)", "state_after": "no goals"}, {"tactic": "rw [withDensity_apply' \u03ba hf a MeasurableSet.univ]", "annotated_tactic": ["rw [<a>withDensity_apply'</a> \u03ba hf a <a>MeasurableSet.univ</a>]", [{"full_name": "ProbabilityTheory.kernel.withDensity_apply'", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}, {"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\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 : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : Measurable (Function.uncurry f)\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(withDensity \u03ba f) a) Set.univ \u2264 B * IsFiniteKernel.bound \u03ba", "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 : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : Measurable (Function.uncurry f)\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2) in Set.univ, f a b \u2202\u2191\u03ba a \u2264 B * IsFiniteKernel.bound \u03ba"}, {"tactic": "calc\n  \u222b\u207b b in Set.univ, f a b \u2202\u03ba a \u2264 \u222b\u207b _ in Set.univ, B \u2202\u03ba a := lintegral_mono (hf_B a)\n  _ = B * \u03ba a Set.univ := by\n    simp only [Measure.restrict_univ, MeasureTheory.lintegral_const]\n  _ \u2264 B * IsFiniteKernel.bound \u03ba := mul_le_mul_left' (measure_le_bound \u03ba a Set.univ) _", "annotated_tactic": ["calc\n          \u222b\u207b b in <a>Set.univ</a>, f a b \u2202\u03ba a \u2264 \u222b\u207b _ in <a>Set.univ</a>, B \u2202\u03ba a := <a>lintegral_mono</a> (hf_B a)\n          _ = B * \u03ba a <a>Set.univ</a> := by\n            simp only [<a>Measure.restrict_univ</a>, <a>MeasureTheory.lintegral_const</a>]\n          _ \u2264 B * <a>IsFiniteKernel.bound</a> \u03ba := <a>mul_le_mul_left'</a> (<a>measure_le_bound</a> \u03ba a <a>Set.univ</a>) _", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}, {"full_name": "ProbabilityTheory.IsFiniteKernel.bound", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [128, 19], "def_end_pos": [128, 39]}, {"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": "ProbabilityTheory.kernel.measure_le_bound", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [142, 9], "def_end_pos": [142, 32]}, {"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\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 : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : Measurable (Function.uncurry f)\na : \u03b1\n\u22a2 \u222b\u207b (b : \u03b2) in Set.univ, f a b \u2202\u2191\u03ba a \u2264 B * IsFiniteKernel.bound \u03ba", "state_after": "no goals"}, {"tactic": "simp only [Measure.restrict_univ, MeasureTheory.lintegral_const]", "annotated_tactic": ["simp only [<a>Measure.restrict_univ</a>, <a>MeasureTheory.lintegral_const</a>]", [{"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}, {"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [136, 9], "def_end_pos": [136, 24]}]], "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 : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : Measurable (Function.uncurry f)\na : \u03b1\n\u22a2 \u222b\u207b (x : \u03b2) in Set.univ, B \u2202\u2191\u03ba a = B * \u2191\u2191(\u2191\u03ba a) Set.univ", "state_after": "no goals"}, {"tactic": "rw [withDensity_of_not_measurable _ hf]", "annotated_tactic": ["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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsFiniteKernel 0"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nB : \u211d\u22650\u221e\nhB_top : B \u2260 \u22a4\nhf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsFiniteKernel 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Function.Injective.comp_injOn", "start": [1631, 1], "end": [1632, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.reaches_eval", "start": [863, 1], "end": [869, 39], "traced_tactics": [{"tactic": "refine' Part.ext fun _ \u21a6 \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9", "annotated_tactic": ["refine' <a>Part.ext</a> fun _ \u21a6 \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9", [{"full_name": "Part.ext", "def_path": "Mathlib/Data/Part.lean", "def_pos": [116, 9], "def_end_pos": [116, 12]}]], "state_before": "\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\n\u22a2 eval f a = eval f b", "state_after": "case refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f a\n\u22a2 x\u271d \u2208 eval f b\n\ncase refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f b\n\u22a2 x\u271d \u2208 eval f a"}, {"tactic": "have \u27e8ac, c0\u27e9 := mem_eval.1 h", "annotated_tactic": ["have \u27e8ac, c0\u27e9 := <a>mem_eval</a>.1 h", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "state_before": "case refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f a\n\u22a2 x\u271d \u2208 eval f b", "state_after": "case refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f a\nac : Reaches f a x\u271d\nc0 : f x\u271d = none\n\u22a2 x\u271d \u2208 eval f b"}, {"tactic": "exact mem_eval.2 \u27e8(or_iff_left_of_imp fun cb \u21a6 (eval_maximal h).1 cb \u25b8 ReflTransGen.refl).1\n  (reaches_total ab ac), c0\u27e9", "annotated_tactic": ["exact <a>mem_eval</a>.2 \u27e8(<a>or_iff_left_of_imp</a> fun cb \u21a6 (<a>eval_maximal</a> h).1 cb \u25b8 <a>ReflTransGen.refl</a>).1\n      (<a>reaches_total</a> ab ac), c0\u27e9", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}, {"full_name": "or_iff_left_of_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [291, 9], "def_end_pos": [291, 27]}, {"full_name": "Turing.eval_maximal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [858, 9], "def_end_pos": [858, 21]}, {"full_name": "Relation.ReflTransGen.refl", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [223, 5], "def_end_pos": [223, 9]}, {"full_name": "Turing.reaches_total", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [760, 9], "def_end_pos": [760, 22]}]], "state_before": "case refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f a\nac : Reaches f a x\u271d\nc0 : f x\u271d = none\n\u22a2 x\u271d \u2208 eval f b", "state_after": "no goals"}, {"tactic": "have \u27e8bc, c0\u27e9 := mem_eval.1 h", "annotated_tactic": ["have \u27e8bc, c0\u27e9 := <a>mem_eval</a>.1 h", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "state_before": "case refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f b\n\u22a2 x\u271d \u2208 eval f a", "state_after": "case refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f b\nbc : Reaches f b x\u271d\nc0 : f x\u271d = none\n\u22a2 x\u271d \u2208 eval f a"}, {"tactic": "exact mem_eval.2 \u27e8ab.trans bc, c0\u27e9", "annotated_tactic": ["exact <a>mem_eval</a>.2 \u27e8ab.trans bc, c0\u27e9", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "state_before": "case refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f b\nbc : Reaches f b x\u271d\nc0 : f x\u271d = none\n\u22a2 x\u271d \u2208 eval f a", "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_iInter", "start": [105, 1], "end": [107, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_add_ncard_compl", "start": [925, 1], "end": [927, 93], "traced_tactics": [{"tactic": "rw [\u2190 ncard_univ, \u2190 ncard_union_eq (@disjoint_compl_right _ _ s) hs hsc, union_compl_self]", "annotated_tactic": ["rw [\u2190 <a>ncard_univ</a>, \u2190 <a>ncard_union_eq</a> (@<a>disjoint_compl_right</a> _ _ s) hs hsc, <a>union_compl_self</a>]", [{"full_name": "Set.ncard_univ", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [526, 9], "def_end_pos": [526, 19]}, {"full_name": "Set.ncard_union_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 29]}, {"full_name": "Set.union_compl_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1739, 9], "def_end_pos": [1739, 25]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nhsc : autoParam (Set.Finite s\u1d9c) _auto\u271d\n\u22a2 ncard s + ncard s\u1d9c = Nat.card \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.continuous_of_dominated_interval", "start": [1105, 1], "end": [1114, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr\u2082_bisim_tail", "start": [205, 1], "end": [211, 37], "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\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh :\n  \u2203 R, R s\u2081 s\u2082 \u2227 \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\n\u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2", "state_after": "case intro.intro\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\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) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\n\u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2"}, {"tactic": "exact (mapAccumr\u2082_bisim R h\u2080 hR).2", "annotated_tactic": ["exact (<a>mapAccumr\u2082_bisim</a> R h\u2080 hR).2", [{"full_name": "Vector.mapAccumr\u2082_bisim", "def_path": "Mathlib/Data/Vector/MapLemmas.lean", "def_pos": [192, 9], "def_end_pos": [192, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\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) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2\n\u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2", "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.forIn_eq_forIn_toList", "start": [644, 1], "end": [645, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.BorelCantelli.integrable_process", "start": [314, 1], "end": [317, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.outerMeasure_caratheodory", "start": [327, 1], "end": [333, 27], "traced_tactics": [{"tactic": "rw [Opens.forall]", "annotated_tactic": ["rw [<a>Opens.forall</a>]", [{"full_name": "TopologicalSpace.Opens.forall", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [81, 9], "def_end_pos": [81, 17]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 MeasurableSet A \u2194\n    \u2200 (U : Opens G),\n      \u2191(Content.outerMeasure \u03bc) (\u2191U \u2229 A) + \u2191(Content.outerMeasure \u03bc) (\u2191U \\ A) \u2264 \u2191(Content.outerMeasure \u03bc) \u2191U", "state_after": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 MeasurableSet A \u2194\n    \u2200 (U : Set G) (hU : IsOpen U),\n      \u2191(Content.outerMeasure \u03bc) (\u2191{ carrier := U, is_open' := hU } \u2229 A) +\n          \u2191(Content.outerMeasure \u03bc) (\u2191{ carrier := U, is_open' := hU } \\ A) \u2264\n        \u2191(Content.outerMeasure \u03bc) \u2191{ carrier := U, is_open' := hU }"}, {"tactic": "apply inducedOuterMeasure_caratheodory", "annotated_tactic": ["apply <a>inducedOuterMeasure_caratheodory</a>", [{"full_name": "MeasureTheory.inducedOuterMeasure_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1513, 9], "def_end_pos": [1513, 41]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 MeasurableSet A \u2194\n    \u2200 (U : Set G) (hU : IsOpen U),\n      \u2191(Content.outerMeasure \u03bc) (\u2191{ carrier := U, is_open' := hU } \u2229 A) +\n          \u2191(Content.outerMeasure \u03bc) (\u2191{ carrier := U, is_open' := hU } \\ A) \u2264\n        \u2191(Content.outerMeasure \u03bc) \u2191{ carrier := U, is_open' := hU }", "state_after": "case msU\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set G\u2984 (hm : \u2200 (i : \u2115), IsOpen (f i)),\n    innerContent \u03bc { carrier := \u22c3 i, f i, is_open' := (_ : ?P (\u22c3 i, f i)) } \u2264\n      \u2211' (i : \u2115), innerContent \u03bc { carrier := f i, is_open' := (_ : IsOpen (f i)) }\n\ncase m_mono\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983s\u2081 s\u2082 : Set G\u2984 (hs\u2081 : IsOpen s\u2081) (hs\u2082 : IsOpen s\u2082),\n    s\u2081 \u2286 s\u2082 \u2192 innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 } \u2264 innerContent \u03bc { carrier := s\u2082, is_open' := hs\u2082 }\n\ncase PU\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set G\u2984, (\u2200 (i : \u2115), IsOpen (f i)) \u2192 IsOpen (\u22c3 i, f i)"}, {"tactic": "apply innerContent_iUnion_nat", "annotated_tactic": ["apply <a>innerContent_iUnion_nat</a>", [{"full_name": "MeasureTheory.Content.innerContent_iUnion_nat", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [199, 9], "def_end_pos": [199, 32]}]], "state_before": "case msU\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set G\u2984 (hm : \u2200 (i : \u2115), IsOpen (f i)),\n    innerContent \u03bc { carrier := \u22c3 i, f i, is_open' := (_ : ?P (\u22c3 i, f i)) } \u2264\n      \u2211' (i : \u2115), innerContent \u03bc { carrier := f i, is_open' := (_ : IsOpen (f i)) }\n\ncase m_mono\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983s\u2081 s\u2082 : Set G\u2984 (hs\u2081 : IsOpen s\u2081) (hs\u2082 : IsOpen s\u2082),\n    s\u2081 \u2286 s\u2082 \u2192 innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 } \u2264 innerContent \u03bc { carrier := s\u2082, is_open' := hs\u2082 }\n\ncase PU\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set G\u2984, (\u2200 (i : \u2115), IsOpen (f i)) \u2192 IsOpen (\u22c3 i, f i)", "state_after": "case m_mono\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983s\u2081 s\u2082 : Set G\u2984 (hs\u2081 : IsOpen s\u2081) (hs\u2082 : IsOpen s\u2082),\n    s\u2081 \u2286 s\u2082 \u2192 innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 } \u2264 innerContent \u03bc { carrier := s\u2082, is_open' := hs\u2082 }"}, {"tactic": "apply innerContent_mono'", "annotated_tactic": ["apply <a>innerContent_mono'</a>", [{"full_name": "MeasureTheory.Content.innerContent_mono'", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [240, 9], "def_end_pos": [240, 27]}]], "state_before": "case m_mono\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nA : Set G\n\u22a2 \u2200 \u2983s\u2081 s\u2082 : Set G\u2984 (hs\u2081 : IsOpen s\u2081) (hs\u2082 : IsOpen s\u2082),\n    s\u2081 \u2286 s\u2082 \u2192 innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 } \u2264 innerContent \u03bc { carrier := s\u2082, is_open' := hs\u2082 }", "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_image_left_anticomm", "start": [492, 1], "end": [495, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.pi_update_of_mem", "start": [831, 1], "end": [838, 81], "traced_tactics": [{"tactic": "rw [union_diff_self, union_eq_self_of_subset_left (singleton_subset_iff.2 hi)]", "annotated_tactic": ["rw [<a>union_diff_self</a>, <a>union_eq_self_of_subset_left</a> (<a>singleton_subset_iff</a>.2 hi)]", [{"full_name": "Set.union_diff_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2043, 9], "def_end_pos": [2043, 24]}, {"full_name": "Set.union_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [820, 9], "def_end_pos": [820, 37]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "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\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 (pi s fun j => t j (update f i a j)) = pi ({i} \u222a s \\ {i}) fun j => t j (update f i a j)", "state_after": "no goals"}, {"tactic": "rw [union_pi, singleton_pi', update_same, pi_update_of_not_mem]", "annotated_tactic": ["rw [<a>union_pi</a>, <a>singleton_pi'</a>, <a>update_same</a>, <a>pi_update_of_not_mem</a>]", [{"full_name": "Set.union_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [815, 9], "def_end_pos": [815, 17]}, {"full_name": "Set.singleton_pi'", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [790, 9], "def_end_pos": [790, 22]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Set.pi_update_of_not_mem", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [824, 9], "def_end_pos": [824, 29]}]], "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\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 (pi ({i} \u222a s \\ {i}) fun j => t j (update f i a j)) = {x | x i \u2208 t i a} \u2229 pi (s \\ {i}) fun j => t j (f j)", "state_after": "case hi\n\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\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 \u00aci \u2208 s \\ {i}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case hi\n\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\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 \u00aci \u2208 s \\ {i}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "full_name": "VitaliFamily.fineSubfamilyOn_of_frequently", "start": [289, 1], "end": [294, 26], "traced_tactics": [{"tactic": "intro x hx \u03b5 \u03b5pos", "annotated_tactic": ["intro x hx \u03b5 \u03b5pos", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv\u271d v : VitaliFamily \u03bc\nf : \u03b1 \u2192 Set (Set \u03b1)\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x\n\u22a2 FineSubfamilyOn v f s", "state_after": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv\u271d v : VitaliFamily \u03bc\nf : \u03b1 \u2192 Set (Set \u03b1)\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "obtain \u27e8a, av, ha, af\u27e9 : \u2203 (a : Set \u03b1) , a \u2208 v.setsAt x \u2227 a \u2286 closedBall x \u03b5 \u2227 a \u2208 f x :=\n  v.frequently_filterAt_iff.1 (h x hx) \u03b5 \u03b5pos", "annotated_tactic": ["obtain \u27e8a, av, ha, af\u27e9 : \u2203 (a : <a>Set</a> \u03b1) , a \u2208 v.setsAt x \u2227 a \u2286 <a>closedBall</a> x \u03b5 \u2227 a \u2208 f x :=\n    v.frequently_filterAt_iff.1 (h x hx) \u03b5 \u03b5pos", [{"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": "\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv\u271d v : VitaliFamily \u03bc\nf : \u03b1 \u2192 Set (Set \u03b1)\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv\u271d v : VitaliFamily \u03bc\nf : \u03b1 \u2192 Set (Set \u03b1)\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\na : Set \u03b1\nav : a \u2208 setsAt v x\nha : a \u2286 closedBall x \u03b5\naf : a \u2208 f x\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "exact \u27e8a, \u27e8av, af\u27e9, ha\u27e9", "annotated_tactic": ["exact \u27e8a, \u27e8av, af\u27e9, ha\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv\u271d v : VitaliFamily \u03bc\nf : \u03b1 \u2192 Set (Set \u03b1)\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\na : Set \u03b1\nav : a \u2208 setsAt v x\nha : a \u2286 closedBall x \u03b5\naf : a \u2208 f x\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.mul_X_divMonomial", "start": [174, 1], "end": [176, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.inf_gt_preCdf", "start": [511, 1], "end": [517, 76], "traced_tactics": [{"tactic": "rw [ae_all_iff]", "annotated_tactic": ["rw [<a>ae_all_iff</a>]", [{"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, \u2200 (t : \u211a), \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t 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 : \u211a), \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 i a"}, {"tactic": "refine' fun t => ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite _ measurable_preCdf _", "annotated_tactic": ["refine' fun t => <a>ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite</a> _ <a>measurable_preCdf</a> _", [{"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": "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 : \u211a), \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 i a", "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\n\u22a2 Measurable fun a => \u2a05 r, preCdf \u03c1 (\u2191r) 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\nt : \u211a\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u2192\n        \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1"}, {"tactic": "intro s hs _", "annotated_tactic": ["intro s hs _", []], "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\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u2192\n        \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t 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\na\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1"}, {"tactic": "rw [set_lintegral_iInf_gt_preCdf \u03c1 t hs, set_lintegral_preCdf_fst \u03c1 t hs]", "annotated_tactic": ["rw [<a>set_lintegral_iInf_gt_preCdf</a> \u03c1 t hs, <a>set_lintegral_preCdf_fst</a> \u03c1 t hs]", [{"full_name": "ProbabilityTheory.set_lintegral_iInf_gt_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [329, 9], "def_end_pos": [329, 37]}, {"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": "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\na\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1", "state_after": "no goals"}, {"tactic": "exact measurable_iInf fun i => measurable_preCdf", "annotated_tactic": ["exact <a>measurable_iInf</a> fun i => <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\n\u22a2 Measurable fun a => \u2a05 r, preCdf \u03c1 (\u2191r) a", "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_filter", "start": [215, 9], "end": [218, 28], "traced_tactics": [{"tactic": "rw [count, countP_filter]", "annotated_tactic": ["rw [<a>count</a>, <a>countP_filter</a>]", [{"full_name": "List.count", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [804, 15], "def_end_pos": [804, 20]}, {"full_name": "List.countP_filter", "def_path": "lake-packages/std/Std/Data/List/Count.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\n\u22a2 count a (filter p l) = count a l", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\n\u22a2 countP (fun a_1 => decide ((a_1 == a) = true \u2227 p a_1 = true)) l = countP (fun x => x == a) l"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\n\u22a2 countP (fun a_1 => decide ((a_1 == a) = true \u2227 p a_1 = true)) l = countP (fun x => x == a) l", "state_after": "case e_p\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\n\u22a2 (fun a_1 => decide ((a_1 == a) = true \u2227 p a_1 = true)) = fun x => x == a"}, {"tactic": "funext b", "annotated_tactic": ["funext b", []], "state_before": "case e_p\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\n\u22a2 (fun a_1 => decide ((a_1 == a) = true \u2227 p a_1 = true)) = fun x => x == a", "state_after": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 decide ((b == a) = true \u2227 p b = true) = (b == a)"}, {"tactic": "rw [(by rfl : (b == a) = decide (b = a)), decide_eq_decide]", "annotated_tactic": ["rw [(by rfl : (b == a) = decide (b = a)), <a>decide_eq_decide</a>]", [{"full_name": "decide_eq_decide", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [542, 17], "def_end_pos": [542, 33]}]], "state_before": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 decide ((b == a) = true \u2227 p b = true) = (b == a)", "state_after": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 decide (b = a) = true \u2227 p b = true \u2194 b = a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 decide (b = a) = true \u2227 p b = true \u2194 b = a", "state_after": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 b = a \u2192 p b = true"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 b = a \u2192 p b = true", "state_after": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nb : \u03b1\nh : p b = true\n\u22a2 p b = true"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case e_p.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nb : \u03b1\nh : p b = true\n\u22a2 p b = true", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nh : p a = true\nb : \u03b1\n\u22a2 (b == a) = decide (b = a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.diag_insert", "start": [419, 1], "end": [420, 56], "traced_tactics": [{"tactic": "rw [insert_eq, insert_eq, diag_union, diag_singleton]", "annotated_tactic": ["rw [<a>insert_eq</a>, <a>insert_eq</a>, <a>diag_union</a>, <a>diag_singleton</a>]", [{"full_name": "Finset.insert_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 18]}, {"full_name": "Finset.insert_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 18]}, {"full_name": "Finset.diag_union", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [395, 9], "def_end_pos": [395, 19]}, {"full_name": "Finset.diag_singleton", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [415, 9], "def_end_pos": [415, 23]}]], "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\na : \u03b1\n\u22a2 diag (insert a s) = insert (a, a) (diag s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "not_and_self_iff", "start": [241, 1], "end": [241, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.withDensity_limRatioMeas_eq", "start": [665, 1], "end": [686, 40], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\n\u22a2 withDensity \u03bc (limRatioMeas v h\u03c1) = \u03c1", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s = \u2191\u2191\u03c1 s"}, {"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": "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s = \u2191\u2191\u03c1 s", "state_after": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191\u2191\u03c1 s\n\ncase h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s"}, {"tactic": "simp only [one_pow, one_mul, ENNReal.coe_one] at this", "annotated_tactic": ["simp only [<a>one_pow</a>, <a>one_mul</a>, <a>ENNReal.coe_one</a>] at this", [{"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]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}]], "state_before": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2 * \u2191\u2191\u03c1 s))\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191\u2191\u03c1 s", "state_after": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s))\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191\u2191\u03c1 s"}, {"tactic": "refine' ge_of_tendsto this _", "annotated_tactic": ["refine' <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": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s))\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191\u2191\u03c1 s", "state_after": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 1] 1, \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191c ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with _ ht", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with _ ht", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 1] 1, \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191c ^ 2 * \u2191\u2191\u03c1 s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s))\na\u271d : \u211d\u22650\nht : a\u271d \u2208 Ioi 1\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191a\u271d ^ 2 * \u2191\u2191\u03c1 s"}, {"tactic": "exact v.withDensity_le_mul h\u03c1 hs ht", "annotated_tactic": ["exact v.withDensity_le_mul h\u03c1 hs ht", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s))\na\u271d : \u211d\u22650\nht : a\u271d \u2208 Ioi 1\n\u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191a\u271d ^ 2 * \u2191\u2191\u03c1 s", "state_after": "no goals"}, {"tactic": "refine' ENNReal.Tendsto.mul _ _ tendsto_const_nhds _", "annotated_tactic": ["refine' <a>ENNReal.Tendsto.mul</a> _ _ <a>tendsto_const_nhds</a> _", [{"full_name": "ENNReal.Tendsto.mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [353, 19], "def_end_pos": [353, 30]}, {"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\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2 * \u2191\u2191\u03c1 s))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun t => \u2191t ^ 2) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2))\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 1 ^ 2 \u2260 0 \u2228 \u2191\u2191\u03c1 s \u2260 \u22a4\n\ncase refine'_3\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s \u2260 0 \u2228 1 ^ 2 \u2260 \u22a4"}, {"tactic": "exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds)", "annotated_tactic": ["exact <a>ENNReal.Tendsto.pow</a> (<a>ENNReal.tendsto_coe</a>.2 <a>nhdsWithin_le_nhds</a>)", [{"full_name": "ENNReal.Tendsto.pow", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [471, 19], "def_end_pos": [471, 30]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun t => \u2191t ^ 2) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2))", "state_after": "no goals"}, {"tactic": "simp only [one_pow, ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero]", "annotated_tactic": ["simp only [<a>one_pow</a>, <a>ENNReal.coe_one</a>, <a>true_or_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>one_ne_zero</a>]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"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]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 1 ^ 2 \u2260 0 \u2228 \u2191\u2191\u03c1 s \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [one_pow, ENNReal.coe_one, Ne.def, or_true_iff, ENNReal.one_ne_top, not_false_iff]", "annotated_tactic": ["simp only [<a>one_pow</a>, <a>ENNReal.coe_one</a>, <a>Ne.def</a>, <a>or_true_iff</a>, <a>ENNReal.one_ne_top</a>, <a>not_false_iff</a>]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"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": "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]}]], "state_before": "case refine'_3\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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s \u2260 0 \u2228 1 ^ 2 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "have :\n  Tendsto (fun t : \u211d\u22650 => (t : \u211d\u22650\u221e) * \u03bc.withDensity (v.limRatioMeas h\u03c1) s) (\ud835\udcdd[>] 1)\n    (\ud835\udcdd ((1 : \u211d\u22650\u221e) * \u03bc.withDensity (v.limRatioMeas h\u03c1) s)) := by\n  refine' ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) _\n  simp only [ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero]", "annotated_tactic": ["have :\n      <a>Tendsto</a> (fun t : \u211d\u22650 => (t : \u211d\u22650\u221e) * \u03bc.withDensity (v.limRatioMeas h\u03c1) s) (\ud835\udcdd[>] 1)\n        (\ud835\udcdd ((1 : \u211d\u22650\u221e) * \u03bc.withDensity (v.limRatioMeas h\u03c1) s)) := by\n      refine' <a>ENNReal.Tendsto.mul_const</a> (<a>ENNReal.tendsto_coe</a>.2 <a>nhdsWithin_le_nhds</a>) _\n      simp only [<a>ENNReal.coe_one</a>, <a>true_or_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>one_ne_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.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": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"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]}]], "state_before": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s", "state_after": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (1 * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s"}, {"tactic": "simp only [one_mul, ENNReal.coe_one] at this", "annotated_tactic": ["simp only [<a>one_mul</a>, <a>ENNReal.coe_one</a>] at this", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}]], "state_before": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (1 * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s", "state_after": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s"}, {"tactic": "refine' ge_of_tendsto this _", "annotated_tactic": ["refine' <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": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s", "state_after": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 1] 1, \u2191\u2191\u03c1 s \u2264 \u2191c * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with _ ht", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with _ ht", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "case h.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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 1] 1, \u2191\u2191\u03c1 s \u2264 \u2191c * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\na\u271d : \u211d\u22650\nht : a\u271d \u2208 Ioi 1\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191a\u271d * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s"}, {"tactic": "exact v.le_mul_withDensity h\u03c1 hs ht", "annotated_tactic": ["exact v.le_mul_withDensity h\u03c1 hs ht", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nthis :\n  Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))\na\u271d : \u211d\u22650\nht : a\u271d \u2208 Ioi 1\n\u22a2 \u2191\u2191\u03c1 s \u2264 \u2191a\u271d * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s", "state_after": "no goals"}, {"tactic": "refine' ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) _", "annotated_tactic": ["refine' <a>ENNReal.Tendsto.mul_const</a> (<a>ENNReal.tendsto_coe</a>.2 <a>nhdsWithin_le_nhds</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": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)\n    (\ud835\udcdd (1 * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s))", "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 1 \u2260 0 \u2228 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2260 \u22a4"}, {"tactic": "simp only [ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero]", "annotated_tactic": ["simp only [<a>ENNReal.coe_one</a>, <a>true_or_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>one_ne_zero</a>]", [{"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}, {"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]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 1 \u2260 0 \u2228 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.ae_eq_of_ae_le_of_lintegral_le", "start": [871, 1], "end": [884, 84], "traced_tactics": [{"tactic": "have : \u2200 n : \u2115, \u2200\u1d50 x \u2202\u03bc, g x < f x + (n : \u211d\u22650\u221e)\u207b\u00b9 := by\n  intro n\n  simp only [ae_iff, not_lt]\n  have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc { x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x } \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n    (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n\u207b\u00b9).trans hgf\n  rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this\n  exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _))", "annotated_tactic": ["have : \u2200 n : \u2115, \u2200\u1d50 x \u2202\u03bc, g x < f x + (n : \u211d\u22650\u221e)\u207b\u00b9 := by\n    intro n\n    simp only [<a>ae_iff</a>, <a>not_lt</a>]\n    have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc { x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x } \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n      (<a>lintegral_add_mul_meas_add_le_le_lintegral</a> hfg hg n\u207b\u00b9).<a>trans</a> hgf\n    rw [(<a>ENNReal.cancel_of_ne</a> hf).<a>add_le_iff_nonpos_right</a>, <a>nonpos_iff_eq_zero</a>, <a>mul_eq_zero</a>] at this\n    exact this.resolve_left (<a>ENNReal.inv_ne_zero</a>.2 (<a>ENNReal.nat_ne_top</a> _))", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [807, 9], "def_end_pos": [807, 51]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "ENNReal.cancel_of_ne", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1101, 9], "def_end_pos": [1101, 21]}, {"full_name": "AddLECancellable.add_le_iff_nonpos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1670, 3], "def_end_pos": [1670, 14]}, {"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": "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": "\u03b1 : Type u_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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g", "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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\n\u22a2 f =\u1d50[\u03bc] g"}, {"tactic": "refine' hfg.mp ((ae_all_iff.2 this).mono fun x hlt hle => hle.antisymm _)", "annotated_tactic": ["refine' hfg.mp ((<a>ae_all_iff</a>.2 this).<a>mono</a> fun x hlt hle => hle.antisymm _)", [{"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": "\u03b1 : Type u_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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\n\u22a2 f =\u1d50[\u03bc] g", "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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 g x \u2264 f x"}, {"tactic": "suffices Tendsto (fun n : \u2115 => f x + (n : \u211d\u22650\u221e)\u207b\u00b9) atTop (\ud835\udcdd (f x)) from\n  ge_of_tendsto' this fun i => (hlt i).le", "annotated_tactic": ["suffices <a>Tendsto</a> (fun n : \u2115 => f x + (n : \u211d\u22650\u221e)\u207b\u00b9) <a>atTop</a> (\ud835\udcdd (f x)) from\n    <a>ge_of_tendsto'</a> this fun i => (hlt i).<a>le</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": "ge_of_tendsto'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 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\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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 g x \u2264 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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 Tendsto (fun n => f x + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (f x))"}, {"tactic": "simpa only [inv_top, add_zero] using\n  tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top)", "annotated_tactic": ["simpa only [<a>inv_top</a>, <a>add_zero</a>] using\n    tendsto_const_nhds.add (<a>ENNReal.tendsto_inv_iff</a>.2 <a>ENNReal.tendsto_nat_nhds_top</a>)", [{"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": "ENNReal.tendsto_inv_iff", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [533, 19], "def_end_pos": [533, 34]}, {"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_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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 Tendsto (fun n => f x + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (f x))", "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\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9", "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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9"}, {"tactic": "simp only [ae_iff, not_lt]", "annotated_tactic": ["simp only [<a>ae_iff</a>, <a>not_lt</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"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\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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9", "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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0"}, {"tactic": "have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc { x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x } \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n  (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n\u207b\u00b9).trans hgf", "annotated_tactic": ["have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc { x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x } \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n      (<a>lintegral_add_mul_meas_add_le_le_lintegral</a> hfg hg n\u207b\u00b9).<a>trans</a> hgf", [{"full_name": "MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [807, 9], "def_end_pos": [807, 51]}, {"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\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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 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 g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\nthis : \u222b\u207b (x : \u03b1), f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0"}, {"tactic": "rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this", "annotated_tactic": ["rw [(<a>ENNReal.cancel_of_ne</a> hf).<a>add_le_iff_nonpos_right</a>, <a>nonpos_iff_eq_zero</a>, <a>mul_eq_zero</a>] at this", [{"full_name": "ENNReal.cancel_of_ne", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1101, 9], "def_end_pos": [1101, 21]}, {"full_name": "AddLECancellable.add_le_iff_nonpos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1670, 3], "def_end_pos": [1670, 14]}, {"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]}]], "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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\nthis : \u222b\u207b (x : \u03b1), f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 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 g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\nthis : (\u2191n)\u207b\u00b9 = 0 \u2228 \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} = 0\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0"}, {"tactic": "exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _))", "annotated_tactic": ["exact this.resolve_left (<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": "\u03b1 : Type u_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\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\nthis : (\u2191n)\u207b\u00b9 = 0 \u2228 \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} = 0\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aeval_eq_zero", "start": [1578, 1], "end": [1580, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "equivalence_of_oneOneEquiv", "start": [197, 1], "end": [198, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.X_mul_divMonomial", "start": [161, 1], "end": [163, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.FinMeasAdditive.smul_measure_iff", "start": [139, 1], "end": [141, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_mul_Iio_of_neg", "start": [638, 1], "end": [640, 64], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Iio_of_neg a h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Iio_of_neg</a> a 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_Iio_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [552, 9], "def_end_pos": [552, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a c : \u03b1\nh : c < 0\n\u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Iio a = Ioi (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measureUnivNNReal_pos", "start": [2963, 1], "end": [2965, 58], "traced_tactics": [{"tactic": "contrapose! h\u03bc", "annotated_tactic": ["contrapose! h\u03bc", []], "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 : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\n\u22a2 0 < measureUnivNNReal \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\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 : IsFiniteMeasure \u03bc\nh\u03bc : measureUnivNNReal \u03bc \u2264 0\n\u22a2 \u03bc = 0"}, {"tactic": "simpa [measureUnivNNReal_eq_zero, le_zero_iff] using h\u03bc", "annotated_tactic": ["simpa [<a>measureUnivNNReal_eq_zero</a>, <a>le_zero_iff</a>] using h\u03bc", [{"full_name": "MeasureTheory.measureUnivNNReal_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2958, 9], "def_end_pos": [2958, 34]}, {"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\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 : IsFiniteMeasure \u03bc\nh\u03bc : measureUnivNNReal \u03bc \u2264 0\n\u22a2 \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.exists_positive_of_not_mutuallySingular", "start": [389, 1], "end": [449, 54], "traced_tactics": [{"tactic": "have :\n  \u2200 n : \u2115, \u2203 i : Set \u03b1,\n    MeasurableSet i \u2227\n      0 \u2264[i] \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).toSignedMeasure \u2227\n        \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).toSignedMeasure \u2264[i\u1d9c] 0 := by\n  intro; exact exists_compl_positive_negative _", "annotated_tactic": ["have :\n    \u2200 n : \u2115, \u2203 i : <a>Set</a> \u03b1,\n      <a>MeasurableSet</a> i \u2227\n        0 \u2264[i] \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).<a>toSignedMeasure</a> \u2227\n          \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).<a>toSignedMeasure</a> \u2264[i\u1d9c] 0 := by\n    intro; exact <a>exists_compl_positive_negative</a> _", [{"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.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [410, 5], "def_end_pos": [410, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [410, 5], "def_end_pos": [410, 20]}, {"full_name": "MeasureTheory.SignedMeasure.exists_compl_positive_negative", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [370, 9], "def_end_pos": [370, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nthis :\n  \u2200 (n : \u2115),\n    \u2203 i,\n      MeasurableSet i \u2227\n        VectorMeasure.restrict 0 i \u2264\n            VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i \u2227\n          VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i\u1d9c \u2264\n            VectorMeasure.restrict 0 i\u1d9c\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "choose f hf\u2081 hf\u2082 hf\u2083 using this", "annotated_tactic": ["choose f hf\u2081 hf\u2082 hf\u2083 using this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nthis :\n  \u2200 (n : \u2115),\n    \u2203 i,\n      MeasurableSet i \u2227\n        VectorMeasure.restrict 0 i \u2264\n            VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i \u2227\n          VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i\u1d9c \u2264\n            VectorMeasure.restrict 0 i\u1d9c\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "set A := \u22c2 n, (f n)\u1d9c with hA\u2081", "annotated_tactic": ["set A := \u22c2 n, (f n)\u1d9c with hA\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "have hAmeas : MeasurableSet A := MeasurableSet.iInter fun n => (hf\u2081 n).compl", "annotated_tactic": ["have hAmeas : <a>MeasurableSet</a> A := <a>MeasurableSet.iInter</a> fun n => (hf\u2081 n).<a>compl</a>", [{"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.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\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "have hA\u2082 : \u2200 n : \u2115, \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).toSignedMeasure \u2264[A] 0 := by\n  intro n; exact restrict_le_restrict_subset _ _ (hf\u2081 n).compl (hf\u2083 n) (iInter_subset _ _)", "annotated_tactic": ["have hA\u2082 : \u2200 n : \u2115, \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).<a>toSignedMeasure</a> \u2264[A] 0 := by\n    intro n; exact <a>restrict_le_restrict_subset</a> _ _ (hf\u2081 n).<a>compl</a> (hf\u2083 n) (<a>iInter_subset</a> _ _)", [{"full_name": "MeasureTheory.Measure.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [410, 5], "def_end_pos": [410, 20]}, {"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [883, 9], "def_end_pos": [883, 36]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "Set.iInter_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [435, 9], "def_end_pos": [435, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "have hA\u2083 : \u2200 n : \u2115, \u03bc A \u2264 (1 / (n + 1) : \u211d\u22650) * \u03bd A := by\n  intro n\n  have := nonpos_of_restrict_le_zero _ (hA\u2082 n)\n  rwa [toSignedMeasure_sub_apply hAmeas, sub_nonpos, ENNReal.toReal_le_toReal] at this\n  exacts [ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)]", "annotated_tactic": ["have hA\u2083 : \u2200 n : \u2115, \u03bc A \u2264 (1 / (n + 1) : \u211d\u22650) * \u03bd A := by\n    intro n\n    have := <a>nonpos_of_restrict_le_zero</a> _ (hA\u2082 n)\n    rwa [<a>toSignedMeasure_sub_apply</a> hAmeas, <a>sub_nonpos</a>, <a>ENNReal.toReal_le_toReal</a>] at this\n    exacts [<a>ne_of_lt</a> (<a>measure_lt_top</a> _ _), <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)]", [{"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.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 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": "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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "rw [MutuallySingular] at h", "annotated_tactic": ["rw [<a>MutuallySingular</a>] at h", [{"full_name": "MeasureTheory.Measure.MutuallySingular", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [39, 5], "def_end_pos": [39, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u2203 s, MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0 \u2227 \u2191\u2191\u03bd s\u1d9c = 0\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "push_neg at h", "annotated_tactic": ["push_neg at h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u2203 s, MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0 \u2227 \u2191\u2191\u03bd s\u1d9c = 0\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "have := h _ hAmeas h\u03bc", "annotated_tactic": ["have := h _ hAmeas h\u03bc", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd A\u1d9c \u2260 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "simp_rw [compl_iInter, compl_compl] at this", "annotated_tactic": ["simp_rw [<a>compl_iInter</a>, <a>compl_compl</a>] at this", [{"full_name": "Set.compl_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [615, 9], "def_end_pos": [615, 21]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd A\u1d9c \u2260 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "obtain \u27e8n, hn\u27e9 := exists_measure_pos_of_not_measure_iUnion_null this", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := <a>exists_measure_pos_of_not_measure_iUnion_null</a> this", [{"full_name": "MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [336, 9], "def_end_pos": [336, 54]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0\nn : \u2115\nhn : 0 < \u2191\u2191\u03bd (f n)\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E"}, {"tactic": "exact \u27e81 / (n + 1), by simp, f n, hf\u2081 n, hn, hf\u2082 n\u27e9", "annotated_tactic": ["exact \u27e81 / (n + 1), by simp, f n, hf\u2081 n, hn, hf\u2082 n\u27e9", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0\nn : \u2115\nhn : 0 < \u2191\u2191\u03bd (f n)\n\u22a2 \u2203 \u03b5,\n    0 < \u03b5 \u2227\n      \u2203 E,\n        MeasurableSet E \u2227\n          0 < \u2191\u2191\u03bd E \u2227\n            VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\n\u22a2 \u2200 (n : \u2115),\n    \u2203 i,\n      MeasurableSet i \u2227\n        VectorMeasure.restrict 0 i \u2264\n            VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i \u2227\n          VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i\u1d9c \u2264\n            VectorMeasure.restrict 0 i\u1d9c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nn\u271d : \u2115\n\u22a2 \u2203 i,\n    MeasurableSet i \u2227\n      VectorMeasure.restrict 0 i \u2264\n          VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n\u271d + 1)) \u2022 \u03bd)) i \u2227\n        VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n\u271d + 1)) \u2022 \u03bd)) i\u1d9c \u2264\n          VectorMeasure.restrict 0 i\u1d9c"}, {"tactic": "exact exists_compl_positive_negative _", "annotated_tactic": ["exact <a>exists_compl_positive_negative</a> _", [{"full_name": "MeasureTheory.SignedMeasure.exists_compl_positive_negative", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [370, 9], "def_end_pos": [370, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nn\u271d : \u2115\n\u22a2 \u2203 i,\n    MeasurableSet i \u2227\n      VectorMeasure.restrict 0 i \u2264\n          VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n\u271d + 1)) \u2022 \u03bd)) i \u2227\n        VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n\u271d + 1)) \u2022 \u03bd)) i\u1d9c \u2264\n          VectorMeasure.restrict 0 i\u1d9c", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\n\u22a2 \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nn : \u2115\n\u22a2 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A"}, {"tactic": "exact restrict_le_restrict_subset _ _ (hf\u2081 n).compl (hf\u2083 n) (iInter_subset _ _)", "annotated_tactic": ["exact <a>restrict_le_restrict_subset</a> _ _ (hf\u2081 n).<a>compl</a> (hf\u2083 n) (<a>iInter_subset</a> _ _)", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [883, 9], "def_end_pos": [883, 36]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "Set.iInter_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [435, 9], "def_end_pos": [435, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nn : \u2115\n\u22a2 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\n\u22a2 \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A"}, {"tactic": "have := nonpos_of_restrict_le_zero _ (hA\u2082 n)", "annotated_tactic": ["have := <a>nonpos_of_restrict_le_zero</a> _ (hA\u2082 n)", [{"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]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\n\u22a2 \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\nthis : \u2191(toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 0\n\u22a2 \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A"}, {"tactic": "rwa [toSignedMeasure_sub_apply hAmeas, sub_nonpos, ENNReal.toReal_le_toReal] at this", "annotated_tactic": ["rwa [<a>toSignedMeasure_sub_apply</a> hAmeas, <a>sub_nonpos</a>, <a>ENNReal.toReal_le_toReal</a>] at this", [{"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": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"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\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\nthis : \u2191(toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 0\n\u22a2 \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A", "state_after": "case ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\nthis : ENNReal.toReal (\u2191\u2191\u03bc A) \u2264 ENNReal.toReal (\u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A)\n\u22a2 \u2191\u2191\u03bc A \u2260 \u22a4\n\ncase hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\nthis : ENNReal.toReal (\u2191\u2191\u03bc A) \u2264 ENNReal.toReal (\u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A)\n\u22a2 \u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A \u2260 \u22a4"}, {"tactic": "exacts [ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)]", "annotated_tactic": ["exacts [<a>ne_of_lt</a> (<a>measure_lt_top</a> _ _), <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]}, {"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\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\nthis : ENNReal.toReal (\u2191\u2191\u03bc A) \u2264 ENNReal.toReal (\u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A)\n\u22a2 \u2191\u2191\u03bc A \u2260 \u22a4\n\ncase hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nn : \u2115\nthis : ENNReal.toReal (\u2191\u2191\u03bc A) \u2264 ENNReal.toReal (\u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A)\n\u22a2 \u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "lift \u03bc A to \u211d\u22650 using ne_of_lt (measure_lt_top _ _) with \u03bcA", "annotated_tactic": ["lift \u03bc A to \u211d\u22650 using <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _) with \u03bcA", [{"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\n\u22a2 \u2191\u2191\u03bc A = 0", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u22a2 \u2191\u03bcA = 0"}, {"tactic": "lift \u03bd A to \u211d\u22650 using ne_of_lt (measure_lt_top _ _) with \u03bdA", "annotated_tactic": ["lift \u03bd A to \u211d\u22650 using <a>ne_of_lt</a> (<a>measure_lt_top</a> _ _) with \u03bdA", [{"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 intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u22a2 \u2191\u03bcA = 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\n\u22a2 \u2191\u03bcA = 0"}, {"tactic": "rw [ENNReal.coe_eq_zero]", "annotated_tactic": ["rw [<a>ENNReal.coe_eq_zero</a>]", [{"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 intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\n\u22a2 \u2191\u03bcA = 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\n\u22a2 \u03bcA = 0"}, {"tactic": "by_cases hb : 0 < \u03bdA", "annotated_tactic": ["by_cases hb : 0 < \u03bdA", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\n\u22a2 \u03bcA = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\n\u22a2 \u03bcA = 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u00ac0 < \u03bdA\n\u22a2 \u03bcA = 0"}, {"tactic": "suffices \u2200 b, 0 < b \u2192 \u03bcA \u2264 b by\n  by_contra h\n  have h' := this (\u03bcA / 2) (half_pos (zero_lt_iff.2 h))\n  rw [\u2190 @Classical.not_not (\u03bcA \u2264 \u03bcA / 2)] at h'\n  exact h' (not_le.2 (NNReal.half_lt_self h))", "annotated_tactic": ["suffices \u2200 b, 0 < b \u2192 \u03bcA \u2264 b by\n        by_contra h\n        have h' := this (\u03bcA / 2) (<a>half_pos</a> (<a>zero_lt_iff</a>.2 h))\n        rw [\u2190 @<a>Classical.not_not</a> (\u03bcA \u2264 \u03bcA / 2)] at h'\n        exact h' (<a>not_le</a>.2 (<a>NNReal.half_lt_self</a> h))", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "NNReal.half_lt_self", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [888, 16], "def_end_pos": [888, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\n\u22a2 \u03bcA = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\n\u22a2 \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b"}, {"tactic": "intro c hc", "annotated_tactic": ["intro c hc", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\n\u22a2 \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 \u03bcA \u2264 c"}, {"tactic": "have : \u2203 n : \u2115, 1 / (n + 1 : \u211d) < c * (\u03bdA : \u211d)\u207b\u00b9", "annotated_tactic": ["have : \u2203 n : \u2115, 1 / (n + 1 : \u211d) < c * (\u03bdA : \u211d)\u207b\u00b9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 \u03bcA \u2264 c", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c"}, {"tactic": "refine' exists_nat_one_div_lt _", "annotated_tactic": ["refine' <a>exists_nat_one_div_lt</a> _", [{"full_name": "exists_nat_one_div_lt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [312, 9], "def_end_pos": [312, 30]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 0 < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c"}, {"tactic": "rcases this with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases this with \u27e8n, hn\u27e9", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c"}, {"tactic": "have hb\u2081 : (0 : \u211d) < (\u03bdA : \u211d)\u207b\u00b9 := by rw [_root_.inv_pos]; exact hb", "annotated_tactic": ["have hb\u2081 : (0 : \u211d) < (\u03bdA : \u211d)\u207b\u00b9 := by rw [<a>_root_.inv_pos</a>]; exact hb", [{"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bcA \u2264 c"}, {"tactic": "refine' le_trans _ (le_of_lt h')", "annotated_tactic": ["refine' <a>le_trans</a> _ (<a>le_of_lt</a> h')", [{"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]}]], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\nh' : 1 / (\u2191n + 1) * \u03bdA < c\n\u22a2 \u03bcA \u2264 c", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\nh' : 1 / (\u2191n + 1) * \u03bdA < c\n\u22a2 \u03bcA \u2264 1 / (\u2191n + 1) * \u03bdA"}, {"tactic": "rw [\u2190 ENNReal.coe_le_coe, ENNReal.coe_mul]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_le_coe</a>, <a>ENNReal.coe_mul</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": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\nh' : 1 / (\u2191n + 1) * \u03bdA < c\n\u22a2 \u03bcA \u2264 1 / (\u2191n + 1) * \u03bdA", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\nh' : 1 / (\u2191n + 1) * \u03bdA < c\n\u22a2 \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA"}, {"tactic": "exact hA\u2083 n", "annotated_tactic": ["exact hA\u2083 n", []], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\nh' : 1 / (\u2191n + 1) * \u03bdA < c\n\u22a2 \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\n\u22a2 \u03bcA = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\nh : \u00ac\u03bcA = 0\n\u22a2 False"}, {"tactic": "have h' := this (\u03bcA / 2) (half_pos (zero_lt_iff.2 h))", "annotated_tactic": ["have h' := this (\u03bcA / 2) (<a>half_pos</a> (<a>zero_lt_iff</a>.2 h))", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\nh : \u00ac\u03bcA = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\nh : \u00ac\u03bcA = 0\nh' : \u03bcA \u2264 \u03bcA / 2\n\u22a2 False"}, {"tactic": "rw [\u2190 @Classical.not_not (\u03bcA \u2264 \u03bcA / 2)] at h'", "annotated_tactic": ["rw [\u2190 @<a>Classical.not_not</a> (\u03bcA \u2264 \u03bcA / 2)] at h'", [{"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\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\nh : \u00ac\u03bcA = 0\nh' : \u03bcA \u2264 \u03bcA / 2\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\nh : \u00ac\u03bcA = 0\nh' : \u00ac\u00ac\u03bcA \u2264 \u03bcA / 2\n\u22a2 False"}, {"tactic": "exact h' (not_le.2 (NNReal.half_lt_self h))", "annotated_tactic": ["exact h' (<a>not_le</a>.2 (<a>NNReal.half_lt_self</a> h))", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "NNReal.half_lt_self", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [888, 16], "def_end_pos": [888, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nthis : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b\nh : \u00ac\u03bcA = 0\nh' : \u00ac\u00ac\u03bcA \u2264 \u03bcA / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' mul_pos hc _", "annotated_tactic": ["refine' <a>mul_pos</a> hc _", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 0 < \u2191c * (\u2191\u03bdA)\u207b\u00b9", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 0 < (\u2191\u03bdA)\u207b\u00b9"}, {"tactic": "rw [_root_.inv_pos]", "annotated_tactic": ["rw [<a>_root_.inv_pos</a>]", [{"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 0 < (\u2191\u03bdA)\u207b\u00b9", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 0 < \u2191\u03bdA"}, {"tactic": "exact hb", "annotated_tactic": ["exact hb", []], "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 0 < \u2191\u03bdA", "state_after": "no goals"}, {"tactic": "rw [_root_.inv_pos]", "annotated_tactic": ["rw [<a>_root_.inv_pos</a>]", [{"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 0 < (\u2191\u03bdA)\u207b\u00b9", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 0 < \u2191\u03bdA"}, {"tactic": "exact hb", "annotated_tactic": ["exact hb", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\u22a2 0 < \u2191\u03bdA", "state_after": "no goals"}, {"tactic": "rw [\u2190 NNReal.coe_lt_coe, \u2190 mul_lt_mul_right hb\u2081, NNReal.coe_mul, mul_assoc, \u2190\n  NNReal.coe_inv, \u2190 NNReal.coe_mul, _root_.mul_inv_cancel, \u2190 NNReal.coe_mul, mul_one,\n  NNReal.coe_inv]", "annotated_tactic": ["rw [\u2190 <a>NNReal.coe_lt_coe</a>, \u2190 <a>mul_lt_mul_right</a> hb\u2081, <a>NNReal.coe_mul</a>, <a>mul_assoc</a>, \u2190\n          <a>NNReal.coe_inv</a>, \u2190 <a>NNReal.coe_mul</a>, <a>_root_.mul_inv_cancel</a>, \u2190 <a>NNReal.coe_mul</a>, <a>mul_one</a>,\n          <a>NNReal.coe_inv</a>]", [{"full_name": "NNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [371, 19], "def_end_pos": [371, 29]}, {"full_name": "mul_lt_mul_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [186, 19], "def_end_pos": [186, 26]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "NNReal.coe_inv", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [191, 19], "def_end_pos": [191, 26]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [186, 19], "def_end_pos": [186, 26]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [186, 19], "def_end_pos": [186, 26]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "NNReal.coe_inv", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [191, 19], "def_end_pos": [191, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\n\u22a2 1 / (\u2191n + 1) * \u03bdA < c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u2191(1 / (\u2191n + 1)) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bdA \u2260 0"}, {"tactic": "exact hn", "annotated_tactic": ["exact hn", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u2191(1 / (\u2191n + 1)) < \u2191c * (\u2191\u03bdA)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "exact Ne.symm (ne_of_lt hb)", "annotated_tactic": ["exact <a>Ne.symm</a> (<a>ne_of_lt</a> hb)", [{"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": "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\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : 0 < \u03bdA\nc : \u211d\u22650\nhc : 0 < c\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9\nhb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9\n\u22a2 \u03bdA \u2260 0", "state_after": "no goals"}, {"tactic": "rw [not_lt, le_zero_iff] at hb", "annotated_tactic": ["rw [<a>not_lt</a>, <a>le_zero_iff</a>] at hb", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u00ac0 < \u03bdA\n\u22a2 \u03bcA = 0", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u03bdA = 0\n\u22a2 \u03bcA = 0"}, {"tactic": "specialize hA\u2083 0", "annotated_tactic": ["specialize hA\u2083 0", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u03bdA = 0\n\u22a2 \u03bcA = 0", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u03bdA = 0\nhA\u2083 : \u2191\u03bcA \u2264 \u2191(1 / (\u21910 + 1)) * \u2191\u03bdA\n\u22a2 \u03bcA = 0"}, {"tactic": "simp [hb, le_zero_iff] at hA\u2083", "annotated_tactic": ["simp [hb, <a>le_zero_iff</a>] at hA\u2083", [{"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u03bdA = 0\nhA\u2083 : \u2191\u03bcA \u2264 \u2191(1 / (\u21910 + 1)) * \u2191\u03bdA\n\u22a2 \u03bcA = 0", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u03bdA = 0\nhA\u2083 : \u03bcA = 0\n\u22a2 \u03bcA = 0"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nh : \u00ac\u03bc \u27c2\u2098 \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c)\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264\n      VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c)\n\u03bcA : \u211d\u22650\nhA\u2081\u271d hA\u2081 : True\nhA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c)\n\u03bdA : \u211d\u22650\nhA\u2083\u271d : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA\nhb : \u03bdA = 0\nhA\u2083 : \u03bcA = 0\n\u22a2 \u03bcA = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n)\nhf\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict 0 (f n) \u2264\n      VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\nhf\u2083 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264\n      VectorMeasure.restrict 0 (f n)\u1d9c\nA : Set \u03b1 := \u22c2 n, (f n)\u1d9c\nhA\u2081 : A = \u22c2 n, (f n)\u1d9c\nhAmeas : MeasurableSet A\nhA\u2082 :\n  \u2200 (n : \u2115),\n    VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A\nhA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A\nh\u03bc : \u2191\u2191\u03bc A = 0\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0\nthis : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0\nn : \u2115\nhn : 0 < \u2191\u2191\u03bd (f n)\n\u22a2 0 < 1 / (\u2191n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_smul", "start": [194, 1], "end": [196, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "generatePiSystem_mono", "start": [262, 1], "end": [266, 63], "traced_tactics": [{"tactic": "induction' ht with s h_s s u _ _ h_nonempty h_s h_u", "annotated_tactic": ["induction' ht with s h_s s u _ _ h_nonempty h_s h_u", []], "state_before": "\u03b1 : Type u_1\nS T : Set (Set \u03b1)\nhST : S \u2286 T\nt : Set \u03b1\nht : t \u2208 generatePiSystem S\n\u22a2 t \u2208 generatePiSystem T", "state_after": "case base\n\u03b1 : Type u_1\nS T : Set (Set \u03b1)\nhST : S \u2286 T\nt s : Set \u03b1\nh_s : s \u2208 S\n\u22a2 s \u2208 generatePiSystem T\n\ncase inter\n\u03b1 : Type u_1\nS T : Set (Set \u03b1)\nhST : S \u2286 T\nt s u : Set \u03b1\nh_s\u271d : generatePiSystem S s\nh_t\u271d : generatePiSystem S u\nh_nonempty : Set.Nonempty (s \u2229 u)\nh_s : s \u2208 generatePiSystem T\nh_u : u \u2208 generatePiSystem T\n\u22a2 s \u2229 u \u2208 generatePiSystem T"}, {"tactic": "exact generatePiSystem.base (Set.mem_of_subset_of_mem hST h_s)", "annotated_tactic": ["exact <a>generatePiSystem.base</a> (<a>Set.mem_of_subset_of_mem</a> hST h_s)", [{"full_name": "generatePiSystem.base", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [238, 5], "def_end_pos": [238, 9]}, {"full_name": "Set.mem_of_subset_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 29]}]], "state_before": "case base\n\u03b1 : Type u_1\nS T : Set (Set \u03b1)\nhST : S \u2286 T\nt s : Set \u03b1\nh_s : s \u2208 S\n\u22a2 s \u2208 generatePiSystem T", "state_after": "no goals"}, {"tactic": "exact isPiSystem_generatePiSystem T _ h_s _ h_u h_nonempty", "annotated_tactic": ["exact <a>isPiSystem_generatePiSystem</a> T _ h_s _ h_u h_nonempty", [{"full_name": "isPiSystem_generatePiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [243, 9], "def_end_pos": [243, 36]}]], "state_before": "case inter\n\u03b1 : Type u_1\nS T : Set (Set \u03b1)\nhST : S \u2286 T\nt s u : Set \u03b1\nh_s\u271d : generatePiSystem S s\nh_t\u271d : generatePiSystem S u\nh_nonempty : Set.Nonempty (s \u2229 u)\nh_s : s \u2208 generatePiSystem T\nh_u : u \u2208 generatePiSystem T\n\u22a2 s \u2229 u \u2208 generatePiSystem T", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_inter_add_diff", "start": [120, 1], "end": [122, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/AddCircle.lean", "full_name": "AddCircle.isAddFundamentalDomain_of_ae_ball", "start": [54, 1], "end": [92, 22], "traced_tactics": [{"tactic": "set G := AddSubgroup.zmultiples u", "annotated_tactic": ["set G := <a>AddSubgroup.zmultiples</a> u", [{"full_name": "AddSubgroup.zmultiples", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\n\u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.zmultiples u } I", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I"}, {"tactic": "set n := addOrderOf u", "annotated_tactic": ["set n := <a>addOrderOf</a> u", [{"full_name": "addOrderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [133, 3], "def_end_pos": [133, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u)))\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191n))\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I"}, {"tactic": "set B := ball x (T / (2 * n))", "annotated_tactic": ["set B := <a>ball</a> x (T / (2 * n))", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nhI : I =\u1da0[ae volume] ball x (T / (2 * \u2191n))\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I"}, {"tactic": "have hn : 1 \u2264 (n : \u211d) := by norm_cast; linarith [addOrderOf_pos' hu]", "annotated_tactic": ["have hn : 1 \u2264 (n : \u211d) := by norm_cast; linarith [<a>addOrderOf_pos'</a> hu]", [{"full_name": "addOrderOf_pos'", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [151, 3], "def_end_pos": [151, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I"}, {"tactic": "refine' IsAddFundamentalDomain.mk_of_measure_univ_le _ _ _ _", "annotated_tactic": ["refine' <a>IsAddFundamentalDomain.mk_of_measure_univ_le</a> _ _ _ _", [{"full_name": "MeasureTheory.IsAddFundamentalDomain.mk_of_measure_univ_le", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [107, 3], "def_end_pos": [107, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 IsAddFundamentalDomain { x // x \u2208 G } I", "state_after": "case refine'_1\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 NullMeasurableSet I\n\ncase refine'_2\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 \u2200 (g : { x // x \u2208 G }), g \u2260 0 \u2192 AEDisjoint volume (g +\u1d65 I) I\n\ncase refine'_3\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 \u2200 (g : { x // x \u2208 G }), QuasiMeasurePreserving ((fun x x_1 => x +\u1d65 x_1) g)\n\ncase refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\n\u22a2 1 \u2264 \u2191n", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\n\u22a2 1 \u2264 n"}, {"tactic": "linarith [addOrderOf_pos' hu]", "annotated_tactic": ["linarith [<a>addOrderOf_pos'</a> hu]", [{"full_name": "addOrderOf_pos'", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [151, 3], "def_end_pos": [151, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\n\u22a2 1 \u2264 n", "state_after": "no goals"}, {"tactic": "exact measurableSet_ball.nullMeasurableSet.congr hI.symm", "annotated_tactic": ["exact measurableSet_ball.nullMeasurableSet.congr hI.symm", []], "state_before": "case refine'_1\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 NullMeasurableSet I", "state_after": "no goals"}, {"tactic": "rintro \u27e8g, hg\u27e9 hg'", "annotated_tactic": ["rintro \u27e8g, hg\u27e9 hg'", []], "state_before": "case refine'_2\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 \u2200 (g : { x // x \u2208 G }), g \u2260 0 \u2192 AEDisjoint volume (g +\u1d65 I) I", "state_after": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : { val := g, property := hg } \u2260 0\n\u22a2 AEDisjoint volume ({ val := g, property := hg } +\u1d65 I) I"}, {"tactic": "replace hg' : g \u2260 0", "annotated_tactic": ["replace hg' : g \u2260 0", []], "state_before": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : { val := g, property := hg } \u2260 0\n\u22a2 AEDisjoint volume ({ val := g, property := hg } +\u1d65 I) I", "state_after": "case hg'\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : { val := g, property := hg } \u2260 0\n\u22a2 g \u2260 0\n\ncase refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 AEDisjoint volume ({ val := g, property := hg } +\u1d65 I) I"}, {"tactic": "change AEDisjoint volume (g +\u1d65 I) I", "annotated_tactic": ["change <a>AEDisjoint</a> <a>volume</a> (g +\u1d65 I) I", [{"full_name": "MeasureTheory.AEDisjoint", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [26, 5], "def_end_pos": [26, 15]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}]], "state_before": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 AEDisjoint volume ({ val := g, property := hg } +\u1d65 I) I", "state_after": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 AEDisjoint volume (g +\u1d65 I) I"}, {"tactic": "refine' AEDisjoint.congr (Disjoint.aedisjoint _)\n  ((quasiMeasurePreserving_add_left volume (-g)).vadd_ae_eq_of_ae_eq g hI) hI", "annotated_tactic": ["refine' <a>AEDisjoint.congr</a> (<a>Disjoint.aedisjoint</a> _)\n      ((<a>quasiMeasurePreserving_add_left</a> <a>volume</a> (-g)).<a>vadd_ae_eq_of_ae_eq</a> g hI) hI", [{"full_name": "MeasureTheory.AEDisjoint.congr", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [87, 19], "def_end_pos": [87, 24]}, {"full_name": "Disjoint.aedisjoint", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [65, 19], "def_end_pos": [65, 45]}, {"full_name": "MeasureTheory.quasiMeasurePreserving_add_left", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [495, 3], "def_end_pos": [495, 14]}, {"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.QuasiMeasurePreserving.vadd_ae_eq_of_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2348, 3], "def_end_pos": [2348, 14]}]], "state_before": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 AEDisjoint volume (g +\u1d65 I) I", "state_after": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 Disjoint (g +\u1d65 B) B"}, {"tactic": "have hBg : g +\u1d65 B = ball (g + x) (T / (2 * n)) := by\n  rw [add_comm g x, \u2190 singleton_add_ball _ x g, add_ball, thickening_singleton]", "annotated_tactic": ["have hBg : g +\u1d65 B = <a>ball</a> (g + x) (T / (2 * n)) := by\n      rw [<a>add_comm</a> g x, \u2190 <a>singleton_add_ball</a> _ x g, <a>add_ball</a>, <a>thickening_singleton</a>]", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "singleton_add_ball", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "add_ball", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [228, 3], "def_end_pos": [228, 14]}, {"full_name": "Metric.thickening_singleton", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [988, 9], "def_end_pos": [988, 29]}]], "state_before": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 Disjoint (g +\u1d65 B) B", "state_after": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 Disjoint (g +\u1d65 B) B"}, {"tactic": "rw [hBg]", "annotated_tactic": ["rw [hBg]", []], "state_before": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 Disjoint (g +\u1d65 B) B", "state_after": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 Disjoint (ball (g + x) (T / (2 * \u2191n))) B"}, {"tactic": "apply ball_disjoint_ball", "annotated_tactic": ["apply <a>ball_disjoint_ball</a>", [{"full_name": "Metric.ball_disjoint_ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [550, 9], "def_end_pos": [550, 27]}]], "state_before": "case refine'_2.mk\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 Disjoint (ball (g + x) (T / (2 * \u2191n))) B", "state_after": "case refine'_2.mk.h\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 T / (2 * \u2191n) + T / (2 * \u2191n) \u2264 dist (g + x) x"}, {"tactic": "rw [dist_eq_norm, add_sub_cancel, div_mul_eq_div_div, \u2190 add_div, \u2190 add_div, add_self_div_two,\n  div_le_iff' (by positivity : 0 < (n : \u211d)), \u2190 nsmul_eq_mul]", "annotated_tactic": ["rw [<a>dist_eq_norm</a>, <a>add_sub_cancel</a>, <a>div_mul_eq_div_div</a>, \u2190 <a>add_div</a>, \u2190 <a>add_div</a>, <a>add_self_div_two</a>,\n      <a>div_le_iff'</a> (by positivity : 0 < (n : \u211d)), \u2190 <a>nsmul_eq_mul</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": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}, {"full_name": "div_mul_eq_div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [547, 9], "def_end_pos": [547, 27]}, {"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "add_self_div_two", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [500, 9], "def_end_pos": [500, 25]}, {"full_name": "div_le_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 20]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}]], "state_before": "case refine'_2.mk.h\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 T / (2 * \u2191n) + T / (2 * \u2191n) \u2264 dist (g + x) x", "state_after": "case refine'_2.mk.h\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 T \u2264 n \u2022 \u2016g\u2016"}, {"tactic": "refine' (le_add_order_smul_norm_of_isOfFinAddOrder (hu.of_mem_zmultiples hg) hg').trans\n  (nsmul_le_nsmul (norm_nonneg g) _)", "annotated_tactic": ["refine' (<a>le_add_order_smul_norm_of_isOfFinAddOrder</a> (hu.of_mem_zmultiples hg) hg').<a>trans</a>\n      (<a>nsmul_le_nsmul</a> (<a>norm_nonneg</a> g) _)", [{"full_name": "AddCircle.le_add_order_smul_norm_of_isOfFinAddOrder", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [261, 9], "def_end_pos": [261, 50]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "nsmul_le_nsmul", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [65, 15], "def_end_pos": [65, 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 refine'_2.mk.h\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 T \u2264 n \u2022 \u2016g\u2016", "state_after": "case refine'_2.mk.h\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 addOrderOf g \u2264 n"}, {"tactic": "exact Nat.le_of_dvd (addOrderOf_pos_iff.mpr hu) (addOrderOf_dvd_of_mem_zmultiples hg)", "annotated_tactic": ["exact <a>Nat.le_of_dvd</a> (addOrderOf_pos_iff.mpr hu) (<a>addOrderOf_dvd_of_mem_zmultiples</a> hg)", [{"full_name": "Nat.le_of_dvd", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [907, 9], "def_end_pos": [907, 18]}, {"full_name": "addOrderOf_dvd_of_mem_zmultiples", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [611, 15], "def_end_pos": [611, 47]}]], "state_before": "case refine'_2.mk.h\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 addOrderOf g \u2264 n", "state_after": "no goals"}, {"tactic": "simpa only [Ne.def, AddSubgroup.mk_eq_zero_iff] using hg'", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>AddSubgroup.mk_eq_zero_iff</a>] using hg'", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "AddSubgroup.mk_eq_zero_iff", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [743, 3], "def_end_pos": [743, 14]}]], "state_before": "case hg'\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : { val := g, property := hg } \u2260 0\n\u22a2 g \u2260 0", "state_after": "no goals"}, {"tactic": "rw [add_comm g x, \u2190 singleton_add_ball _ x g, add_ball, thickening_singleton]", "annotated_tactic": ["rw [<a>add_comm</a> g x, \u2190 <a>singleton_add_ball</a> _ x g, <a>add_ball</a>, <a>thickening_singleton</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "singleton_add_ball", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "add_ball", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [228, 3], "def_end_pos": [228, 14]}, {"full_name": "Metric.thickening_singleton", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [988, 9], "def_end_pos": [988, 29]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\n\u22a2 g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\ng : AddCircle T\nhg : g \u2208 G\nhg' : g \u2260 0\nhBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n))\n\u22a2 0 < \u2191n", "state_after": "no goals"}, {"tactic": "exact fun g => quasiMeasurePreserving_add_left (G := AddCircle T) volume g", "annotated_tactic": ["exact fun g => <a>quasiMeasurePreserving_add_left</a> (G := <a>AddCircle</a> T) <a>volume</a> g", [{"full_name": "MeasureTheory.quasiMeasurePreserving_add_left", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [495, 3], "def_end_pos": [495, 14]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [124, 8], "def_end_pos": [124, 17]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}]], "state_before": "case refine'_3\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 \u2200 (g : { x // x \u2208 G }), QuasiMeasurePreserving ((fun x x_1 => x +\u1d65 x_1) g)", "state_after": "no goals"}, {"tactic": "replace hI := hI.trans closedBall_ae_eq_ball.symm", "annotated_tactic": ["replace hI := hI.trans closedBall_ae_eq_ball.symm", []], "state_before": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhI : I =\u1da0[ae volume] B\nhn : 1 \u2264 \u2191n\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)", "state_after": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)"}, {"tactic": "haveI : Fintype G := @Fintype.ofFinite _ hu.finite_zmultiples", "annotated_tactic": ["haveI : <a>Fintype</a> G := @<a>Fintype.ofFinite</a> _ hu.finite_zmultiples", [{"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [54, 7], "def_end_pos": [54, 14]}, {"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [448, 19], "def_end_pos": [448, 35]}]], "state_before": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)", "state_after": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)"}, {"tactic": "have hG_card : (Finset.univ : Finset G).card = n := by\n  show _ = addOrderOf u\n  rw [add_order_eq_card_zmultiples', Nat.card_eq_fintype_card]; rfl", "annotated_tactic": ["have hG_card : (<a>Finset.univ</a> : <a>Finset</a> G).<a>card</a> = n := by\n      show _ = <a>addOrderOf</a> u\n      rw [<a>add_order_eq_card_zmultiples'</a>, <a>Nat.card_eq_fintype_card</a>]; rfl", [{"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "addOrderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [133, 3], "def_end_pos": [133, 14]}, {"full_name": "add_order_eq_card_zmultiples'", "def_path": "Mathlib/Data/ZMod/Quotient.lean", "def_pos": [207, 15], "def_end_pos": [207, 44]}, {"full_name": "Nat.card_eq_fintype_card", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [42, 9], "def_end_pos": [42, 29]}]], "state_before": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)", "state_after": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\nhG_card : Finset.card Finset.univ = n\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)"}, {"tactic": "simp_rw [measure_vadd]", "annotated_tactic": ["simp_rw [<a>measure_vadd</a>]", [{"full_name": "MeasureTheory.measure_vadd", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [214, 3], "def_end_pos": [214, 14]}]], "state_before": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\nhG_card : Finset.card Finset.univ = n\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I)", "state_after": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\nhG_card : Finset.card Finset.univ = n\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 AddSubgroup.zmultiples u }), \u2191\u2191volume I"}, {"tactic": "rw [AddCircle.measure_univ, tsum_fintype, Finset.sum_const, measure_congr hI,\n  volume_closedBall, \u2190 ENNReal.ofReal_nsmul, mul_div, mul_div_mul_comm,\n  div_self, one_mul, min_eq_right (div_le_self hT.out.le hn), hG_card,\n  nsmul_eq_mul, mul_div_cancel' T (lt_of_lt_of_le zero_lt_one hn).ne.symm]", "annotated_tactic": ["rw [<a>AddCircle.measure_univ</a>, <a>tsum_fintype</a>, <a>Finset.sum_const</a>, <a>measure_congr</a> hI,\n      <a>volume_closedBall</a>, \u2190 <a>ENNReal.ofReal_nsmul</a>, <a>mul_div</a>, <a>mul_div_mul_comm</a>,\n      <a>div_self</a>, <a>one_mul</a>, <a>min_eq_right</a> (<a>div_le_self</a> hT.out.le hn), hG_card,\n      <a>nsmul_eq_mul</a>, <a>mul_div_cancel'</a> T (<a>lt_of_lt_of_le</a> <a>zero_lt_one</a> hn).ne.symm]", [{"full_name": "AddCircle.measure_univ", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [69, 19], "def_end_pos": [69, 31]}, {"full_name": "tsum_fintype", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 21]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "AddCircle.volume_closedBall", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [96, 9], "def_end_pos": [96, 26]}, {"full_name": "ENNReal.ofReal_nsmul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2239, 9], "def_end_pos": [2239, 21]}, {"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "mul_div_mul_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 25]}, {"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "min_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}, {"full_name": "div_le_self", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 20]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"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 refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\nhG_card : Finset.card Finset.univ = n\n\u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 AddSubgroup.zmultiples u }), \u2191\u2191volume I", "state_after": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\nhG_card : Finset.card Finset.univ = n\n\u22a2 2 \u2260 0"}, {"tactic": "exact two_ne_zero", "annotated_tactic": ["exact <a>two_ne_zero</a>", [{"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}]], "state_before": "case refine'_4\nT : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\nhG_card : Finset.card Finset.univ = n\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "show _ = addOrderOf u", "annotated_tactic": ["show _ = <a>addOrderOf</a> u", [{"full_name": "addOrderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [133, 3], "def_end_pos": [133, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 Finset.card Finset.univ = n", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 Finset.card Finset.univ = addOrderOf u"}, {"tactic": "rw [add_order_eq_card_zmultiples', Nat.card_eq_fintype_card]", "annotated_tactic": ["rw [<a>add_order_eq_card_zmultiples'</a>, <a>Nat.card_eq_fintype_card</a>]", [{"full_name": "add_order_eq_card_zmultiples'", "def_path": "Mathlib/Data/ZMod/Quotient.lean", "def_pos": [207, 15], "def_end_pos": [207, 44]}, {"full_name": "Nat.card_eq_fintype_card", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [42, 9], "def_end_pos": [42, 29]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 Finset.card Finset.univ = addOrderOf u", "state_after": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 Finset.card Finset.univ = Fintype.card { x // x \u2208 AddSubgroup.zmultiples u }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : \u2115 := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * \u2191n))\nhn : 1 \u2264 \u2191n\nhI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n))\nthis : Fintype { x // x \u2208 G }\n\u22a2 Finset.card Finset.univ = Fintype.card { x // x \u2208 AddSubgroup.zmultiples u }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "full_name": "MvPolynomial.pderiv_monomial", "start": [71, 1], "end": [79, 11], "traced_tactics": [{"tactic": "simp only [pderiv_def, mkDerivation_monomial, Finsupp.smul_sum, smul_eq_mul, \u2190 smul_mul_assoc,\n  \u2190 (monomial _).map_smul]", "annotated_tactic": ["simp only [<a>pderiv_def</a>, <a>mkDerivation_monomial</a>, <a>Finsupp.smul_sum</a>, <a>smul_eq_mul</a>, \u2190 <a>smul_mul_assoc</a>,\n      \u2190 (<a>monomial</a> _).<a>map_smul</a>]", [{"full_name": "MvPolynomial.pderiv_def", "def_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "def_pos": [66, 9], "def_end_pos": [66, 19]}, {"full_name": "MvPolynomial.mkDerivation_monomial", "def_path": "Mathlib/Data/MvPolynomial/Derivation.lean", "def_pos": [145, 9], "def_end_pos": [145, 30]}, {"full_name": "Finsupp.smul_sum", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 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_mul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [429, 9], "def_end_pos": [429, 23]}, {"full_name": "MvPolynomial.monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [167, 5], "def_end_pos": [167, 13]}, {"full_name": "LinearMap.map_smul", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [350, 19], "def_end_pos": [350, 27]}]], "state_before": "R : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\n\u22a2 \u2191(pderiv i) (\u2191(monomial s) a) = \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i))", "state_after": "R : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\n\u22a2 (sum s fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) =\n    \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i))"}, {"tactic": "refine' (Finset.sum_eq_single i (fun j _ hne => _) fun hi => _).trans _", "annotated_tactic": ["refine' (<a>Finset.sum_eq_single</a> i (fun j _ hne => _) fun hi => _).<a>trans</a> _", [{"full_name": "Finset.sum_eq_single", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [798, 3], "def_end_pos": [798, 14]}, {"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": "R : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\n\u22a2 (sum s fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) =\n    \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i))", "state_after": "case refine'_1\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni j : \u03c3\nx\u271d : j \u2208 s.support\nhne : j \u2260 i\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) j (\u2191s j) = 0\n\ncase refine'_2\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\nhi : \u00aci \u2208 s.support\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) = 0\n\ncase refine'_3\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) =\n    \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i))"}, {"tactic": "simp [Pi.single_eq_of_ne hne]", "annotated_tactic": ["simp [<a>Pi.single_eq_of_ne</a> hne]", [{"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 refine'_1\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni j : \u03c3\nx\u271d : j \u2208 s.support\nhne : j \u2260 i\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) j (\u2191s j) = 0", "state_after": "no goals"}, {"tactic": "rw [Finsupp.not_mem_support_iff] at hi", "annotated_tactic": ["rw [<a>Finsupp.not_mem_support_iff</a>] at hi", [{"full_name": "Finsupp.not_mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [195, 9], "def_end_pos": [195, 28]}]], "state_before": "case refine'_2\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\nhi : \u00aci \u2208 s.support\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) = 0", "state_after": "case refine'_2\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\nhi : \u2191s i = 0\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) = 0"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "case refine'_2\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\nhi : \u2191s i = 0\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine'_3\nR : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\ni : \u03c3\n\u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) =\n    \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.coe_le_one", "start": [136, 1], "end": [138, 55], "traced_tactics": [{"tactic": "refine' hasSum_le (fun b => _) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)", "annotated_tactic": ["refine' <a>hasSum_le</a> (fun b => _) (<a>hasSum_ite_eq</a> a (p a)) (<a>hasSum_coe_one</a> p)", [{"full_name": "hasSum_le", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [43, 9], "def_end_pos": [43, 18]}, {"full_name": "hasSum_ite_eq", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 22]}, {"full_name": "PMF.hasSum_coe_one", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 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 \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na b : \u03b1\n\u22a2 (if b = a then \u2191p a else 0) \u2264 \u2191p b"}, {"tactic": "split_ifs with h <;> simp only [h, zero_le', le_rfl]", "annotated_tactic": ["split_ifs with h <;> simp only [h, <a>zero_le'</a>, <a>le_rfl</a>]", [{"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 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\np : PMF \u03b1\na b : \u03b1\n\u22a2 (if b = a then \u2191p a else 0) \u2264 \u2191p b", "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.bind_eq_bindTR", "start": [81, 10], "end": [86, 25], "traced_tactics": [{"tactic": "funext \u03b1 \u03b2 as f", "annotated_tactic": ["funext \u03b1 \u03b2 as f", []], "state_before": "\u22a2 @List.bind = @bindTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nas : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 List.bind as f = bindTR as f"}, {"tactic": "exact (go as #[]).symm", "annotated_tactic": ["exact (go as #[]).<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\nas : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 List.bind as f = bindTR as f", "state_after": "no goals"}, {"tactic": "simp [bindTR.go, bind]", "annotated_tactic": ["simp [<a>bindTR.go</a>, <a>bind</a>]", [{"full_name": "List.bindTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [77, 17], "def_end_pos": [77, 19]}, {"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": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nas : List \u03b1\nf : \u03b1 \u2192 List \u03b2\nacc : Array \u03b2\n\u22a2 bindTR.go f [] acc = acc.data ++ List.bind [] f", "state_after": "no goals"}, {"tactic": "simp [bindTR.go, bind, go xs]", "annotated_tactic": ["simp [<a>bindTR.go</a>, <a>bind</a>, go xs]", [{"full_name": "List.bindTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [77, 17], "def_end_pos": [77, 19]}, {"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": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nas : List \u03b1\nf : \u03b1 \u2192 List \u03b2\nx : \u03b1\nxs : List \u03b1\nacc : Array \u03b2\n\u22a2 bindTR.go f (x :: xs) acc = acc.data ++ List.bind (x :: xs) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.cast_zero'", "start": [1028, 1], "end": [1029, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.lookup_insert", "start": [498, 1], "end": [499, 95], "traced_tactics": [{"tactic": "simp only [insert_toFinmap, lookup_toFinmap, AList.lookup_insert]", "annotated_tactic": ["simp only [<a>insert_toFinmap</a>, <a>lookup_toFinmap</a>, <a>AList.lookup_insert</a>]", [{"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", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [310, 9], "def_end_pos": [310, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u271d : Finmap \u03b2\ns : AList \u03b2\n\u22a2 lookup a (insert a b \u27e6s\u27e7) = some b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_ret_halt", "start": [1144, 1], "end": [1144, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.IsMetric.finset_iUnion_of_pairwise_separated", "start": [144, 1], "end": [154, 74], "traced_tactics": [{"tactic": "induction' I using Finset.induction_on with i I hiI ihI hI", "annotated_tactic": ["induction' I using <a>Finset.induction_on</a> with i I hiI ihI hI", [{"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\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)", "state_after": "case empty\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\nI : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\nhI : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i \u2208 \u2205, s i) = \u2211 i in \u2205, \u2191\u03bc (s i)\n\ncase insert\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 \u2200 (j : \u03b9), j \u2208 insert i I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i_1 \u2208 insert i I, s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i)"}, {"tactic": "simp only [Finset.mem_insert] at hI", "annotated_tactic": ["simp only [<a>Finset.mem_insert</a>] at hI", [{"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\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 \u2200 (j : \u03b9), j \u2208 insert i I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i_1 \u2208 insert i I, s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i)", "state_after": "case insert\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i_1 \u2208 insert i I, s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i)"}, {"tactic": "rw [Finset.set_biUnion_insert, hm, ihI, Finset.sum_insert hiI]", "annotated_tactic": ["rw [<a>Finset.set_biUnion_insert</a>, hm, ihI, <a>Finset.sum_insert</a> hiI]", [{"full_name": "Finset.set_biUnion_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2136, 9], "def_end_pos": [2136, 27]}, {"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\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i_1 \u2208 insert i I, s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i)", "state_after": "case insert\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\ncase insert.a\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 IsMetricSeparated (s i) (\u22c3 x \u2208 I, s x)"}, {"tactic": "exacts [fun i hi j hj hij => hI i (Or.inr hi) j (Or.inr hj) hij,\n  IsMetricSeparated.finset_iUnion_right fun j hj =>\n    hI i (Or.inl rfl) j (Or.inr hj) (ne_of_mem_of_not_mem hj hiI).symm]", "annotated_tactic": ["exacts [fun i hi j hj hij => hI i (<a>Or.inr</a> hi) j (<a>Or.inr</a> hj) hij,\n    <a>IsMetricSeparated.finset_iUnion_right</a> fun j hj =>\n      hI i (<a>Or.inl</a> <a>rfl</a>) j (<a>Or.inr</a> hj) (<a>ne_of_mem_of_not_mem</a> hj hiI).<a>symm</a>]", [{"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.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "IsMetricSeparated.finset_iUnion_right", "def_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "def_pos": [135, 11], "def_end_pos": [135, 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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"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": "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\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\ncase insert.a\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\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 IsMetricSeparated (s i) (\u22c3 x \u2208 I, s x)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\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\nI : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\nhI : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\u22a2 \u2191\u03bc (\u22c3 i \u2208 \u2205, s i) = \u2211 i in \u2205, \u2191\u03bc (s i)", "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_of_ne", "start": [405, 1], "end": [407, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "generateFrom_generatePiSystem_eq", "start": [283, 1], "end": [287, 78], "traced_tactics": [{"tactic": "apply le_antisymm <;> apply generateFrom_le", "annotated_tactic": ["apply <a>le_antisymm</a> <;> apply <a>generateFrom_le</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]}]], "state_before": "\u03b1 : Type u_1\ng : Set (Set \u03b1)\n\u22a2 generateFrom (generatePiSystem g) = generateFrom g", "state_after": "case a.h\n\u03b1 : Type u_1\ng : Set (Set \u03b1)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 generatePiSystem g \u2192 MeasurableSet t\n\ncase a.h\n\u03b1 : Type u_1\ng : Set (Set \u03b1)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 g \u2192 MeasurableSet t"}, {"tactic": "exact fun t h_t => generateFrom_measurableSet_of_generatePiSystem t h_t", "annotated_tactic": ["exact fun t h_t => <a>generateFrom_measurableSet_of_generatePiSystem</a> t h_t", [{"full_name": "generateFrom_measurableSet_of_generatePiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [277, 9], "def_end_pos": [277, 55]}]], "state_before": "case a.h\n\u03b1 : Type u_1\ng : Set (Set \u03b1)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 generatePiSystem g \u2192 MeasurableSet t", "state_after": "no goals"}, {"tactic": "exact fun t h_t => measurableSet_generateFrom (generatePiSystem.base h_t)", "annotated_tactic": ["exact fun t h_t => <a>measurableSet_generateFrom</a> (<a>generatePiSystem.base</a> h_t)", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"full_name": "generatePiSystem.base", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [238, 5], "def_end_pos": [238, 9]}]], "state_before": "case a.h\n\u03b1 : Type u_1\ng : Set (Set \u03b1)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 g \u2192 MeasurableSet t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/BorelCantelli.lean", "full_name": "ProbabilityTheory.iIndepFun.condexp_natural_ae_eq_of_lt", "start": [50, 1], "end": [54, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "ne_self_iff_false", "start": [84, 1], "end": [84, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.iUnion_smul_eq_setOf_exists", "start": [1003, 1], "end": [1004, 76], "traced_tactics": [{"tactic": "simp_rw [\u2190 iUnion_setOf, \u2190 iUnion_inv_smul, \u2190 preimage_smul, preimage]", "annotated_tactic": ["simp_rw [\u2190 <a>iUnion_setOf</a>, \u2190 <a>iUnion_inv_smul</a>, \u2190 <a>preimage_smul</a>, <a>preimage</a>]", [{"full_name": "Set.iUnion_setOf", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [546, 9], "def_end_pos": [546, 21]}, {"full_name": "Set.iUnion_inv_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [997, 9], "def_end_pos": [997, 24]}, {"full_name": "Set.preimage_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [899, 9], "def_end_pos": [899, 22]}, {"full_name": "Set.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t A B : Set \u03b2\na : \u03b1\nx : \u03b2\ns : Set \u03b2\n\u22a2 \u22c3 g, g \u2022 s = {a | \u2203 g, g \u2022 a \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "set_integral_re_add_im", "start": [1225, 1], "end": [1228, 24], "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_right", "start": [1113, 11], "end": [1114, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.exists_mk'", "start": [576, 1], "end": [577, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.neg_mkRat", "start": [217, 1], "end": [218, 79], "traced_tactics": [{"tactic": "if z : d = 0 then simp [z] else simp [\u2190 normalize_eq_mkRat z, neg_normalize]", "annotated_tactic": ["if z : d = 0 then simp [z] else simp [\u2190 <a>normalize_eq_mkRat</a> z, <a>neg_normalize</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.neg_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 22]}]], "state_before": "n : Int\nd : Nat\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}, {"tactic": "simp [z]", "annotated_tactic": ["simp [z]", []], "state_before": "n : Int\nd : Nat\nz : d = 0\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}, {"tactic": "simp [\u2190 normalize_eq_mkRat z, neg_normalize]", "annotated_tactic": ["simp [\u2190 <a>normalize_eq_mkRat</a> z, <a>neg_normalize</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.neg_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 22]}]], "state_before": "n : Int\nd : Nat\nz : \u00acd = 0\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.integral_compProd", "start": [245, 1], "end": [268, 62], "traced_tactics": [{"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": "\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 {f : \u03b2 \u00d7 \u03b3 \u2192 E}, Integrable f \u2192 \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h_ind\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'\n\u22a2 \u2200 (c : E) \u2983s : Set (\u03b2 \u00d7 \u03b3)\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u2192\n        \u222b (z : \u03b2 \u00d7 \u03b3), indicator s (fun x => c) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a =\n          \u222b (x : \u03b2), \u222b (y : \u03b3), indicator s (fun x => c) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\ncase h_add\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'\n\u22a2 \u2200 \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984,\n    Disjoint (support f) (support g) \u2192\n      Integrable f \u2192\n        Integrable g \u2192\n          \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192\n            \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192\n              \u222b (z : \u03b2 \u00d7 \u03b3), (f + g) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), (f + g) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\ncase h_closed\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'\n\u22a2 IsClosed {f | \u222b (z : \u03b2 \u00d7 \u03b3), \u2191\u2191f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a}\n\ncase h_ae\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'\n\u22a2 \u2200 \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984,\n    f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g \u2192\n      Integrable f \u2192\n        \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192\n          \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a"}, {"tactic": "intro c s hs h2s", "annotated_tactic": ["intro c s hs h2s", []], "state_before": "case h_ind\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'\n\u22a2 \u2200 (c : E) \u2983s : Set (\u03b2 \u00d7 \u03b3)\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u2192\n        \u222b (z : \u03b2 \u00d7 \u03b3), indicator s (fun x => c) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a =\n          \u222b (x : \u03b2), \u222b (y : \u03b3), indicator s (fun x => c) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h_ind\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), indicator s (fun x => c) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a =\n    \u222b (x : \u03b2), \u222b (y : \u03b3), indicator s (fun x => c) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a"}, {"tactic": "simp_rw [integral_indicator hs, \u2190 indicator_comp_right, Function.comp,\n  integral_indicator (measurable_prod_mk_left hs), MeasureTheory.set_integral_const,\n  integral_smul_const]", "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>MeasureTheory.set_integral_const</a>,\n      <a>integral_smul_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": "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]}]], "state_before": "case h_ind\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), indicator s (fun x => c) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a =\n    \u222b (x : \u03b2), \u222b (y : \u03b3), indicator s (fun x => c) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h_ind\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) \u2022 c = (\u222b (x : \u03b2), ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)) \u2202\u2191\u03ba a) \u2022 c"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h_ind\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) \u2022 c = (\u222b (x : \u03b2), ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)) \u2202\u2191\u03ba a) \u2022 c", "state_after": "case h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = \u222b (x : \u03b2), ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)) \u2202\u2191\u03ba a"}, {"tactic": "rw [integral_toReal]", "annotated_tactic": ["rw [<a>integral_toReal</a>]", [{"full_name": "MeasureTheory.integral_toReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 24]}]], "state_before": "case h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = \u222b (x : \u03b2), ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)) \u2202\u2191\u03ba a", "state_after": "case h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)\n\ncase h_ind.e_a.hfm\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 AEMeasurable fun x => \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)\n\ncase h_ind.e_a.hf\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) < \u22a4"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "state_before": "case h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)\n\ncase h_ind.e_a.hfm\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 AEMeasurable fun x => \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)\n\ncase h_ind.e_a.hf\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) < \u22a4", "state_after": "case h_ind.e_a.hfm\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 AEMeasurable fun x => \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)\n\ncase h_ind.e_a.hf\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) < \u22a4\n\ncase h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)"}, {"tactic": "rw [kernel.compProd_apply _ _ _ hs]", "annotated_tactic": ["rw [<a>kernel.compProd_apply</a> _ _ _ hs]", [{"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_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)", "state_after": "case h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a) =\n    ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h_ind.e_a\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a) =\n    ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)", "state_after": "no goals"}, {"tactic": "exact (kernel.measurable_kernel_prod_mk_left' hs _).aemeasurable", "annotated_tactic": ["exact (<a>kernel.measurable_kernel_prod_mk_left'</a> hs _).<a>aemeasurable</a>", [{"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left'", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [111, 9], "def_end_pos": [111, 40]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}]], "state_before": "case h_ind.e_a.hfm\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 AEMeasurable fun x => \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "exact ae_kernel_lt_top a h2s.ne", "annotated_tactic": ["exact <a>ae_kernel_lt_top</a> a h2s.ne", [{"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": "case h_ind.e_a.hf\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'\nc : E\ns : Set (\u03b2 \u00d7 \u03b3)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) < \u22a4", "state_after": "no goals"}, {"tactic": "intro f g _ i_f i_g hf hg", "annotated_tactic": ["intro f g _ i_f i_g hf hg", []], "state_before": "case h_add\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'\n\u22a2 \u2200 \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984,\n    Disjoint (support f) (support g) \u2192\n      Integrable f \u2192\n        Integrable g \u2192\n          \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192\n            \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192\n              \u222b (z : \u03b2 \u00d7 \u03b3), (f + g) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), (f + g) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h_add\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\na\u271d : Disjoint (support f) (support g)\ni_f : Integrable f\ni_g : Integrable g\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\nhg : \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), (f + g) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), (f + g) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a"}, {"tactic": "simp_rw [integral_add' i_f i_g, kernel.integral_integral_add' i_f i_g, hf, hg]", "annotated_tactic": ["simp_rw [<a>integral_add'</a> i_f i_g, <a>kernel.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": "ProbabilityTheory.kernel.integral_integral_add'", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [194, 9], "def_end_pos": [194, 38]}]], "state_before": "case h_add\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\na\u271d : Disjoint (support f) (support g)\ni_f : Integrable f\ni_g : Integrable g\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\nhg : \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), (f + g) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), (f + g) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "no goals"}, {"tactic": "exact isClosed_eq continuous_integral kernel.continuous_integral_integral", "annotated_tactic": ["exact <a>isClosed_eq</a> <a>continuous_integral</a> <a>kernel.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": "ProbabilityTheory.kernel.continuous_integral_integral", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [217, 9], "def_end_pos": [217, 44]}]], "state_before": "case h_closed\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'\n\u22a2 IsClosed {f | \u222b (z : \u03b2 \u00d7 \u03b3), \u2191\u2191f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a}", "state_after": "no goals"}, {"tactic": "intro f g hfg _ hf", "annotated_tactic": ["intro f g hfg _ hf", []], "state_before": "case h_ae\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'\n\u22a2 \u2200 \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984,\n    f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g \u2192\n      Integrable f \u2192\n        \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192\n          \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h_ae\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a"}, {"tactic": "convert hf using 1", "annotated_tactic": ["convert hf using 1", []], "state_before": "case h_ae\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h.e'_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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a\n\ncase h.e'_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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a"}, {"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\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a", "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\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "case h.e'_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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 (fun x => \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x)) =\u1d50[\u2191\u03ba a] fun x => \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x)"}, {"tactic": "refine' (ae_ae_of_ae_compProd hfg).mp (eventually_of_forall _)", "annotated_tactic": ["refine' (<a>ae_ae_of_ae_compProd</a> hfg).<a>mp</a> (<a>eventually_of_forall</a> _)", [{"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": "Filter.Eventually.mp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 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 h.e'_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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 (fun x => \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x)) =\u1d50[\u2191\u03ba a] fun x => \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x)", "state_after": "case h.e'_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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u2200 (x : \u03b2),\n    (\u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, x), f (x, c) = g (x, c)) \u2192\n      (fun x => \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x)) x = (fun x => \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x)) x"}, {"tactic": "exact fun x hfgx => integral_congr_ae (ae_eq_symm hfgx)", "annotated_tactic": ["exact fun x hfgx => <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\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'\nf g : \u03b2 \u00d7 \u03b3 \u2192 E\nhfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g\na\u271d : Integrable f\nhf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a\n\u22a2 \u2200 (x : \u03b2),\n    (\u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, x), f (x, c) = g (x, c)) \u2192\n      (fun x => \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x)) x = (fun x => \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.lookup_union_right", "start": [588, 1], "end": [589, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.neg_lt_sub_left_of_lt_add", "start": [1109, 11], "end": [1111, 26], "traced_tactics": [{"tactic": "have h := Int.lt_neg_add_of_add_lt (Int.sub_left_lt_of_lt_add h)", "annotated_tactic": ["have h := <a>Int.lt_neg_add_of_add_lt</a> (<a>Int.sub_left_lt_of_lt_add</a> h)", [{"full_name": "Int.lt_neg_add_of_add_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1046, 19], "def_end_pos": [1046, 39]}, {"full_name": "Int.sub_left_lt_of_lt_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1078, 19], "def_end_pos": [1078, 40]}]], "state_before": "a b c : Int\nh : c < a + b\n\u22a2 -a < b - c", "state_after": "a b c : Int\nh\u271d : c < a + b\nh : -a < -c + b\n\u22a2 -a < b - c"}, {"tactic": "rwa [Int.add_comm] at h", "annotated_tactic": ["rwa [<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\u271d : c < a + b\nh : -a < -c + b\n\u22a2 -a < b - c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector3.lean", "full_name": "vectorEx_iff_exists", "start": [242, 1], "end": [245, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.range_loop_range'", "start": [2060, 1], "end": [2062, 101], "traced_tactics": [{"tactic": "rw [\u2190 Nat.add_assoc, Nat.add_right_comm n s 1]", "annotated_tactic": ["rw [\u2190 <a>Nat.add_assoc</a>, <a>Nat.add_right_comm</a> n s 1]", [{"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_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 n : Nat\n\u22a2 range.loop (s + 1) (range' (s + 1) n) = range' 0 (n + (s + 1))", "state_after": "s n : Nat\n\u22a2 range.loop (s + 1) (range' (s + 1) n) = range' 0 (n + 1 + s)"}, {"tactic": "exact range_loop_range' s (n + 1)", "annotated_tactic": ["exact range_loop_range' s (n + 1)", []], "state_before": "s n : Nat\n\u22a2 range.loop (s + 1) (range' (s + 1) n) = range' 0 (n + 1 + s)", "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.erase", "start": [1171, 1], "end": [1172, 49], "traced_tactics": [{"tactic": "simp [erase_eq_eraseP]", "annotated_tactic": ["simp [<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\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 List.erase l\u2081 a <+ List.erase l\u2082 a", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 List.eraseP (fun b => decide (a = b)) l\u2081 <+ List.eraseP (fun b => decide (a = b)) l\u2082"}, {"tactic": "exact Sublist.eraseP h", "annotated_tactic": ["exact <a>Sublist.eraseP</a> h", [{"full_name": "List.Sublist.eraseP", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 List.eraseP (fun b => decide (a = b)) l\u2081 <+ List.eraseP (fun b => decide (a = b)) l\u2082", "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.le_size", "start": [73, 1], "end": [81, 67], "traced_tactics": [{"tactic": "rw [size, Nat.add_right_comm (size _), Nat.add_assoc, depthLB, Nat.pow_succ, Nat.mul_two]", "annotated_tactic": ["rw [<a>size</a>, <a>Nat.add_right_comm</a> (<a>size</a> _), <a>Nat.add_assoc</a>, <a>depthLB</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": "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.RBNode.size", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [346, 13], "def_end_pos": [346, 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": "Std.RBNode.depthLB", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [43, 5], "def_end_pos": [43, 12]}, {"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\nt : RBNode \u03b1\nc : RBColor\nn\u271d n : Nat\nx\u271d y\u271d : RBNode \u03b1\nv\u271d : \u03b1\nhl : Balanced x\u271d black n\nhr : Balanced y\u271d black n\n\u22a2 2 ^ depthLB red n \u2264 size (node red x\u271d v\u271d y\u271d) + 1", "state_after": "\u03b1 : Type u_1\nt : RBNode \u03b1\nc : RBColor\nn\u271d n : Nat\nx\u271d y\u271d : RBNode \u03b1\nv\u271d : \u03b1\nhl : Balanced x\u271d black n\nhr : Balanced y\u271d black n\n\u22a2 2 ^ n + 2 ^ n \u2264 size x\u271d + 1 + (size y\u271d + 1)"}, {"tactic": "exact Nat.add_le_add hl.le_size hr.le_size", "annotated_tactic": ["exact <a>Nat.add_le_add</a> hl.le_size hr.le_size", [{"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]}]], "state_before": "\u03b1 : Type u_1\nt : RBNode \u03b1\nc : RBColor\nn\u271d n : Nat\nx\u271d y\u271d : RBNode \u03b1\nv\u271d : \u03b1\nhl : Balanced x\u271d black n\nhr : Balanced y\u271d black n\n\u22a2 2 ^ n + 2 ^ n \u2264 size x\u271d + 1 + (size y\u271d + 1)", "state_after": "no goals"}, {"tactic": "rw [size, Nat.add_right_comm (size _), Nat.add_assoc, depthLB, Nat.pow_succ, Nat.mul_two]", "annotated_tactic": ["rw [<a>size</a>, <a>Nat.add_right_comm</a> (<a>size</a> _), <a>Nat.add_assoc</a>, <a>depthLB</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": "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.RBNode.size", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [346, 13], "def_end_pos": [346, 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": "Std.RBNode.depthLB", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [43, 5], "def_end_pos": [43, 12]}, {"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\nt : RBNode \u03b1\nc : RBColor\nn : Nat\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\n\u22a2 2 ^ depthLB black (n\u271d + 1) \u2264 size (node black x\u271d v\u271d y\u271d) + 1", "state_after": "\u03b1 : Type u_1\nt : RBNode \u03b1\nc : RBColor\nn : Nat\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\n\u22a2 2 ^ n\u271d + 2 ^ n\u271d \u2264 size x\u271d + 1 + (size y\u271d + 1)"}, {"tactic": "refine Nat.add_le_add (Nat.le_trans ?_ hl.le_size) (Nat.le_trans ?_ hr.le_size) <;>\n  exact Nat.pow_le_pow_of_le_right (by decide) (depthLB_le ..)", "annotated_tactic": ["refine <a>Nat.add_le_add</a> (<a>Nat.le_trans</a> ?_ hl.le_size) (<a>Nat.le_trans</a> ?_ hr.le_size) <;>\n      exact <a>Nat.pow_le_pow_of_le_right</a> (by decide) (<a>depthLB_le</a> ..)", [{"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.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"full_name": "Nat.le_trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1592, 19], "def_end_pos": [1592, 31]}, {"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": "Std.RBNode.depthLB_le", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [47, 9], "def_end_pos": [47, 19]}]], "state_before": "\u03b1 : Type u_1\nt : RBNode \u03b1\nc : RBColor\nn : Nat\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\n\u22a2 2 ^ n\u271d + 2 ^ n\u271d \u2264 size x\u271d + 1 + (size y\u271d + 1)", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1 : Type u_1\nt : RBNode \u03b1\nc : RBColor\nn : Nat\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\n\u22a2 2 > 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.aemeasurable_fderivWithin", "start": [701, 1], "end": [755, 12], "traced_tactics": [{"tactic": "refine' aemeasurable_of_unif_approx fun \u03b5 \u03b5pos => _", "annotated_tactic": ["refine' <a>aemeasurable_of_unif_approx</a> fun \u03b5 \u03b5pos => _", [{"full_name": "aemeasurable_of_unif_approx", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [122, 9], "def_end_pos": [122, 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\n\u22a2 AEMeasurable f'", "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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5"}, {"tactic": "let \u03b4 : \u211d\u22650 := \u27e8\u03b5, le_of_lt \u03b5pos\u27e9", "annotated_tactic": ["let \u03b4 : \u211d\u22650 := \u27e8\u03b5, <a>le_of_lt</a> \u03b5pos\u27e9", [{"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 : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\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\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5"}, {"tactic": "have \u03b4pos : 0 < \u03b4 := \u03b5pos", "annotated_tactic": ["have \u03b4pos : 0 < \u03b4 := \u03b5pos", []], "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\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5"}, {"tactic": "obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, _\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) \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' (fun _ => \u03b4) fun _ =>\n    \u03b4pos.ne'", "annotated_tactic": ["obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, _\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) \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' (fun _ => \u03b4) fun _ =>\n      \u03b4pos.ne'", [{"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]}]], "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\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5", "state_after": "case 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5"}, {"tactic": "obtain \u27e8g, g_meas, hg\u27e9 :\n    \u2203 g : E \u2192 E \u2192L[\u211d] E, Measurable g \u2227 \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n :=\n  exists_measurable_piecewise t t_meas (fun n _ => A n) (fun n => measurable_const) <|\n    t_disj.mono fun i j h => by simp only [h.inter_eq, eqOn_empty]", "annotated_tactic": ["obtain \u27e8g, g_meas, hg\u27e9 :\n      \u2203 g : E \u2192 E \u2192L[\u211d] E, <a>Measurable</a> g \u2227 \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n :=\n    <a>exists_measurable_piecewise</a> t t_meas (fun n _ => A n) (fun n => <a>measurable_const</a>) <|\n      t_disj.mono fun i j h => by simp only [h.inter_eq, <a>eqOn_empty</a>]", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "exists_measurable_piecewise", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [847, 9], "def_end_pos": [847, 36]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Set.eqOn_empty", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}]], "state_before": "case 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5", "state_after": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5"}, {"tactic": "refine' \u27e8g, g_meas.aemeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8g, g_meas.aemeasurable, _\u27e9", []], "state_before": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5", "state_after": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "suffices H : \u2200\u1d50 x : E \u2202sum fun n => \u03bc.restrict (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5", "annotated_tactic": ["suffices H : \u2200\u1d50 x : E \u2202<a>sum</a> fun n => \u03bc.restrict (s \u2229 t n), <a>dist</a> (g x) (f' x) \u2264 \u03b5", [{"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}]], "state_before": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5", "state_after": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "refine' ae_sum_iff.2 fun n => _", "annotated_tactic": ["refine' <a>ae_sum_iff</a>.2 fun n => _", [{"full_name": "MeasureTheory.Measure.ae_sum_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2021, 9], "def_end_pos": [2021, 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "have E\u2081 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 :=\n  (ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n    (hf' x hx.1).mono (inter_subset_left _ _)", "annotated_tactic": ["have E\u2081 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 :=\n    (ht n).<a>norm_fderiv_sub_le</a> \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n      (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _)", [{"full_name": "ApproximatesLinearOn.norm_fderiv_sub_le", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [465, 9], "def_end_pos": [465, 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": "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "have E\u2082 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), g x = A n := by\n  suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A n\n  exact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H\n  filter_upwards [ae_restrict_mem (t_meas n)]\n  exact hg n", "annotated_tactic": ["have E\u2082 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), g x = A n := by\n    suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A n\n    exact <a>ae_mono</a> (<a>restrict_mono</a> (<a>inter_subset_right</a> _ _) <a>le_rfl</a>) H\n    filter_upwards [<a>ae_restrict_mem</a> (t_meas n)]\n    exact hg n", [{"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.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]}, {"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\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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "filter_upwards [E\u2081, E\u2082] with x hx1 hx2", "annotated_tactic": ["filter_upwards [E\u2081, E\u2082] with x hx1 hx2", []], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nhx2 : g x = A n\n\u22a2 dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "rw [\u2190 nndist_eq_nnnorm] at hx1", "annotated_tactic": ["rw [\u2190 <a>nndist_eq_nnnorm</a>] at hx1", [{"full_name": "nndist_eq_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [929, 7], "def_end_pos": [929, 23]}]], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nhx2 : g x = A n\n\u22a2 dist (g x) (f' x) \u2264 \u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : nndist (f' x) (A n) \u2264 \u03b4\nhx2 : g x = A n\n\u22a2 dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "rw [hx2, dist_comm]", "annotated_tactic": ["rw [hx2, <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 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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : nndist (f' x) (A n) \u2264 \u03b4\nhx2 : g x = A n\n\u22a2 dist (g x) (f' x) \u2264 \u03b5", "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : nndist (f' x) (A n) \u2264 \u03b4\nhx2 : g x = A n\n\u22a2 dist (f' x) (A n) \u2264 \u03b5"}, {"tactic": "exact hx1", "annotated_tactic": ["exact hx1", []], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : nndist (f' x) (A n) \u2264 \u03b4\nhx2 : g x = A n\n\u22a2 dist (f' x) (A n) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp only [h.inter_eq, eqOn_empty]", "annotated_tactic": ["simp only [h.inter_eq, <a>eqOn_empty</a>]", [{"full_name": "Set.eqOn_empty", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [185, 9], "def_end_pos": [185, 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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ni j : \u2115\nh : (Disjoint on t) i j\n\u22a2 EqOn ((fun n x => A n) i) ((fun n x => A n) j) (t i \u2229 t j)", "state_after": "no goals"}, {"tactic": "have : \u03bc.restrict s \u2264 sum fun n => \u03bc.restrict (s \u2229 t n) := by\n  have : s = \u22c3 n, s \u2229 t n := by\n    rw [\u2190 inter_iUnion]\n    exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n  conv_lhs => rw [this]\n  exact restrict_iUnion_le", "annotated_tactic": ["have : \u03bc.restrict s \u2264 <a>sum</a> fun n => \u03bc.restrict (s \u2229 t n) := by\n      have : s = \u22c3 n, s \u2229 t n := by\n        rw [\u2190 <a>inter_iUnion</a>]\n        exact <a>Subset.antisymm</a> (<a>subset_inter</a> <a>Subset.rfl</a> t_cover) (<a>inter_subset_left</a> _ _)\n      conv_lhs => rw [this]\n      exact <a>restrict_iUnion_le</a>", [{"full_name": "MeasureTheory.Measure.sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1978, 5], "def_end_pos": [1978, 8]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"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]}, {"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.Measure.restrict_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2099, 9], "def_end_pos": [2099, 27]}]], "state_before": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5", "state_after": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5"}, {"tactic": "exact ae_mono this H", "annotated_tactic": ["exact <a>ae_mono</a> this H", [{"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}]], "state_before": "case intro.intro.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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "have : s = \u22c3 n, s \u2229 t n := by\n  rw [\u2190 inter_iUnion]\n  exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)", "annotated_tactic": ["have : s = \u22c3 n, s \u2229 t n := by\n        rw [\u2190 <a>inter_iUnion</a>]\n        exact <a>Subset.antisymm</a> (<a>subset_inter</a> <a>Subset.rfl</a> t_cover) (<a>inter_subset_left</a> _ _)", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}, {"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"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]}, {"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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 n, s \u2229 t n\n\u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)"}, {"tactic": "conv_lhs => rw [this]", "annotated_tactic": ["conv_lhs => rw [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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 n, s \u2229 t n\n\u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 n, s \u2229 t n\n\u22a2 Measure.restrict \u03bc (\u22c3 n, s \u2229 t n) \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)"}, {"tactic": "exact restrict_iUnion_le", "annotated_tactic": ["exact <a>restrict_iUnion_le</a>", [{"full_name": "MeasureTheory.Measure.restrict_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2099, 9], "def_end_pos": [2099, 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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 n, s \u2229 t n\n\u22a2 Measure.restrict \u03bc (\u22c3 n, s \u2229 t n) \u2264 sum fun n => Measure.restrict \u03bc (s \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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 s = \u22c3 n, 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 s = s \u2229 \u22c3 i, t i"}, {"tactic": "exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)", "annotated_tactic": ["exact <a>Subset.antisymm</a> (<a>subset_inter</a> <a>Subset.rfl</a> t_cover) (<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.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]}, {"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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 s = s \u2229 \u22c3 i, t i", "state_after": "no goals"}, {"tactic": "suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A n", "annotated_tactic": ["suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A 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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A 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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nH : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n"}, {"tactic": "exact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H", "annotated_tactic": ["exact <a>ae_mono</a> (<a>restrict_mono</a> (<a>inter_subset_right</a> _ _) <a>le_rfl</a>) H", [{"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.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": "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\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nH : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n"}, {"tactic": "filter_upwards [ae_restrict_mem (t_meas n)]", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> (t_meas n)]", [{"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\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\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200 (a : E), a \u2208 t n \u2192 g a = A n"}, {"tactic": "exact hg n", "annotated_tactic": ["exact hg 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\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\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200 (a : E), a \u2208 t n \u2192 g a = A n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_\u2112p", "start": [1418, 1], "end": [1451, 26], "traced_tactics": [{"tactic": "let B := fun n : \u2115 => ((1 : \u211d) / 2) ^ n", "annotated_tactic": ["let B := fun n : \u2115 => ((1 : \u211d) / 2) ^ n", []], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 CompleteSpace { x // x \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\n\u22a2 CompleteSpace { x // x \u2208 Lp E p }"}, {"tactic": "have hB_pos : \u2200 n, 0 < B n := fun n => pow_pos (div_pos zero_lt_one zero_lt_two) n", "annotated_tactic": ["have hB_pos : \u2200 n, 0 < B n := fun n => <a>pow_pos</a> (<a>div_pos</a> <a>zero_lt_one</a> <a>zero_lt_two</a>) n", [{"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 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": "\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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\n\u22a2 CompleteSpace { x // x \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\n\u22a2 CompleteSpace { x // x \u2208 Lp E p }"}, {"tactic": "refine' Metric.complete_of_convergent_controlled_sequences B hB_pos fun f hf => _", "annotated_tactic": ["refine' <a>Metric.complete_of_convergent_controlled_sequences</a> B hB_pos fun f hf => _", [{"full_name": "Metric.complete_of_convergent_controlled_sequences", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1343, 9], "def_end_pos": [1343, 59]}]], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\n\u22a2 CompleteSpace { x // x \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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\n\u22a2 \u2203 x, Tendsto f atTop (\ud835\udcdd x)"}, {"tactic": "rsuffices \u27e8f_lim, hf_lim_meas, h_tendsto\u27e9 :\n  \u2203 (f_lim : \u03b1 \u2192 E), Mem\u2112p f_lim p \u03bc \u2227\n    atTop.Tendsto (fun n => snorm (\u21d1(f n) - f_lim) p \u03bc) (\ud835\udcdd 0)", "annotated_tactic": ["rsuffices \u27e8f_lim, hf_lim_meas, h_tendsto\u27e9 :\n    \u2203 (f_lim : \u03b1 \u2192 E), <a>Mem\u2112p</a> f_lim p \u03bc \u2227\n      atTop.Tendsto (fun n => <a>snorm</a> (\u21d1(f n) - f_lim) p \u03bc) (\ud835\udcdd 0)", [{"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", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}]], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\n\u22a2 \u2203 x, Tendsto f atTop (\ud835\udcdd x)", "state_after": "case intro.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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nf_lim : \u03b1 \u2192 E\nhf_lim_meas : Mem\u2112p f_lim p\nh_tendsto : Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 x, Tendsto f atTop (\ud835\udcdd x)\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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "obtain \u27e8M, hB\u27e9 : Summable B := summable_geometric_two", "annotated_tactic": ["obtain \u27e8M, hB\u27e9 : <a>Summable</a> B := <a>summable_geometric_two</a>", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "summable_geometric_two", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [228, 9], "def_end_pos": [228, 31]}]], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "let B1 n := ENNReal.ofReal (B n)", "annotated_tactic": ["let B1 n := <a>ENNReal.ofReal</a> (B n)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have hB1_has : HasSum B1 (ENNReal.ofReal M) := by\n  have h_tsum_B1 : \u2211' i, B1 i = ENNReal.ofReal M := by\n    change (\u2211' n : \u2115, ENNReal.ofReal (B n)) = ENNReal.ofReal M\n    rw [\u2190 hB.tsum_eq]\n    exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).symm\n  have h_sum := (@ENNReal.summable _ B1).hasSum\n  rwa [h_tsum_B1] at h_sum", "annotated_tactic": ["have hB1_has : <a>HasSum</a> B1 (<a>ENNReal.ofReal</a> M) := by\n    have h_tsum_B1 : \u2211' i, B1 i = <a>ENNReal.ofReal</a> M := by\n      change (\u2211' n : \u2115, <a>ENNReal.ofReal</a> (B n)) = <a>ENNReal.ofReal</a> M\n      rw [\u2190 hB.tsum_eq]\n      exact (<a>ENNReal.ofReal_tsum_of_nonneg</a> (fun n => <a>le_of_lt</a> (hB_pos n)) hB.summable).<a>symm</a>\n    have h_sum := (@<a>ENNReal.summable</a> _ B1).<a>hasSum</a>\n    rwa [h_tsum_B1] at h_sum", [{"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"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", "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_tsum_of_nonneg", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 38]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 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": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}, {"full_name": "Summable.hasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 24]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have hB1 : \u2211' i, B1 i < \u221e := by\n  rw [hB1_has.tsum_eq]\n  exact ENNReal.ofReal_lt_top", "annotated_tactic": ["have hB1 : \u2211' i, B1 i < \u221e := by\n    rw [hB1_has.tsum_eq]\n    exact <a>ENNReal.ofReal_lt_top</a>", [{"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 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "let f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => f n", "annotated_tactic": ["let f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => f n", []], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "refine' H f1 (fun n => Lp.mem\u2112p (f n)) B1 hB1 fun N n m hn hm => _", "annotated_tactic": ["refine' H f1 (fun n => <a>Lp.mem\u2112p</a> (f n)) B1 hB1 fun N n m hn hm => _", [{"full_name": "MeasureTheory.Lp.mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [216, 19], "def_end_pos": [216, 24]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\n\u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N"}, {"tactic": "specialize hf N n m hn hm", "annotated_tactic": ["specialize hf N n m hn hm", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : dist (f n) (f m) < B N\n\u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N"}, {"tactic": "rw [dist_def] at hf", "annotated_tactic": ["rw [<a>dist_def</a>] at hf", [{"full_name": "MeasureTheory.Lp.dist_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [290, 9], "def_end_pos": [290, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : dist (f n) (f m) < B N\n\u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc < ENNReal.ofReal ((1 / 2) ^ N)"}, {"tactic": "rwa [ENNReal.lt_ofReal_iff_toReal_lt]", "annotated_tactic": ["rwa [<a>ENNReal.lt_ofReal_iff_toReal_lt</a>]", [{"full_name": "ENNReal.lt_ofReal_iff_toReal_lt", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2216, 9], "def_end_pos": [2216, 32]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc < ENNReal.ofReal ((1 / 2) ^ N)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) 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": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc \u2260 \u22a4", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (\u2191\u2191(f n - f m)) p \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": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\nhB1 : \u2211' (i : \u2115), B1 i < \u22a4\nf1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n)\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nhf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N\n\u22a2 snorm (\u2191\u2191(f n - f m)) p \u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact \u27e8hf_lim_meas.toLp f_lim, tendsto_Lp_of_tendsto_\u2112p f_lim hf_lim_meas h_tendsto\u27e9", "annotated_tactic": ["exact \u27e8hf_lim_meas.toLp f_lim, <a>tendsto_Lp_of_tendsto_\u2112p</a> f_lim hf_lim_meas h_tendsto\u27e9", [{"full_name": "MeasureTheory.Lp.tendsto_Lp_of_tendsto_\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1402, 9], "def_end_pos": [1402, 33]}]], "state_before": "case intro.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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nf_lim : \u03b1 \u2192 E\nhf_lim_meas : Mem\u2112p f_lim p\nh_tendsto : Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 x, Tendsto f atTop (\ud835\udcdd x)", "state_after": "no goals"}, {"tactic": "have h_tsum_B1 : \u2211' i, B1 i = ENNReal.ofReal M := by\n  change (\u2211' n : \u2115, ENNReal.ofReal (B n)) = ENNReal.ofReal M\n  rw [\u2190 hB.tsum_eq]\n  exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).symm", "annotated_tactic": ["have h_tsum_B1 : \u2211' i, B1 i = <a>ENNReal.ofReal</a> M := by\n      change (\u2211' n : \u2115, <a>ENNReal.ofReal</a> (B n)) = <a>ENNReal.ofReal</a> M\n      rw [\u2190 hB.tsum_eq]\n      exact (<a>ENNReal.ofReal_tsum_of_nonneg</a> (fun n => <a>le_of_lt</a> (hB_pos n)) hB.summable).<a>symm</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": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal_tsum_of_nonneg", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 38]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 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 : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 HasSum B1 (ENNReal.ofReal M)", "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nh_tsum_B1 : \u2211' (i : \u2115), B1 i = ENNReal.ofReal M\n\u22a2 HasSum B1 (ENNReal.ofReal M)"}, {"tactic": "have h_sum := (@ENNReal.summable _ B1).hasSum", "annotated_tactic": ["have h_sum := (@<a>ENNReal.summable</a> _ B1).<a>hasSum</a>", [{"full_name": "ENNReal.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 27]}, {"full_name": "Summable.hasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nh_tsum_B1 : \u2211' (i : \u2115), B1 i = ENNReal.ofReal M\n\u22a2 HasSum B1 (ENNReal.ofReal M)", "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nh_tsum_B1 : \u2211' (i : \u2115), B1 i = ENNReal.ofReal M\nh_sum : HasSum B1 (\u2211' (b : \u2115), B1 b)\n\u22a2 HasSum B1 (ENNReal.ofReal M)"}, {"tactic": "rwa [h_tsum_B1] at h_sum", "annotated_tactic": ["rwa [h_tsum_B1] at h_sum", []], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nh_tsum_B1 : \u2211' (i : \u2115), B1 i = ENNReal.ofReal M\nh_sum : HasSum B1 (\u2211' (b : \u2115), B1 b)\n\u22a2 HasSum B1 (ENNReal.ofReal M)", "state_after": "no goals"}, {"tactic": "change (\u2211' n : \u2115, ENNReal.ofReal (B n)) = ENNReal.ofReal M", "annotated_tactic": ["change (\u2211' n : \u2115, <a>ENNReal.ofReal</a> (B n)) = <a>ENNReal.ofReal</a> M", [{"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": "\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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2211' (i : \u2115), B1 i = ENNReal.ofReal M", "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2211' (n : \u2115), ENNReal.ofReal (B n) = ENNReal.ofReal M"}, {"tactic": "rw [\u2190 hB.tsum_eq]", "annotated_tactic": ["rw [\u2190 hB.tsum_eq]", []], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2211' (n : \u2115), ENNReal.ofReal (B n) = ENNReal.ofReal M", "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2211' (n : \u2115), ENNReal.ofReal (B n) = ENNReal.ofReal (\u2211' (b : \u2115), B b)"}, {"tactic": "exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).symm", "annotated_tactic": ["exact (<a>ENNReal.ofReal_tsum_of_nonneg</a> (fun n => <a>le_of_lt</a> (hB_pos n)) hB.summable).<a>symm</a>", [{"full_name": "ENNReal.ofReal_tsum_of_nonneg", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 38]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 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 : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\n\u22a2 \u2211' (n : \u2115), ENNReal.ofReal (B n) = ENNReal.ofReal (\u2211' (b : \u2115), B b)", "state_after": "no goals"}, {"tactic": "rw [hB1_has.tsum_eq]", "annotated_tactic": ["rw [hB1_has.tsum_eq]", []], "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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\n\u22a2 \u2211' (i : \u2115), B1 i < \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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\n\u22a2 ENNReal.ofReal M < \u22a4"}, {"tactic": "exact ENNReal.ofReal_lt_top", "annotated_tactic": ["exact <a>ENNReal.ofReal_lt_top</a>", [{"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [314, 17], "def_end_pos": [314, 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\nhp : Fact (1 \u2264 p)\nH :\n  \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),\n    (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192\n      \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),\n        \u2211' (i : \u2115), B i < \u22a4 \u2192\n          (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192\n            \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0)\nB : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n\nhB_pos : \u2200 (n : \u2115), 0 < B n\nf : \u2115 \u2192 { x // x \u2208 Lp E p }\nhf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N\nM : \u211d\nhB : HasSum B M\nB1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n)\nhB1_has : HasSum B1 (ENNReal.ofReal M)\n\u22a2 ENNReal.ofReal M < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "OneOneReducible.disjoin_left", "start": [298, 1], "end": [300, 80], "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.Buckets.update_update", "start": [29, 1], "end": [31, 45], "traced_tactics": [{"tactic": "simp [update]", "annotated_tactic": ["simp [<a>update</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.update", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [32, 5], "def_end_pos": [32, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nself : Buckets \u03b1 \u03b2\ni : USize\nd d' : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size self.val\nh' : USize.toNat i < Array.size (update self i d h).val\n\u22a2 update (update self i d h) i d' h' = update self i d' h", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nself : Buckets \u03b1 \u03b2\ni : USize\nd d' : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size self.val\nh' : USize.toNat i < Array.size (update self i d h).val\n\u22a2 { val := Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' } d',\n      property :=\n        (_ :\n          (fun b => 0 < Array.size b)\n            (Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' }\n              d')) } =\n    { val := Array.set self.val { val := USize.toNat i, isLt := h } d',\n      property := (_ : (fun b => 0 < Array.size b) (Array.set self.val { val := USize.toNat i, isLt := h } d')) }"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nself : Buckets \u03b1 \u03b2\ni : USize\nd d' : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size self.val\nh' : USize.toNat i < Array.size (update self i d h).val\n\u22a2 { val := Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' } d',\n      property :=\n        (_ :\n          (fun b => 0 < Array.size b)\n            (Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' }\n              d')) } =\n    { val := Array.set self.val { val := USize.toNat i, isLt := h } d',\n      property := (_ : (fun b => 0 < Array.size b) (Array.set self.val { val := USize.toNat i, isLt := h } d')) }", "state_after": "case e_val\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nself : Buckets \u03b1 \u03b2\ni : USize\nd d' : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size self.val\nh' : USize.toNat i < Array.size (update self i d h).val\n\u22a2 Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' } d' =\n    Array.set self.val { val := USize.toNat i, isLt := h } d'"}, {"tactic": "rw [Array.set_set]", "annotated_tactic": ["rw [<a>Array.set_set</a>]", [{"full_name": "Array.set_set", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [110, 9], "def_end_pos": [110, 16]}]], "state_before": "case e_val\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nself : Buckets \u03b1 \u03b2\ni : USize\nd d' : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size self.val\nh' : USize.toNat i < Array.size (update self i d h).val\n\u22a2 Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' } d' =\n    Array.set self.val { val := USize.toNat i, isLt := h } d'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.MeasurablySeparable.iUnion", "start": [379, 1], "end": [390, 47], "traced_tactics": [{"tactic": "choose u hsu htu hu using h", "annotated_tactic": ["choose u hsu htu hu using h", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nh : \u2200 (m n : \u03b9), MeasurablySeparable (s m) (t n)\n\u22a2 MeasurablySeparable (\u22c3 n, s n) (\u22c3 m, t m)", "state_after": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 MeasurablySeparable (\u22c3 n, s n) (\u22c3 m, t m)"}, {"tactic": "refine' \u27e8\u22c3 m, \u22c2 n, u m n, _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8\u22c3 m, \u22c2 n, u m n, _, _, _\u27e9", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 MeasurablySeparable (\u22c3 n, s n) (\u22c3 m, t m)", "state_after": "case refine'_1\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 \u22c3 n, s n \u2286 \u22c3 m, \u22c2 n, u m n\n\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 Disjoint (\u22c3 m, t m) (\u22c3 m, \u22c2 n, u m n)\n\ncase refine'_3\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 MeasurableSet (\u22c3 m, \u22c2 n, u m n)"}, {"tactic": "refine' iUnion_subset fun m => subset_iUnion_of_subset m _", "annotated_tactic": ["refine' <a>iUnion_subset</a> fun m => <a>subset_iUnion_of_subset</a> m _", [{"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.subset_iUnion_of_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [451, 9], "def_end_pos": [451, 32]}]], "state_before": "case refine'_1\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 \u22c3 n, s n \u2286 \u22c3 m, \u22c2 n, u m n", "state_after": "case refine'_1\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nm : \u03b9\n\u22a2 s m \u2286 \u22c2 n, u m n"}, {"tactic": "exact subset_iInter fun n => hsu m n", "annotated_tactic": ["exact <a>subset_iInter</a> fun n => hsu m n", [{"full_name": "Set.subset_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [400, 9], "def_end_pos": [400, 22]}]], "state_before": "case refine'_1\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nm : \u03b9\n\u22a2 s m \u2286 \u22c2 n, u m n", "state_after": "no goals"}, {"tactic": "simp_rw [disjoint_iUnion_left, disjoint_iUnion_right]", "annotated_tactic": ["simp_rw [<a>disjoint_iUnion_left</a>, <a>disjoint_iUnion_right</a>]", [{"full_name": "Set.disjoint_iUnion_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2218, 9], "def_end_pos": [2218, 29]}, {"full_name": "Set.disjoint_iUnion_right", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2224, 9], "def_end_pos": [2224, 30]}]], "state_before": "case refine'_2\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 Disjoint (\u22c3 m, t m) (\u22c3 m, \u22c2 n, u m n)", "state_after": "case refine'_2\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 \u2200 (i i_1 : \u03b9), Disjoint (t i) (\u22c2 n, u i_1 n)"}, {"tactic": "intro n m", "annotated_tactic": ["intro n m", []], "state_before": "case refine'_2\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 \u2200 (i i_1 : \u03b9), Disjoint (t i) (\u22c2 n, u i_1 n)", "state_after": "case refine'_2\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nn m : \u03b9\n\u22a2 Disjoint (t n) (\u22c2 n, u m n)"}, {"tactic": "apply Disjoint.mono_right _ (htu m n)", "annotated_tactic": ["apply <a>Disjoint.mono_right</a> _ (htu m n)", [{"full_name": "Disjoint.mono_right", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [74, 9], "def_end_pos": [74, 28]}]], "state_before": "case refine'_2\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nn m : \u03b9\n\u22a2 Disjoint (t n) (\u22c2 n, u m n)", "state_after": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nn m : \u03b9\n\u22a2 \u22c2 n, u m n \u2264 u m n"}, {"tactic": "apply iInter_subset", "annotated_tactic": ["apply <a>iInter_subset</a>", [{"full_name": "Set.iInter_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [435, 9], "def_end_pos": [435, 22]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nn m : \u03b9\n\u22a2 \u22c2 n, u m n \u2264 u m n", "state_after": "no goals"}, {"tactic": "refine' MeasurableSet.iUnion fun m => _", "annotated_tactic": ["refine' <a>MeasurableSet.iUnion</a> fun m => _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case refine'_3\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 MeasurableSet (\u22c3 m, \u22c2 n, u m n)", "state_after": "case refine'_3\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nm : \u03b9\n\u22a2 MeasurableSet (\u22c2 n, u m n)"}, {"tactic": "exact MeasurableSet.iInter fun n => hu m n", "annotated_tactic": ["exact <a>MeasurableSet.iInter</a> fun n => hu m n", [{"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}]], "state_before": "case refine'_3\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nm : \u03b9\n\u22a2 MeasurableSet (\u22c2 n, u m n)", "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_caratheodory", "start": [286, 1], "end": [298, 69], "traced_tactics": [{"tactic": "refine' iSup_le _", "annotated_tactic": ["refine' <a>iSup_le</a> _", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "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)\n\u22a2 MeasurableSpace.pi \u2264 OuterMeasure.caratheodory (OuterMeasure.pi fun i => \u2191(\u03bc i))", "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)\n\u22a2 \u2200 (i : \u03b9),\n    MeasurableSpace.comap (fun b => b i) ((fun a => inst\u271d a) i) \u2264\n      OuterMeasure.caratheodory (OuterMeasure.pi fun i => \u2191(\u03bc i))"}, {"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)\n\u22a2 \u2200 (i : \u03b9),\n    MeasurableSpace.comap (fun b => b i) ((fun a => inst\u271d a) i) \u2264\n      OuterMeasure.caratheodory (OuterMeasure.pi fun i => \u2191(\u03bc i))", "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)\ni : \u03b9\ns : Set ((a : \u03b9) \u2192 \u03b1 a)\nhs : MeasurableSet s\n\u22a2 MeasurableSet s"}, {"tactic": "rcases hs with \u27e8s, hs, rfl\u27e9", "annotated_tactic": ["rcases hs with \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)\ni : \u03b9\ns : Set ((a : \u03b9) \u2192 \u03b1 a)\nhs : MeasurableSet s\n\u22a2 MeasurableSet 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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\n\u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s)"}, {"tactic": "apply boundedBy_caratheodory", "annotated_tactic": ["apply <a>boundedBy_caratheodory</a>", [{"full_name": "MeasureTheory.OuterMeasure.boundedBy_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 31]}]], "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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\n\u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s)", "state_after": "case intro.intro.hs\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set ((a : \u03b9) \u2192 \u03b1 a)),\n    piPremeasure (fun i => \u2191(\u03bc i)) (t \u2229 (fun b => b i) \u207b\u00b9' s) +\n        piPremeasure (fun i => \u2191(\u03bc i)) (t \\ (fun b => b i) \u207b\u00b9' s) \u2264\n      piPremeasure (fun i => \u2191(\u03bc i)) t"}, {"tactic": "intro t", "annotated_tactic": ["intro t", []], "state_before": "case intro.intro.hs\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set ((a : \u03b9) \u2192 \u03b1 a)),\n    piPremeasure (fun i => \u2191(\u03bc i)) (t \u2229 (fun b => b i) \u207b\u00b9' s) +\n        piPremeasure (fun i => \u2191(\u03bc i)) (t \\ (fun b => b i) \u207b\u00b9' s) \u2264\n      piPremeasure (fun i => \u2191(\u03bc i)) t", "state_after": "case intro.intro.hs\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 piPremeasure (fun i => \u2191(\u03bc i)) (t \u2229 (fun b => b i) \u207b\u00b9' s) +\n      piPremeasure (fun i => \u2191(\u03bc i)) (t \\ (fun b => b i) \u207b\u00b9' s) \u2264\n    piPremeasure (fun i => \u2191(\u03bc i)) t"}, {"tactic": "simp_rw [piPremeasure]", "annotated_tactic": ["simp_rw [<a>piPremeasure</a>]", [{"full_name": "MeasureTheory.piPremeasure", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [159, 5], "def_end_pos": [159, 17]}]], "state_before": "case intro.intro.hs\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 piPremeasure (fun i => \u2191(\u03bc i)) (t \u2229 (fun b => b i) \u207b\u00b9' s) +\n      piPremeasure (fun i => \u2191(\u03bc i)) (t \\ (fun b => b i) \u207b\u00b9' s) \u2264\n    piPremeasure (fun i => \u2191(\u03bc i)) t", "state_after": "case intro.intro.hs\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) + \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264\n    \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' t)"}, {"tactic": "refine' Finset.prod_add_prod_le' (Finset.mem_univ i) _ _ _", "annotated_tactic": ["refine' <a>Finset.prod_add_prod_le'</a> (<a>Finset.mem_univ</a> i) _ _ _", [{"full_name": "Finset.prod_add_prod_le'", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [667, 9], "def_end_pos": [667, 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 intro.intro.hs\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) + \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264\n    \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' t)", "state_after": "case intro.intro.hs.refine'_1\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u2191\u2191(\u03bc i) (eval i '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) + \u2191\u2191(\u03bc i) (eval i '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264\n    \u2191\u2191(\u03bc i) (eval i '' t)\n\ncase intro.intro.hs.refine'_2\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u2200 (j : \u03b9), j \u2208 Finset.univ \u2192 j \u2260 i \u2192 \u2191\u2191(\u03bc j) (eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)\n\ncase intro.intro.hs.refine'_3\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u2200 (j : \u03b9), j \u2208 Finset.univ \u2192 j \u2260 i \u2192 \u2191\u2191(\u03bc j) (eval j '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)"}, {"tactic": "simp [image_inter_preimage, image_diff_preimage, measure_inter_add_diff _ hs, le_refl]", "annotated_tactic": ["simp [<a>image_inter_preimage</a>, <a>image_diff_preimage</a>, <a>measure_inter_add_diff</a> _ hs, <a>le_refl</a>]", [{"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.image_diff_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [548, 9], "def_end_pos": [548, 28]}, {"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": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case intro.intro.hs.refine'_1\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u2191\u2191(\u03bc i) (eval i '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) + \u2191\u2191(\u03bc i) (eval i '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264\n    \u2191\u2191(\u03bc i) (eval i '' t)", "state_after": "no goals"}, {"tactic": "rintro j - _", "annotated_tactic": ["rintro j - _", []], "state_before": "case intro.intro.hs.refine'_2\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u2200 (j : \u03b9), j \u2208 Finset.univ \u2192 j \u2260 i \u2192 \u2191\u2191(\u03bc j) (eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)", "state_after": "case intro.intro.hs.refine'_2\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 \u2191\u2191(\u03bc j) (eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)"}, {"tactic": "apply mono'", "annotated_tactic": ["apply <a>mono'</a>", [{"full_name": "MeasureTheory.OuterMeasure.mono'", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [97, 9], "def_end_pos": [97, 14]}]], "state_before": "case intro.intro.hs.refine'_2\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 \u2191\u2191(\u03bc j) (eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)", "state_after": "case intro.intro.hs.refine'_2.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s) \u2286 eval j '' t"}, {"tactic": "apply image_subset", "annotated_tactic": ["apply <a>image_subset</a>", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "state_before": "case intro.intro.hs.refine'_2.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s) \u2286 eval j '' t", "state_after": "case intro.intro.hs.refine'_2.h.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 t \u2229 (fun b => b i) \u207b\u00b9' s \u2286 t"}, {"tactic": "apply inter_subset_left", "annotated_tactic": ["apply <a>inter_subset_left</a>", [{"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.hs.refine'_2.h.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 t \u2229 (fun b => b i) \u207b\u00b9' s \u2286 t", "state_after": "no goals"}, {"tactic": "rintro j - _", "annotated_tactic": ["rintro j - _", []], "state_before": "case intro.intro.hs.refine'_3\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\n\u22a2 \u2200 (j : \u03b9), j \u2208 Finset.univ \u2192 j \u2260 i \u2192 \u2191\u2191(\u03bc j) (eval j '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)", "state_after": "case intro.intro.hs.refine'_3\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 \u2191\u2191(\u03bc j) (eval j '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)"}, {"tactic": "apply mono'", "annotated_tactic": ["apply <a>mono'</a>", [{"full_name": "MeasureTheory.OuterMeasure.mono'", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [97, 9], "def_end_pos": [97, 14]}]], "state_before": "case intro.intro.hs.refine'_3\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 \u2191\u2191(\u03bc j) (eval j '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t)", "state_after": "case intro.intro.hs.refine'_3.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 eval j '' (t \\ (fun b => b i) \u207b\u00b9' s) \u2286 eval j '' t"}, {"tactic": "apply image_subset", "annotated_tactic": ["apply <a>image_subset</a>", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "state_before": "case intro.intro.hs.refine'_3.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 eval j '' (t \\ (fun b => b i) \u207b\u00b9' s) \u2286 eval j '' t", "state_after": "case intro.intro.hs.refine'_3.h.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 t \\ (fun b => b i) \u207b\u00b9' s \u2286 t"}, {"tactic": "apply diff_subset", "annotated_tactic": ["apply <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.hs.refine'_3.h.h\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)\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : MeasurableSet s\nt : Set ((a : \u03b9) \u2192 \u03b1 a)\nj : \u03b9\na\u271d : j \u2260 i\n\u22a2 t \\ (fun b => b i) \u207b\u00b9' s \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae", "start": [853, 1], "end": [910, 53], "traced_tactics": [{"tactic": "let g x := f x \u2229 Ioo 0 (R x)", "annotated_tactic": ["let g x := f x \u2229 <a>Ioo</a> 0 (R x)", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "have hg : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (g x \u2229 Ioo 0 \u03b4).Nonempty := by\n  intro x hx \u03b4 \u03b4pos\n  rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9\n  exact\n    \u27e8r,\n      \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9,\n        \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\u27e9\u27e9", "annotated_tactic": ["have hg : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (g x \u2229 <a>Ioo</a> 0 \u03b4).<a>Nonempty</a> := by\n    intro x hx \u03b4 \u03b4pos\n    rcases hf x hx (<a>min</a> \u03b4 (R x)) (<a>lt_min</a> \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9\n    exact\n      \u27e8r,\n        \u27e8\u27e8hr.1, hr.2.1, hr.2.2.<a>trans_le</a> (<a>min_le_right</a> _ _)\u27e9,\n          \u27e8hr.2.1, hr.2.2.<a>trans_le</a> (<a>min_le_left</a> _ _)\u27e9\u27e9\u27e9", [{"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]}, {"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": "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": "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": "\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "rcases exists_disjoint_closedBall_covering_ae_aux \u03bc g s hg with \u27e8v, v_count, vs, vg, \u03bcv, v_disj\u27e9", "annotated_tactic": ["rcases <a>exists_disjoint_closedBall_covering_ae_aux</a> \u03bc g s hg with \u27e8v, v_count, vs, vg, \u03bcv, v_disj\u27e9", [{"full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae_aux", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [831, 9], "def_end_pos": [831, 51]}]], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "let t := Prod.fst '' v", "annotated_tactic": ["let t := <a>Prod.fst</a> '' v", [{"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 intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "have : \u2200 x \u2208 t, \u2203 r : \u211d, (x, r) \u2208 v := by\n  intro x hx\n  rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n  exact \u27e8q, hp\u27e9", "annotated_tactic": ["have : \u2200 x \u2208 t, \u2203 r : \u211d, (x, r) \u2208 v := by\n    intro x hx\n    rcases (<a>mem_image</a> _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n    exact \u27e8q, hp\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.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nthis : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "choose! r hr using this", "annotated_tactic": ["choose! r hr using this", []], "state_before": "case intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nthis : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "refine' \u27e8t, r, v_count.image _, _, _, _, _\u27e9", "annotated_tactic": ["refine' \u27e8t, r, v_count.image _, _, _, _, _\u27e9", []], "state_before": "case intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = 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 \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 t \u2286 s\n\ncase intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)\n\ncase intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0\n\ncase intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "intro x hx \u03b4 \u03b4pos", "annotated_tactic": ["intro x hx \u03b4 \u03b4pos", []], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)"}, {"tactic": "rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9", "annotated_tactic": ["rcases hf x hx (<a>min</a> \u03b4 (R x)) (<a>lt_min</a> \u03b4pos (hR x hx)) 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]}]], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)", "state_after": "case 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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nhr : r \u2208 f x \u2229 Ioo 0 (min \u03b4 (R x))\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)"}, {"tactic": "exact\n  \u27e8r,\n    \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9,\n      \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\u27e9\u27e9", "annotated_tactic": ["exact\n      \u27e8r,\n        \u27e8\u27e8hr.1, hr.2.1, hr.2.2.<a>trans_le</a> (<a>min_le_right</a> _ _)\u27e9,\n          \u27e8hr.2.1, hr.2.2.<a>trans_le</a> (<a>min_le_left</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]}, {"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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nhr : r \u2208 f x \u2229 Ioo 0 (min \u03b4 (R x))\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)", "state_after": "no goals"}, {"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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 r, (x, r) \u2208 v"}, {"tactic": "rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, 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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 r, (x, r) \u2208 v", "state_after": "case intro.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).1 \u2208 t\n\u22a2 \u2203 r, ((p, q).1, r) \u2208 v"}, {"tactic": "exact \u27e8q, hp\u27e9", "annotated_tactic": ["exact \u27e8q, hp\u27e9", []], "state_before": "case intro.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).1 \u2208 t\n\u22a2 \u2203 r, ((p, q).1, r) \u2208 v", "state_after": "no goals"}, {"tactic": "have I : \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 v \u2192 0 \u2264 p.2 := fun p hp => (vg p hp).2.1.le", "annotated_tactic": ["have I : \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 v \u2192 0 \u2264 p.2 := fun p hp => (vg p hp).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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\n\u22a2 (fun x => (x, r x)) '' t = v", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 (fun x => (x, r x)) '' t = v"}, {"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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 (fun x => (x, r x)) '' t = v", "state_after": "case h\u2081\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 (fun x => (x, r x)) '' t \u2286 v\n\ncase h\u2082\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 v \u2286 (fun x => (x, r x)) '' t"}, {"tactic": "simp only [image_subset_iff]", "annotated_tactic": ["simp only [<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 h\u2081\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 (fun x => (x, r x)) '' t \u2286 v", "state_after": "case h\u2081\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 v \u2286 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)"}, {"tactic": "rintro \u27e8x, p\u27e9 hxp", "annotated_tactic": ["rintro \u27e8x, p\u27e9 hxp", []], "state_before": "case h\u2081\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 v \u2286 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)", "state_after": "case h\u2081.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)"}, {"tactic": "simp only [mem_preimage]", "annotated_tactic": ["simp only [<a>mem_preimage</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}]], "state_before": "case h\u2081.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)", "state_after": "case h\u2081.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, r x) \u2208 v"}, {"tactic": "exact hr _ (mem_image_of_mem _ hxp)", "annotated_tactic": ["exact hr _ (<a>mem_image_of_mem</a> _ hxp)", [{"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 h\u2081.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, r x) \u2208 v", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, p\u27e9 hxp", "annotated_tactic": ["rintro \u27e8x, p\u27e9 hxp", []], "state_before": "case h\u2082\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\n\u22a2 v \u2286 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "have hxrx : (x, r x) \u2208 v := hr _ (mem_image_of_mem _ hxp)", "annotated_tactic": ["have hxrx : (x, r x) \u2208 v := hr _ (<a>mem_image_of_mem</a> _ hxp)", [{"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 h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "have : p = r x := by\n  by_contra h\n  have A : (x, p) \u2260 (x, r x) := by\n    simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n  have H := v_disj hxp hxrx A\n  contrapose H\n  rw [not_disjoint_iff_nonempty_inter]\n  refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9", "annotated_tactic": ["have : p = r x := by\n        by_contra h\n        have A : (x, p) \u2260 (x, r x) := by\n          simpa only [<a>true_and_iff</a>, <a>Prod.mk.inj_iff</a>, <a>eq_self_iff_true</a>, <a>Ne.def</a>] using h\n        have H := v_disj hxp hxrx A\n        contrapose H\n        rw [<a>not_disjoint_iff_nonempty_inter</a>]\n        refine' \u27e8x, by simp (config := { proj := <a>false</a> }) [I _ hxp, I _ hxrx]\u27e9", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"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": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"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": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "state_before": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, r x) \u2208 (fun x => (x, r x)) '' t"}, {"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": "case h\u2082.mk\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, r x) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 x \u2208 t"}, {"tactic": "exact mem_image_of_mem _ hxp", "annotated_tactic": ["exact <a>mem_image_of_mem</a> _ hxp", [{"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 h\u2082.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 x \u2208 t", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 p = r x", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 False"}, {"tactic": "have A : (x, p) \u2260 (x, r x) := by\n  simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h", "annotated_tactic": ["have A : (x, p) \u2260 (x, r x) := by\n          simpa only [<a>true_and_iff</a>, <a>Prod.mk.inj_iff</a>, <a>eq_self_iff_true</a>, <a>Ne.def</a>] using h", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"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": "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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 False", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\n\u22a2 False"}, {"tactic": "have H := v_disj hxp hxrx A", "annotated_tactic": ["have H := v_disj hxp hxrx A", []], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\n\u22a2 False", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : (Disjoint on fun p => closedBall p.1 p.2) (x, p) (x, r x)\n\u22a2 False"}, {"tactic": "contrapose H", "annotated_tactic": ["contrapose H", []], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : (Disjoint on fun p => closedBall p.1 p.2) (x, p) (x, r x)\n\u22a2 False", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 \u00ac(Disjoint on fun p => closedBall p.1 p.2) (x, p) (x, r x)"}, {"tactic": "rw [not_disjoint_iff_nonempty_inter]", "annotated_tactic": ["rw [<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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 \u00ac(Disjoint on fun p => closedBall p.1 p.2) (x, p) (x, r x)", "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 Set.Nonempty ((fun p => closedBall p.1 p.2) (x, p) \u2229 (fun p => closedBall p.1 p.2) (x, r x))"}, {"tactic": "refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9", "annotated_tactic": ["refine' \u27e8x, by simp (config := { proj := <a>false</a> }) [I _ hxp, I _ hxrx]\u27e9", [{"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 Set.Nonempty ((fun p => closedBall p.1 p.2) (x, p) \u2229 (fun p => closedBall p.1 p.2) (x, r x))", "state_after": "no goals"}, {"tactic": "simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h", "annotated_tactic": ["simpa only [<a>true_and_iff</a>, <a>Prod.mk.inj_iff</a>, <a>eq_self_iff_true</a>, <a>Ne.def</a>] using h", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"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": "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\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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 (x, p) \u2260 (x, r x)", "state_after": "no goals"}, {"tactic": "simp (config := { proj := false }) [I _ hxp, I _ hxrx]", "annotated_tactic": ["simp (config := { proj := <a>false</a> }) [I _ hxp, I _ hxrx]", [{"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 x \u2208 (fun p => closedBall p.1 p.2) (x, p) \u2229 (fun p => closedBall p.1 p.2) (x, r x)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 t \u2286 s", "state_after": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s"}, {"tactic": "rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, 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.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s", "state_after": "case intro.intro.intro.intro.intro.refine'_1.intro.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).1 \u2208 t\n\u22a2 (p, q).1 \u2208 s"}, {"tactic": "exact vs _ hp", "annotated_tactic": ["exact vs _ hp", []], "state_before": "case intro.intro.intro.intro.intro.refine'_1.intro.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).1 \u2208 t\n\u22a2 (p, q).1 \u2208 s", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)", "state_after": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 r x \u2208 f x \u2229 Ioo 0 (R x)"}, {"tactic": "rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, _, rfl\u27e9", "annotated_tactic": ["rcases (<a>mem_image</a> _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, _, 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.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 r x \u2208 f x \u2229 Ioo 0 (R x)", "state_after": "case intro.intro.intro.intro.intro.refine'_2.intro.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nleft\u271d : (p, q) \u2208 v\nhx : (p, q).1 \u2208 t\n\u22a2 r (p, q).1 \u2208 f (p, q).1 \u2229 Ioo 0 (R (p, q).1)"}, {"tactic": "exact vg _ (hr _ hx)", "annotated_tactic": ["exact vg _ (hr _ hx)", []], "state_before": "case intro.intro.intro.intro.intro.refine'_2.intro.mk.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nleft\u271d : (p, q) \u2208 v\nhx : (p, q).1 \u2208 t\n\u22a2 r (p, q).1 \u2208 f (p, q).1 \u2229 Ioo 0 (R (p, q).1)", "state_after": "no goals"}, {"tactic": "have :\n  \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) =\n    \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.1 p.2 :=\n  by conv_rhs => rw [biUnion_image]", "annotated_tactic": ["have :\n      \u22c3 (x : \u03b1) (_ : x \u2208 t), <a>closedBall</a> x (r x) =\n        \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), <a>closedBall</a> p.1 p.2 :=\n      by conv_rhs => rw [<a>biUnion_image</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.biUnion_image", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1838, 9], "def_end_pos": [1838, 22]}]], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0", "state_after": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis : \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0"}, {"tactic": "rw [this, im_t]", "annotated_tactic": ["rw [this, im_t]", []], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis : \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0", "state_after": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis : \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0"}, {"tactic": "exact \u03bcv", "annotated_tactic": ["exact \u03bcv", []], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis : \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [biUnion_image]", "annotated_tactic": ["conv_rhs => rw [<a>biUnion_image</a>]", [{"full_name": "Set.biUnion_image", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1838, 9], "def_end_pos": [1838, 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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2", "state_after": "no goals"}, {"tactic": "have A : InjOn (fun x : \u03b1 => (x, r x)) t := by\n  simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff,\n    eq_self_iff_true]", "annotated_tactic": ["have A : <a>InjOn</a> (fun x : \u03b1 => (x, r x)) t := by\n      simp (config := { contextual := <a>true</a> }) only [<a>InjOn</a>, <a>Prod.mk.inj_iff</a>, <a>imp_true_iff</a>,\n        <a>eq_self_iff_true</a>]", [{"full_name": "Set.InjOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [603, 5], "def_end_pos": [603, 10]}, {"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.InjOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [603, 5], "def_end_pos": [603, 10]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"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]}]], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nA : InjOn (fun x => (x, r x)) t\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "rwa [\u2190 im_t, A.pairwiseDisjoint_image] at v_disj", "annotated_tactic": ["rwa [\u2190 im_t, A.pairwiseDisjoint_image] at v_disj", []], "state_before": "case intro.intro.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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nA : InjOn (fun x => (x, r x)) t\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff,\n  eq_self_iff_true]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) only [<a>InjOn</a>, <a>Prod.mk.inj_iff</a>, <a>imp_true_iff</a>,\n        <a>eq_self_iff_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]}, {"full_name": "Set.InjOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [603, 5], "def_end_pos": [603, 10]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"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]}]], "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 : SigmaFinite \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)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 InjOn (fun x => (x, r x)) t", "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\u2081", "start": [2655, 1], "end": [2664, 74], "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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn : \u2115\nH : n \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn : \u2115\nH\u271d : n \u2264 List.length S\nH : Nat.zero \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }"}, {"tactic": "apply (IH (le_of_lt H)).tail", "annotated_tactic": ["apply (IH (<a>le_of_lt</a> H)).<a>tail</a>", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Turing.Reaches\u2080.tail", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [797, 9], "def_end_pos": [797, 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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n    { l := some (go k o 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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) } \u2208\n    TM1.step (tr M) { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }"}, {"tactic": "rw [iterate_succ_apply']", "annotated_tactic": ["rw [<a>iterate_succ_apply'</a>]", [{"full_name": "Function.iterate_succ_apply'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [190, 9], "def_end_pos": [190, 28]}]], "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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) } \u2208\n    TM1.step (tr M) { l := some (go k o 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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 { l := some (go k o q), var := v,\n      Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) } \u2208\n    TM1.step (tr M) { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }"}, {"tactic": "simp only [TM1.step, TM1.stepAux, tr, Tape.mk'_nth_nat, Tape.move_right_n_head,\n  addBottom_nth_snd, Option.mem_def]", "annotated_tactic": ["simp only [<a>TM1.step</a>, <a>TM1.stepAux</a>, <a>tr</a>, <a>Tape.mk'_nth_nat</a>, <a>Tape.move_right_n_head</a>,\n    <a>addBottom_nth_snd</a>, <a>Option.mem_def</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]}, {"full_name": "Turing.TM2to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2637, 5], "def_end_pos": [2637, 7]}, {"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.Tape.move_right_n_head", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [649, 9], "def_end_pos": [649, 31]}, {"full_name": "Turing.TM2to1.addBottom_nth_snd", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2386, 9], "def_end_pos": [2386, 26]}, {"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 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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 { l := some (go k o q), var := v,\n      Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) } \u2208\n    TM1.step (tr M) { l := some (go k o 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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 some\n      (bif Option.isNone (ListBlank.nth L n k) then\n        TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))\n      else\n        { l := some (go k o q), var := v,\n          Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =\n    some\n      { l := some (go k o q), var := v,\n        Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }"}, {"tactic": "rw [stk_nth_val _ hL, List.get?_eq_get]", "annotated_tactic": ["rw [<a>stk_nth_val</a> _ hL, <a>List.get?_eq_get</a>]", [{"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?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}]], "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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 some\n      (bif Option.isNone (ListBlank.nth L n k) then\n        TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))\n      else\n        { l := some (go k o q), var := v,\n          Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =\n    some\n      { l := some (go k o q), var := v,\n        Tape := Tape.move Dir.right ((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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 some\n      (bif Option.isNone (some (List.get (List.reverse S) { val := n, isLt := ?succ })) then\n        TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))\n      else\n        { l := some (go k o q), var := v,\n          Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =\n    some\n      { l := some (go k o q), var := v,\n        Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 n < List.length (List.reverse S)"}, {"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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 some\n      (bif Option.isNone (some (List.get (List.reverse S) { val := n, isLt := ?succ })) then\n        TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))\n      else\n        { l := some (go k o q), var := v,\n          Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =\n    some\n      { l := some (go k o q), var := v,\n        Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 n < List.length (List.reverse S)", "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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 n < List.length (List.reverse S)"}, {"tactic": "rwa [List.length_reverse]", "annotated_tactic": ["rwa [<a>List.length_reverse</a>]", [{"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]}]], "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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n  n \u2264 List.length S \u2192\n    Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n      { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 n < List.length (List.reverse S)", "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\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn : \u2115\nH\u271d : n \u2264 List.length S\nH : Nat.zero \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) }", "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_preCdf_fst", "start": [307, 1], "end": [315, 35], "traced_tactics": [{"tactic": "have : \u2200 r, \u222b\u207b x in s, preCdf \u03c1 r x \u2202\u03c1.fst = \u222b\u207b x in s, (preCdf \u03c1 r * 1) x \u2202\u03c1.fst := by\n  simp only [mul_one, eq_self_iff_true, forall_const]", "annotated_tactic": ["have : \u2200 r, \u222b\u207b x in s, <a>preCdf</a> \u03c1 r x \u2202\u03c1.fst = \u222b\u207b x in s, (<a>preCdf</a> \u03c1 r * 1) x \u2202\u03c1.fst := by\n    simp only [<a>mul_one</a>, <a>eq_self_iff_true</a>, <a>forall_const</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": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"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": "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\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s"}, {"tactic": "rw [this, \u2190 set_lintegral_withDensity_eq_set_lintegral_mul _ measurable_preCdf _ hs]", "annotated_tactic": ["rw [this, \u2190 <a>set_lintegral_withDensity_eq_set_lintegral_mul</a> _ <a>measurable_preCdf</a> _ hs]", [{"full_name": "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [288, 9], "def_end_pos": [288, 55]}, {"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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, OfNat.ofNat 1 x \u2202Measure.withDensity (Measure.fst \u03c1) (preCdf \u03c1 r) = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s\n\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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 Measurable 1"}, {"tactic": "simp only [mul_one, eq_self_iff_true, forall_const]", "annotated_tactic": ["simp only [<a>mul_one</a>, <a>eq_self_iff_true</a>, <a>forall_const</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"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": "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\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1", "state_after": "no goals"}, {"tactic": "simp only [withDensity_preCdf \u03c1 r, Pi.one_apply, lintegral_one, Measure.restrict_apply,\n  MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>withDensity_preCdf</a> \u03c1 r, <a>Pi.one_apply</a>, <a>lintegral_one</a>, <a>Measure.restrict_apply</a>,\n      <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "ProbabilityTheory.withDensity_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [302, 9], "def_end_pos": [302, 27]}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, OfNat.ofNat 1 x \u2202Measure.withDensity (Measure.fst \u03c1) (preCdf \u03c1 r) = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s", "state_after": "no goals"}, {"tactic": "rw [(_ : (1 : \u03b1 \u2192 \u211d\u22650\u221e) = fun _ => 1)]", "annotated_tactic": ["rw [(_ : (1 : \u03b1 \u2192 \u211d\u22650\u221e) = fun _ => 1)]", []], "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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 Measurable 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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 Measurable fun x => 1\n\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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 1 = fun x => 1"}, {"tactic": "exacts [measurable_const, rfl]", "annotated_tactic": ["exacts [<a>measurable_const</a>, <a>rfl</a>]", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 Measurable fun x => 1\n\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)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 1 = fun x => 1", "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_compQuasiMeasurePreserving", "start": [237, 1], "end": [240, 17], "traced_tactics": [{"tactic": "rw [compQuasiMeasurePreserving_eq_mk]", "annotated_tactic": ["rw [<a>compQuasiMeasurePreserving_eq_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [232, 9], "def_end_pos": [232, 41]}]], "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 \u2191(compQuasiMeasurePreserving g f hf) =\u1d50[\u03bc] \u2191g \u2218 f", "state_after": "\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 \u2191(mk (\u2191g \u2218 f) (_ : AEStronglyMeasurable (\u2191g \u2218 f) \u03bc)) =\u1d50[\u03bc] \u2191g \u2218 f"}, {"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\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 \u2191(mk (\u2191g \u2218 f) (_ : AEStronglyMeasurable (\u2191g \u2218 f) \u03bc)) =\u1d50[\u03bc] \u2191g \u2218 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.smul_extend", "start": [1322, 1], "end": [1329, 35], "traced_tactics": [{"tactic": "ext1 s", "annotated_tactic": ["ext1 s", []], "state_before": "\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\n\u22a2 c \u2022 extend m = extend fun s h => c \u2022 m s h", "state_after": "case h\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\n\u22a2 (c \u2022 extend m) s = extend (fun s h => c \u2022 m s h) s"}, {"tactic": "dsimp [extend]", "annotated_tactic": ["dsimp [<a>extend</a>]", [{"full_name": "MeasureTheory.extend", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1312, 5], "def_end_pos": [1312, 11]}]], "state_before": "case h\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\n\u22a2 (c \u2022 extend m) s = extend (fun s h => c \u2022 m s h) s", "state_after": "case h\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h"}, {"tactic": "by_cases h : P s", "annotated_tactic": ["by_cases h : P s", []], "state_before": "case h\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h", "state_after": "case pos\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\nh : P s\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h\n\ncase neg\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\nh : \u00acP s\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case pos\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\nh : P s\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h", "state_after": "no goals"}, {"tactic": "simp [h, ENNReal.smul_top, hc]", "annotated_tactic": ["simp [h, <a>ENNReal.smul_top</a>, hc]", [{"full_name": "ENNReal.smul_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [593, 9], "def_end_pos": [593, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\nh : \u00acP s\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h", "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_smul", "start": [407, 1], "end": [409, 80], "traced_tactics": [{"tactic": "simp_rw [condexpIndSMul]", "annotated_tactic": ["simp_rw [<a>condexpIndSMul</a>]", [{"full_name": "MeasureTheory.condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [384, 19], "def_end_pos": [384, 33]}]], "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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 condexpIndSMul hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndSMul hm hs h\u03bcs 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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (c \u2022 x))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\n    c \u2022 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))"}, {"tactic": "rw [toSpanSingleton_smul, smul_compLpL, smul_apply]", "annotated_tactic": ["rw [<a>toSpanSingleton_smul</a>, <a>smul_compLpL</a>, <a>smul_apply</a>]", [{"full_name": "ContinuousLinearMap.toSpanSingleton_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1264, 9], "def_end_pos": [1264, 29]}, {"full_name": "ContinuousLinearMap.smul_compLpL", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1191, 9], "def_end_pos": [1191, 21]}, {"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\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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (c \u2022 x))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\n    c \u2022 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "MeasureTheory.meas_le_ae_eq_meas_lt", "start": [84, 1], "end": [87, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.diag_sub_val", "start": [646, 1], "end": [650, 23], "traced_tactics": [{"tactic": "ext i x", "annotated_tactic": ["ext i x", []], "state_before": "n\u271d n : \u2115\n\u03b1 : TypeVec.{u} n\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a diagSub = prod.diag", "state_after": "case a.h\nn\u271d n : \u2115\n\u03b1 : TypeVec.{u} n\ni : Fin2 n\nx : \u03b1 i\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) i x = prod.diag i x"}, {"tactic": "induction' i with _ _ _ i_ih", "annotated_tactic": ["induction' i with _ _ _ i_ih", []], "state_before": "case a.h\nn\u271d n : \u2115\n\u03b1 : TypeVec.{u} n\ni : Fin2 n\nx : \u03b1 i\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) i x = prod.diag i x", "state_after": "case a.h.fz\nn\u271d\u00b9 n : \u2115\n\u03b1\u271d : TypeVec.{u} n\ni : Fin2 n\nx\u271d : \u03b1\u271d i\nn\u271d : \u2115\n\u03b1 : TypeVec.{u} (succ n\u271d)\nx : \u03b1 Fin2.fz\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) Fin2.fz x = prod.diag Fin2.fz x\n\ncase a.h.fs\nn\u271d\u00b9 n : \u2115\n\u03b1\u271d : TypeVec.{u} n\ni : Fin2 n\nx\u271d : \u03b1\u271d i\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\ni_ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : \u03b1 a\u271d), (subtypeVal (repeatEq \u03b1) \u229a diagSub) a\u271d x = prod.diag a\u271d x\n\u03b1 : TypeVec.{u} (succ n\u271d)\nx : \u03b1 (Fin2.fs a\u271d)\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) (Fin2.fs a\u271d) x = prod.diag (Fin2.fs a\u271d) x"}, {"tactic": "simp only [comp, subtypeVal, repeatEq._eq_2, diagSub, prod.diag]", "annotated_tactic": ["simp only [<a>comp</a>, <a>subtypeVal</a>, repeatEq._eq_2, <a>diagSub</a>, <a>prod.diag</a>]", [{"full_name": "TypeVec.comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [73, 5], "def_end_pos": [73, 9]}, {"full_name": "TypeVec.subtypeVal", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [598, 5], "def_end_pos": [598, 15]}, {"full_name": "TypeVec.diagSub", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [636, 5], "def_end_pos": [636, 12]}, {"full_name": "TypeVec.prod.diag", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [516, 5], "def_end_pos": [516, 14]}]], "state_before": "case a.h.fz\nn\u271d\u00b9 n : \u2115\n\u03b1\u271d : TypeVec.{u} n\ni : Fin2 n\nx\u271d : \u03b1\u271d i\nn\u271d : \u2115\n\u03b1 : TypeVec.{u} (succ n\u271d)\nx : \u03b1 Fin2.fz\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) Fin2.fz x = prod.diag Fin2.fz x\n\ncase a.h.fs\nn\u271d\u00b9 n : \u2115\n\u03b1\u271d : TypeVec.{u} n\ni : Fin2 n\nx\u271d : \u03b1\u271d i\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\ni_ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : \u03b1 a\u271d), (subtypeVal (repeatEq \u03b1) \u229a diagSub) a\u271d x = prod.diag a\u271d x\n\u03b1 : TypeVec.{u} (succ n\u271d)\nx : \u03b1 (Fin2.fs a\u271d)\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) (Fin2.fs a\u271d) x = prod.diag (Fin2.fs a\u271d) x", "state_after": "case a.h.fs\nn\u271d\u00b9 n : \u2115\n\u03b1\u271d : TypeVec.{u} n\ni : Fin2 n\nx\u271d : \u03b1\u271d i\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\ni_ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : \u03b1 a\u271d), (subtypeVal (repeatEq \u03b1) \u229a diagSub) a\u271d x = prod.diag a\u271d x\n\u03b1 : TypeVec.{u} (succ n\u271d)\nx : \u03b1 (Fin2.fs a\u271d)\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) (Fin2.fs a\u271d) x = prod.diag (Fin2.fs a\u271d) x"}, {"tactic": "apply @i_ih (drop \u03b1)", "annotated_tactic": ["apply @i_ih (<a>drop</a> \u03b1)", [{"full_name": "TypeVec.drop", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [102, 5], "def_end_pos": [102, 9]}]], "state_before": "case a.h.fs\nn\u271d\u00b9 n : \u2115\n\u03b1\u271d : TypeVec.{u} n\ni : Fin2 n\nx\u271d : \u03b1\u271d i\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\ni_ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : \u03b1 a\u271d), (subtypeVal (repeatEq \u03b1) \u229a diagSub) a\u271d x = prod.diag a\u271d x\n\u03b1 : TypeVec.{u} (succ n\u271d)\nx : \u03b1 (Fin2.fs a\u271d)\n\u22a2 (subtypeVal (repeatEq \u03b1) \u229a 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/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_smul_measure", "start": [1571, 1], "end": [1584, 80], "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\nc : \u211d\u22650\u221e\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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\nc : \u211d\u22650\u221e\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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\nc : \u211d\u22650\u221e\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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\nc : \u211d\u22650\u221e\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rcases eq_or_ne c \u221e with (rfl | hc)", "annotated_tactic": ["rcases <a>eq_or_ne</a> c \u221e with (rfl | hc)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.inl\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\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202\u22a4 \u2022 \u03bc = ENNReal.toReal \u22a4 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc\n\ncase pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "simp_rw [integral_eq_setToFun, \u2190 setToFun_smul_left]", "annotated_tactic": ["simp_rw [<a>integral_eq_setToFun</a>, \u2190 <a>setToFun_smul_left</a>]", [{"full_name": "MeasureTheory.integral_eq_setToFun", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [825, 9], "def_end_pos": [825, 29]}, {"full_name": "MeasureTheory.setToFun_smul_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1329, 9], "def_end_pos": [1329, 27]}]], "state_before": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\n\u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>\n      f a) =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a"}, {"tactic": "have hdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc) : Set \u03b1 \u2192 G \u2192L[\u211d] G) c.toReal :=\n  mul_one c.toReal \u25b8 (dominatedFinMeasAdditive_weightedSMul (c \u2022 \u03bc)).of_smul_measure c hc", "annotated_tactic": ["have hdfma : <a>DominatedFinMeasAdditive</a> \u03bc (<a>weightedSMul</a> (c \u2022 \u03bc) : <a>Set</a> \u03b1 \u2192 G \u2192L[\u211d] G) c.toReal :=\n    <a>mul_one</a> c.toReal \u25b8 (<a>dominatedFinMeasAdditive_weightedSMul</a> (c \u2022 \u03bc)).<a>of_smul_measure</a> c hc", [{"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": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}, {"full_name": "MeasureTheory.DominatedFinMeasAdditive.of_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}]], "state_before": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\n\u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>\n      f a) =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a", "state_after": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\nhdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c)\n\u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>\n      f a) =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a"}, {"tactic": "have hdfma_smul := dominatedFinMeasAdditive_weightedSMul (F := G) (c \u2022 \u03bc)", "annotated_tactic": ["have hdfma_smul := <a>dominatedFinMeasAdditive_weightedSMul</a> (F := G) (c \u2022 \u03bc)", [{"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.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\nhdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c)\n\u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>\n      f a) =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a", "state_after": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\nhdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c)\nhdfma_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1\n\u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>\n      f a) =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a"}, {"tactic": "rw [\u2190 setToFun_congr_smul_measure c hc hdfma hdfma_smul f]", "annotated_tactic": ["rw [\u2190 <a>setToFun_congr_smul_measure</a> c hc hdfma hdfma_smul f]", [{"full_name": "MeasureTheory.setToFun_congr_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 36]}]], "state_before": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\nhdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c)\nhdfma_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1\n\u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>\n      f a) =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a", "state_after": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\nhdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c)\nhdfma_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1\n\u22a2 setToFun \u03bc (weightedSMul (c \u2022 \u03bc)) hdfma f =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a"}, {"tactic": "exact setToFun_congr_left' _ _ (fun s _ _ => weightedSMul_smul_measure \u03bc c) f", "annotated_tactic": ["exact <a>setToFun_congr_left'</a> _ _ (fun s _ _ => <a>weightedSMul_smul_measure</a> \u03bc c) f", [{"full_name": "MeasureTheory.setToFun_congr_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1304, 9], "def_end_pos": [1304, 29]}, {"full_name": "MeasureTheory.weightedSMul_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [194, 9], "def_end_pos": [194, 34]}]], "state_before": "case pos.inr\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\nc : \u211d\u22650\u221e\nhG : CompleteSpace G\nhc : c \u2260 \u22a4\nhdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c)\nhdfma_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1\n\u22a2 setToFun \u03bc (weightedSMul (c \u2022 \u03bc)) hdfma f =\n    setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)\n      (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>\n      f a", "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\nc : \u211d\u22650\u221e\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [ENNReal.top_toReal, zero_smul, integral_eq_setToFun, setToFun_top_smul_measure]", "annotated_tactic": ["rw [<a>ENNReal.top_toReal</a>, <a>zero_smul</a>, <a>integral_eq_setToFun</a>, <a>setToFun_top_smul_measure</a>]", [{"full_name": "ENNReal.top_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [227, 17], "def_end_pos": [227, 27]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "MeasureTheory.integral_eq_setToFun", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [825, 9], "def_end_pos": [825, 29]}, {"full_name": "MeasureTheory.setToFun_top_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1666, 9], "def_end_pos": [1666, 34]}]], "state_before": "case pos.inl\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\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b1), f x \u2202\u22a4 \u2022 \u03bc = ENNReal.toReal \u22a4 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "exists_measurable_piecewise", "start": [847, 1], "end": [866, 57], "traced_tactics": [{"tactic": "inhabit \u03b9", "annotated_tactic": ["inhabit \u03b9", []], "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\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)"}, {"tactic": "set g' : (i : \u03b9) \u2192 t i \u2192 \u03b2 := fun i => g i \u2218 (\u2191)", "annotated_tactic": ["set g' : (i : \u03b9) \u2192 t i \u2192 \u03b2 := fun i => g i \u2218 (\u2191)", []], "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\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)"}, {"tactic": "have ht' : \u2200 (i j) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j), g' i \u27e8x, hxi\u27e9 = g' j \u27e8x, hxj\u27e9", "annotated_tactic": ["have ht' : \u2200 (i j) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j), g' i \u27e8x, hxi\u27e9 = g' j \u27e8x, hxj\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "state_after": "case ht'\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\n\u22a2 \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\n\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)"}, {"tactic": "set f : (\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' _ Subset.rfl", "annotated_tactic": ["set f : (\u22c3 i, t i) \u2192 \u03b2 := <a>iUnionLift</a> t g' ht' _ <a>Subset.rfl</a>", [{"full_name": "Set.iUnionLift", "def_path": "Mathlib/Data/Set/UnionLift.lean", "def_pos": [53, 19], "def_end_pos": [53, 29]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "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\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)"}, {"tactic": "have hfm : Measurable f := measurable_iUnionLift _ _ t_meas\n  (fun i => (hg i).comp measurable_subtype_coe)", "annotated_tactic": ["have hfm : <a>Measurable</a> f := <a>measurable_iUnionLift</a> _ _ t_meas\n    (fun i => (hg i).<a>comp</a> <a>measurable_subtype_coe</a>)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_iUnionLift", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [824, 9], "def_end_pos": [824, 30]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)"}, {"tactic": "classical\n  refine \u27e8fun x => if hx : x \u2208 \u22c3 i, t i then f \u27e8x, hx\u27e9 else g default x,\n    hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_\u27e9\n  simp only [dif_pos (mem_iUnion.2 \u27e8i, hx\u27e9)]\n  exact iUnionLift_of_mem \u27e8x, mem_iUnion.2 \u27e8i, hx\u27e9\u27e9 hx", "annotated_tactic": ["classical\n    refine \u27e8fun x => if hx : x \u2208 \u22c3 i, t i then f \u27e8x, hx\u27e9 else g <a>default</a> x,\n      hfm.dite ((hg <a>default</a>).<a>comp</a> <a>measurable_subtype_coe</a>) (.iUnion t_meas), fun i x hx => ?_\u27e9\n    simp only [<a>dif_pos</a> (<a>mem_iUnion</a>.2 \u27e8i, hx\u27e9)]\n    exact <a>iUnionLift_of_mem</a> \u27e8x, <a>mem_iUnion</a>.2 \u27e8i, hx\u27e9\u27e9 hx", [{"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": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}, {"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": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.iUnionLift_of_mem", "def_path": "Mathlib/Data/Set/UnionLift.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"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\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "state_after": "no goals"}, {"tactic": "intro i j x hxi hxj", "annotated_tactic": ["intro i j x hxi hxj", []], "state_before": "case ht'\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\n\u22a2 \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }", "state_after": "case ht'\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\ni j : \u03b9\nx : \u03b1\nhxi : x \u2208 t i\nhxj : x \u2208 t j\n\u22a2 g' i { val := x, property := hxi } = g' j { val := x, property := hxj }"}, {"tactic": "rcases eq_or_ne i j with rfl | hij", "annotated_tactic": ["rcases <a>eq_or_ne</a> i j with rfl | hij", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case ht'\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\ni j : \u03b9\nx : \u03b1\nhxi : x \u2208 t i\nhxj : x \u2208 t j\n\u22a2 g' i { val := x, property := hxi } = g' j { val := x, property := hxj }", "state_after": "case ht'.inl\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\ni : \u03b9\nx : \u03b1\nhxi hxj : x \u2208 t i\n\u22a2 g' i { val := x, property := hxi } = g' i { val := x, property := hxj }\n\ncase ht'.inr\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\ni j : \u03b9\nx : \u03b1\nhxi : x \u2208 t i\nhxj : x \u2208 t j\nhij : i \u2260 j\n\u22a2 g' i { val := x, property := hxi } = g' j { val := x, property := hxj }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case ht'.inl\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\ni : \u03b9\nx : \u03b1\nhxi hxj : x \u2208 t i\n\u22a2 g' i { val := x, property := hxi } = g' i { val := x, property := hxj }", "state_after": "no goals"}, {"tactic": "exact ht hij \u27e8hxi, hxj\u27e9", "annotated_tactic": ["exact ht hij \u27e8hxi, hxj\u27e9", []], "state_before": "case ht'.inr\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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\ni j : \u03b9\nx : \u03b1\nhxi : x \u2208 t i\nhxj : x \u2208 t j\nhij : i \u2260 j\n\u22a2 g' i { val := x, property := hxi } = g' j { val := x, property := hxj }", "state_after": "no goals"}, {"tactic": "refine \u27e8fun x => if hx : x \u2208 \u22c3 i, t i then f \u27e8x, hx\u27e9 else g default x,\n  hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun x => if hx : x \u2208 \u22c3 i, t i then f \u27e8x, hx\u27e9 else g <a>default</a> x,\n      hfm.dite ((hg <a>default</a>).<a>comp</a> <a>measurable_subtype_coe</a>) (.iUnion t_meas), fun i x hx => ?_\u27e9", [{"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": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\n\u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n)", "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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\ni : \u03b9\nx : \u03b1\nhx : x \u2208 t i\n\u22a2 (fun x => if hx : x \u2208 \u22c3 i, t i then f { val := x, property := hx } else g default x) x = g i x"}, {"tactic": "simp only [dif_pos (mem_iUnion.2 \u27e8i, hx\u27e9)]", "annotated_tactic": ["simp only [<a>dif_pos</a> (<a>mem_iUnion</a>.2 \u27e8i, hx\u27e9)]", [{"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": "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\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\ni : \u03b9\nx : \u03b1\nhx : x \u2208 t i\n\u22a2 (fun x => if hx : x \u2208 \u22c3 i, t i then f { val := x, property := hx } else g default x) x = g i x", "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\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\ni : \u03b9\nx : \u03b1\nhx : x \u2208 t i\n\u22a2 iUnionLift t (fun i => g i \u2218 Subtype.val) ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\n      { val := x, property := (_ : x \u2208 \u22c3 i, t i) } =\n    g i x"}, {"tactic": "exact iUnionLift_of_mem \u27e8x, mem_iUnion.2 \u27e8i, hx\u27e9\u27e9 hx", "annotated_tactic": ["exact <a>iUnionLift_of_mem</a> \u27e8x, <a>mem_iUnion</a>.2 \u27e8i, hx\u27e9\u27e9 hx", [{"full_name": "Set.iUnionLift_of_mem", "def_path": "Mathlib/Data/Set/UnionLift.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"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\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u\u03b9\ns t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : Nonempty \u03b9\nt : \u03b9 \u2192 Set \u03b1\nt_meas : \u2200 (n : \u03b9), MeasurableSet (t n)\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (n : \u03b9), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j)\ninhabited_h : Inhabited \u03b9\ng' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val\nht' :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),\n    g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\nhfm : Measurable f\ni : \u03b9\nx : \u03b1\nhx : x \u2208 t i\n\u22a2 iUnionLift t (fun i => g i \u2218 Subtype.val) ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)\n      { val := x, property := (_ : x \u2208 \u22c3 i, t i) } =\n    g i x", "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.get_push_lt", "start": [126, 1], "end": [129, 88], "traced_tactics": [{"tactic": "simp [*, Nat.lt_succ_of_le, Nat.le_of_lt]", "annotated_tactic": ["simp [*, <a>Nat.lt_succ_of_le</a>, <a>Nat.le_of_lt</a>]", [{"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]}, {"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": "\u03b1 : Type ?u.45557\na : Array \u03b1\nx : \u03b1\ni : Nat\nh : i < size a\n\u22a2 i < size (push a x)", "state_after": "no goals"}, {"tactic": "simp only [push, getElem_eq_data_get, List.concat_eq_append, List.get_append_left, h]", "annotated_tactic": ["simp only [<a>push</a>, <a>getElem_eq_data_get</a>, <a>List.concat_eq_append</a>, <a>List.get_append_left</a>, h]", [{"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": "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.concat_eq_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [850, 9], "def_end_pos": [850, 25]}, {"full_name": "List.get_append_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/BasicAux.lean", "def_pos": [94, 9], "def_end_pos": [94, 24]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nx : \u03b1\ni : Nat\nh : i < size a\n\u22a2 (push a x)[i] = a[i]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.martingale_zero", "start": [84, 1], "end": [85, 82], "traced_tactics": [{"tactic": "rw [Pi.zero_apply, condexp_zero]", "annotated_tactic": ["rw [<a>Pi.zero_apply</a>, <a>condexp_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": "MeasureTheory.condexp_zero", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 21]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 \u03bc[OfNat.ofNat 0 j|\u2191\u2131 i] =\u1d50[\u03bc] OfNat.ofNat 0 i", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 0 =\u1d50[\u03bc] OfNat.ofNat 0 i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 0 =\u1d50[\u03bc] OfNat.ofNat 0 i", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 0 =\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 0 =\u1d50[\u03bc] 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.comp_snd_map_prod_id", "start": [52, 1], "end": [57, 11], "traced_tactics": [{"tactic": "rw [\u2190 integrable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf", "annotated_tactic": ["rw [\u2190 <a>integrable_comp_snd_map_prod_mk_iff</a> (<a>measurable_id''</a> hm)] at hf", [{"full_name": "ProbabilityTheory.integrable_comp_snd_map_prod_mk_iff", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [332, 9], "def_end_pos": [332, 44]}, {"full_name": "measurable_id''", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [239, 9], "def_end_pos": [239, 24]}]], "state_before": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : NormedAddCommGroup F\nhm : m \u2264 m\u03a9\nhf : Integrable f\n\u22a2 Integrable fun x => f x.2", "state_after": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : NormedAddCommGroup F\nhm : m \u2264 m\u03a9\nhf : Integrable fun x => f x.2\n\u22a2 Integrable fun x => f x.2"}, {"tactic": "exact hf", "annotated_tactic": ["exact hf", []], "state_before": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : NormedAddCommGroup F\nhm : m \u2264 m\u03a9\nhf : Integrable fun x => f x.2\n\u22a2 Integrable fun x => f x.2", "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_atBot", "start": [179, 1], "end": [183, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector3.lean", "full_name": "vectorAll_iff_forall", "start": [248, 1], "end": [252, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2.stmts_trans", "start": [2227, 1], "end": [2231, 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.TM2.stmts", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2223, 19], "def_end_pos": [2223, 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": "K : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\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": "K : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\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.TM2.stmts\u2081_trans", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2192, 9], "def_end_pos": [2192, 21]}]], "state_before": "K : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\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/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.Indep.symm", "start": [229, 1], "end": [230, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.range_add", "start": [2102, 1], "end": [2104, 75], "traced_tactics": [{"tactic": "rw [\u2190 range'_eq_map_range]", "annotated_tactic": ["rw [\u2190 <a>range'_eq_map_range</a>]", [{"full_name": "List.range'_eq_map_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2071, 9], "def_end_pos": [2071, 28]}]], "state_before": "a b : Nat\n\u22a2 range (a + b) = range a ++ map (fun x => a + x) (range b)", "state_after": "a b : Nat\n\u22a2 range (a + b) = range a ++ range' a b"}, {"tactic": "simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm", "annotated_tactic": ["simpa [<a>range_eq_range'</a>, <a>Nat.add_comm</a>] using (<a>range'_append_1</a> 0 a b).<a>symm</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": "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": "List.range'_append_1", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2028, 17], "def_end_pos": [2028, 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": "a b : Nat\n\u22a2 range (a + b) = range a ++ range' a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "QuotientGroup.measurable_coe", "start": [550, 1], "end": [552, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegrable_sub_zpow_iff", "start": [301, 1], "end": [331, 74], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R \u2194 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|", "state_after": "case mp\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R \u2192 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|\n\ncase mpr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\n\u22a2 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| \u2192 CircleIntegrable (fun z => (z - w) ^ n) c R"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R \u2192 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|", "state_after": "case mp\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nh : CircleIntegrable (fun z => (z - w) ^ n) c R\n\u22a2 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case mp\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nh : CircleIntegrable (fun z => (z - w) ^ n) c R\n\u22a2 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|", "state_after": "case mp\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nh : R \u2260 0 \u2227 n < 0 \u2227 w \u2208 sphere c |R|\n\u22a2 \u00acCircleIntegrable (fun z => (z - w) ^ n) c R"}, {"tactic": "rcases h with \u27e8hR, hn, hw\u27e9", "annotated_tactic": ["rcases h with \u27e8hR, hn, hw\u27e9", []], "state_before": "case mp\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nh : R \u2260 0 \u2227 n < 0 \u2227 w \u2208 sphere c |R|\n\u22a2 \u00acCircleIntegrable (fun z => (z - w) ^ n) c R", "state_after": "case mp.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\nhw : w \u2208 sphere c |R|\n\u22a2 \u00acCircleIntegrable (fun z => (z - w) ^ n) c R"}, {"tactic": "simp only [circleIntegrable_iff R, deriv_circleMap]", "annotated_tactic": ["simp only [<a>circleIntegrable_iff</a> R, <a>deriv_circleMap</a>]", [{"full_name": "circleIntegrable_iff", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [273, 9], "def_end_pos": [273, 29]}, {"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}]], "state_before": "case mp.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\nhw : w \u2208 sphere c |R|\n\u22a2 \u00acCircleIntegrable (fun z => (z - w) ^ n) c R", "state_after": "case mp.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\nhw : w \u2208 sphere c |R|\n\u22a2 \u00acIntervalIntegrable (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (circleMap c R \u03b8 - w) ^ n) volume 0 (2 * \u03c0)"}, {"tactic": "rw [\u2190 image_circleMap_Ioc] at hw", "annotated_tactic": ["rw [\u2190 <a>image_circleMap_Ioc</a>] at hw", [{"full_name": "image_circleMap_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [154, 9], "def_end_pos": [154, 28]}]], "state_before": "case mp.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\nhw : w \u2208 sphere c |R|\n\u22a2 \u00acIntervalIntegrable (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (circleMap c R \u03b8 - w) ^ n) volume 0 (2 * \u03c0)", "state_after": "case mp.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\nhw : w \u2208 circleMap c R '' Ioc 0 (2 * \u03c0)\n\u22a2 \u00acIntervalIntegrable (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (circleMap c R \u03b8 - w) ^ n) volume 0 (2 * \u03c0)"}, {"tactic": "rcases hw with \u27e8\u03b8, h\u03b8, rfl\u27e9", "annotated_tactic": ["rcases hw with \u27e8\u03b8, h\u03b8, rfl\u27e9", []], "state_before": "case mp.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\nhw : w \u2208 circleMap c R '' Ioc 0 (2 * \u03c0)\n\u22a2 \u00acIntervalIntegrable (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (circleMap c R \u03b8 - w) ^ n) volume 0 (2 * \u03c0)", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0)"}, {"tactic": "replace h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]", "annotated_tactic": ["replace h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]", []], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0)", "state_after": "case h\u03b8\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 \u03b8 \u2208 [[0, 2 * \u03c0]]\n\ncase mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0)"}, {"tactic": "exact Icc_subset_uIcc (Ioc_subset_Icc_self h\u03b8)", "annotated_tactic": ["exact <a>Icc_subset_uIcc</a> (<a>Ioc_subset_Icc_self</a> h\u03b8)", [{"full_name": "Set.Icc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [92, 7], "def_end_pos": [92, 22]}, {"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": "case h\u03b8\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 \u03b8 \u2208 [[0, 2 * \u03c0]]\n\ncase mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0)", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0)"}, {"tactic": "refine' not_intervalIntegrable_of_sub_inv_isBigO_punctured _ Real.two_pi_pos.ne h\u03b8", "annotated_tactic": ["refine' <a>not_intervalIntegrable_of_sub_inv_isBigO_punctured</a> _ Real.two_pi_pos.ne h\u03b8", [{"full_name": "not_intervalIntegrable_of_sub_inv_isBigO_punctured", "def_path": "Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean", "def_pos": [140, 9], "def_end_pos": [140, 59]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0)", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n"}, {"tactic": "set f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8", "annotated_tactic": ["set f : \u211d \u2192 \u2102 := fun \u03b8' => <a>circleMap</a> c R \u03b8' - <a>circleMap</a> c R \u03b8", [{"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\n\u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n"}, {"tactic": "have : \u2200\u1da0 \u03b8' in \ud835\udcdd[\u2260] \u03b8, f \u03b8' \u2208 ball (0 : \u2102) 1 \\ {0} := by\n  suffices : \u2200\u1da0 z in \ud835\udcdd[\u2260] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball (0 : \u2102) 1 \\ {0}\n  exact ((differentiable_circleMap c R \u03b8).hasDerivAt.tendsto_punctured_nhds\n    (deriv_circleMap_ne_zero hR)).eventually this\n  filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (ball_mem_nhds _ zero_lt_one)]\n  simp_all [dist_eq, sub_eq_zero]", "annotated_tactic": ["have : \u2200\u1da0 \u03b8' in \ud835\udcdd[\u2260] \u03b8, f \u03b8' \u2208 <a>ball</a> (0 : \u2102) 1 \\ {0} := by\n      suffices : \u2200\u1da0 z in \ud835\udcdd[\u2260] <a>circleMap</a> c R \u03b8, z - <a>circleMap</a> c R \u03b8 \u2208 <a>ball</a> (0 : \u2102) 1 \\ {0}\n      exact ((<a>differentiable_circleMap</a> c R \u03b8).hasDerivAt.tendsto_punctured_nhds\n        (<a>deriv_circleMap_ne_zero</a> hR)).<a>eventually</a> this\n      filter_upwards [<a>self_mem_nhdsWithin</a>, <a>mem_nhdsWithin_of_mem_nhds</a> (<a>ball_mem_nhds</a> _ <a>zero_lt_one</a>)]\n      simp_all [<a>dist_eq</a>, <a>sub_eq_zero</a>]", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "differentiable_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [180, 9], "def_end_pos": [180, 33]}, {"full_name": "deriv_circleMap_ne_zero", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [203, 9], "def_end_pos": [203, 32]}, {"full_name": "Filter.Tendsto.eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2953, 9], "def_end_pos": [2953, 27]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Complex.dist_eq", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n"}, {"tactic": "refine' (((hasDerivAt_circleMap c R \u03b8).isBigO_sub.mono inf_le_left).inv_rev\n  (this.mono fun \u03b8' h\u2081 h\u2082 => absurd h\u2082 h\u2081.2)).trans _", "annotated_tactic": ["refine' (((<a>hasDerivAt_circleMap</a> c R \u03b8).isBigO_sub.mono <a>inf_le_left</a>).<a>inv_rev</a>\n      (this.mono fun \u03b8' h\u2081 h\u2082 => <a>absurd</a> h\u2082 h\u2081.2)).<a>trans</a> _", [{"full_name": "hasDerivAt_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [172, 9], "def_end_pos": [172, 29]}, {"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [388, 9], "def_end_pos": [388, 20]}, {"full_name": "Asymptotics.IsBigO.inv_rev", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1718, 9], "def_end_pos": [1718, 23]}, {"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"full_name": "Asymptotics.IsBigO.trans", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [505, 9], "def_end_pos": [505, 21]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u22a2 (fun x => (circleMap c R x - circleMap c R \u03b8)\u207b\u00b9) =O[\ud835\udcdd \u03b8 \u2293 \ud835\udcdf {\u03b8}\u1d9c] fun \u03b8_1 =>\n    (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n"}, {"tactic": "refine' IsBigO.of_bound |R|\u207b\u00b9 (this.mono fun \u03b8' h\u03b8' => _)", "annotated_tactic": ["refine' <a>IsBigO.of_bound</a> |R|\u207b\u00b9 (this.mono fun \u03b8' h\u03b8' => _)", [{"full_name": "Asymptotics.IsBigO.of_bound", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [155, 9], "def_end_pos": [155, 24]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u22a2 (fun x => (circleMap c R x - circleMap c R \u03b8)\u207b\u00b9) =O[\ud835\udcdd \u03b8 \u2293 \ud835\udcdf {\u03b8}\u1d9c] fun \u03b8_1 =>\n    (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\n\u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016"}, {"tactic": "set x := abs (f \u03b8')", "annotated_tactic": ["set x := <a>abs</a> (f \u03b8')", [{"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\n\u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\n\u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016"}, {"tactic": "suffices x\u207b\u00b9 \u2264 x ^ n by\n  simpa only [inv_mul_cancel_left\u2080, abs_eq_zero.not.2 hR, norm_eq_abs, map_inv\u2080,\n    Algebra.id.smul_eq_mul, map_mul, abs_circleMap_zero, abs_I, mul_one, abs_zpow, Ne.def,\n    not_false_iff] using this", "annotated_tactic": ["suffices x\u207b\u00b9 \u2264 x ^ n by\n      simpa only [<a>inv_mul_cancel_left\u2080</a>, abs_eq_zero.not.2 hR, <a>norm_eq_abs</a>, <a>map_inv\u2080</a>,\n        <a>Algebra.id.smul_eq_mul</a>, <a>map_mul</a>, <a>abs_circleMap_zero</a>, <a>abs_I</a>, <a>mul_one</a>, <a>abs_zpow</a>, <a>Ne.def</a>,\n        <a>not_false_iff</a>] using this", [{"full_name": "inv_mul_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 29]}, {"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_inv\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [238, 9], "def_end_pos": [238, 17]}, {"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": "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": "Complex.abs_zpow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1057, 9], "def_end_pos": [1057, 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]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\n\u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\n\u22a2 x\u207b\u00b9 \u2264 x ^ n"}, {"tactic": "have : x \u2208 Ioo (0 : \u211d) 1 := by simpa [and_comm] using h\u03b8'", "annotated_tactic": ["have : x \u2208 <a>Ioo</a> (0 : \u211d) 1 := by simpa [<a>and_comm</a>] using h\u03b8'", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\n\u22a2 x\u207b\u00b9 \u2264 x ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x \u2208 Ioo 0 1\n\u22a2 x\u207b\u00b9 \u2264 x ^ n"}, {"tactic": "rw [\u2190 zpow_neg_one]", "annotated_tactic": ["rw [\u2190 <a>zpow_neg_one</a>]", [{"full_name": "zpow_neg_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x \u2208 Ioo 0 1\n\u22a2 x\u207b\u00b9 \u2264 x ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x \u2208 Ioo 0 1\n\u22a2 x ^ (-1) \u2264 x ^ n"}, {"tactic": "refine' (zpow_strictAnti this.1 this.2).le_iff_le.2 (Int.lt_add_one_iff.1 _)", "annotated_tactic": ["refine' (<a>zpow_strictAnti</a> this.1 this.2).<a>le_iff_le</a>.2 (<a>Int.lt_add_one_iff</a>.1 _)", [{"full_name": "zpow_strictAnti", "def_path": "Mathlib/Algebra/Order/Field/Power.lean", "def_pos": [70, 9], "def_end_pos": [70, 24]}, {"full_name": "StrictAnti.le_iff_le", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [845, 9], "def_end_pos": [845, 29]}, {"full_name": "Int.lt_add_one_iff", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}]], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x \u2208 Ioo 0 1\n\u22a2 x ^ (-1) \u2264 x ^ n", "state_after": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x \u2208 Ioo 0 1\n\u22a2 n < -1 + 1"}, {"tactic": "exact hn", "annotated_tactic": ["exact hn", []], "state_before": "case mp.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x \u2208 Ioo 0 1\n\u22a2 n < -1 + 1", "state_after": "no goals"}, {"tactic": "suffices : \u2200\u1da0 z in \ud835\udcdd[\u2260] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball (0 : \u2102) 1 \\ {0}", "annotated_tactic": ["suffices : \u2200\u1da0 z in \ud835\udcdd[\u2260] <a>circleMap</a> c R \u03b8, z - <a>circleMap</a> c R \u03b8 \u2208 <a>ball</a> (0 : \u2102) 1 \\ {0}", [{"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}\n\u22a2 \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\ncase this\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}"}, {"tactic": "exact ((differentiable_circleMap c R \u03b8).hasDerivAt.tendsto_punctured_nhds\n  (deriv_circleMap_ne_zero hR)).eventually this", "annotated_tactic": ["exact ((<a>differentiable_circleMap</a> c R \u03b8).hasDerivAt.tendsto_punctured_nhds\n        (<a>deriv_circleMap_ne_zero</a> hR)).<a>eventually</a> this", [{"full_name": "differentiable_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [180, 9], "def_end_pos": [180, 33]}, {"full_name": "deriv_circleMap_ne_zero", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [203, 9], "def_end_pos": [203, 32]}, {"full_name": "Filter.Tendsto.eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2953, 9], "def_end_pos": [2953, 27]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}\n\u22a2 \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\ncase this\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}", "state_after": "case this\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}"}, {"tactic": "filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (ball_mem_nhds _ zero_lt_one)]", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>, <a>mem_nhdsWithin_of_mem_nhds</a> (<a>ball_mem_nhds</a> _ <a>zero_lt_one</a>)]", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case this\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}", "state_after": "case h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200 (a : \u2102), a \u2208 {circleMap c R \u03b8}\u1d9c \u2192 a \u2208 ball (circleMap c R \u03b8) 1 \u2192 a - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}"}, {"tactic": "simp_all [dist_eq, sub_eq_zero]", "annotated_tactic": ["simp_all [<a>dist_eq</a>, <a>sub_eq_zero</a>]", [{"full_name": "Complex.dist_eq", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "case h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\n\u22a2 \u2200 (a : \u2102), a \u2208 {circleMap c R \u03b8}\u1d9c \u2192 a \u2208 ball (circleMap c R \u03b8) 1 \u2192 a - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0}", "state_after": "no goals"}, {"tactic": "simpa only [inv_mul_cancel_left\u2080, abs_eq_zero.not.2 hR, norm_eq_abs, map_inv\u2080,\n  Algebra.id.smul_eq_mul, map_mul, abs_circleMap_zero, abs_I, mul_one, abs_zpow, Ne.def,\n  not_false_iff] using this", "annotated_tactic": ["simpa only [<a>inv_mul_cancel_left\u2080</a>, abs_eq_zero.not.2 hR, <a>norm_eq_abs</a>, <a>map_inv\u2080</a>,\n        <a>Algebra.id.smul_eq_mul</a>, <a>map_mul</a>, <a>abs_circleMap_zero</a>, <a>abs_I</a>, <a>mul_one</a>, <a>abs_zpow</a>, <a>Ne.def</a>,\n        <a>not_false_iff</a>] using this", [{"full_name": "inv_mul_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 29]}, {"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_inv\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [238, 9], "def_end_pos": [238, 17]}, {"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": "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": "Complex.abs_zpow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1057, 9], "def_end_pos": [1057, 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]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\nthis : x\u207b\u00b9 \u2264 x ^ n\n\u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016", "state_after": "no goals"}, {"tactic": "simpa [and_comm] using h\u03b8'", "annotated_tactic": ["simpa [<a>and_comm</a>] using h\u03b8'", [{"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\nn : \u2124\nhR : R \u2260 0\nhn : n < 0\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]]\nf : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8\nthis : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}\n\u03b8' : \u211d\nh\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0}\nx : \u211d := \u2191Complex.abs (f \u03b8')\n\u22a2 x \u2208 Ioo 0 1", "state_after": "no goals"}, {"tactic": "rintro (rfl | H)", "annotated_tactic": ["rintro (rfl | H)", []], "state_before": "case mpr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\n\u22a2 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| \u2192 CircleIntegrable (fun z => (z - w) ^ n) c R", "state_after": "case mpr.inl\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nn : \u2124\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c 0\n\ncase mpr.inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nH : 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R"}, {"tactic": "exacts [circleIntegrable_zero_radius,\n  ((continuousOn_id.sub continuousOn_const).zpow\u2080 _ fun z hz =>\n    H.symm.imp_left fun (hw : w \u2209 sphere c |R|) =>\n      sub_ne_zero.2 <| ne_of_mem_of_not_mem hz hw).circleIntegrable']", "annotated_tactic": ["exacts [<a>circleIntegrable_zero_radius</a>,\n      ((continuousOn_id.sub <a>continuousOn_const</a>).<a>zpow\u2080</a> _ fun z hz =>\n        H.symm.imp_left fun (hw : w \u2209 <a>sphere</a> c |R|) =>\n          <a>sub_ne_zero</a>.2 <| <a>ne_of_mem_of_not_mem</a> hz hw).<a>circleIntegrable'</a>]", [{"full_name": "circleIntegrable_zero_radius", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [269, 9], "def_end_pos": [269, 37]}, {"full_name": "continuousOn_const", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 27]}, {"full_name": "ContinuousOn.zpow\u2080", "def_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "def_pos": [343, 9], "def_end_pos": [343, 27]}, {"full_name": "Metric.sphere", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [485, 5], "def_end_pos": [485, 11]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}, {"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": "ContinuousOn.circleIntegrable'", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [288, 9], "def_end_pos": [288, 39]}]], "state_before": "case mpr.inl\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nn : \u2124\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c 0\n\ncase mpr.inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc w : \u2102\nR : \u211d\nn : \u2124\nH : 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R|\n\u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.elim_apply", "start": [429, 1], "end": [430, 89], "traced_tactics": [{"tactic": "rw [elim_comp fun f : \u03b1 \u2192 \u03b2 => f y]", "annotated_tactic": ["rw [<a>elim_comp</a> fun f : \u03b1 \u2192 \u03b2 => f y]", [{"full_name": "Option.elim_comp", "def_path": "Mathlib/Data/Option/Basic.lean", "def_pos": [422, 9], "def_end_pos": [422, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nf : \u03b3 \u2192 \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192 \u03b2\ni : Option \u03b3\ny : \u03b1\n\u22a2 Option.elim i x f y = Option.elim i (x y) fun j => f j y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "Real.hasPDF_iff_of_measurable", "start": [273, 8], "end": [276, 36], "traced_tactics": [{"tactic": "rw [hasPDF_iff_of_measurable hX]", "annotated_tactic": ["rw [<a>hasPDF_iff_of_measurable</a> hX]", [{"full_name": "MeasureTheory.pdf.hasPDF_iff_of_measurable", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [232, 9], "def_end_pos": [232, 33]}]], "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\nhX : Measurable X\n\u22a2 HasPDF X \u2119 \u2194 map X \u2119 \u226a volume", "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\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Measurable X\n\u22a2 HaveLebesgueDecomposition (map X \u2119) volume \u2227 map X \u2119 \u226a volume \u2194 map X \u2119 \u226a volume"}, {"tactic": "exact and_iff_right inferInstance", "annotated_tactic": ["exact <a>and_iff_right</a> <a>inferInstance</a>", [{"full_name": "and_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [206, 9], "def_end_pos": [206, 22]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "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\nhX : Measurable X\n\u22a2 HaveLebesgueDecomposition (map X \u2119) volume \u2227 map X \u2119 \u226a volume \u2194 map X \u2119 \u226a volume", "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_antitone", "start": [454, 1], "end": [458, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.halting_problem_not_re", "start": [266, 1], "end": [267, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_eq_zero", "start": [1050, 1], "end": [1054, 35], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "n : \u2115\nx : ZMod n\n\u22a2 valMinAbs x = 0 \u2194 x = 0", "state_after": "case zero\nx : ZMod Nat.zero\n\u22a2 valMinAbs x = 0 \u2194 x = 0\n\ncase succ\nn : \u2115\nx : ZMod (Nat.succ n)\n\u22a2 valMinAbs x = 0 \u2194 x = 0"}, {"tactic": "rw [\u2190 valMinAbs_zero n.succ]", "annotated_tactic": ["rw [\u2190 <a>valMinAbs_zero</a> n.succ]", [{"full_name": "ZMod.valMinAbs_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 23]}]], "state_before": "case succ\nn : \u2115\nx : ZMod (Nat.succ n)\n\u22a2 valMinAbs x = 0 \u2194 x = 0", "state_after": "case succ\nn : \u2115\nx : ZMod (Nat.succ n)\n\u22a2 valMinAbs x = valMinAbs 0 \u2194 x = 0"}, {"tactic": "apply injective_valMinAbs.eq_iff", "annotated_tactic": ["apply injective_valMinAbs.eq_iff", []], "state_before": "case succ\nn : \u2115\nx : ZMod (Nat.succ n)\n\u22a2 valMinAbs x = valMinAbs 0 \u2194 x = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nx : ZMod Nat.zero\n\u22a2 valMinAbs x = 0 \u2194 x = 0", "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_le", "start": [962, 1], "end": [963, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_ae", "start": [819, 1], "end": [839, 20], "traced_tactics": [{"tactic": "have A : \u2200 i, Integrable (X i) := fun i \u21a6 (hident i).integrable_iff.2 hint", "annotated_tactic": ["have A : \u2200 i, <a>Integrable</a> (X i) := fun i \u21a6 (hident i).<a>integrable_iff</a>.2 hint", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "ProbabilityTheory.IdentDistrib.integrable_iff", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}]], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "let Y : \u2115 \u2192 \u03a9 \u2192 E := fun i \u21a6 (A i).1.mk (X i)", "annotated_tactic": ["let Y : \u2115 \u2192 \u03a9 \u2192 E := fun i \u21a6 (A i).1.<a>mk</a> (X i)", [{"full_name": "MeasureTheory.AEStronglyMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1202, 29], "def_end_pos": [1202, 31]}]], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "have B : \u2200\u1d50 \u03c9, \u2200 n, X n \u03c9 = Y n \u03c9 :=\n  ae_all_iff.2 (fun i \u21a6 AEStronglyMeasurable.ae_eq_mk (A i).1)", "annotated_tactic": ["have B : \u2200\u1d50 \u03c9, \u2200 n, X n \u03c9 = Y n \u03c9 :=\n    <a>ae_all_iff</a>.2 (fun i \u21a6 <a>AEStronglyMeasurable.ae_eq_mk</a> (A i).1)", [{"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.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}]], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "have Yint: Integrable (Y 0) := Integrable.congr hint (AEStronglyMeasurable.ae_eq_mk (A 0).1)", "annotated_tactic": ["have Yint: <a>Integrable</a> (Y 0) := <a>Integrable.congr</a> hint (<a>AEStronglyMeasurable.ae_eq_mk</a> (A 0).1)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 25]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}]], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "have C : \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 \u21a6 (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i in range n, Y i \u03c9)) atTop (\ud835\udcdd \ud835\udd3c[Y 0]) := by\n  apply strong_law_ae_of_measurable Y Yint ((A 0).1.stronglyMeasurable_mk)\n    (fun i j hij \u21a6 IndepFun.ae_eq (hindep hij) (A i).1.ae_eq_mk (A j).1.ae_eq_mk)\n    (fun i \u21a6 ((A i).1.identDistrib_mk.symm.trans (hident i)).trans (A 0).1.identDistrib_mk)", "annotated_tactic": ["have C : \u2200\u1d50 \u03c9, <a>Tendsto</a> (fun n : \u2115 \u21a6 (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i in <a>range</a> n, Y i \u03c9)) <a>atTop</a> (\ud835\udcdd \ud835\udd3c[Y 0]) := by\n    apply <a>strong_law_ae_of_measurable</a> Y Yint ((A 0).1.<a>stronglyMeasurable_mk</a>)\n      (fun i j hij \u21a6 <a>IndepFun.ae_eq</a> (hindep hij) (A i).1.<a>ae_eq_mk</a> (A j).1.<a>ae_eq_mk</a>)\n      (fun i \u21a6 ((A i).1.identDistrib_mk.symm.trans (hident i)).<a>trans</a> (A 0).1.<a>identDistrib_mk</a>)", [{"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_ae_of_measurable", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [730, 7], "def_end_pos": [730, 34]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "ProbabilityTheory.IndepFun.ae_eq", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 23]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "ProbabilityTheory.IdentDistrib.trans", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [94, 19], "def_end_pos": [94, 24]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.identDistrib_mk", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [127, 7], "def_end_pos": [127, 64]}]], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "filter_upwards [B, C] with \u03c9 h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [B, C] with \u03c9 h\u2081 h\u2082", []], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case 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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "have : \ud835\udd3c[X 0] = \ud835\udd3c[Y 0] := integral_congr_ae (AEStronglyMeasurable.ae_eq_mk (A 0).1)", "annotated_tactic": ["have : \ud835\udd3c[X 0] = \ud835\udd3c[Y 0] := <a>integral_congr_ae</a> (<a>AEStronglyMeasurable.ae_eq_mk</a> (A 0).1)", [{"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.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}]], "state_before": "case 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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case 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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case 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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))", "state_after": "case 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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))"}, {"tactic": "apply Tendsto.congr (fun n \u21a6 ?_) h\u2082", "annotated_tactic": ["apply <a>Tendsto.congr</a> (fun n \u21a6 ?_) h\u2082", [{"full_name": "Filter.Tendsto.congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3019, 9], "def_end_pos": [3019, 22]}]], "state_before": "case 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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))", "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\nn : \u2115\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9 = (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9"}, {"tactic": "congr with i", "annotated_tactic": ["congr with i", []], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\nn : \u2115\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9 = (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9", "state_after": "case e_a.e_f.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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\nn i : \u2115\n\u22a2 Y i \u03c9 = X i \u03c9"}, {"tactic": "exact (h\u2081 i).symm", "annotated_tactic": ["exact (h\u2081 i).<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 e_a.e_f.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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\nC : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\n\u03c9 : \u03a9\nh\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nh\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))\nthis : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a\nn i : \u2115\n\u22a2 Y i \u03c9 = X i \u03c9", "state_after": "no goals"}, {"tactic": "apply strong_law_ae_of_measurable Y Yint ((A 0).1.stronglyMeasurable_mk)\n  (fun i j hij \u21a6 IndepFun.ae_eq (hindep hij) (A i).1.ae_eq_mk (A j).1.ae_eq_mk)\n  (fun i \u21a6 ((A i).1.identDistrib_mk.symm.trans (hident i)).trans (A 0).1.identDistrib_mk)", "annotated_tactic": ["apply <a>strong_law_ae_of_measurable</a> Y Yint ((A 0).1.<a>stronglyMeasurable_mk</a>)\n      (fun i j hij \u21a6 <a>IndepFun.ae_eq</a> (hindep hij) (A i).1.<a>ae_eq_mk</a> (A j).1.<a>ae_eq_mk</a>)\n      (fun i \u21a6 ((A i).1.identDistrib_mk.symm.trans (hident i)).<a>trans</a> (A 0).1.<a>identDistrib_mk</a>)", [{"full_name": "ProbabilityTheory.strong_law_ae_of_measurable", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [730, 7], "def_end_pos": [730, 34]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "ProbabilityTheory.IndepFun.ae_eq", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 23]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "ProbabilityTheory.IdentDistrib.trans", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [94, 19], "def_end_pos": [94, 24]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.identDistrib_mk", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [127, 7], "def_end_pos": [127, 64]}]], "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\nX : \u2115 \u2192 \u03a9 \u2192 E\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nA : \u2200 (i : \u2115), Integrable (X i)\nY : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119)\nB : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9\nYint : Integrable (Y 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monotone.lean", "full_name": "StrictMonoOn.Iic_id_le", "start": [203, 1], "end": [218, 76], "traced_tactics": [{"tactic": "revert h\u03c6", "annotated_tactic": ["revert h\u03c6", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic n)\n\u22a2 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\n\u22a2 StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m"}, {"tactic": "refine'\n  Succ.rec_bot (fun n => StrictMonoOn \u03c6 (Set.Iic n) \u2192 \u2200 m \u2264 n, m \u2264 \u03c6 m)\n    (fun _ _ hm => hm.trans bot_le) _ _", "annotated_tactic": ["refine'\n    <a>Succ.rec_bot</a> (fun n => <a>StrictMonoOn</a> \u03c6 (<a>Set.Iic</a> n) \u2192 \u2200 m \u2264 n, m \u2264 \u03c6 m)\n      (fun _ _ hm => hm.trans <a>bot_le</a>) _ _", [{"full_name": "Succ.rec_bot", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 21]}, {"full_name": "StrictMonoOn", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [108, 5], "def_end_pos": [108, 17]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"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\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\n\u22a2 StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\n\u22a2 \u2200 (a : \u03b1),\n    (fun n => StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m) a \u2192\n      (fun n => StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m) (succ a)"}, {"tactic": "rintro k ih h\u03c6 m hm", "annotated_tactic": ["rintro k ih h\u03c6 m hm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\n\u22a2 \u2200 (a : \u03b1),\n    (fun n => StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m) a \u2192\n      (fun n => StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m) (succ a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\n\u22a2 m \u2264 \u03c6 m"}, {"tactic": "by_cases hk : IsMax k", "annotated_tactic": ["by_cases hk : <a>IsMax</a> k", [{"full_name": "IsMax", "def_path": "Mathlib/Order/Max.lean", "def_pos": [209, 5], "def_end_pos": [209, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\n\u22a2 m \u2264 \u03c6 m", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\nhk : IsMax k\n\u22a2 m \u2264 \u03c6 m\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\nhk : \u00acIsMax k\n\u22a2 m \u2264 \u03c6 m"}, {"tactic": "obtain rfl | h := le_succ_iff_eq_or_le.1 hm", "annotated_tactic": ["obtain rfl | h := <a>le_succ_iff_eq_or_le</a>.1 hm", [{"full_name": "Order.le_succ_iff_eq_or_le", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [440, 9], "def_end_pos": [440, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\nhk : \u00acIsMax k\n\u22a2 m \u2264 \u03c6 m", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\n\u22a2 succ k \u2264 \u03c6 (succ k)\n\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\nhk : \u00acIsMax k\nh : m \u2264 k\n\u22a2 m \u2264 \u03c6 m"}, {"tactic": "rw [succ_eq_iff_isMax.2 hk] at hm", "annotated_tactic": ["rw [<a>succ_eq_iff_isMax</a>.2 hk] at hm", [{"full_name": "Order.succ_eq_iff_isMax", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [406, 9], "def_end_pos": [406, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\nhk : IsMax k\n\u22a2 m \u2264 \u03c6 m", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 k\nhk : IsMax k\n\u22a2 m \u2264 \u03c6 m"}, {"tactic": "exact ih (h\u03c6.mono <| Iic_subset_Iic.2 (le_succ _)) _ hm", "annotated_tactic": ["exact ih (h\u03c6.mono <| <a>Iic_subset_Iic</a>.2 (<a>le_succ</a> _)) _ hm", [{"full_name": "Set.Iic_subset_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [419, 9], "def_end_pos": [419, 23]}, {"full_name": "Order.le_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 k\nhk : IsMax k\n\u22a2 m \u2264 \u03c6 m", "state_after": "no goals"}, {"tactic": "specialize ih (StrictMonoOn.mono h\u03c6 fun x hx => le_trans hx (le_succ _)) k le_rfl", "annotated_tactic": ["specialize ih (<a>StrictMonoOn.mono</a> h\u03c6 fun x hx => <a>le_trans</a> hx (<a>le_succ</a> _)) k <a>le_rfl</a>", [{"full_name": "StrictMonoOn.mono", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Order.le_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 16]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\n\u22a2 succ k \u2264 \u03c6 (succ k)", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\nih : k \u2264 \u03c6 k\n\u22a2 succ k \u2264 \u03c6 (succ k)"}, {"tactic": "refine' le_trans (succ_mono ih) (succ_le_of_lt (h\u03c6 (le_succ _) le_rfl _))", "annotated_tactic": ["refine' <a>le_trans</a> (<a>succ_mono</a> ih) (<a>succ_le_of_lt</a> (h\u03c6 (<a>le_succ</a> _) <a>le_rfl</a> _))", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Order.succ_mono", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [280, 9], "def_end_pos": [280, 18]}, {"full_name": "Order.succ_le_of_lt", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 22]}, {"full_name": "Order.le_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 16]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\nih : k \u2264 \u03c6 k\n\u22a2 succ k \u2264 \u03c6 (succ k)", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\nih : k \u2264 \u03c6 k\n\u22a2 k < succ k"}, {"tactic": "rw [lt_succ_iff_eq_or_lt_of_not_isMax hk]", "annotated_tactic": ["rw [<a>lt_succ_iff_eq_or_lt_of_not_isMax</a> hk]", [{"full_name": "Order.lt_succ_iff_eq_or_lt_of_not_isMax", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [446, 9], "def_end_pos": [446, 42]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\nih : k \u2264 \u03c6 k\n\u22a2 k < succ k", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\nih : k \u2264 \u03c6 k\n\u22a2 k = k \u2228 k < k"}, {"tactic": "exact Or.inl rfl", "annotated_tactic": ["exact <a>Or.inl</a> <a>rfl</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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nhk : \u00acIsMax k\nhm : succ k \u2264 succ k\nih : k \u2264 \u03c6 k\n\u22a2 k = k \u2228 k < k", "state_after": "no goals"}, {"tactic": "exact ih (StrictMonoOn.mono h\u03c6 fun x hx => le_trans hx (le_succ _)) _ h", "annotated_tactic": ["exact ih (<a>StrictMonoOn.mono</a> h\u03c6 fun x hx => <a>le_trans</a> hx (<a>le_succ</a> _)) _ h", [{"full_name": "StrictMonoOn.mono", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Order.le_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 16]}]], "state_before": "case neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : OrderBot \u03b1\nn : \u03b1\n\u03c6 : \u03b1 \u2192 \u03b1\nk : \u03b1\nih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m\nh\u03c6 : StrictMonoOn \u03c6 (Iic (succ k))\nm : \u03b1\nhm : m \u2264 succ k\nhk : \u00acIsMax k\nh : m \u2264 k\n\u22a2 m \u2264 \u03c6 m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.hasFiniteIntegral_neg_iff", "start": [249, 1], "end": [250, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.limsup_preimage_iterate_ae_eq", "start": [2311, 1], "end": [2319, 89], "traced_tactics": [{"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\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\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2200 (n : \u2115), (preimage f)^[n] 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 s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nn : \u2115\n\u22a2 (preimage f)^[n] s =\u1d50[\u03bc] s"}, {"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\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\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nn : \u2115\n\u22a2 (preimage f)^[n] s =\u1d50[\u03bc] s", "state_after": "case zero\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 : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 (preimage f)^[Nat.zero] s =\u1d50[\u03bc] s\n\ncase succ\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 : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nn : \u2115\nih : (preimage f)^[n] s =\u1d50[\u03bc] s\n\u22a2 (preimage f)^[Nat.succ n] s =\u1d50[\u03bc] s"}, {"tactic": "simpa only [iterate_succ', comp_apply] using ae_eq_trans (hf.ae_eq ih) hs", "annotated_tactic": ["simpa only [<a>iterate_succ'</a>, <a>comp_apply</a>] using <a>ae_eq_trans</a> (hf.ae_eq ih) hs", [{"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": "MeasureTheory.ae_eq_trans", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [444, 9], "def_end_pos": [444, 20]}]], "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\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\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nn : \u2115\nih : (preimage f)^[n] s =\u1d50[\u03bc] s\n\u22a2 (preimage f)^[Nat.succ n] s =\u1d50[\u03bc] s", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\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\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 (preimage f)^[Nat.zero] s =\u1d50[\u03bc] s", "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_integral_deterministic", "start": [427, 1], "end": [431, 60], "traced_tactics": [{"tactic": "rw [kernel.deterministic_apply, set_integral_dirac f _ s]", "annotated_tactic": ["rw [<a>kernel.deterministic_apply</a>, <a>set_integral_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_integral_dirac", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1685, 9], "def_end_pos": [1685, 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 : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf : \u03b2 \u2192 E\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 (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/LocallyFinite.lean", "full_name": "Finset.Icc_subset_Ico_iff", "start": [266, 1], "end": [267, 65], "traced_tactics": [{"tactic": "rw [\u2190 coe_subset, coe_Icc, coe_Ico, Set.Icc_subset_Ico_iff h\u2081]", "annotated_tactic": ["rw [\u2190 <a>coe_subset</a>, <a>coe_Icc</a>, <a>coe_Ico</a>, <a>Set.Icc_subset_Ico_iff</a> h\u2081]", [{"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "Finset.coe_Icc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [346, 9], "def_end_pos": [346, 16]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Set.Icc_subset_Ico_iff", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 27]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 Icc a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 < b\u2082", "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.neg_divInt", "start": [220, 1], "end": [221, 83], "traced_tactics": [{"tactic": "rcases Int.eq_nat_or_neg d with \u27e8_, rfl | rfl\u27e9 <;> simp [divInt_neg', neg_mkRat]", "annotated_tactic": ["rcases <a>Int.eq_nat_or_neg</a> d with \u27e8_, rfl | rfl\u27e9 <;> simp [<a>divInt_neg'</a>, <a>neg_mkRat</a>]", [{"full_name": "Int.eq_nat_or_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [166, 9], "def_end_pos": [166, 22]}, {"full_name": "Rat.divInt_neg'", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [139, 9], "def_end_pos": [139, 20]}, {"full_name": "Rat.neg_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [217, 9], "def_end_pos": [217, 18]}]], "state_before": "n d : Int\n\u22a2 -(n /. d) = -n /. d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aeval_X_left_apply", "start": [1492, 1], "end": [1493, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "ApproximatesLinearOn.norm_fderiv_sub_le", "start": [465, 1], "end": [550, 86], "traced_tactics": [{"tactic": "filter_upwards [Besicovitch.ae_tendsto_measure_inter_div \u03bc s, ae_restrict_mem hs]", "annotated_tactic": ["filter_upwards [<a>Besicovitch.ae_tendsto_measure_inter_div</a> \u03bc s, <a>ae_restrict_mem</a> hs]", [{"full_name": "Besicovitch.ae_tendsto_measure_inter_div", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [1186, 9], "def_end_pos": [1186, 37]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, \u2016f' x - A\u2016\u208a \u2264 \u03b4", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2200 (a : E),\n    Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \u2192 a \u2208 s \u2192 \u2016f' a - A\u2016\u208a \u2264 \u03b4"}, {"tactic": "intro x hx xs", "annotated_tactic": ["intro x hx xs", []], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2200 (a : E),\n    Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \u2192 a \u2208 s \u2192 \u2016f' a - A\u2016\u208a \u2264 \u03b4", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\n\u22a2 \u2016f' x - A\u2016\u208a \u2264 \u03b4"}, {"tactic": "apply ContinuousLinearMap.op_norm_le_bound _ \u03b4.2 fun z => ?_", "annotated_tactic": ["apply <a>ContinuousLinearMap.op_norm_le_bound</a> _ \u03b4.2 fun z => ?_", [{"full_name": "ContinuousLinearMap.op_norm_le_bound", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [150, 9], "def_end_pos": [150, 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 : Set E\nf : E \u2192 E\nf'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\n\u22a2 \u2016f' x - A\u2016\u208a \u2264 \u03b4", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016"}, {"tactic": "suffices H : \u2200 \u03b5, 0 < \u03b5 \u2192 \u2016(f' x - A) z\u2016 \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "annotated_tactic": ["suffices H : \u2200 \u03b5, 0 < \u03b5 \u2192 \u2016(f' x - A) z\u2016 \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", []], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "intro \u03b5 \u03b5pos", "annotated_tactic": ["intro \u03b5 \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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "have B\u2081 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), (s \u2229 ({x} + r \u2022 closedBall z \u03b5)).Nonempty :=\n  eventually_nonempty_inter_smul_of_density_one \u03bc s x hx _ measurableSet_closedBall\n    (measure_closedBall_pos \u03bc z \u03b5pos).ne'", "annotated_tactic": ["have B\u2081 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), (s \u2229 ({x} + r \u2022 <a>closedBall</a> z \u03b5)).<a>Nonempty</a> :=\n    <a>eventually_nonempty_inter_smul_of_density_one</a> \u03bc s x hx _ <a>measurableSet_closedBall</a>\n      (<a>measure_closedBall_pos</a> \u03bc z \u03b5pos).<a>ne'</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": "MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [858, 9], "def_end_pos": [858, 54]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 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]}]], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "obtain \u27e8\u03c1, \u03c1pos, h\u03c1\u27e9 :\n  \u2203 \u03c1 > 0, ball x \u03c1 \u2229 s \u2286 {y : E | \u2016f y - f x - (f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} :=\n  mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) \u03b5pos)", "annotated_tactic": ["obtain \u27e8\u03c1, \u03c1pos, h\u03c1\u27e9 :\n    \u2203 \u03c1 > 0, <a>ball</a> x \u03c1 \u2229 s \u2286 {y : E | \u2016f y - f x - (f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} :=\n    <a>mem_nhdsWithin_iff</a>.1 (<a>IsLittleO.def</a> (hf' x xs) \u03b5pos)", [{"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "Metric.mem_nhdsWithin_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1038, 9], "def_end_pos": [1038, 27]}, {"full_name": "Asymptotics.IsLittleO.def", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [198, 9], "def_end_pos": [198, 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 : Set E\nf : E \u2192 E\nf'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "case 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "have B\u2082 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 := by\n  apply nhdsWithin_le_nhds\n  exact eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x \u03c1pos)", "annotated_tactic": ["have B\u2082 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), {x} + r \u2022 <a>closedBall</a> z \u03b5 \u2286 <a>ball</a> x \u03c1 := by\n    apply <a>nhdsWithin_le_nhds</a>\n    exact <a>eventually_singleton_add_smul_subset</a> <a>isBounded_closedBall</a> (<a>ball_mem_nhds</a> x \u03c1pos)", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}, {"full_name": "eventually_singleton_add_smul_subset", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [119, 9], "def_end_pos": [119, 45]}, {"full_name": "Metric.isBounded_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2364, 9], "def_end_pos": [2364, 29]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}]], "state_before": "case 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "case 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "obtain \u27e8r, \u27e8y, \u27e8ys, hy\u27e9\u27e9, r\u03c1, rpos\u27e9 :\n  \u2203 r : \u211d,\n    (s \u2229 ({x} + r \u2022 closedBall z \u03b5)).Nonempty \u2227 {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 \u2227 0 < r :=\n  (B\u2081.and (B\u2082.and self_mem_nhdsWithin)).exists", "annotated_tactic": ["obtain \u27e8r, \u27e8y, \u27e8ys, hy\u27e9\u27e9, r\u03c1, rpos\u27e9 :\n    \u2203 r : \u211d,\n      (s \u2229 ({x} + r \u2022 <a>closedBall</a> z \u03b5)).<a>Nonempty</a> \u2227 {x} + r \u2022 <a>closedBall</a> z \u03b5 \u2286 <a>ball</a> x \u03c1 \u2227 0 < r :=\n    (B\u2081.and (B\u2082.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": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"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 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "case H.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "obtain \u27e8a, az, ya\u27e9 : \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a := by\n  simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy\n  rcases hy with \u27e8a, az, ha\u27e9\n  exact \u27e8a, az, by simp only [ha, add_neg_cancel_left]\u27e9", "annotated_tactic": ["obtain \u27e8a, az, ya\u27e9 : \u2203 a, a \u2208 <a>closedBall</a> z \u03b5 \u2227 y = x + r \u2022 a := by\n    simp only [<a>mem_smul_set</a>, <a>image_add_left</a>, <a>mem_preimage</a>, <a>singleton_add</a>] at hy\n    rcases hy with \u27e8a, az, ha\u27e9\n    exact \u27e8a, az, by simp only [ha, <a>add_neg_cancel_left</a>]\u27e9", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "Set.mem_smul_set", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [318, 9], "def_end_pos": [318, 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": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [402, 3], "def_end_pos": [402, 14]}, {"full_name": "add_neg_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1152, 3], "def_end_pos": [1152, 14]}]], "state_before": "case H.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "case H.intro.intro.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "have norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 :=\n  calc\n    \u2016a\u2016 = \u2016z + (a - z)\u2016 := by simp only [add_sub_cancel'_right]\n    _ \u2264 \u2016z\u2016 + \u2016a - z\u2016 := (norm_add_le _ _)\n    _ \u2264 \u2016z\u2016 + \u03b5 := add_le_add_left (mem_closedBall_iff_norm.1 az) _", "annotated_tactic": ["have norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 :=\n    calc\n      \u2016a\u2016 = \u2016z + (a - z)\u2016 := by simp only [<a>add_sub_cancel'_right</a>]\n      _ \u2264 \u2016z\u2016 + \u2016a - z\u2016 := (<a>norm_add_le</a> _ _)\n      _ \u2264 \u2016z\u2016 + \u03b5 := <a>add_le_add_left</a> (<a>mem_closedBall_iff_norm</a>.1 az) _", [{"full_name": "add_sub_cancel'_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [953, 3], "def_end_pos": [953, 14]}, {"full_name": "norm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [482, 15], "def_end_pos": [482, 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": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [646, 15], "def_end_pos": [646, 38]}]], "state_before": "case H.intro.intro.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "case H.intro.intro.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "have I : r * \u2016(f' x - A) a\u2016 \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n  calc\n    r * \u2016(f' x - A) a\u2016 = \u2016(f' x - A) (r \u2022 a)\u2016 := by\n      simp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n    _ = \u2016f y - f x - A (y - x) - (f y - f x - (f' x) (y - x))\u2016 := by\n      congr 1\n      simp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, ContinuousLinearMap.coe_sub',\n        eq_self_iff_true, sub_left_inj, Pi.sub_apply, ContinuousLinearMap.map_smul, smul_sub]\n    _ \u2264 \u2016f y - f x - A (y - x)\u2016 + \u2016f y - f x - (f' x) (y - x)\u2016 := (norm_sub_le _ _)\n    _ \u2264 \u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 := (add_le_add (hf _ ys _ xs) (h\u03c1 \u27e8r\u03c1 hy, ys\u27e9))\n    _ = r * (\u03b4 + \u03b5) * \u2016a\u2016 := by\n      simp only [ya, add_sub_cancel', norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n      ring\n    _ \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n      mul_le_mul_of_nonneg_left norm_a (mul_nonneg rpos.le (add_nonneg \u03b4.2 \u03b5pos.le))", "annotated_tactic": ["have I : r * \u2016(f' x - A) a\u2016 \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n    calc\n      r * \u2016(f' x - A) a\u2016 = \u2016(f' x - A) (r \u2022 a)\u2016 := by\n        simp only [<a>ContinuousLinearMap.map_smul</a>, <a>norm_smul</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> rpos.le]\n      _ = \u2016f y - f x - A (y - x) - (f y - f x - (f' x) (y - x))\u2016 := by\n        congr 1\n        simp only [ya, <a>add_sub_cancel'</a>, <a>sub_sub_sub_cancel_left</a>, <a>ContinuousLinearMap.coe_sub'</a>,\n          <a>eq_self_iff_true</a>, <a>sub_left_inj</a>, <a>Pi.sub_apply</a>, <a>ContinuousLinearMap.map_smul</a>, <a>smul_sub</a>]\n      _ \u2264 \u2016f y - f x - A (y - x)\u2016 + \u2016f y - f x - (f' x) (y - x)\u2016 := (<a>norm_sub_le</a> _ _)\n      _ \u2264 \u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 := (<a>add_le_add</a> (hf _ ys _ xs) (h\u03c1 \u27e8r\u03c1 hy, ys\u27e9))\n      _ = r * (\u03b4 + \u03b5) * \u2016a\u2016 := by\n        simp only [ya, <a>add_sub_cancel'</a>, <a>norm_smul</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> rpos.le]\n        ring\n      _ \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n        <a>mul_le_mul_of_nonneg_left</a> norm_a (<a>mul_nonneg</a> rpos.le (<a>add_nonneg</a> \u03b4.2 \u03b5pos.le))", [{"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [520, 19], "def_end_pos": [520, 27]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 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": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "sub_sub_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [997, 3], "def_end_pos": [997, 14]}, {"full_name": "ContinuousLinearMap.coe_sub'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 17]}, {"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": "sub_left_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [782, 3], "def_end_pos": [782, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [520, 19], "def_end_pos": [520, 27]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}, {"full_name": "norm_sub_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [553, 3], "def_end_pos": [553, 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": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 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_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}, {"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}]], "state_before": "case H.intro.intro.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "case H.intro.intro.intro.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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"tactic": "have :\n  Tendsto (fun \u03b5 : \u211d => ((\u03b4 : \u211d) + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd ((\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0)) :=\n  Tendsto.mono_left (Continuous.tendsto (by continuity) 0) nhdsWithin_le_nhds", "annotated_tactic": ["have :\n      <a>Tendsto</a> (fun \u03b5 : \u211d => ((\u03b4 : \u211d) + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd ((\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0)) :=\n      <a>Tendsto.mono_left</a> (<a>Continuous.tendsto</a> (by continuity) 0) <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": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3036, 9], "def_end_pos": [3036, 26]}, {"full_name": "Continuous.tendsto", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 27]}, {"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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd ((\u2191\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016"}, {"tactic": "simp only [add_zero, mul_zero] at this", "annotated_tactic": ["simp only [<a>add_zero</a>, <a>mul_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.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd ((\u2191\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016"}, {"tactic": "apply le_of_tendsto_of_tendsto tendsto_const_nhds this", "annotated_tactic": ["apply <a>le_of_tendsto_of_tendsto</a> <a>tendsto_const_nhds</a> this", [{"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": "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 (fun x_1 => \u2016\u2191(f' x - A) z\u2016) \u2264\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5"}, {"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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 (fun x_1 => \u2016\u2191(f' x - A) z\u2016) \u2264\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + a) * (\u2016z\u2016 + a) + \u2016f' x - A\u2016 * a"}, {"tactic": "exact H", "annotated_tactic": ["exact H", []], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + a) * (\u2016z\u2016 + a) + \u2016f' x - A\u2016 * a", "state_after": "no goals"}, {"tactic": "continuity", "annotated_tactic": ["continuity", []], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n\u22a2 Continuous fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5", "state_after": "no goals"}, {"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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 {x_1 | (fun r => {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1) x_1} \u2208 \ud835\udcdd 0"}, {"tactic": "exact eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x \u03c1pos)", "annotated_tactic": ["exact <a>eventually_singleton_add_smul_subset</a> <a>isBounded_closedBall</a> (<a>ball_mem_nhds</a> x \u03c1pos)", [{"full_name": "eventually_singleton_add_smul_subset", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [119, 9], "def_end_pos": [119, 45]}, {"full_name": "Metric.isBounded_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2364, 9], "def_end_pos": [2364, 29]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}]], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 {x_1 | (fun r => {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1) x_1} \u2208 \ud835\udcdd 0", "state_after": "no goals"}, {"tactic": "simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy", "annotated_tactic": ["simp only [<a>mem_smul_set</a>, <a>image_add_left</a>, <a>mem_preimage</a>, <a>singleton_add</a>] at hy", [{"full_name": "Set.mem_smul_set", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [318, 9], "def_end_pos": [318, 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": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"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\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 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 : Set E\nf : E \u2192 E\nf'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\nhy : \u2203 y_1, y_1 \u2208 closedBall z \u03b5 \u2227 r \u2022 y_1 = -x + y\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a"}, {"tactic": "rcases hy with \u27e8a, az, ha\u27e9", "annotated_tactic": ["rcases hy with \u27e8a, az, ha\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 : Set E\nf : E \u2192 E\nf'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\nhy : \u2203 y_1, y_1 \u2208 closedBall z \u03b5 \u2227 r \u2022 y_1 = -x + y\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nha : r \u2022 a = -x + y\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a"}, {"tactic": "exact \u27e8a, az, by simp only [ha, add_neg_cancel_left]\u27e9", "annotated_tactic": ["exact \u27e8a, az, by simp only [ha, <a>add_neg_cancel_left</a>]\u27e9", [{"full_name": "add_neg_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1152, 3], "def_end_pos": [1152, 14]}]], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nha : r \u2022 a = -x + y\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a", "state_after": "no goals"}, {"tactic": "simp only [ha, add_neg_cancel_left]", "annotated_tactic": ["simp only [ha, <a>add_neg_cancel_left</a>]", [{"full_name": "add_neg_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1152, 3], "def_end_pos": [1152, 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nha : r \u2022 a = -x + y\n\u22a2 y = x + r \u2022 a", "state_after": "no goals"}, {"tactic": "simp only [add_sub_cancel'_right]", "annotated_tactic": ["simp only [<a>add_sub_cancel'_right</a>]", [{"full_name": "add_sub_cancel'_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [953, 3], "def_end_pos": [953, 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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\n\u22a2 \u2016a\u2016 = \u2016z + (a - z)\u2016", "state_after": "no goals"}, {"tactic": "simp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]", "annotated_tactic": ["simp only [<a>ContinuousLinearMap.map_smul</a>, <a>norm_smul</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> rpos.le]", [{"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [520, 19], "def_end_pos": [520, 27]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 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]}]], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 r * \u2016\u2191(f' x - A) a\u2016 = \u2016\u2191(f' x - A) (r \u2022 a)\u2016", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2016\u2191(f' x - A) (r \u2022 a)\u2016 = \u2016f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x))\u2016", "state_after": "case e_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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191(f' x - A) (r \u2022 a) = f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x))"}, {"tactic": "simp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, ContinuousLinearMap.coe_sub',\n  eq_self_iff_true, sub_left_inj, Pi.sub_apply, ContinuousLinearMap.map_smul, smul_sub]", "annotated_tactic": ["simp only [ya, <a>add_sub_cancel'</a>, <a>sub_sub_sub_cancel_left</a>, <a>ContinuousLinearMap.coe_sub'</a>,\n          <a>eq_self_iff_true</a>, <a>sub_left_inj</a>, <a>Pi.sub_apply</a>, <a>ContinuousLinearMap.map_smul</a>, <a>smul_sub</a>]", [{"full_name": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "sub_sub_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [997, 3], "def_end_pos": [997, 14]}, {"full_name": "ContinuousLinearMap.coe_sub'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 17]}, {"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": "sub_left_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [782, 3], "def_end_pos": [782, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [520, 19], "def_end_pos": [520, 27]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}]], "state_before": "case e_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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191(f' x - A) (r \u2022 a) = f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x))", "state_after": "no goals"}, {"tactic": "simp only [ya, add_sub_cancel', norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]", "annotated_tactic": ["simp only [ya, <a>add_sub_cancel'</a>, <a>norm_smul</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> rpos.le]", [{"full_name": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 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]}]], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191\u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016", "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191\u03b4 * (r * \u2016a\u2016) + \u03b5 * (r * \u2016a\u2016) = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016"}, {"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 : Set E\nf : E \u2192 E\nf'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191\u03b4 * (r * \u2016a\u2016) + \u03b5 * (r * \u2016a\u2016) = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) z\u2016 = \u2016\u2191(f' x - A) a + \u2191(f' x - A) (z - a)\u2016", "state_after": "case e_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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x - A) z = \u2191(f' x - A) a + \u2191(f' x - A) (z - a)"}, {"tactic": "simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]", "annotated_tactic": ["simp only [<a>ContinuousLinearMap.coe_sub'</a>, <a>map_sub</a>, <a>Pi.sub_apply</a>]", [{"full_name": "ContinuousLinearMap.coe_sub'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 17]}, {"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": "case e_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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x - A) z = \u2191(f' x - A) a + \u2191(f' x - A) (z - a)", "state_after": "case e_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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x) z - \u2191A z = \u2191(f' x) a - \u2191A a + (\u2191(f' x) z - \u2191A z - (\u2191(f' x) a - \u2191A a))"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case e_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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x) z - \u2191A z = \u2191(f' x) a - \u2191A a + (\u2191(f' x) z - \u2191A z - (\u2191(f' x) a - \u2191A a))", "state_after": "no goals"}, {"tactic": "apply add_le_add", "annotated_tactic": ["apply <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": "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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) a\u2016 + \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u2016z - a\u2016", "state_after": "case h\u2081\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\ncase h\u2082\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 \u2016f' x - A\u2016 * \u2016z - a\u2016"}, {"tactic": "rw [mul_assoc] at I", "annotated_tactic": ["rw [<a>mul_assoc</a>] at I", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case h\u2081\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)", "state_after": "case h\u2081\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * ((\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5))\n\u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)"}, {"tactic": "exact (mul_le_mul_left rpos).1 I", "annotated_tactic": ["exact (<a>mul_le_mul_left</a> rpos).1 I", [{"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 24]}]], "state_before": "case h\u2081\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * ((\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5))\n\u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)", "state_after": "no goals"}, {"tactic": "apply ContinuousLinearMap.le_op_norm", "annotated_tactic": ["apply <a>ContinuousLinearMap.le_op_norm</a>", [{"full_name": "ContinuousLinearMap.le_op_norm", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [200, 9], "def_end_pos": [200, 19]}]], "state_before": "case h\u2082\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'\u271d : 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\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 \u2016f' x - A\u2016 * \u2016z - a\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.isHaarMeasure_map", "start": [793, 1], "end": [802, 59], "traced_tactics": [{"tactic": "intro K hK", "annotated_tactic": ["intro K hK", []], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH\u271d : Type u_3\ninst\u271d\u00b9\u00b2 : MeasurableSpace G\ninst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : TopologicalSpace G\n\u03bc : Measure G\ninst\u271d\u2078 : IsHaarMeasure \u03bc\ninst\u271d\u2077 : BorelSpace G\ninst\u271d\u2076 : TopologicalGroup G\nH : Type u_4\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : BorelSpace H\ninst\u271d\u00b9 : T2Space H\ninst\u271d : TopologicalGroup H\nf : G \u2192* H\nhf : Continuous \u2191f\nh_surj : Surjective \u2191f\nh_prop : Tendsto (\u2191f) (cocompact G) (cocompact H)\n\u22a2 \u2200 \u2983K : Set H\u2984, IsCompact K \u2192 \u2191\u2191(map (\u2191f) \u03bc) K < \u22a4", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH\u271d : Type u_3\ninst\u271d\u00b9\u00b2 : MeasurableSpace G\ninst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : TopologicalSpace G\n\u03bc : Measure G\ninst\u271d\u2078 : IsHaarMeasure \u03bc\ninst\u271d\u2077 : BorelSpace G\ninst\u271d\u2076 : TopologicalGroup G\nH : Type u_4\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : BorelSpace H\ninst\u271d\u00b9 : T2Space H\ninst\u271d : TopologicalGroup H\nf : G \u2192* H\nhf : Continuous \u2191f\nh_surj : Surjective \u2191f\nh_prop : Tendsto (\u2191f) (cocompact G) (cocompact H)\nK : Set H\nhK : IsCompact K\n\u22a2 \u2191\u2191(map (\u2191f) \u03bc) K < \u22a4"}, {"tactic": "rw [map_apply hf.measurable hK.measurableSet]", "annotated_tactic": ["rw [<a>map_apply</a> hf.measurable hK.measurableSet]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH\u271d : Type u_3\ninst\u271d\u00b9\u00b2 : MeasurableSpace G\ninst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : TopologicalSpace G\n\u03bc : Measure G\ninst\u271d\u2078 : IsHaarMeasure \u03bc\ninst\u271d\u2077 : BorelSpace G\ninst\u271d\u2076 : TopologicalGroup G\nH : Type u_4\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : BorelSpace H\ninst\u271d\u00b9 : T2Space H\ninst\u271d : TopologicalGroup H\nf : G \u2192* H\nhf : Continuous \u2191f\nh_surj : Surjective \u2191f\nh_prop : Tendsto (\u2191f) (cocompact G) (cocompact H)\nK : Set H\nhK : IsCompact K\n\u22a2 \u2191\u2191(map (\u2191f) \u03bc) K < \u22a4", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH\u271d : Type u_3\ninst\u271d\u00b9\u00b2 : MeasurableSpace G\ninst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : TopologicalSpace G\n\u03bc : Measure G\ninst\u271d\u2078 : IsHaarMeasure \u03bc\ninst\u271d\u2077 : BorelSpace G\ninst\u271d\u2076 : TopologicalGroup G\nH : Type u_4\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : BorelSpace H\ninst\u271d\u00b9 : T2Space H\ninst\u271d : TopologicalGroup H\nf : G \u2192* H\nhf : Continuous \u2191f\nh_surj : Surjective \u2191f\nh_prop : Tendsto (\u2191f) (cocompact G) (cocompact H)\nK : Set H\nhK : IsCompact K\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' K) < \u22a4"}, {"tactic": "exact IsCompact.measure_lt_top ((\u27e8\u27e8f, hf\u27e9, h_prop\u27e9 : CocompactMap G H).isCompact_preimage hK)", "annotated_tactic": ["exact <a>IsCompact.measure_lt_top</a> ((\u27e8\u27e8f, hf\u27e9, h_prop\u27e9 : <a>CocompactMap</a> G H).<a>isCompact_preimage</a> hK)", [{"full_name": "IsCompact.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3858, 9], "def_end_pos": [3858, 40]}, {"full_name": "CocompactMap", "def_path": "Mathlib/Topology/ContinuousFunction/CocompactMap.lean", "def_pos": [33, 11], "def_end_pos": [33, 23]}, {"full_name": "CocompactMap.isCompact_preimage", "def_path": "Mathlib/Topology/ContinuousFunction/CocompactMap.lean", "def_pos": [188, 9], "def_end_pos": [188, 27]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH\u271d : Type u_3\ninst\u271d\u00b9\u00b2 : MeasurableSpace G\ninst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : TopologicalSpace G\n\u03bc : Measure G\ninst\u271d\u2078 : IsHaarMeasure \u03bc\ninst\u271d\u2077 : BorelSpace G\ninst\u271d\u2076 : TopologicalGroup G\nH : Type u_4\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : MeasurableSpace H\ninst\u271d\u00b2 : BorelSpace H\ninst\u271d\u00b9 : T2Space H\ninst\u271d : TopologicalGroup H\nf : G \u2192* H\nhf : Continuous \u2191f\nh_surj : Surjective \u2191f\nh_prop : Tendsto (\u2191f) (cocompact G) (cocompact H)\nK : Set H\nhK : IsCompact K\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' K) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.mem_lpMeasSubgroup_toLp_of_trim", "start": [300, 1], "end": [306, 34], "traced_tactics": [{"tactic": "let hf_mem_\u2112p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)", "annotated_tactic": ["let hf_mem_\u2112p := <a>mem\u2112p_of_mem\u2112p_trim</a> hm (<a>Lp.mem\u2112p</a> f)", [{"full_name": "MeasureTheory.mem\u2112p_of_mem\u2112p_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1035, 9], "def_end_pos": [1035, 28]}, {"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\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 Lp F p }\n\u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc", "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\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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc"}, {"tactic": "rw [mem_lpMeasSubgroup_iff_aeStronglyMeasurable']", "annotated_tactic": ["rw [<a>mem_lpMeasSubgroup_iff_aeStronglyMeasurable'</a>]", [{"full_name": "MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [232, 9], "def_end_pos": [232, 53]}]], "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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc", "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\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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p))) \u03bc"}, {"tactic": "refine' AEStronglyMeasurable'.congr _ (Mem\u2112p.coeFn_toLp hf_mem_\u2112p).symm", "annotated_tactic": ["refine' <a>AEStronglyMeasurable'.congr</a> _ (<a>Mem\u2112p.coeFn_toLp</a> hf_mem_\u2112p).<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.Mem\u2112p.coeFn_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"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\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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p))) \u03bc", "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\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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191f) \u03bc"}, {"tactic": "refine' aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm _", "annotated_tactic": ["refine' <a>aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim</a> hm _", [{"full_name": "MeasureTheory.aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [132, 9], "def_end_pos": [132, 60]}]], "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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191f) \u03bc", "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\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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191f) (Measure.trim \u03bc hm)"}, {"tactic": "exact Lp.aestronglyMeasurable f", "annotated_tactic": ["exact <a>Lp.aestronglyMeasurable</a> f", [{"full_name": "MeasureTheory.Lp.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [212, 19], "def_end_pos": [212, 39]}]], "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 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191f) (Measure.trim \u03bc hm)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aevalTower_toAlgHom", "start": [1651, 1], "end": [1653, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.tendsto_sum_indicator_atTop_iff", "start": [326, 1], "end": [352, 52], "traced_tactics": [{"tactic": "have h\u2081 := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atTop\n  (martingalePart_bdd_difference \u2131 hbdd)", "annotated_tactic": ["have h\u2081 := (<a>martingale_martingalePart</a> hf hint).<a>ae_not_tendsto_atTop_atTop</a>\n    (<a>martingalePart_bdd_difference</a> \u2131 hbdd)", [{"full_name": "MeasureTheory.martingale_martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [93, 9], "def_end_pos": [93, 34]}, {"full_name": "MeasureTheory.Martingale.ae_not_tendsto_atTop_atTop", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [257, 9], "def_end_pos": [257, 46]}, {"full_name": "MeasureTheory.martingalePart_bdd_difference", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [174, 9], "def_end_pos": [174, 38]}]], "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop", "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop"}, {"tactic": "have h\u2082 := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atBot\n  (martingalePart_bdd_difference \u2131 hbdd)", "annotated_tactic": ["have h\u2082 := (<a>martingale_martingalePart</a> hf hint).<a>ae_not_tendsto_atTop_atBot</a>\n    (<a>martingalePart_bdd_difference</a> \u2131 hbdd)", [{"full_name": "MeasureTheory.martingale_martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [93, 9], "def_end_pos": [93, 34]}, {"full_name": "MeasureTheory.Martingale.ae_not_tendsto_atTop_atBot", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [264, 9], "def_end_pos": [264, 46]}, {"full_name": "MeasureTheory.martingalePart_bdd_difference", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [174, 9], "def_end_pos": [174, 38]}]], "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop", "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop"}, {"tactic": "have h\u2083 : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, 0 \u2264 (\u03bc[f (n + 1) - f n|\u2131 n]) \u03c9 := by\n  refine' ae_all_iff.2 fun n => condexp_nonneg _\n  filter_upwards [ae_all_iff.1 hfmono n] with \u03c9 h\u03c9 using sub_nonneg.2 h\u03c9", "annotated_tactic": ["have h\u2083 : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, 0 \u2264 (\u03bc[f (n + 1) - f n|\u2131 n]) \u03c9 := by\n    refine' <a>ae_all_iff</a>.2 fun n => <a>condexp_nonneg</a> _\n    filter_upwards [<a>ae_all_iff</a>.1 hfmono n] with \u03c9 h\u03c9 using <a>sub_nonneg</a>.2 h\u03c9", [{"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.condexp_nonneg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 23]}, {"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 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\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop", "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop"}, {"tactic": "filter_upwards [h\u2081, h\u2082, h\u2083, hfmono] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 h\u03c9\u2083 h\u03c9\u2084", "annotated_tactic": ["filter_upwards [h\u2081, h\u2082, h\u2083, hfmono] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 h\u03c9\u2083 h\u03c9\u2084", []], "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop", "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop"}, {"tactic": "constructor <;> intro ht", "annotated_tactic": ["constructor <;> intro ht", []], "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop", "state_after": "case h.mp\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\ncase h.mpr\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop"}, {"tactic": "refine' ae_all_iff.2 fun n => condexp_nonneg _", "annotated_tactic": ["refine' <a>ae_all_iff</a>.2 fun n => <a>condexp_nonneg</a> _", [{"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.condexp_nonneg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 23]}]], "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9", "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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nn : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] f (n + 1) - f n"}, {"tactic": "filter_upwards [ae_all_iff.1 hfmono n] with \u03c9 h\u03c9 using sub_nonneg.2 h\u03c9", "annotated_tactic": ["filter_upwards [<a>ae_all_iff</a>.1 hfmono n] with \u03c9 h\u03c9 using <a>sub_nonneg</a>.2 h\u03c9", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 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\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nn : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] f (n + 1) - f n", "state_after": "no goals"}, {"tactic": "refine' tendsto_atTop_atTop_of_monotone' _ _", "annotated_tactic": ["refine' <a>tendsto_atTop_atTop_of_monotone'</a> _ _", [{"full_name": "Filter.tendsto_atTop_atTop_of_monotone'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1734, 9], "def_end_pos": [1734, 41]}]], "state_before": "case h.mp\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop", "state_after": "case h.mp.refine'_1\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 Monotone fun n => predictablePart f \u2131 \u03bc n \u03c9\n\ncase h.mp.refine'_2\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 \u00acBddAbove (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)"}, {"tactic": "rintro \u27e8b, hbdd\u27e9", "annotated_tactic": ["rintro \u27e8b, hbdd\u27e9", []], "state_before": "case h.mp.refine'_2\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 \u00acBddAbove (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)", "state_after": "case h.mp.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n\u22a2 False"}, {"tactic": "rw [\u2190 tendsto_neg_atBot_iff] at ht", "annotated_tactic": ["rw [\u2190 <a>tendsto_neg_atBot_iff</a>] at ht", [{"full_name": "Filter.tendsto_neg_atBot_iff", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [909, 9], "def_end_pos": [909, 30]}]], "state_before": "case h.mp.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n\u22a2 False", "state_after": "case h.mp.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun x => -f x \u03c9) atTop atBot\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n\u22a2 False"}, {"tactic": "simp only [martingalePart, sub_eq_add_neg] at h\u03c9\u2081", "annotated_tactic": ["simp only [<a>martingalePart</a>, <a>sub_eq_add_neg</a>] at h\u03c9\u2081", [{"full_name": "MeasureTheory.martingalePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [66, 19], "def_end_pos": [66, 33]}, {"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.mp.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun x => -f x \u03c9) atTop atBot\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n\u22a2 False", "state_after": "case h.mp.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun x => -f x \u03c9) atTop atBot\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\nh\u03c9\u2081 : \u00acTendsto (fun n => (f n + -predictablePart f \u2131 \u03bc n) \u03c9) atTop atTop\n\u22a2 False"}, {"tactic": "exact h\u03c9\u2081 (tendsto_atTop_add_right_of_le _ (-b) (tendsto_neg_atBot_iff.1 ht) fun n =>\n  neg_le_neg (hbdd \u27e8n, rfl\u27e9))", "annotated_tactic": ["exact h\u03c9\u2081 (<a>tendsto_atTop_add_right_of_le</a> _ (-b) (<a>tendsto_neg_atBot_iff</a>.1 ht) fun n =>\n      <a>neg_le_neg</a> (hbdd \u27e8n, <a>rfl</a>\u27e9))", [{"full_name": "Filter.tendsto_atTop_add_right_of_le", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [845, 9], "def_end_pos": [845, 38]}, {"full_name": "Filter.tendsto_neg_atBot_iff", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [909, 9], "def_end_pos": [909, 30]}, {"full_name": "neg_le_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1238, 15], "def_end_pos": [1238, 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.mp.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun x => -f x \u03c9) atTop atBot\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\nh\u03c9\u2081 : \u00acTendsto (fun n => (f n + -predictablePart f \u2131 \u03bc n) \u03c9) atTop atTop\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro n m hnm", "annotated_tactic": ["intro n m hnm", []], "state_before": "case h.mp.refine'_1\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 Monotone fun n => predictablePart f \u2131 \u03bc n \u03c9", "state_after": "case h.mp.refine'_1\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 (fun n => predictablePart f \u2131 \u03bc n \u03c9) n \u2264 (fun n => predictablePart f \u2131 \u03bc n \u03c9) m"}, {"tactic": "simp only [predictablePart, Finset.sum_apply]", "annotated_tactic": ["simp only [<a>predictablePart</a>, <a>Finset.sum_apply</a>]", [{"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"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.mp.refine'_1\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 (fun n => predictablePart f \u2131 \u03bc n \u03c9) n \u2264 (fun n => predictablePart f \u2131 \u03bc n \u03c9) m", "state_after": "case h.mp.refine'_1\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2211 c in Finset.range n, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9 \u2264 \u2211 c in Finset.range m, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9"}, {"tactic": "refine' Finset.sum_mono_set_of_nonneg h\u03c9\u2083 (Finset.range_mono hnm)", "annotated_tactic": ["refine' <a>Finset.sum_mono_set_of_nonneg</a> h\u03c9\u2083 (<a>Finset.range_mono</a> hnm)", [{"full_name": "Finset.sum_mono_set_of_nonneg", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [167, 15], "def_end_pos": [167, 37]}, {"full_name": "Finset.range_mono", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3079, 9], "def_end_pos": [3079, 19]}]], "state_before": "case h.mp.refine'_1\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2211 c in Finset.range n, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9 \u2264 \u2211 c in Finset.range m, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9", "state_after": "no goals"}, {"tactic": "refine' tendsto_atTop_atTop_of_monotone' (monotone_nat_of_le_succ h\u03c9\u2084) _", "annotated_tactic": ["refine' <a>tendsto_atTop_atTop_of_monotone'</a> (<a>monotone_nat_of_le_succ</a> h\u03c9\u2084) _", [{"full_name": "Filter.tendsto_atTop_atTop_of_monotone'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1734, 9], "def_end_pos": [1734, 41]}, {"full_name": "monotone_nat_of_le_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 32]}]], "state_before": "case h.mpr\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop", "state_after": "case h.mpr\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u00acBddAbove (Set.range fun n => f n \u03c9)"}, {"tactic": "rintro \u27e8b, hbdd\u27e9", "annotated_tactic": ["rintro \u27e8b, hbdd\u27e9", []], "state_before": "case h.mpr\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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u00acBddAbove (Set.range fun n => f n \u03c9)", "state_after": "case h.mpr.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\n\u22a2 False"}, {"tactic": "exact h\u03c9\u2082 ((tendsto_atBot_add_left_of_ge _ b fun n =>\n  hbdd \u27e8n, rfl\u27e9) <| tendsto_neg_atBot_iff.2 ht)", "annotated_tactic": ["exact h\u03c9\u2082 ((<a>tendsto_atBot_add_left_of_ge</a> _ b fun n =>\n      hbdd \u27e8n, <a>rfl</a>\u27e9) <| <a>tendsto_neg_atBot_iff</a>.2 ht)", [{"full_name": "Filter.tendsto_atBot_add_left_of_ge", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [829, 9], "def_end_pos": [829, 37]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Filter.tendsto_neg_atBot_iff", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [909, 9], "def_end_pos": [909, 30]}]], "state_before": "case h.mpr.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\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.eval\u2082Hom_bind\u2081", "start": [303, 1], "end": [305, 34], "traced_tactics": [{"tactic": "rw [hom_bind\u2081, eval\u2082Hom_comp_C]", "annotated_tactic": ["rw [<a>hom_bind\u2081</a>, <a>eval\u2082Hom_comp_C</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.eval\u2082Hom_comp_C", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [297, 9], "def_end_pos": [297, 24]}]], "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 : \u03c4 \u2192 S\nh : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(eval\u2082Hom f g) (\u2191(bind\u2081 h) \u03c6) = \u2191(eval\u2082Hom f fun i => \u2191(eval\u2082Hom f g) (h i)) \u03c6", "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_neg_iff", "start": [273, 1], "end": [274, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.trStmts\u2081_supports", "start": [1968, 1], "end": [1990, 49], "traced_tactics": [{"tactic": "have W := fun {q} => trStmts\u2081_self q", "annotated_tactic": ["have W := fun {q} => <a>trStmts\u2081_self</a> q", [{"full_name": "Turing.PartrecToTM2.trStmts\u2081_self", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 22]}]], "state_before": "S : Finset \u039b'\nq : \u039b'\nH\u2081 : \u039b'.Supports S q\nHS\u2081 : trStmts\u2081 q \u2286 S\n\u22a2 Supports (trStmts\u2081 q) S", "state_after": "S : Finset \u039b'\nq : \u039b'\nH\u2081 : \u039b'.Supports S q\nHS\u2081 : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\n\u22a2 Supports (trStmts\u2081 q) S"}, {"tactic": "induction' q with _ _ _ q q_ih _ _ q q_ih q q_ih _ _ q q_ih q q_ih q q_ih q\u2081 q\u2082 q\u2081_ih q\u2082_ih _ <;>\n  simp [trStmts\u2081, -Finset.singleton_subset_iff] at HS\u2081 \u22a2", "annotated_tactic": ["induction' q with _ _ _ q q_ih _ _ q q_ih q q_ih _ _ q q_ih q q_ih q q_ih q\u2081 q\u2082 q\u2081_ih q\u2082_ih _ <;>\n    simp [<a>trStmts\u2081</a>, -<a>Finset.singleton_subset_iff</a>] at HS\u2081 \u22a2", [{"full_name": "Turing.PartrecToTM2.trStmts\u2081", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1738, 5], "def_end_pos": [1738, 13]}, {"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [772, 9], "def_end_pos": [772, 29]}]], "state_before": "S : Finset \u039b'\nq : \u039b'\nH\u2081 : \u039b'.Supports S q\nHS\u2081 : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\n\u22a2 Supports (trStmts\u2081 q) S", "state_after": "case move\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u2081\u271d k\u2082\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q)\nHS\u2081 : insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)) S\n\ncase clear\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.clear p\u271d k\u271d q)\nHS\u2081 : insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)) S\n\ncase copy\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.copy q)\nHS\u2081 : insert (\u039b'.copy q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.copy q) (trStmts\u2081 q)) S\n\ncase push\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q)\nHS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S\n\ncase read\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S\nH\u2081 : \u039b'.Supports S (\u039b'.read q)\nHS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S\n\u22a2 Supports (insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))) S\n\ncase succ\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.succ q)\nHS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S\n\u22a2 Supports (insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) S\n\ncase pred\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S\n\ncase ret\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : Cont'\nH\u2081 : \u039b'.Supports S (\u039b'.ret k\u271d)\nHS\u2081 : {\u039b'.ret k\u271d} \u2286 S\n\u22a2 Supports {\u039b'.ret k\u271d} S"}, {"tactic": "any_goals\n  cases' Finset.insert_subset_iff.1 HS\u2081 with h\u2081 h\u2082\n  first | have h\u2083 := h\u2082 W | try simp [Finset.subset_iff] at h\u2082", "annotated_tactic": ["any_goals\n    cases' <a>Finset.insert_subset_iff</a>.1 HS\u2081 with h\u2081 h\u2082\n    first | have h\u2083 := h\u2082 W | try simp [<a>Finset.subset_iff</a>] at h\u2082", [{"full_name": "Finset.insert_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1187, 9], "def_end_pos": [1187, 26]}, {"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "case move\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u2081\u271d k\u2082\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q)\nHS\u2081 : insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)) S\n\ncase clear\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.clear p\u271d k\u271d q)\nHS\u2081 : insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)) S\n\ncase copy\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.copy q)\nHS\u2081 : insert (\u039b'.copy q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.copy q) (trStmts\u2081 q)) S\n\ncase push\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q)\nHS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S\n\u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S\n\ncase read\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S\nH\u2081 : \u039b'.Supports S (\u039b'.read q)\nHS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S\n\u22a2 Supports (insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))) S\n\ncase succ\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.succ q)\nHS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S\n\u22a2 Supports (insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) S\n\ncase pred\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S\n\ncase ret\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : Cont'\nH\u2081 : \u039b'.Supports S (\u039b'.ret k\u271d)\nHS\u2081 : {\u039b'.ret k\u271d} \u2286 S\n\u22a2 Supports {\u039b'.ret k\u271d} S", "state_after": "case move.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u2081\u271d k\u2082\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q)\nHS\u2081 : insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)) S\n\ncase clear.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.clear p\u271d k\u271d q)\nHS\u2081 : insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.clear p\u271d k\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)) S\n\ncase copy.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.copy q)\nHS\u2081 : insert (\u039b'.copy q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.copy q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.copy q) (trStmts\u2081 q)) S\n\ncase push.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q)\nHS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.push k\u271d s\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S\n\ncase read.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S\nH\u2081 : \u039b'.Supports S (\u039b'.read q)\nHS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S\nh\u2081 : \u039b'.read q \u2208 S\nh\u2082 : \u2200 \u2983x : \u039b'\u2984 (x_1 : Option \u0393'), x \u2208 trStmts\u2081 (q x_1) \u2192 x \u2208 S\n\u22a2 Supports (insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))) S\n\ncase succ.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.succ q)\nHS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S\nh\u2081 : \u039b'.succ q \u2208 S\nh\u2082 : unrev q \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) S\n\ncase pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S\n\ncase ret\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : Cont'\nH\u2081 : \u039b'.Supports S (\u039b'.ret k\u271d)\nHS\u2081 : {\u039b'.ret k\u271d} \u2286 S\n\u22a2 Supports {\u039b'.ret k\u271d} S"}, {"tactic": "cases' Finset.insert_subset_iff.1 HS\u2081 with h\u2081 h\u2082", "annotated_tactic": ["cases' <a>Finset.insert_subset_iff</a>.1 HS\u2081 with h\u2081 h\u2082", [{"full_name": "Finset.insert_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1187, 9], "def_end_pos": [1187, 26]}]], "state_before": "case pred\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S", "state_after": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S"}, {"tactic": "first | have h\u2083 := h\u2082 W | try simp [Finset.subset_iff] at h\u2082", "annotated_tactic": ["first | have h\u2083 := h\u2082 W | try simp [<a>Finset.subset_iff</a>] at h\u2082", [{"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S", "state_after": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S"}, {"tactic": "have h\u2083 := h\u2082 W", "annotated_tactic": ["have h\u2083 := h\u2082 W", []], "state_before": "case push.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q)\nHS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.push k\u271d s\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\n\u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S", "state_after": "case push.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q)\nHS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.push k\u271d s\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S"}, {"tactic": "try simp [Finset.subset_iff] at h\u2082", "annotated_tactic": ["try simp [<a>Finset.subset_iff</a>] at h\u2082", [{"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S", "state_after": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S"}, {"tactic": "simp [Finset.subset_iff] at h\u2082", "annotated_tactic": ["simp [<a>Finset.subset_iff</a>] at h\u2082", [{"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S", "state_after": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S"}, {"tactic": "exact supports_insert.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2081\u27e9, q_ih H\u2081 h\u2082\u27e9", "annotated_tactic": ["exact <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2081\u27e9, q_ih H\u2081 h\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case move.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u2081\u271d k\u2082\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q)\nHS\u2081 : insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)) S", "state_after": "no goals"}, {"tactic": "exact supports_insert.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2081\u27e9, q_ih H\u2081 h\u2082\u27e9", "annotated_tactic": ["exact <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2081\u27e9, q_ih H\u2081 h\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case clear.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\np\u271d : \u0393' \u2192 Bool\nk\u271d : K'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.clear p\u271d k\u271d q)\nHS\u2081 : insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.clear p\u271d k\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)) S", "state_after": "no goals"}, {"tactic": "exact supports_insert.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2083\u27e9, q_ih H\u2081 h\u2082\u27e9", "annotated_tactic": ["exact <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2083\u27e9, q_ih H\u2081 h\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case copy.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.copy q)\nHS\u2081 : insert (\u039b'.copy q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.copy q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.copy q) (trStmts\u2081 q)) S", "state_after": "no goals"}, {"tactic": "exact supports_insert.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2083\u27e9, q_ih H\u2081 h\u2082\u27e9", "annotated_tactic": ["exact <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2083\u27e9, q_ih H\u2081 h\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case push.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q)\nHS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S\nh\u2081 : \u039b'.push k\u271d s\u271d q \u2208 S\nh\u2082 : trStmts\u2081 q \u2286 S\nh\u2083 : q \u2208 S\n\u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S", "state_after": "no goals"}, {"tactic": "refine' supports_insert.2 \u27e8fun _ => h\u2082 _ W, _\u27e9", "annotated_tactic": ["refine' <a>supports_insert</a>.2 \u27e8fun _ => h\u2082 _ W, _\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case read.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S\nH\u2081 : \u039b'.Supports S (\u039b'.read q)\nHS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S\nh\u2081 : \u039b'.read q \u2208 S\nh\u2082 : \u2200 \u2983x : \u039b'\u2984 (x_1 : Option \u0393'), x \u2208 trStmts\u2081 (q x_1) \u2192 x \u2208 S\n\u22a2 Supports (insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))) S", "state_after": "case read.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S\nH\u2081 : \u039b'.Supports S (\u039b'.read q)\nHS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S\nh\u2081 : \u039b'.read q \u2208 S\nh\u2082 : \u2200 \u2983x : \u039b'\u2984 (x_1 : Option \u0393'), x \u2208 trStmts\u2081 (q x_1) \u2192 x \u2208 S\n\u22a2 Supports (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) S"}, {"tactic": "exact supports_biUnion.2 fun _ => q_ih _ (H\u2081 _) fun _ h => h\u2082 _ h", "annotated_tactic": ["exact <a>supports_biUnion</a>.2 fun _ => q_ih _ (H\u2081 _) fun _ h => h\u2082 _ h", [{"full_name": "Turing.PartrecToTM2.supports_biUnion", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1944, 9], "def_end_pos": [1944, 25]}]], "state_before": "case read.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S\nH\u2081 : \u039b'.Supports S (\u039b'.read q)\nHS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S\nh\u2081 : \u039b'.read q \u2208 S\nh\u2082 : \u2200 \u2983x : \u039b'\u2984 (x_1 : Option \u0393'), x \u2208 trStmts\u2081 (q x_1) \u2192 x \u2208 S\n\u22a2 Supports (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) S", "state_after": "no goals"}, {"tactic": "refine' supports_insert.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2082.1, fun _ => h\u2082.1\u27e9, _\u27e9", "annotated_tactic": ["refine' <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2082.1, fun _ => h\u2082.1\u27e9, _\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case succ.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.succ q)\nHS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S\nh\u2081 : \u039b'.succ q \u2208 S\nh\u2082 : unrev q \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) S", "state_after": "case succ.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.succ q)\nHS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S\nh\u2081 : \u039b'.succ q \u2208 S\nh\u2082 : unrev q \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q \u2192 a \u2208 S\n\u22a2 Supports (insert (unrev q) (trStmts\u2081 q)) S"}, {"tactic": "exact supports_insert.2 \u27e8\u27e8fun _ => h\u2082.2 _ W, fun _ => h\u2082.1\u27e9, q_ih H\u2081 h\u2082.2\u27e9", "annotated_tactic": ["exact <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2082.2 _ W, fun _ => h\u2082.1\u27e9, q_ih H\u2081 h\u2082.2\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}]], "state_before": "case succ.intro\nS : Finset \u039b'\nq\u271d : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\u271d\nHS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq : \u039b'\nq_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S\nH\u2081 : \u039b'.Supports S (\u039b'.succ q)\nHS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S\nh\u2081 : \u039b'.succ q \u2208 S\nh\u2082 : unrev q \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q \u2192 a \u2208 S\n\u22a2 Supports (insert (unrev q) (trStmts\u2081 q)) S", "state_after": "no goals"}, {"tactic": "refine' supports_insert.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2082.2 _ (Or.inl W), fun _ => h\u2082.1, fun _ => h\u2082.1\u27e9, _\u27e9", "annotated_tactic": ["refine' -- pred\n      <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2082.2 _ (<a>Or.inl</a> W), fun _ => h\u2082.1, fun _ => h\u2082.1\u27e9, _\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}, {"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 pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S", "state_after": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) S"}, {"tactic": "refine' supports_insert.2 \u27e8\u27e8fun _ => h\u2082.2 _ (Or.inr W), fun _ => h\u2082.1\u27e9, _\u27e9", "annotated_tactic": ["refine' <a>supports_insert</a>.2 \u27e8\u27e8fun _ => h\u2082.2 _ (<a>Or.inr</a> W), fun _ => h\u2082.1\u27e9, _\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_insert", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1933, 9], "def_end_pos": [1933, 24]}, {"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 pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) S", "state_after": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) S"}, {"tactic": "refine' supports_union.2 \u27e8_, _\u27e9", "annotated_tactic": ["refine' <a>supports_union</a>.2 \u27e8_, _\u27e9", [{"full_name": "Turing.PartrecToTM2.supports_union", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1940, 9], "def_end_pos": [1940, 23]}]], "state_before": "case pred.intro\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) S", "state_after": "case pred.intro.refine'_1\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (trStmts\u2081 q\u2081) S\n\ncase pred.intro.refine'_2\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (trStmts\u2081 q\u2082) S"}, {"tactic": "exact q\u2081_ih H\u2081.1 fun _ h => h\u2082.2 _ (Or.inl h)", "annotated_tactic": ["exact q\u2081_ih H\u2081.1 fun _ h => h\u2082.2 _ (<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 pred.intro.refine'_1\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (trStmts\u2081 q\u2081) S", "state_after": "no goals"}, {"tactic": "exact q\u2082_ih H\u2081.2 fun _ h => h\u2082.2 _ (Or.inr h)", "annotated_tactic": ["exact q\u2082_ih H\u2081.2 fun _ h => h\u2082.2 _ (<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 pred.intro.refine'_2\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nq\u2081 q\u2082 : \u039b'\nq\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S\nq\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S\nH\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082)\nHS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S\nh\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S\nh\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S\n\u22a2 Supports (trStmts\u2081 q\u2082) S", "state_after": "no goals"}, {"tactic": "exact supports_singleton.2 (ret_supports H\u2081)", "annotated_tactic": ["exact <a>supports_singleton</a>.2 (<a>ret_supports</a> H\u2081)", [{"full_name": "Turing.PartrecToTM2.supports_singleton", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1937, 9], "def_end_pos": [1937, 27]}, {"full_name": "Turing.PartrecToTM2.ret_supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1953, 9], "def_end_pos": [1953, 21]}]], "state_before": "case ret\nS : Finset \u039b'\nq : \u039b'\nH\u2081\u271d : \u039b'.Supports S q\nHS\u2081\u271d : trStmts\u2081 q \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nk\u271d : Cont'\nH\u2081 : \u039b'.Supports S (\u039b'.ret k\u271d)\nHS\u2081 : {\u039b'.ret k\u271d} \u2286 S\n\u22a2 Supports {\u039b'.ret k\u271d} S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.to_re", "start": [187, 1], "end": [194, 30], "traced_tactics": [{"tactic": "obtain \u27e8f, hf, rfl\u27e9 := computable_iff.1 hp", "annotated_tactic": ["obtain \u27e8f, hf, rfl\u27e9 := <a>computable_iff</a>.1 hp", [{"full_name": "ComputablePred.computable_iff", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [172, 9], "def_end_pos": [172, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nhp : ComputablePred p\n\u22a2 RePred p", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 RePred fun a => f a = true"}, {"tactic": "unfold RePred", "annotated_tactic": ["unfold <a>RePred</a>", [{"full_name": "RePred", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [143, 5], "def_end_pos": [143, 11]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 RePred fun a => f a = true", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 Partrec fun a => Part.assert ((fun a => f a = true) a) fun x => Part.some ()"}, {"tactic": "dsimp only []", "annotated_tactic": ["dsimp only []", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 Partrec fun a => Part.assert ((fun a => f a = true) a) fun x => Part.some ()", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 Partrec fun a => Part.assert (f a = true) fun x => Part.some ()"}, {"tactic": "refine'\n  (Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n =>\n    Part.ext fun a => _", "annotated_tactic": ["refine'\n    (<a>Partrec.cond</a> hf (<a>Decidable.Partrec.const'</a> (<a>Part.some</a> ())) <a>Partrec.none</a>).<a>of_eq</a> fun n =>\n      <a>Part.ext</a> fun a => _", [{"full_name": "Partrec.cond", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [116, 9], "def_end_pos": [116, 13]}, {"full_name": "Decidable.Partrec.const'", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [446, 9], "def_end_pos": [446, 40]}, {"full_name": "Part.some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}, {"full_name": "Partrec.none", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [438, 9], "def_end_pos": [438, 13]}, {"full_name": "Partrec.of_eq", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [430, 9], "def_end_pos": [430, 14]}, {"full_name": "Part.ext", "def_path": "Mathlib/Data/Part.lean", "def_pos": [116, 9], "def_end_pos": [116, 12]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 Partrec fun a => Part.assert (f a = true) fun x => Part.some ()", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\na : Unit\n\u22a2 (a \u2208 bif f n then Part.some () else Part.none) \u2194 a \u2208 Part.assert (f n = true) fun x => Part.some ()"}, {"tactic": "cases a", "annotated_tactic": ["cases a", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\na : Unit\n\u22a2 (a \u2208 bif f n then Part.some () else Part.none) \u2194 a \u2208 Part.assert (f n = true) fun x => Part.some ()", "state_after": "case intro.intro.unit\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (PUnit.unit \u2208 bif f n then Part.some () else Part.none) \u2194 PUnit.unit \u2208 Part.assert (f n = true) fun x => Part.some ()"}, {"tactic": "cases f n <;> simp", "annotated_tactic": ["cases f n <;> simp", []], "state_before": "case intro.intro.unit\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (PUnit.unit \u2208 bif f n then Part.some () else Part.none) \u2194 PUnit.unit \u2208 Part.assert (f n = true) fun x => Part.some ()", "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_inter_right", "start": [202, 1], "end": [206, 32], "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\nhf : Injective2 f\n\u22a2 \u2191(image\u2082 f s (t \u2229 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 \u03b2\nhf : Injective2 f\n\u22a2 image2 f (\u2191s) (\u2191t \u2229 \u2191t') = image2 f \u2191s \u2191t \u2229 image2 f \u2191s \u2191t'"}, {"tactic": "exact image2_inter_right hf", "annotated_tactic": ["exact <a>image2_inter_right</a> hf", [{"full_name": "Set.image2_inter_right", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [141, 7], "def_end_pos": [141, 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\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\nhf : Injective2 f\n\u22a2 image2 f (\u2191s) (\u2191t \u2229 \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/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.lowerCrossingTime_stabilize'", "start": [279, 1], "end": [281, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_mem", "start": [460, 1], "end": [462, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.integral_integral_sub", "start": [405, 1], "end": [409, 61], "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", "start": [158, 1], "end": [179, 49], "traced_tactics": [{"tactic": "let R\u2081 i := max 0 (r\u2081 i)", "annotated_tactic": ["let R\u2081 i := <a>max</a> 0 (r\u2081 i)", [{"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 : 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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 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"}, {"tactic": "let R\u2082 i := max 0 (r\u2082 i)", "annotated_tactic": ["let R\u2082 i := <a>max</a> 0 (r\u2082 i)", [{"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 : 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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (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"}, {"tactic": "have hRp : 0 \u2264 R\u2081 := fun i => le_max_left 0 (r\u2081 i)", "annotated_tactic": ["have hRp : 0 \u2264 R\u2081 := fun i => <a>le_max_left</a> 0 (r\u2081 i)", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\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"}, {"tactic": "replace hMr : \u2200\u1da0 i in atTop, M * R\u2081 i \u2264 R\u2082 i", "annotated_tactic": ["replace hMr : \u2200\u1da0 i in <a>atTop</a>, M * R\u2081 i \u2264 R\u2082 i", [{"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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\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": "case hMr\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\n\u22a2 \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\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"}, {"tactic": "simp_rw [\u2190 cthickening_max_zero (r\u2081 _), \u2190 cthickening_max_zero (r\u2082 _)]", "annotated_tactic": ["simp_rw [\u2190 <a>cthickening_max_zero</a> (r\u2081 _), \u2190 <a>cthickening_max_zero</a> (r\u2082 _)]", [{"full_name": "Metric.cthickening_max_zero", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1084, 9], "def_end_pos": [1084, 29]}, {"full_name": "Metric.cthickening_max_zero", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1084, 9], "def_end_pos": [1084, 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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p"}, {"tactic": "cases' le_or_lt 1 M with hM' hM'", "annotated_tactic": ["cases' <a>le_or_lt</a> 1 M with hM' hM'", [{"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\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p\n\ncase 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : M < 1\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p"}, {"tactic": "refine' hMr.mono fun i hi => _", "annotated_tactic": ["refine' hMr.mono fun i hi => _", []], "state_before": "case hMr\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\n\u22a2 \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i", "state_after": "case hMr\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\ni : \u2115\nhi : M * r\u2081 i \u2264 r\u2082 i\n\u22a2 M * R\u2081 i \u2264 R\u2082 i"}, {"tactic": "rw [mul_max_of_nonneg _ _ hM.le, mul_zero]", "annotated_tactic": ["rw [<a>mul_max_of_nonneg</a> _ _ hM.le, <a>mul_zero</a>]", [{"full_name": "mul_max_of_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [968, 9], "def_end_pos": [968, 26]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case hMr\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\ni : \u2115\nhi : M * r\u2081 i \u2264 r\u2082 i\n\u22a2 M * R\u2081 i \u2264 R\u2082 i", "state_after": "case hMr\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\ni : \u2115\nhi : M * r\u2081 i \u2264 r\u2082 i\n\u22a2 max 0 (M * r\u2081 i) \u2264 R\u2082 i"}, {"tactic": "exact max_le_max (le_refl 0) hi", "annotated_tactic": ["exact <a>max_le_max</a> (<a>le_refl</a> 0) hi", [{"full_name": "max_le_max", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [73, 9], "def_end_pos": [73, 19]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case hMr\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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\ni : \u2115\nhi : M * r\u2081 i \u2264 r\u2082 i\n\u22a2 max 0 (M * r\u2081 i) \u2264 R\u2082 i", "state_after": "no goals"}, {"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 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p", "state_after": "case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2286\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p"}, {"tactic": "change _ \u2264 _", "annotated_tactic": ["change _ \u2264 _", []], "state_before": "case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2286\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p", "state_after": "case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p"}, {"tactic": "refine' mono_blimsup' (hMr.mono fun i hi _ => cthickening_mono _ (s i))", "annotated_tactic": ["refine' <a>mono_blimsup'</a> (hMr.mono fun i hi _ => <a>cthickening_mono</a> _ (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_mono", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 25]}]], "state_before": "case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p", "state_after": "case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\ni : \u2115\nhi : M * R\u2081 i \u2264 R\u2082 i\nx\u271d : p i\n\u22a2 max 0 (r\u2081 i) \u2264 max 0 (r\u2082 i)"}, {"tactic": "exact (le_mul_of_one_le_left (hRp i) hM').trans hi", "annotated_tactic": ["exact (<a>le_mul_of_one_le_left</a> (hRp i) hM').<a>trans</a> hi", [{"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": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : 1 \u2264 M\ni : \u2115\nhi : M * R\u2081 i \u2264 R\u2082 i\nx\u271d : p i\n\u22a2 max 0 (r\u2081 i) \u2264 max 0 (r\u2082 i)", "state_after": "no goals"}, {"tactic": "simp only [\u2190 @cthickening_closure _ _ _ (s _)]", "annotated_tactic": ["simp only [\u2190 @<a>cthickening_closure</a> _ _ _ (s _)]", [{"full_name": "Metric.cthickening_closure", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 28]}]], "state_before": "case 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : M < 1\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : M < 1\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (closure (s i))) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (closure (s i))) atTop p"}, {"tactic": "have hs : \u2200 i, IsClosed (closure (s i)) := fun i => isClosed_closure", "annotated_tactic": ["have hs : \u2200 i, <a>IsClosed</a> (<a>closure</a> (s i)) := fun i => <a>isClosed_closure</a>", [{"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "isClosed_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 25]}]], "state_before": "case 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : M < 1\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (closure (s i))) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (closure (s i))) atTop p", "state_after": "case 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : M < 1\nhs : \u2200 (i : \u2115), IsClosed (closure (s i))\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (closure (s i))) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (closure (s i))) atTop p"}, {"tactic": "exact blimsup_cthickening_ae_le_of_eventually_mul_le_aux \u03bc p hs\n  (tendsto_nhds_max_right hr) hRp hM hM' hMr", "annotated_tactic": ["exact <a>blimsup_cthickening_ae_le_of_eventually_mul_le_aux</a> \u03bc p hs\n      (<a>tendsto_nhds_max_right</a> hr) hRp hM hM' hMr", [{"full_name": "blimsup_cthickening_ae_le_of_eventually_mul_le_aux", "def_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "def_pos": [42, 9], "def_end_pos": [42, 59]}, {"full_name": "Filter.tendsto_nhds_max_right", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [754, 9], "def_end_pos": [754, 38]}]], "state_before": "case 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\nM : \u211d\nhM : 0 < M\nr\u2081 r\u2082 : \u2115 \u2192 \u211d\nhr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0)\nR\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i)\nR\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i)\nhRp : 0 \u2264 R\u2081\nhMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i\nhM' : M < 1\nhs : \u2200 (i : \u2115), IsClosed (closure (s i))\n\u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (closure (s i))) atTop p \u2264\u1d50[\u03bc]\n    blimsup (fun i => cthickening (max 0 (r\u2082 i)) (closure (s i))) atTop p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.toMeasure_ofFinset_apply", "start": [190, 1], "end": [192, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.QuasiMeasurePreserving.prod_of_left", "start": [770, 1], "end": [778, 59], "traced_tactics": [{"tactic": "convert (QuasiMeasurePreserving.prod_of_right (hf.comp measurable_swap) h2f).comp\n    ((measurable_swap.measurePreserving (\u03bd.prod \u03bc)).symm\n        MeasurableEquiv.prodComm).quasiMeasurePreserving", "annotated_tactic": ["convert (<a>QuasiMeasurePreserving.prod_of_right</a> (hf.comp <a>measurable_swap</a>) h2f).<a>comp</a>\n      ((measurable_swap.measurePreserving (\u03bd.prod \u03bc)).<a>symm</a>\n          <a>MeasurableEquiv.prodComm</a>).<a>quasiMeasurePreserving</a>", [{"full_name": "MeasureTheory.QuasiMeasurePreserving.prod_of_right", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [760, 9], "def_end_pos": [760, 22]}, {"full_name": "measurable_swap", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [760, 9], "def_end_pos": [760, 24]}, {"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.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}, {"full_name": "MeasurableEquiv.prodComm", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1500, 5], "def_end_pos": [1500, 13]}, {"full_name": "MeasureTheory.MeasurePreserving.quasiMeasurePreserving", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [97, 19], "def_end_pos": [97, 41]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\u271d\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4\u271d : Measure \u03b3\u271d\ninst\u271d\u2075 : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, QuasiMeasurePreserving fun x => f (x, y)\n\u22a2 QuasiMeasurePreserving 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.exists_subset_restrict_nonpos'", "start": [224, 9], "end": [263, 40], "traced_tactics": [{"tactic": "by_cases s \u2264[i] 0", "annotated_tactic": ["by_cases s \u2264[i] 0", []], "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case pos\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh : restrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0\n\ncase neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "push_neg at hn", "annotated_tactic": ["push_neg at hn", []], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh : \u00acrestrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "set k := Nat.find hn", "annotated_tactic": ["set k := <a>Nat.find</a> hn", [{"full_name": "Nat.find", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [713, 15], "def_end_pos": [713, 19]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have hk\u2082 : s \u2264[i \\ \u22c3 l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn", "annotated_tactic": ["have hk\u2082 : s \u2264[i \\ \u22c3 l < k, <a>restrictNonposSeq</a> s i l] 0 := <a>Nat.find_spec</a> hn", [{"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]}, {"full_name": "Nat.find_spec", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [717, 19], "def_end_pos": [717, 28]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "have hmeas : MeasurableSet (\u22c3 (l : \u2115) (_ : l < k), restrictNonposSeq s i l) :=\n  MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _", "annotated_tactic": ["have hmeas : <a>MeasurableSet</a> (\u22c3 (l : \u2115) (_ : l < k), <a>restrictNonposSeq</a> s i l) :=\n    <a>MeasurableSet.iUnion</a> fun _ => <a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</a> _", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"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]}, {"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": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0"}, {"tactic": "refine' \u27e8i \\ \u22c3 l < k, restrictNonposSeq s i l, hi\u2081.diff hmeas, Set.diff_subset _ _, hk\u2082, _\u27e9", "annotated_tactic": ["refine' \u27e8i \\ \u22c3 l < k, <a>restrictNonposSeq</a> s i l, hi\u2081.diff hmeas, <a>Set.diff_subset</a> _ _, hk\u2082, _\u27e9", [{"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]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) < 0"}, {"tactic": "rw [of_diff hmeas hi\u2081, s.of_disjoint_iUnion_nat]", "annotated_tactic": ["rw [<a>of_diff</a> hmeas hi\u2081, s.of_disjoint_iUnion_nat]", [{"full_name": "MeasureTheory.VectorMeasure.of_diff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [196, 9], "def_end_pos": [196, 16]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0\n\ncase neg.hf\u2081\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), MeasurableSet (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)\n\ncase neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Pairwise (Disjoint on fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\ncase neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2286 i"}, {"tactic": "exact \u27e8i, hi\u2081, Set.Subset.refl _, h, hi\u2082\u27e9", "annotated_tactic": ["exact \u27e8i, hi\u2081, <a>Set.Subset.refl</a> _, h, hi\u2082\u27e9", [{"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "case pos\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nhn :\n  \u00ac\u2200 (n : \u2115),\n      \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n          restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh : restrict s i \u2264 restrict 0 i\n\u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0", "state_after": "no goals"}, {"tactic": "have h\u2081 : \u2200 l < k, 0 \u2264 s (restrictNonposSeq s i l) := by\n  intro l hl\n  refine' le_of_lt (measure_of_restrictNonposSeq h _ _)\n  refine' mt (restrict_le_zero_subset _ (hi\u2081.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)\n  exact\n    MeasurableSet.iUnion fun _ =>\n      MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _", "annotated_tactic": ["have h\u2081 : \u2200 l < k, 0 \u2264 s (<a>restrictNonposSeq</a> s i l) := by\n      intro l hl\n      refine' <a>le_of_lt</a> (<a>measure_of_restrictNonposSeq</a> h _ _)\n      refine' <a>mt</a> (<a>restrict_le_zero_subset</a> _ (hi\u2081.diff _) (<a>Set.Subset.refl</a> _)) (<a>Nat.find_min</a> hn hl)\n      exact\n        <a>MeasurableSet.iUnion</a> fun _ =>\n          <a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</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]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [185, 17], "def_end_pos": [185, 45]}, {"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.VectorMeasure.restrict_le_zero_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 32]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Nat.find_min", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [721, 19], "def_end_pos": [721, 27]}, {"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": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0"}, {"tactic": "suffices 0 \u2264 \u2211' l : \u2115, s (\u22c3 _ : l < k, restrictNonposSeq s i l) by\n  rw [sub_neg]\n  exact lt_of_lt_of_le hi\u2082 this", "annotated_tactic": ["suffices 0 \u2264 \u2211' l : \u2115, s (\u22c3 _ : l < k, <a>restrictNonposSeq</a> s i l) by\n      rw [<a>sub_neg</a>]\n      exact <a>lt_of_lt_of_le</a> hi\u2082 this", [{"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]}, {"full_name": "sub_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [893, 30], "def_end_pos": [893, 37]}, {"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\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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "refine' tsum_nonneg _", "annotated_tactic": ["refine' <a>tsum_nonneg</a> _", [{"full_name": "tsum_nonneg", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), 0 \u2264 \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)"}, {"tactic": "intro l", "annotated_tactic": ["intro l", []], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), 0 \u2264 \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)", "state_after": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\n\u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "by_cases l < k", "annotated_tactic": ["by_cases l < k", []], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\n\u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case pos\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\n\u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\ncase neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\n\u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "intro l hl", "annotated_tactic": ["intro l hl", []], "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nhl : l < k\n\u22a2 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)"}, {"tactic": "refine' le_of_lt (measure_of_restrictNonposSeq h _ _)", "annotated_tactic": ["refine' <a>le_of_lt</a> (<a>measure_of_restrictNonposSeq</a> h _ _)", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [185, 17], "def_end_pos": [185, 45]}]], "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nhl : l < k\n\u22a2 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nhl : l < k\n\u22a2 \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264\n      restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)"}, {"tactic": "refine' mt (restrict_le_zero_subset _ (hi\u2081.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)", "annotated_tactic": ["refine' <a>mt</a> (<a>restrict_le_zero_subset</a> _ (hi\u2081.diff _) (<a>Set.Subset.refl</a> _)) (<a>Nat.find_min</a> hn hl)", [{"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.VectorMeasure.restrict_le_zero_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 32]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "Nat.find_min", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [721, 19], "def_end_pos": [721, 27]}]], "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nhl : l < k\n\u22a2 \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264\n      restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)", "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nhl : l < k\n\u22a2 MeasurableSet (\u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)"}, {"tactic": "exact\n  MeasurableSet.iUnion fun _ =>\n    MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _", "annotated_tactic": ["exact\n        <a>MeasurableSet.iUnion</a> fun _ =>\n          <a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</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": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}]], "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nhl : l < k\n\u22a2 MeasurableSet (\u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)", "state_after": "no goals"}, {"tactic": "rw [sub_neg]", "annotated_tactic": ["rw [<a>sub_neg</a>]", [{"full_name": "sub_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [893, 30], "def_end_pos": [893, 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 : SignedMeasure \u03b1\ni j : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nthis : 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0", "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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nthis : 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i < \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)"}, {"tactic": "exact lt_of_lt_of_le hi\u2082 this", "annotated_tactic": ["exact <a>lt_of_lt_of_le</a> hi\u2082 this", [{"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": "\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\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nthis : 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2191s i < \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)", "state_after": "no goals"}, {"tactic": "convert h\u2081 _ h", "annotated_tactic": ["convert h\u2081 _ h", []], "state_before": "case pos\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\n\u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case h.e'_4.h.e'_7\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\n\u22a2 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l =\n    MeasureTheory.SignedMeasure.restrictNonposSeq s i l"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_4.h.e'_7\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\n\u22a2 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l =\n    MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194\n    x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l"}, {"tactic": "rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]", "annotated_tactic": ["rw [<a>Set.mem_iUnion</a>, <a>exists_prop</a>, <a>and_iff_right_iff_imp</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]}, {"full_name": "and_iff_right_iff_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [211, 17], "def_end_pos": [211, 38]}]], "state_before": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194\n    x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\nx : \u03b1\n\u22a2 x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192 l < k"}, {"tactic": "exact fun _ => h", "annotated_tactic": ["exact fun _ => h", []], "state_before": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : l < k\nx : \u03b1\n\u22a2 x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192 l < k", "state_after": "no goals"}, {"tactic": "convert le_of_eq s.empty.symm", "annotated_tactic": ["convert <a>le_of_eq</a> s.empty.symm", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\n\u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case h.e'_4.h.e'_7\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\n\u22a2 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l = \u2205"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h.e'_4.h.e'_7\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\n\u22a2 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l = \u2205", "state_after": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194 x\u271d \u2208 \u2205"}, {"tactic": "simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>Set.mem_empty_iff_false</a>, <a>Set.mem_iUnion</a>, <a>not_and</a>, <a>iff_false_iff</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_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194 x\u271d \u2208 \u2205", "state_after": "case h.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\nx\u271d : \u03b1\n\u22a2 l < Nat.find hn \u2192 \u00acx\u271d \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l"}, {"tactic": "exact fun h' => False.elim (h h')", "annotated_tactic": ["exact fun h' => <a>False.elim</a> (h h')", [{"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.e'_4.h.e'_7.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh\u271d : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nh\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nl : \u2115\nh : \u00acl < k\nx\u271d : \u03b1\n\u22a2 l < Nat.find hn \u2192 \u00acx\u271d \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case neg.hf\u2081\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), MeasurableSet (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)", "state_after": "case neg.hf\u2081\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\ni\u271d : \u2115\n\u22a2 MeasurableSet (\u22c3 (_ : i\u271d < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i\u271d)"}, {"tactic": "exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> fun _ => <a>restrictNonposSeq_measurableSet</a> _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [201, 17], "def_end_pos": [201, 48]}]], "state_before": "case neg.hf\u2081\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\ni\u271d : \u2115\n\u22a2 MeasurableSet (\u22c3 (_ : i\u271d < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i\u271d)", "state_after": "no goals"}, {"tactic": "intro a b hab", "annotated_tactic": ["intro a b hab", []], "state_before": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 Pairwise (Disjoint on fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "state_after": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na b : \u2115\nhab : a \u2260 b\n\u22a2 (Disjoint on fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b"}, {"tactic": "refine' Set.disjoint_iUnion_left.mpr fun _ => _", "annotated_tactic": ["refine' Set.disjoint_iUnion_left.mpr fun _ => _", []], "state_before": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na b : \u2115\nhab : a \u2260 b\n\u22a2 (Disjoint on fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b", "state_after": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na b : \u2115\nhab : a \u2260 b\nx\u271d : a < k\n\u22a2 Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a)\n    ((fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) b)"}, {"tactic": "refine' Set.disjoint_iUnion_right.mpr fun _ => _", "annotated_tactic": ["refine' Set.disjoint_iUnion_right.mpr fun _ => _", []], "state_before": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na b : \u2115\nhab : a \u2260 b\nx\u271d : a < k\n\u22a2 Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a)\n    ((fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) b)", "state_after": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na b : \u2115\nhab : a \u2260 b\nx\u271d\u00b9 : a < k\nx\u271d : b < k\n\u22a2 Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a) (MeasureTheory.SignedMeasure.restrictNonposSeq s i b)"}, {"tactic": "exact restrictNonposSeq_disjoint hab", "annotated_tactic": ["exact <a>restrictNonposSeq_disjoint</a> hab", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [217, 17], "def_end_pos": [217, 43]}]], "state_before": "case neg.hf\u2082\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na b : \u2115\nhab : a \u2260 b\nx\u271d\u00b9 : a < k\nx\u271d : b < k\n\u22a2 Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a) (MeasureTheory.SignedMeasure.restrictNonposSeq s i b)", "state_after": "no goals"}, {"tactic": "apply Set.iUnion_subset", "annotated_tactic": ["apply <a>Set.iUnion_subset</a>", [{"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}]], "state_before": "case neg\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2286 i", "state_after": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), \u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i"}, {"tactic": "intro a x", "annotated_tactic": ["intro a x", []], "state_before": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\n\u22a2 \u2200 (i_1 : \u2115), \u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i", "state_after": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na : \u2115\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (_ : a < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u2192 x \u2208 i"}, {"tactic": "simp only [and_imp, exists_prop, Set.mem_iUnion]", "annotated_tactic": ["simp only [<a>and_imp</a>, <a>exists_prop</a>, <a>Set.mem_iUnion</a>]", [{"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": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na : \u2115\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (_ : a < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u2192 x \u2208 i", "state_after": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na : \u2115\nx : \u03b1\n\u22a2 a < Nat.find hn \u2192 x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u2192 x \u2208 i"}, {"tactic": "intro _ hx", "annotated_tactic": ["intro _ hx", []], "state_before": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na : \u2115\nx : \u03b1\n\u22a2 a < Nat.find hn \u2192 x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u2192 x \u2208 i", "state_after": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na : \u2115\nx : \u03b1\na\u271d : a < Nat.find hn\nhx : x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i a\n\u22a2 x \u2208 i"}, {"tactic": "exact restrictNonposSeq_subset _ hx", "annotated_tactic": ["exact <a>restrictNonposSeq_subset</a> _ hx", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_subset", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [176, 17], "def_end_pos": [176, 41]}]], "state_before": "case neg.h\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\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191s i < 0\nh : \u00acrestrict s i \u2264 restrict 0 i\nhn :\n  \u2203 n,\n    restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n      restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nk : \u2115 := Nat.find hn\nhk\u2082 :\n  restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264\n    restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\nhmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)\na : \u2115\nx : \u03b1\na\u271d : a < Nat.find hn\nhx : x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i a\n\u22a2 x \u2208 i", "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_left_Icc", "start": [1137, 1], "end": [1138, 78], "traced_tactics": [{"tactic": "rw [\u2190 map_add_left_Icc, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_left_Icc</a>, <a>map_eq_image</a>, <a>addLeftEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_left_Icc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 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) (Icc a b) = Icc (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "full_name": "MeasureTheory.lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero", "start": [49, 1], "end": [67, 27], "traced_tactics": [{"tactic": "obtain \u27e8g, hg_sm, hfg\u27e9 := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top", "annotated_tactic": ["obtain \u27e8g, hg_sm, hfg\u27e9 := <a>lpMeas.ae_fin_strongly_measurable'</a> hm f hp_ne_zero hp_ne_top", [{"full_name": "MeasureTheory.lpMeas.ae_fin_strongly_measurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [524, 9], "def_end_pos": [524, 43]}]], "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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\n\u22a2 \u2191\u2191\u2191f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2191\u2191\u2191f =\u1d50[\u03bc] g\n\u22a2 \u2191\u2191\u2191f =\u1d50[\u03bc] 0"}, {"tactic": "refine' hfg.trans _", "annotated_tactic": ["refine' hfg.trans _", []], "state_before": "case intro.intro\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2191\u2191\u2191f =\u1d50[\u03bc] g\n\u22a2 \u2191\u2191\u2191f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2191\u2191\u2191f =\u1d50[\u03bc] g\n\u22a2 g =\u1d50[\u03bc] 0"}, {"tactic": "unfold Filter.EventuallyEq at hfg", "annotated_tactic": ["unfold <a>Filter.EventuallyEq</a> at hfg", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}]], "state_before": "case intro.intro\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2191\u2191\u2191f =\u1d50[\u03bc] g\n\u22a2 g =\u1d50[\u03bc] 0", "state_after": "case intro.intro\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\n\u22a2 g =\u1d50[\u03bc] 0"}, {"tactic": "refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm", "annotated_tactic": ["refine' <a>ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim</a> hm _ _ hg_sm", [{"full_name": "MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [491, 9], "def_end_pos": [491, 75]}]], "state_before": "case intro.intro\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\n\u22a2 g =\u1d50[\u03bc] 0", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\n\ncase intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, g 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\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn g s"}, {"tactic": "have hfg_restrict : f =\u1d50[\u03bc.restrict s] g := ae_restrict_of_ae hfg", "annotated_tactic": ["have hfg_restrict : f =\u1d50[\u03bc.restrict s] g := <a>ae_restrict_of_ae</a> hfg", [{"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 intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn g s", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 IntegrableOn g 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 intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 IntegrableOn g s", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 Integrable \u2191\u2191\u2191f"}, {"tactic": "exact hf_int_finite s hs h\u03bcs", "annotated_tactic": ["exact hf_int_finite s hs h\u03bcs", []], "state_before": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 Integrable \u2191\u2191\u2191f", "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\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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0"}, {"tactic": "have hfg_restrict : f =\u1d50[\u03bc.restrict s] g := ae_restrict_of_ae hfg", "annotated_tactic": ["have hfg_restrict : f =\u1d50[\u03bc.restrict s] g := <a>ae_restrict_of_ae</a> hfg", [{"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 intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 \u222b (x : \u03b1) in s, g 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 intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0", "state_after": "case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 \u222b (a : \u03b1) in s, \u2191\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 intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc }\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\u2191\u2191f) s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0\ng : \u03b1 \u2192 E'\nhg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm)\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nhfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g\n\u22a2 \u222b (a : \u03b1) in s, \u2191\u2191\u2191f a \u2202\u03bc = 0", "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_average", "start": [938, 1], "end": [951, 92], "traced_tactics": [{"tactic": "filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x", "annotated_tactic": ["filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2a0d (y : \u03b1) in a, f y \u2202\u03bc) (filterAt v x) (\ud835\udcdd (f x))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 Tendsto (fun a => \u2a0d (y : \u03b1) in a, f y \u2202\u03bc) (filterAt v x) (\ud835\udcdd (f x))"}, {"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": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 Tendsto (fun a => \u2a0d (y : \u03b1) in a, f y \u2202\u03bc) (filterAt v x) (\ud835\udcdd (f x))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 Tendsto (fun e => \u2016\u2a0d (y : \u03b1) in e, f y \u2202\u03bc - f x\u2016) (filterAt v x) (\ud835\udcdd 0)"}, {"tactic": "refine' squeeze_zero' (eventually_of_forall fun a => norm_nonneg _) _ hx", "annotated_tactic": ["refine' <a>squeeze_zero'</a> (<a>eventually_of_forall</a> fun a => <a>norm_nonneg</a> _) _ hx", [{"full_name": "squeeze_zero'", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1453, 9], "def_end_pos": [1453, 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": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 Tendsto (fun e => \u2016\u2a0d (y : \u03b1) in e, f y \u2202\u03bc - f x\u2016) (filterAt v x) (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 \u2200\u1da0 (t : Set \u03b1) in filterAt v x, \u2016\u2a0d (y : \u03b1) in t, f y \u2202\u03bc - f x\u2016 \u2264 \u2a0d (y : \u03b1) in t, \u2016f y - f x\u2016 \u2202\u03bc"}, {"tactic": "filter_upwards [h'x, v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf]\n  with a ha h'a h''a", "annotated_tactic": ["filter_upwards [h'x, v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf]\n    with a ha h'a h''a", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\n\u22a2 \u2200\u1da0 (t : Set \u03b1) in filterAt v x, \u2016\u2a0d (y : \u03b1) in t, f y \u2202\u03bc - f x\u2016 \u2264 \u2a0d (y : \u03b1) in t, \u2016f y - f x\u2016 \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u2a0d (y : \u03b1) in a, f y \u2202\u03bc - f x\u2016 \u2264 \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc"}, {"tactic": "nth_rw 1 [\u2190 setAverage_const ha.ne' h'a.ne (f x)]", "annotated_tactic": ["nth_rw 1 [\u2190 <a>setAverage_const</a> ha.ne' h'a.ne (f x)]", [{"full_name": "MeasureTheory.setAverage_const", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [393, 9], "def_end_pos": [393, 25]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u2a0d (y : \u03b1) in a, f y \u2202\u03bc - f x\u2016 \u2264 \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u2a0d (y : \u03b1) in a, f y \u2202\u03bc - \u2a0d (x_1 : \u03b1) in a, f x \u2202\u03bc\u2016 \u2264 \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc"}, {"tactic": "simp_rw [setAverage_eq']", "annotated_tactic": ["simp_rw [<a>setAverage_eq'</a>]", [{"full_name": "MeasureTheory.setAverage_eq'", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [303, 9], "def_end_pos": [303, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u2a0d (y : \u03b1) in a, f y \u2202\u03bc - \u2a0d (x_1 : \u03b1) in a, f x \u2202\u03bc\u2016 \u2264 \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u222b (y : \u03b1), f y \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a - \u222b (x_1 : \u03b1), f x \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\u2016 \u2264\n    \u222b (y : \u03b1), \u2016f y - f x\u2016 \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a"}, {"tactic": "rw [\u2190 integral_sub]", "annotated_tactic": ["rw [\u2190 <a>integral_sub</a>]", [{"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u222b (y : \u03b1), f y \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a - \u222b (x_1 : \u03b1), f x \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\u2016 \u2264\n    \u222b (y : \u03b1), \u2016f y - f x\u2016 \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u222b (a : \u03b1), f a - f x \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\u2016 \u2264 \u222b (y : \u03b1), \u2016f y - f x\u2016 \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\n\ncase h.hf\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 Integrable fun y => f y\n\ncase h.hg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 Integrable fun x_1 => f x"}, {"tactic": "exact norm_integral_le_integral_norm _", "annotated_tactic": ["exact <a>norm_integral_le_integral_norm</a> _", [{"full_name": "MeasureTheory.norm_integral_le_integral_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1376, 9], "def_end_pos": [1376, 39]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 \u2016\u222b (a : \u03b1), f a - f x \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\u2016 \u2264 \u222b (y : \u03b1), \u2016f y - f x\u2016 \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a", "state_after": "no goals"}, {"tactic": "exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 h''a", "annotated_tactic": ["exact (<a>integrable_inv_smul_measure</a> ha.ne' h'a.ne).2 h''a", [{"full_name": "MeasureTheory.integrable_inv_smul_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [591, 9], "def_end_pos": [591, 36]}]], "state_before": "case h.hf\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 Integrable fun y => f y", "state_after": "no goals"}, {"tactic": "exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 (integrableOn_const.2 (Or.inr h'a))", "annotated_tactic": ["exact (<a>integrable_inv_smul_measure</a> ha.ne' h'a.ne).2 (<a>integrableOn_const</a>.2 (<a>Or.inr</a> h'a))", [{"full_name": "MeasureTheory.integrable_inv_smul_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [591, 9], "def_end_pos": [591, 36]}, {"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": "case h.hg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nx : \u03b1\nhx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0)\nh'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a\na : Set \u03b1\nha : 0 < \u2191\u2191\u03bc a\nh'a : \u2191\u2191\u03bc a < \u22a4\nh''a : IntegrableOn f a\n\u22a2 Integrable fun x_1 => 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.mul_eq_one_iff", "start": [1039, 11], "end": [1050, 51], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, _\u27e9", []], "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 s * t = 1 \u2194 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "state_after": "case refine'_1\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\nh : s * t = 1\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1\n\ncase refine'_2\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 b, s = {a} \u2227 t = {b} \u2227 a * b = 1) \u2192 s * t = 1"}, {"tactic": "have hst : (s * t).Nonempty := h.symm.subst one_nonempty", "annotated_tactic": ["have hst : (s * t).<a>Nonempty</a> := h.symm.subst <a>one_nonempty</a>", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.one_nonempty", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}]], "state_before": "case refine'_1\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\nh : s * t = 1\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "state_after": "case refine'_1\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1"}, {"tactic": "obtain \u27e8a, ha\u27e9 := hst.of_image2_left", "annotated_tactic": ["obtain \u27e8a, ha\u27e9 := hst.of_image2_left", []], "state_before": "case refine'_1\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "state_after": "case refine'_1.intro\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1"}, {"tactic": "obtain \u27e8b, hb\u27e9 := hst.of_image2_right", "annotated_tactic": ["obtain \u27e8b, hb\u27e9 := hst.of_image2_right", []], "state_before": "case refine'_1.intro\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "state_after": "case refine'_1.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\ns t : Set \u03b1\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1"}, {"tactic": "have H : \u2200 {a b}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = (1 : \u03b1) := fun {a b} ha hb =>\n  h.subset <| mem_image2_of_mem ha hb", "annotated_tactic": ["have H : \u2200 {a b}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = (1 : \u03b1) := fun {a b} ha hb =>\n      h.subset <| <a>mem_image2_of_mem</a> ha hb", [{"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 refine'_1.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\ns t : Set \u03b1\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "state_after": "case refine'_1.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\ns t : Set \u03b1\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nH : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1"}, {"tactic": "refine' \u27e8a, b, _, _, H ha hb\u27e9 <;> refine' eq_singleton_iff_unique_mem.2 \u27e8\u2039_\u203a, fun x hx => _\u27e9", "annotated_tactic": ["refine' \u27e8a, b, _, _, H ha hb\u27e9 <;> refine' <a>eq_singleton_iff_unique_mem</a>.2 \u27e8\u2039_\u203a, fun x hx => _\u27e9", [{"full_name": "Set.eq_singleton_iff_unique_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1379, 9], "def_end_pos": [1379, 36]}]], "state_before": "case refine'_1.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\ns t : Set \u03b1\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nH : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1\n\u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "state_after": "case refine'_1.intro.intro.refine'_1\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nH : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x = a\n\ncase refine'_1.intro.intro.refine'_2\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nH : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x = b"}, {"tactic": "exact (eq_inv_of_mul_eq_one_left <| H hx hb).trans (inv_eq_of_mul_eq_one_left <| H ha hb)", "annotated_tactic": ["exact (<a>eq_inv_of_mul_eq_one_left</a> <| H hx hb).<a>trans</a> (<a>inv_eq_of_mul_eq_one_left</a> <| H ha hb)", [{"full_name": "eq_inv_of_mul_eq_one_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1068, 9], "def_end_pos": [1068, 34]}, {"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": "inv_eq_of_mul_eq_one_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 34]}]], "state_before": "case refine'_1.intro.intro.refine'_1\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nH : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x = a", "state_after": "no goals"}, {"tactic": "exact (eq_inv_of_mul_eq_one_right <| H ha hx).trans (inv_eq_of_mul_eq_one_right <| H ha hb)", "annotated_tactic": ["exact (<a>eq_inv_of_mul_eq_one_right</a> <| H ha hx).<a>trans</a> (<a>inv_eq_of_mul_eq_one_right</a> <| H ha hb)", [{"full_name": "eq_inv_of_mul_eq_one_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 35]}, {"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": "inv_eq_of_mul_eq_one_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1056, 9], "def_end_pos": [1056, 35]}]], "state_before": "case refine'_1.intro.intro.refine'_2\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\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nH : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x = b", "state_after": "no goals"}, {"tactic": "rintro \u27e8b, c, rfl, rfl, h\u27e9", "annotated_tactic": ["rintro \u27e8b, c, rfl, rfl, h\u27e9", []], "state_before": "case refine'_2\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 b, s = {a} \u2227 t = {b} \u2227 a * b = 1) \u2192 s * t = 1", "state_after": "case refine'_2.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\nb c : \u03b1\nh : b * c = 1\n\u22a2 {b} * {c} = 1"}, {"tactic": "rw [singleton_mul_singleton, h, singleton_one]", "annotated_tactic": ["rw [<a>singleton_mul_singleton</a>, h, <a>singleton_one</a>]", [{"full_name": "Set.singleton_mul_singleton", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 32]}, {"full_name": "Set.singleton_one", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 22]}]], "state_before": "case refine'_2.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\nb c : \u03b1\nh : b * c = 1\n\u22a2 {b} * {c} = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "full_name": "IsUnifLocDoublingMeasure.exists_eventually_forall_measure_closedBall_le_mul", "start": [72, 1], "end": [103, 84], "traced_tactics": [{"tactic": "let C := doublingConstant \u03bc", "annotated_tactic": ["let C := <a>doublingConstant</a> \u03bc", [{"full_name": "IsUnifLocDoublingMeasure.doublingConstant", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [63, 5], "def_end_pos": [63, 21]}]], "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\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\nK : \u211d\nC : \u211d\u22650 := doublingConstant \u03bc\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "rcases lt_or_le K 1 with (hK | hK)", "annotated_tactic": ["rcases <a>lt_or_le</a> K 1 with (hK | hK)", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case inl\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)\n\ncase inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "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\nC : \u211d\u22650 := doublingConstant \u03bc\n\u22a2 \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\nK : \u211d\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case zero\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\nC : \u211d\u22650 := doublingConstant \u03bc\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.zero * \u03b5)) \u2264 \u2191(C ^ Nat.zero) * \u2191\u2191\u03bc (closedBall x \u03b5)\n\ncase succ\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "replace ih := eventually_nhdsWithin_pos_mul_left (two_pos : 0 < (2 : \u211d)) ih", "annotated_tactic": ["replace ih := <a>eventually_nhdsWithin_pos_mul_left</a> (<a>two_pos</a> : 0 < (2 : \u211d)) ih", [{"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": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}]], "state_before": "case succ\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case succ\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "refine' (ih.and (exists_measure_closedBall_le_mul' \u03bc)).mono fun \u03b5 h\u03b5 x => _", "annotated_tactic": ["refine' (ih.and (<a>exists_measure_closedBall_le_mul'</a> \u03bc)).<a>mono</a> fun \u03b5 h\u03b5 x => _", [{"full_name": "IsUnifLocDoublingMeasure.exists_measure_closedBall_le_mul'", "def_path": "Mathlib/MeasureTheory/Measure/Doubling.lean", "def_pos": [67, 9], "def_end_pos": [67, 42]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case succ\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case succ\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "calc\n  \u03bc (closedBall x ((2 : \u211d) ^ (n + 1) * \u03b5)) = \u03bc (closedBall x ((2 : \u211d) ^ n * (2 * \u03b5))) := by\n    rw [pow_succ', mul_assoc]\n  _ \u2264 \u2191(C ^ n) * \u03bc (closedBall x (2 * \u03b5)) := (h\u03b5.1 x)\n  _ \u2264 \u2191(C ^ n) * (C * \u03bc (closedBall x \u03b5)) := by gcongr; exact h\u03b5.2 x\n  _ = \u2191(C ^ (n + 1)) * \u03bc (closedBall x \u03b5) := by rw [\u2190 mul_assoc, pow_succ', ENNReal.coe_mul]", "annotated_tactic": ["calc\n      \u03bc (<a>closedBall</a> x ((2 : \u211d) ^ (n + 1) * \u03b5)) = \u03bc (<a>closedBall</a> x ((2 : \u211d) ^ n * (2 * \u03b5))) := by\n        rw [<a>pow_succ'</a>, <a>mul_assoc</a>]\n      _ \u2264 \u2191(C ^ n) * \u03bc (<a>closedBall</a> x (2 * \u03b5)) := (h\u03b5.1 x)\n      _ \u2264 \u2191(C ^ n) * (C * \u03bc (<a>closedBall</a> x \u03b5)) := by gcongr; exact h\u03b5.2 x\n      _ = \u2191(C ^ (n + 1)) * \u03bc (<a>closedBall</a> x \u03b5) := by rw [\u2190 <a>mul_assoc</a>, <a>pow_succ'</a>, <a>ENNReal.coe_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": "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": "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": "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]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "state_before": "case succ\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\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\nC : \u211d\u22650 := doublingConstant \u03bc\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.zero * \u03b5)) \u2264 \u2191(C ^ Nat.zero) * \u2191\u2191\u03bc (closedBall x \u03b5)", "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/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]}]], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191\u2191\u03bc (closedBall x (2 ^ (n + 1) * \u03b5)) = \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5)))", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(C ^ n) * (\u2191C * \u2191\u2191\u03bc (closedBall x \u03b5))", "state_after": "case bc\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "exact h\u03b5.2 x", "annotated_tactic": ["exact h\u03b5.2 x", []], "state_before": "case bc\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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "no goals"}, {"tactic": "rw [\u2190 mul_assoc, pow_succ', ENNReal.coe_mul]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>, <a>pow_succ'</a>, <a>ENNReal.coe_mul</a>]", [{"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]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nn : \u2115\nih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227\n    \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5)\nx : \u03b1\n\u22a2 \u2191(C ^ n) * (\u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)) = \u2191(C ^ (n + 1)) * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "no goals"}, {"tactic": "refine' \u27e81, _\u27e9", "annotated_tactic": ["refine' \u27e81, _\u27e9", []], "state_before": "case inl\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case inl\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u21911 * \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "simp only [ENNReal.coe_one, one_mul]", "annotated_tactic": ["simp only [<a>ENNReal.coe_one</a>, <a>one_mul</a>]", [{"full_name": "ENNReal.coe_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [218, 28], "def_end_pos": [218, 35]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case inl\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u21911 * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case inl\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191\u2191\u03bc (closedBall x \u03b5)"}, {"tactic": "exact\n  eventually_mem_nhdsWithin.mono fun \u03b5 h\u03b5 x t ht =>\n    measure_mono <| closedBall_subset_closedBall (by nlinarith [mem_Ioi.mp h\u03b5])", "annotated_tactic": ["exact\n      eventually_mem_nhdsWithin.mono fun \u03b5 h\u03b5 x t ht =>\n        <a>measure_mono</a> <| <a>closedBall_subset_closedBall</a> (by nlinarith [mem_Ioi.mp h\u03b5])", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}]], "state_before": "case inl\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "no goals"}, {"tactic": "nlinarith [mem_Ioi.mp h\u03b5]", "annotated_tactic": ["nlinarith [mem_Ioi.mp h\u03b5]", []], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : K < 1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 t * \u03b5 \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "refine'\n  \u27e8C ^ \u2308Real.logb 2 K\u2309\u208a,\n    ((h\u03bc \u2308Real.logb 2 K\u2309\u208a).and eventually_mem_nhdsWithin).mono fun \u03b5 h\u03b5 x t ht =>\n      le_trans (measure_mono <| closedBall_subset_closedBall _) (h\u03b5.1 x)\u27e9", "annotated_tactic": ["refine'\n      \u27e8C ^ \u2308<a>Real.logb</a> 2 K\u2309\u208a,\n        ((h\u03bc \u2308<a>Real.logb</a> 2 K\u2309\u208a).<a>and</a> <a>eventually_mem_nhdsWithin</a>).<a>mono</a> fun \u03b5 h\u03b5 x t ht =>\n          <a>le_trans</a> (<a>measure_mono</a> <| <a>closedBall_subset_closedBall</a> _) (h\u03b5.1 x)\u27e9", [{"full_name": "Real.logb", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}, {"full_name": "Real.logb", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "eventually_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [155, 9], "def_end_pos": [155, 34]}, {"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": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}]], "state_before": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)", "state_after": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 t * \u03b5 \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5"}, {"tactic": "refine' mul_le_mul_of_nonneg_right (ht.trans _) (mem_Ioi.mp h\u03b5.2).le", "annotated_tactic": ["refine' <a>mul_le_mul_of_nonneg_right</a> (ht.trans _) (mem_Ioi.mp h\u03b5.2).<a>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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 t * \u03b5 \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5", "state_after": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 K \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a"}, {"tactic": "conv_lhs => rw [\u2190 Real.rpow_logb two_pos (by norm_num) (by linarith : 0 < K)]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>Real.rpow_logb</a> <a>two_pos</a> (by norm_num) (by linarith : 0 < K)]", [{"full_name": "Real.rpow_logb", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [134, 9], "def_end_pos": [134, 18]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}]], "state_before": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 K \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a", "state_after": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 2 ^ Real.logb 2 K \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a"}, {"tactic": "rw [\u2190 Real.rpow_nat_cast]", "annotated_tactic": ["rw [\u2190 <a>Real.rpow_nat_cast</a>]", [{"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 2 ^ Real.logb 2 K \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a", "state_after": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 2 ^ Real.logb 2 K \u2264 2 ^ \u2191\u2308Real.logb 2 K\u2309\u208a"}, {"tactic": "exact Real.rpow_le_rpow_of_exponent_le one_le_two (Nat.le_ceil (Real.logb 2 K))", "annotated_tactic": ["exact <a>Real.rpow_le_rpow_of_exponent_le</a> <a>one_le_two</a> (<a>Nat.le_ceil</a> (<a>Real.logb</a> 2 K))", [{"full_name": "Real.rpow_le_rpow_of_exponent_le", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [491, 9], "def_end_pos": [491, 36]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "Nat.le_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "Real.logb", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [40, 19], "def_end_pos": [40, 23]}]], "state_before": "case inr\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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 2 ^ Real.logb 2 K \u2264 2 ^ \u2191\u2308Real.logb 2 K\u2309\u208a", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 2 \u2260 1", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "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\nC : \u211d\u22650 := doublingConstant \u03bc\nh\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5)\nhK : 1 \u2264 K\n\u03b5 : \u211d\nh\u03b5 :\n  (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227\n    \u03b5 \u2208 Ioi 0\nx : \u03b1\nt : \u211d\nht : t \u2264 K\n\u22a2 0 < K", "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.smul_Lp_ae_eq", "start": [41, 1], "end": [42, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Uniform.lean", "full_name": "PMF.ofMultiset_apply_of_not_mem", "start": [190, 1], "end": [192, 47], "traced_tactics": [{"tactic": "simpa only [ofMultiset_apply, ENNReal.div_eq_zero_iff, Nat.cast_eq_zero, Multiset.count_eq_zero,\n  ENNReal.nat_ne_top, or_false_iff] using ha", "annotated_tactic": ["simpa only [<a>ofMultiset_apply</a>, <a>ENNReal.div_eq_zero_iff</a>, <a>Nat.cast_eq_zero</a>, <a>Multiset.count_eq_zero</a>,\n    <a>ENNReal.nat_ne_top</a>, <a>or_false_iff</a>] using ha", [{"full_name": "PMF.ofMultiset_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Uniform.lean", "def_pos": [177, 9], "def_end_pos": [177, 25]}, {"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": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 21]}, {"full_name": "Multiset.count_eq_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2426, 9], "def_end_pos": [2426, 22]}, {"full_name": "ENNReal.nat_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [717, 17], "def_end_pos": [717, 27]}, {"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\ns : Multiset \u03b1\nhs : s \u2260 0\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 \u2191(ofMultiset s hs) a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_hom", "start": [95, 1], "end": [99, 34], "traced_tactics": [{"tactic": "rw [fold, fold, \u2190 Multiset.fold_hom op hm, Multiset.map_map]", "annotated_tactic": ["rw [<a>fold</a>, <a>fold</a>, \u2190 <a>Multiset.fold_hom</a> op hm, <a>Multiset.map_map</a>]", [{"full_name": "Finset.fold", "def_path": "Mathlib/Data/Finset/Fold.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}, {"full_name": "Finset.fold", "def_path": "Mathlib/Data/Finset/Fold.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}, {"full_name": "Multiset.fold_hom", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}, {"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 16]}]], "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\nop' : \u03b3 \u2192 \u03b3 \u2192 \u03b3\ninst\u271d\u00b9 : IsCommutative \u03b3 op'\ninst\u271d : IsAssociative \u03b3 op'\nm : \u03b2 \u2192 \u03b3\nhm : \u2200 (x y : \u03b2), m (op x y) = op' (m x) (m y)\n\u22a2 fold op' (m b) (fun x => m (f x)) s = m (fold op b f s)", "state_after": "\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\nop' : \u03b3 \u2192 \u03b3 \u2192 \u03b3\ninst\u271d\u00b9 : IsCommutative \u03b3 op'\ninst\u271d : IsAssociative \u03b3 op'\nm : \u03b2 \u2192 \u03b3\nhm : \u2200 (x y : \u03b2), m (op x y) = op' (m x) (m y)\n\u22a2 Multiset.fold op' (m b) (Multiset.map (fun x => m (f x)) s.val) = Multiset.fold op' (m b) (Multiset.map (m \u2218 f) s.val)"}, {"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": "\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\nop' : \u03b3 \u2192 \u03b3 \u2192 \u03b3\ninst\u271d\u00b9 : IsCommutative \u03b3 op'\ninst\u271d : IsAssociative \u03b3 op'\nm : \u03b2 \u2192 \u03b3\nhm : \u2200 (x y : \u03b2), m (op x y) = op' (m x) (m y)\n\u22a2 Multiset.fold op' (m b) (Multiset.map (fun x => m (f x)) s.val) = Multiset.fold op' (m b) (Multiset.map (m \u2218 f) s.val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "full_name": "MeasureTheory.Measure.bind_zero_right'", "start": [169, 1], "end": [170, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Snoc.lean", "full_name": "Vector.replicate_succ_to_snoc", "start": [55, 1], "end": [65, 22], "traced_tactics": [{"tactic": "clear xs", "annotated_tactic": ["clear xs", []], "state_before": "\u03b1 : Type u_1\nn : \u2115\nxs : Vector \u03b1 n\nval : \u03b1\n\u22a2 replicate (n + 1) val = snoc (replicate n val) val", "state_after": "\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (n + 1) val = snoc (replicate n val) val"}, {"tactic": "induction n", "annotated_tactic": ["induction n", []], "state_before": "\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (n + 1) val = snoc (replicate n val) val", "state_after": "case zero\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val\n\ncase succ\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\nn\u271d : \u2115\nn_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val\n\u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val"}, {"tactic": "case zero => rfl", "annotated_tactic": ["case zero => rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val\n\ncase succ\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\nn\u271d : \u2115\nn_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val\n\u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val", "state_after": "case succ\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\nn\u271d : \u2115\nn_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val\n\u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val"}, {"tactic": "case succ n ih =>\n  rw [replicate_succ]\n  conv => {\n    rhs; rw [replicate_succ]\n  }\n  rw[snoc_cons, ih]", "annotated_tactic": ["case succ n ih =>\n    rw [<a>replicate_succ</a>]\n    conv => {\n      rhs; rw [<a>replicate_succ</a>]\n    }\n    rw[<a>snoc_cons</a>, ih]", [{"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [774, 9], "def_end_pos": [774, 23]}, {"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [774, 9], "def_end_pos": [774, 23]}, {"full_name": "Vector.snoc_cons", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [34, 9], "def_end_pos": [34, 18]}]], "state_before": "case succ\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\nn\u271d : \u2115\nn_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val\n\u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val", "state_after": "no goals"}, {"tactic": "rw [replicate_succ]", "annotated_tactic": ["rw [<a>replicate_succ</a>]", [{"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [774, 9], "def_end_pos": [774, 23]}]], "state_before": "\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 replicate (Nat.succ n + 1) val = snoc (replicate (Nat.succ n) val) val", "state_after": "\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val"}, {"tactic": "conv => {\n  rhs; rw [replicate_succ]\n}", "annotated_tactic": ["conv => {\n      rhs; rw [<a>replicate_succ</a>]\n    }", [{"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [774, 9], "def_end_pos": [774, 23]}]], "state_before": "\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val", "state_after": "\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 val ::\u1d65 replicate (n + 1) val = snoc (val ::\u1d65 replicate n val) val"}, {"tactic": "rw[snoc_cons, ih]", "annotated_tactic": ["rw[<a>snoc_cons</a>, ih]", [{"full_name": "Vector.snoc_cons", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [34, 9], "def_end_pos": [34, 18]}]], "state_before": "\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 val ::\u1d65 replicate (n + 1) val = snoc (val ::\u1d65 replicate n val) val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.isMulLeftInvariant_map_smul", "start": [259, 1], "end": [264, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.mem\u2112p_norm_rpow_iff", "start": [912, 1], "end": [922, 15], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun h => h.norm_rpow_div q\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => h.norm_rpow_div q\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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) \u2194 Mem\u2112p f 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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 Mem\u2112p f p"}, {"tactic": "apply (mem\u2112p_norm_iff hf).1", "annotated_tactic": ["apply (<a>mem\u2112p_norm_iff</a> hf).1", [{"full_name": "MeasureTheory.mem\u2112p_norm_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}]], "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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 Mem\u2112p f 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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 Mem\u2112p (fun x => \u2016f x\u2016) p"}, {"tactic": "convert h.norm_rpow_div q\u207b\u00b9 using 1", "annotated_tactic": ["convert h.norm_rpow_div q\u207b\u00b9 using 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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 Mem\u2112p (fun x => \u2016f x\u2016) p", "state_after": "case h.e'_5\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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 (fun x => \u2016f x\u2016) = fun x => \u2016\u2016f x\u2016 ^ ENNReal.toReal q\u2016 ^ ENNReal.toReal q\u207b\u00b9\n\ncase h.e'_6\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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 p = p / q / q\u207b\u00b9"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_5\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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 (fun x => \u2016f x\u2016) = fun x => \u2016\u2016f x\u2016 ^ ENNReal.toReal q\u2016 ^ ENNReal.toReal q\u207b\u00b9", "state_after": "case h.e'_5.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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\nx : \u03b1\n\u22a2 \u2016f x\u2016 = \u2016\u2016f x\u2016 ^ ENNReal.toReal q\u2016 ^ ENNReal.toReal q\u207b\u00b9"}, {"tactic": "rw [Real.norm_eq_abs, Real.abs_rpow_of_nonneg (norm_nonneg _), \u2190 Real.rpow_mul (abs_nonneg _),\n  ENNReal.toReal_inv, mul_inv_cancel, abs_of_nonneg (norm_nonneg _), Real.rpow_one]", "annotated_tactic": ["rw [<a>Real.norm_eq_abs</a>, <a>Real.abs_rpow_of_nonneg</a> (<a>norm_nonneg</a> _), \u2190 <a>Real.rpow_mul</a> (<a>abs_nonneg</a> _),\n      <a>ENNReal.toReal_inv</a>, <a>mul_inv_cancel</a>, <a>abs_of_nonneg</a> (<a>norm_nonneg</a> _), <a>Real.rpow_one</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": "Real.abs_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [146, 9], "def_end_pos": [146, 27]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "ENNReal.toReal_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2381, 9], "def_end_pos": [2381, 19]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"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.e'_5.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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\nx : \u03b1\n\u22a2 \u2016f x\u2016 = \u2016\u2016f x\u2016 ^ ENNReal.toReal q\u2016 ^ ENNReal.toReal q\u207b\u00b9", "state_after": "case h.e'_5.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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\nx : \u03b1\n\u22a2 ENNReal.toReal q \u2260 0"}, {"tactic": "simp [ENNReal.toReal_eq_zero_iff, not_or, q_zero, q_top]", "annotated_tactic": ["simp [<a>ENNReal.toReal_eq_zero_iff</a>, <a>not_or</a>, q_zero, q_top]", [{"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": "case h.e'_5.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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\nx : \u03b1\n\u22a2 ENNReal.toReal q \u2260 0", "state_after": "no goals"}, {"tactic": "rw [div_eq_mul_inv, inv_inv, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel q_zero q_top,\n  mul_one]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>inv_inv</a>, <a>div_eq_mul_inv</a>, <a>mul_assoc</a>, <a>ENNReal.inv_mul_cancel</a> q_zero q_top,\n      <a>mul_one</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": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 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]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case h.e'_6\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\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nq : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nq_zero : q \u2260 0\nq_top : q \u2260 \u22a4\nh : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q)\n\u22a2 p = p / q / q\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.isMulRightInvariant_map_smul", "start": [270, 1], "end": [275, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.Mem\u2112p.of_measure_le_smul", "start": [672, 1], "end": [680, 17], "traced_tactics": [{"tactic": "refine' \u27e8hf.1.mono' (Measure.absolutelyContinuous_of_le_smul h\u03bc'_le), _\u27e9", "annotated_tactic": ["refine' \u27e8hf.1.<a>mono'</a> (<a>Measure.absolutelyContinuous_of_le_smul</a> h\u03bc'_le), _\u27e9", [{"full_name": "MeasureTheory.AEStronglyMeasurable.mono'", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1242, 19], "def_end_pos": [1242, 24]}, {"full_name": "MeasureTheory.Measure.absolutelyContinuous_of_le_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2169, 9], "def_end_pos": [2169, 40]}]], "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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 Mem\u2112p f 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm f p \u03bc' < \u22a4"}, {"tactic": "refine' (snorm_mono_measure f h\u03bc'_le).trans_lt _", "annotated_tactic": ["refine' (<a>snorm_mono_measure</a> f h\u03bc'_le).<a>trans_lt</a> _", [{"full_name": "MeasureTheory.snorm_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [612, 9], "def_end_pos": [612, 27]}, {"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\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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm f p \u03bc' < \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\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm f p (c \u2022 \u03bc) < \u22a4"}, {"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\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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm f p (c \u2022 \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 : \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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : c = 0\n\u22a2 snorm f p (c \u2022 \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 : \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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 snorm f p (c \u2022 \u03bc) < \u22a4"}, {"tactic": "rw [snorm_smul_measure_of_ne_zero hc0, smul_eq_mul]", "annotated_tactic": ["rw [<a>snorm_smul_measure_of_ne_zero</a> hc0, <a>smul_eq_mul</a>]", [{"full_name": "MeasureTheory.snorm_smul_measure_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [651, 9], "def_end_pos": [651, 38]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 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\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 snorm f p (c \u2022 \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 : \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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 c ^ ENNReal.toReal (1 / p) * snorm f p \u03bc < \u22a4"}, {"tactic": "refine' ENNReal.mul_lt_top _ hf.2.ne", "annotated_tactic": ["refine' <a>ENNReal.mul_lt_top</a> _ hf.2.<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\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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 c ^ ENNReal.toReal (1 / p) * 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 : \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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 c ^ ENNReal.toReal (1 / p) \u2260 \u22a4"}, {"tactic": "simp [hc, hc0]", "annotated_tactic": ["simp [hc, hc0]", []], "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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 c ^ ENNReal.toReal (1 / p) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp [hc0]", "annotated_tactic": ["simp [hc0]", []], "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\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : c = 0\n\u22a2 snorm f p (c \u2022 \u03bc) < \u22a4", "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_top_of_bound", "start": [584, 1], "end": [588, 46], "traced_tactics": [{"tactic": "rw [snorm_exponent_top]", "annotated_tactic": ["rw [<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": "\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\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snorm f \u22a4 \u03bc < \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 E\nhf : AEStronglyMeasurable f \u03bc\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snormEssSup f \u03bc < \u22a4"}, {"tactic": "exact snormEssSup_lt_top_of_ae_bound hfC", "annotated_tactic": ["exact <a>snormEssSup_lt_top_of_ae_bound</a> hfC", [{"full_name": "MeasureTheory.snormEssSup_lt_top_of_ae_bound", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [458, 9], "def_end_pos": [458, 39]}]], "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\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snormEssSup f \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.snorm_condexpL2_le", "start": [105, 1], "end": [109, 31], "traced_tactics": [{"tactic": "rw [lpMeas_coe, \u2190 ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _), \u2190\n  Lp.norm_def, \u2190 Lp.norm_def, Submodule.norm_coe]", "annotated_tactic": ["rw [<a>lpMeas_coe</a>, \u2190 <a>ENNReal.toReal_le_toReal</a> (<a>Lp.snorm_ne_top</a> _) (<a>Lp.snorm_ne_top</a> _), \u2190\n    <a>Lp.norm_def</a>, \u2190 <a>Lp.norm_def</a>, <a>Submodule.norm_coe</a>]", [{"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 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.Lp.snorm_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [202, 9], "def_end_pos": [202, 21]}, {"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": "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": "Submodule.norm_coe", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2779, 9], "def_end_pos": [2779, 17]}]], "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 \u2264 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 \u2016\u2191(condexpL2 E \ud835\udd5c hm) f\u2016 \u2264 \u2016f\u2016"}, {"tactic": "exact norm_condexpL2_le hm f", "annotated_tactic": ["exact <a>norm_condexpL2_le</a> hm f", [{"full_name": "MeasureTheory.norm_condexpL2_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [100, 9], "def_end_pos": [100, 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 }\n\u22a2 \u2016\u2191(condexpL2 E \ud835\udd5c hm) f\u2016 \u2264 \u2016f\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/DFA.lean", "full_name": "DFA.eval_append_singleton", "start": [85, 1], "end": [86, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.card_le_of_interleaved", "start": [1671, 1], "end": [1696, 84], "traced_tactics": [{"tactic": "replace h : \u2200 (x) (_ : x \u2208 s) (y) (_ : y \u2208 s), x < y \u2192 \u2203 z \u2208 t, x < z \u2227 z < y", "annotated_tactic": ["replace h : \u2200 (x) (_ : x \u2208 s) (y) (_ : y \u2208 s), x < y \u2192 \u2203 z \u2208 t, x < z \u2227 z < y", []], "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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\n\u22a2 card s \u2264 card t + 1", "state_after": "case h\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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\n\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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\n\u22a2 card s \u2264 card t + 1"}, {"tactic": "set f : \u03b1 \u2192 WithTop \u03b1 := fun x => (t.filter fun y => x < y).min", "annotated_tactic": ["set f : \u03b1 \u2192 <a>WithTop</a> \u03b1 := fun x => (t.filter fun y => x < y).<a>min</a>", [{"full_name": "WithTop", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [567, 5], "def_end_pos": [567, 12]}, {"full_name": "Finset.min", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1329, 15], "def_end_pos": [1329, 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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\n\u22a2 card s \u2264 card t + 1", "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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\n\u22a2 card s \u2264 card t + 1"}, {"tactic": "have f_mono : StrictMonoOn f s := by\n  intro x hx y hy hxy\n  rcases h x hx y hy hxy with \u27e8a, hat, hxa, hay\u27e9\n  calc\n    f x \u2264 a := min_le (mem_filter.2 \u27e8hat, by simpa\u27e9)\n    _ < f y :=\n      (Finset.lt_inf_iff <| WithTop.coe_lt_top a).2 fun b hb =>\n        WithTop.coe_lt_coe.2 <| hay.trans (by simpa using (mem_filter.1 hb).2)", "annotated_tactic": ["have f_mono : <a>StrictMonoOn</a> f s := by\n    intro x hx y hy hxy\n    rcases h x hx y hy hxy with \u27e8a, hat, hxa, hay\u27e9\n    calc\n      f x \u2264 a := <a>min_le</a> (<a>mem_filter</a>.2 \u27e8hat, by simpa\u27e9)\n      _ < f y :=\n        (<a>Finset.lt_inf_iff</a> <| <a>WithTop.coe_lt_top</a> a).2 fun b hb =>\n          <a>WithTop.coe_lt_coe</a>.2 <| hay.trans (by simpa using (<a>mem_filter</a>.1 hb).2)", [{"full_name": "StrictMonoOn", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [108, 5], "def_end_pos": [108, 17]}, {"full_name": "Finset.min_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1375, 9], "def_end_pos": [1375, 15]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "Finset.lt_inf_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [723, 19], "def_end_pos": [723, 29]}, {"full_name": "WithTop.coe_lt_top", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1096, 9], "def_end_pos": [1096, 19]}, {"full_name": "WithTop.coe_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\n\u22a2 card s \u2264 card t + 1", "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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nf_mono : StrictMonoOn f \u2191s\n\u22a2 card s \u2264 card t + 1"}, {"tactic": "calc\n  s.card = (s.image f).card := (card_image_of_injOn f_mono.injOn).symm\n  _ \u2264 (insert \u22a4 (t.image (\u2191)) : Finset (WithTop \u03b1)).card :=\n    card_mono <| image_subset_iff.2 fun x _ =>\n        insert_subset_insert _ (image_subset_image <| filter_subset _ _)\n          (min_mem_insert_top_image_coe _)\n  _ \u2264 t.card + 1 := (card_insert_le _ _).trans (add_le_add_right card_image_le _)", "annotated_tactic": ["calc\n    s.card = (s.image f).<a>card</a> := (<a>card_image_of_injOn</a> f_mono.injOn).<a>symm</a>\n    _ \u2264 (<a>insert</a> \u22a4 (t.image (\u2191)) : <a>Finset</a> (<a>WithTop</a> \u03b1)).<a>card</a> :=\n      <a>card_mono</a> <| <a>image_subset_iff</a>.2 fun x _ =>\n          <a>insert_subset_insert</a> _ (<a>image_subset_image</a> <| <a>filter_subset</a> _ _)\n            (<a>min_mem_insert_top_image_coe</a> _)\n    _ \u2264 t.card + 1 := (<a>card_insert_le</a> _ _).<a>trans</a> (<a>add_le_add_right</a> <a>card_image_le</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_image_of_injOn", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}, {"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": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "WithTop", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [567, 5], "def_end_pos": [567, 12]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card_mono", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}, {"full_name": "Finset.image_subset_iff", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [451, 9], "def_end_pos": [451, 25]}, {"full_name": "Finset.insert_subset_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 29]}, {"full_name": "Finset.image_subset_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [447, 9], "def_end_pos": [447, 27]}, {"full_name": "Finset.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2701, 9], "def_end_pos": [2701, 22]}, {"full_name": "Finset.min_mem_insert_top_image_coe", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1629, 9], "def_end_pos": [1629, 37]}, {"full_name": "Finset.card_insert_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [113, 9], "def_end_pos": [113, 23]}, {"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_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "Finset.card_image_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [228, 9], "def_end_pos": [228, 22]}]], "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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nf_mono : StrictMonoOn f \u2191s\n\u22a2 card s \u2264 card t + 1", "state_after": "no goals"}, {"tactic": "intro x hx y hy hxy", "annotated_tactic": ["intro x hx y hy hxy", []], "state_before": "case h\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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y", "state_after": "case h\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x < y\n\u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y"}, {"tactic": "rcases exists_next_right \u27e8y, hy, hxy\u27e9 with \u27e8a, has, hxa, ha\u27e9", "annotated_tactic": ["rcases <a>exists_next_right</a> \u27e8y, hy, hxy\u27e9 with \u27e8a, has, hxa, ha\u27e9", [{"full_name": "Finset.exists_next_right", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 26]}]], "state_before": "case h\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x < y\n\u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y", "state_after": "case h.intro.intro.intro\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x < y\na : \u03b1\nhas : a \u2208 s\nhxa : x < a\nha : \u2200 (z : \u03b1), z \u2208 s \u2192 x < z \u2192 a \u2264 z\n\u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y"}, {"tactic": "rcases h x hx a has hxa fun z hzs hz => hz.2.not_le <| ha _ hzs hz.1 with \u27e8b, hbt, hxb, hba\u27e9", "annotated_tactic": ["rcases h x hx a has hxa fun z hzs hz => hz.2.<a>not_le</a> <| ha _ hzs hz.1 with \u27e8b, hbt, hxb, hba\u27e9", [{"full_name": "LT.lt.not_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [374, 7], "def_end_pos": [374, 19]}]], "state_before": "case h.intro.intro.intro\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x < y\na : \u03b1\nhas : a \u2208 s\nhxa : x < a\nha : \u2200 (z : \u03b1), z \u2208 s \u2192 x < z \u2192 a \u2264 z\n\u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y", "state_after": "case h.intro.intro.intro.intro.intro.intro\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x < y\na : \u03b1\nhas : a \u2208 s\nhxa : x < a\nha : \u2200 (z : \u03b1), z \u2208 s \u2192 x < z \u2192 a \u2264 z\nb : \u03b1\nhbt : b \u2208 t\nhxb : x < b\nhba : b < a\n\u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y"}, {"tactic": "exact \u27e8b, hbt, hxb, hba.trans_le <| ha _ hy hxy\u27e9", "annotated_tactic": ["exact \u27e8b, hbt, hxb, hba.trans_le <| ha _ hy hxy\u27e9", []], "state_before": "case h.intro.intro.intro.intro.intro.intro\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x < y\na : \u03b1\nhas : a \u2208 s\nhxa : x < a\nha : \u2200 (z : \u03b1), z \u2208 s \u2192 x < z \u2192 a \u2264 z\nb : \u03b1\nhbt : b \u2208 t\nhxb : x < b\nhba : b < a\n\u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y", "state_after": "no goals"}, {"tactic": "intro x hx y hy hxy", "annotated_tactic": ["intro x hx y hy hxy", []], "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 t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\n\u22a2 StrictMonoOn f \u2191s", "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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nhxy : x < y\n\u22a2 f x < f y"}, {"tactic": "rcases h x hx y hy hxy with \u27e8a, hat, hxa, hay\u27e9", "annotated_tactic": ["rcases h x hx y hy hxy with \u27e8a, hat, hxa, hay\u27e9", []], "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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nhxy : x < y\n\u22a2 f x < f y", "state_after": "case intro.intro.intro\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nhxy : x < y\na : \u03b1\nhat : a \u2208 t\nhxa : x < a\nhay : a < y\n\u22a2 f x < f y"}, {"tactic": "calc\n  f x \u2264 a := min_le (mem_filter.2 \u27e8hat, by simpa\u27e9)\n  _ < f y :=\n    (Finset.lt_inf_iff <| WithTop.coe_lt_top a).2 fun b hb =>\n      WithTop.coe_lt_coe.2 <| hay.trans (by simpa using (mem_filter.1 hb).2)", "annotated_tactic": ["calc\n      f x \u2264 a := <a>min_le</a> (<a>mem_filter</a>.2 \u27e8hat, by simpa\u27e9)\n      _ < f y :=\n        (<a>Finset.lt_inf_iff</a> <| <a>WithTop.coe_lt_top</a> a).2 fun b hb =>\n          <a>WithTop.coe_lt_coe</a>.2 <| hay.trans (by simpa using (<a>mem_filter</a>.1 hb).2)", [{"full_name": "Finset.min_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1375, 9], "def_end_pos": [1375, 15]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "Finset.lt_inf_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [723, 19], "def_end_pos": [723, 29]}, {"full_name": "WithTop.coe_lt_top", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1096, 9], "def_end_pos": [1096, 19]}, {"full_name": "WithTop.coe_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case intro.intro.intro\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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nhxy : x < y\na : \u03b1\nhat : a \u2208 t\nhxa : x < a\nhay : a < y\n\u22a2 f x < f y", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nhxy : x < y\na : \u03b1\nhat : a \u2208 t\nhxa : x < a\nhay : a < y\n\u22a2 x < a", "state_after": "no goals"}, {"tactic": "simpa using (mem_filter.1 hb).2", "annotated_tactic": ["simpa using (<a>mem_filter</a>.1 hb).2", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 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\u271d : \u03b1\ns t : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y\nf : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nhxy : x < y\na : \u03b1\nhat : a \u2208 t\nhxa : x < a\nhay : a < y\nb : \u03b1\nhb : b \u2208 filter (fun y_1 => y < y_1) t\n\u22a2 y < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Subtype.lean", "full_name": "Subtype.coind_bijective", "start": [193, 1], "end": [195, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_lipschitz", "start": [1244, 1], "end": [1246, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ", "start": [1922, 1], "end": [1924, 64], "traced_tactics": [{"tactic": "rwa [\u2190 sUnion_eq_biUnion]", "annotated_tactic": ["rwa [\u2190 <a>sUnion_eq_biUnion</a>]", [{"full_name": "Set.sUnion_eq_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 26]}]], "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 : Set (Set \u03b1)\nhc : Set.Countable S\nhs : \u22c3\u2080 S = univ\n\u22a2 \u22c3 i \u2208 S, i = univ", "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_eapproxDiff", "start": [922, 1], "end": [932, 22], "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\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nn : \u2115\na : \u03b1\n\u22a2 \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) 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\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u2211 k in Finset.range (Nat.zero + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f Nat.zero) a\n\ncase succ\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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2211 k in Finset.range (Nat.succ n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f (Nat.succ n)) a"}, {"tactic": "simp only [Nat.zero_eq, Nat.zero_add, Finset.sum_singleton, Finset.range_one]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>Nat.zero_add</a>, <a>Finset.sum_singleton</a>, <a>Finset.range_one</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.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}, {"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "Finset.range_one", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3052, 9], "def_end_pos": [3052, 18]}]], "state_before": "case zero\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\na : \u03b1\n\u22a2 \u2211 k in Finset.range (Nat.zero + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f Nat.zero) 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\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u2191(\u2191(eapproxDiff f 0) a) = \u2191(eapprox f 0) a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\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\na : \u03b1\n\u22a2 \u2191(\u2191(eapproxDiff f 0) a) = \u2191(eapprox f 0) a", "state_after": "no goals"}, {"tactic": "erw [Finset.sum_range_succ, Nat.succ_eq_add_one, IH, eapproxDiff, coe_map, Function.comp_apply,\n  coe_sub, Pi.sub_apply, ENNReal.coe_toNNReal,\n  add_tsub_cancel_of_le (monotone_eapprox f (Nat.le_succ _) _)]", "annotated_tactic": ["erw [<a>Finset.sum_range_succ</a>, <a>Nat.succ_eq_add_one</a>, IH, <a>eapproxDiff</a>, <a>coe_map</a>, <a>Function.comp_apply</a>,\n      <a>coe_sub</a>, <a>Pi.sub_apply</a>, <a>ENNReal.coe_toNNReal</a>,\n      <a>add_tsub_cancel_of_le</a> (<a>monotone_eapprox</a> f (<a>Nat.le_succ</a> _) _)]", [{"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}, {"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]}, {"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.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_sub", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [481, 3], "def_end_pos": [481, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "ENNReal.coe_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [180, 9], "def_end_pos": [180, 21]}, {"full_name": "add_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [24, 9], "def_end_pos": [24, 30]}, {"full_name": "MeasureTheory.SimpleFunc.monotone_eapprox", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [892, 9], "def_end_pos": [892, 25]}, {"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 succ\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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2211 k in Finset.range (Nat.succ n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f (Nat.succ n)) a", "state_after": "case succ\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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a - \u2191(eapprox f (Nat.add n 0)) a \u2260 \u22a4"}, {"tactic": "apply (lt_of_le_of_lt _ (eapprox_lt_top f (n + 1) a)).ne", "annotated_tactic": ["apply (<a>lt_of_le_of_lt</a> _ (<a>eapprox_lt_top</a> f (n + 1) 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": "MeasureTheory.SimpleFunc.eapprox_lt_top", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [877, 9], "def_end_pos": [877, 23]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case succ\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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a - \u2191(eapprox f (Nat.add n 0)) a \u2260 \u22a4", "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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a - \u2191(eapprox f (Nat.add n 0)) a \u2264 \u2191(eapprox f (n + 1)) a"}, {"tactic": "rw [tsub_le_iff_right]", "annotated_tactic": ["rw [<a>tsub_le_iff_right</a>]", [{"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": "\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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a - \u2191(eapprox f (Nat.add n 0)) a \u2264 \u2191(eapprox f (n + 1)) 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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a \u2264 \u2191(eapprox f (n + 1)) a + \u2191(eapprox f (Nat.add n 0)) a"}, {"tactic": "exact le_self_add", "annotated_tactic": ["exact <a>le_self_add</a>", [{"full_name": "le_self_add", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [154, 3], "def_end_pos": [154, 14]}]], "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\na : \u03b1\nn : \u2115\nIH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a\n\u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a \u2264 \u2191(eapprox f (n + 1)) a + \u2191(eapprox f (Nat.add n 0)) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.generateMeasurable_eq_rec", "start": [117, 1], "end": [151, 24], "traced_tactics": [{"tactic": "ext t", "annotated_tactic": ["ext t", []], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 {t | GenerateMeasurable s t} = \u22c3 i, generateMeasurableRec s i", "state_after": "case h\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\n\u22a2 t \u2208 {t | GenerateMeasurable s t} \u2194 t \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "refine' \u27e8fun ht => _, fun ht => _\u27e9", "annotated_tactic": ["refine' \u27e8fun ht => _, fun ht => _\u27e9", []], "state_before": "case h\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\n\u22a2 t \u2208 {t | GenerateMeasurable s t} \u2194 t \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\nht : t \u2208 {t | GenerateMeasurable s t}\n\u22a2 t \u2208 \u22c3 i, generateMeasurableRec s i\n\ncase h.refine'_2\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\nht : t \u2208 \u22c3 i, generateMeasurableRec s i\n\u22a2 t \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "inhabit \u03c9\u2081", "annotated_tactic": ["inhabit \u03c9\u2081", []], "state_before": "case h.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\nht : t \u2208 {t | GenerateMeasurable s t}\n\u22a2 t \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\nht : t \u2208 {t | GenerateMeasurable s t}\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 t \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "induction' ht with u hu u _ IH f _ IH", "annotated_tactic": ["induction' ht with u hu u _ IH f _ IH", []], "state_before": "case h.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\nht : t \u2208 {t | GenerateMeasurable s t}\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 t \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1.basic\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\nhu : u \u2208 s\n\u22a2 u \u2208 \u22c3 i, generateMeasurableRec s i\n\ncase h.refine'_1.empty\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2205 \u2208 \u22c3 i, generateMeasurableRec s i\n\ncase h.refine'_1.compl\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\na\u271d : GenerateMeasurable s u\nIH : u \u2208 \u22c3 i, generateMeasurableRec s i\n\u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i\n\ncase h.refine'_1.iUnion\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\n\u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "exact mem_iUnion.2 \u27e8default, self_subset_generateMeasurableRec s _ hu\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8<a>default</a>, <a>self_subset_generateMeasurableRec</a> s _ hu\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"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": "MeasurableSpace.self_subset_generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [55, 9], "def_end_pos": [55, 42]}]], "state_before": "case h.refine'_1.basic\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\nhu : u \u2208 s\n\u22a2 u \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "no goals"}, {"tactic": "exact mem_iUnion.2 \u27e8default, empty_mem_generateMeasurableRec s _\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8<a>default</a>, <a>empty_mem_generateMeasurableRec</a> s _\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"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": "MeasurableSpace.empty_mem_generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [62, 9], "def_end_pos": [62, 40]}]], "state_before": "case h.refine'_1.empty\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2205 \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "no goals"}, {"tactic": "rcases mem_iUnion.1 IH with \u27e8i, hi\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion</a>.1 IH with \u27e8i, hi\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 h.refine'_1.compl\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\na\u271d : GenerateMeasurable s u\nIH : u \u2208 \u22c3 i, generateMeasurableRec s i\n\u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1.compl.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\na\u271d : GenerateMeasurable s u\nIH : u \u2208 \u22c3 i, generateMeasurableRec s i\ni : (Quotient.out (ord (aleph 1))).\u03b1\nhi : u \u2208 generateMeasurableRec s i\n\u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "obtain \u27e8j, hj\u27e9 := exists_gt i", "annotated_tactic": ["obtain \u27e8j, hj\u27e9 := <a>exists_gt</a> i", [{"full_name": "NoMaxOrder.exists_gt", "def_path": "Mathlib/Order/Max.lean", "def_pos": [61, 3], "def_end_pos": [61, 12]}]], "state_before": "case h.refine'_1.compl.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\na\u271d : GenerateMeasurable s u\nIH : u \u2208 \u22c3 i, generateMeasurableRec s i\ni : (Quotient.out (ord (aleph 1))).\u03b1\nhi : u \u2208 generateMeasurableRec s i\n\u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1.compl.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\na\u271d : GenerateMeasurable s u\nIH : u \u2208 \u22c3 i, generateMeasurableRec s i\ni : (Quotient.out (ord (aleph 1))).\u03b1\nhi : u \u2208 generateMeasurableRec s i\nj : (Quotient.out (ord (aleph 1))).\u03b1\nhj : i < j\n\u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "exact mem_iUnion.2 \u27e8j, compl_mem_generateMeasurableRec hj hi\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8j, <a>compl_mem_generateMeasurableRec</a> hj 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": "MeasurableSpace.compl_mem_generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [68, 9], "def_end_pos": [68, 40]}]], "state_before": "case h.refine'_1.compl.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nu : Set \u03b1\na\u271d : GenerateMeasurable s u\nIH : u \u2208 \u22c3 i, generateMeasurableRec s i\ni : (Quotient.out (ord (aleph 1))).\u03b1\nhi : u \u2208 generateMeasurableRec s i\nj : (Quotient.out (ord (aleph 1))).\u03b1\nhj : i < j\n\u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "no goals"}, {"tactic": "have : \u2200 n, \u2203 i, f n \u2208 generateMeasurableRec s i := fun n => by simpa using IH n", "annotated_tactic": ["have : \u2200 n, \u2203 i, f n \u2208 <a>generateMeasurableRec</a> s i := fun n => by simpa using IH n", [{"full_name": "MeasurableSpace.generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [47, 5], "def_end_pos": [47, 26]}]], "state_before": "case h.refine'_1.iUnion\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\n\u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1.iUnion\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nthis : \u2200 (n : \u2115), \u2203 i, f n \u2208 generateMeasurableRec s i\n\u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "choose I hI using this", "annotated_tactic": ["choose I hI using this", []], "state_before": "case h.refine'_1.iUnion\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nthis : \u2200 (n : \u2115), \u2203 i, f n \u2208 generateMeasurableRec s i\n\u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i", "state_after": "case h.refine'_1.iUnion\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\n\u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i"}, {"tactic": "simpa using IH n", "annotated_tactic": ["simpa using IH n", []], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nn : \u2115\n\u22a2 \u2203 i, f n \u2208 generateMeasurableRec s i", "state_after": "no goals"}, {"tactic": "rw [Ordinal.type_lt]", "annotated_tactic": ["rw [<a>Ordinal.type_lt</a>]", [{"full_name": "Ordinal.type_lt", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 16]}]], "state_before": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < Ordinal.type fun x x_1 => x < x_1", "state_after": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < ord (aleph 1)"}, {"tactic": "refine' Ordinal.lsub_lt_ord_lift _ fun i => Ordinal.typein_lt_self _", "annotated_tactic": ["refine' <a>Ordinal.lsub_lt_ord_lift</a> _ fun i => <a>Ordinal.typein_lt_self</a> _", [{"full_name": "Ordinal.lsub_lt_ord_lift", "def_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "def_pos": [327, 9], "def_end_pos": [327, 25]}, {"full_name": "Ordinal.typein_lt_self", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [444, 9], "def_end_pos": [444, 23]}]], "state_before": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < ord (aleph 1)", "state_after": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 lift.{u, 0} #\u2115 < Ordinal.cof (ord (aleph 1))"}, {"tactic": "rw [mk_denumerable, lift_aleph0, isRegular_aleph_one.cof_eq]", "annotated_tactic": ["rw [<a>mk_denumerable</a>, <a>lift_aleph0</a>, isRegular_aleph_one.cof_eq]", [{"full_name": "Cardinal.mk_denumerable", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1635, 9], "def_end_pos": [1635, 23]}, {"full_name": "Cardinal.lift_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1256, 9], "def_end_pos": [1256, 20]}]], "state_before": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 lift.{u, 0} #\u2115 < Ordinal.cof (ord (aleph 1))", "state_after": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 \u2135\u2080 < aleph 1"}, {"tactic": "exact aleph0_lt_aleph_one", "annotated_tactic": ["exact <a>aleph0_lt_aleph_one</a>", [{"full_name": "Cardinal.aleph0_lt_aleph_one", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [345, 9], "def_end_pos": [345, 28]}]], "state_before": "case h.refine'_1.iUnion.refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\n\u22a2 \u2135\u2080 < aleph 1", "state_after": "no goals"}, {"tactic": "apply Ordinal.lt_lsub fun n : \u2115 => _", "annotated_tactic": ["apply <a>Ordinal.lt_lsub</a> fun n : \u2115 => _", [{"full_name": "Ordinal.lt_lsub", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1596, 9], "def_end_pos": [1596, 16]}]], "state_before": "case h.refine'_1.iUnion.refine'_2\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ninhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1\nf : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n)\nIH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i\nI : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1\nhI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n)\nthis : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1\nn : \u2115\n\u22a2 Ordinal.typein (fun x x_1 => x < x_1) (I n) < Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)", "state_after": "no goals"}, {"tactic": "rcases ht with \u27e8t, \u27e8i, rfl\u27e9, hx\u27e9", "annotated_tactic": ["rcases ht with \u27e8t, \u27e8i, rfl\u27e9, hx\u27e9", []], "state_before": "case h.refine'_2\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\nht : t \u2208 \u22c3 i, generateMeasurableRec s i\n\u22a2 t \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ni : (Quotient.out (ord (aleph 1))).\u03b1\nhx : t \u2208 (fun i => generateMeasurableRec s i) i\n\u22a2 t \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "revert t", "annotated_tactic": ["revert t", []], "state_before": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nt : Set \u03b1\ni : (Quotient.out (ord (aleph 1))).\u03b1\nhx : t \u2208 (fun i => generateMeasurableRec s i) i\n\u22a2 t \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) i \u2192 t \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "apply (aleph 1).ord.out.wo.wf.induction i", "annotated_tactic": ["apply (<a>aleph</a> 1).ord.out.wo.wf.induction i", [{"full_name": "Cardinal.aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [245, 5], "def_end_pos": [245, 10]}]], "state_before": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) i \u2192 t \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2200 (x : (Quotient.out (ord (aleph 1))).\u03b1),\n    (\u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n        WellOrder.r (Quotient.out (ord (aleph 1))) y x \u2192\n          \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}) \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) x \u2192 t \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "intro j H t ht", "annotated_tactic": ["intro j H t ht", []], "state_before": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2200 (x : (Quotient.out (ord (aleph 1))).\u03b1),\n    (\u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n        WellOrder.r (Quotient.out (ord (aleph 1))) y x \u2192\n          \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}) \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) x \u2192 t \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nt : Set \u03b1\nht : t \u2208 (fun i => generateMeasurableRec s i) j\n\u22a2 t \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "unfold generateMeasurableRec at ht", "annotated_tactic": ["unfold <a>generateMeasurableRec</a> at ht", [{"full_name": "MeasurableSpace.generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [47, 5], "def_end_pos": [47, 26]}]], "state_before": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nt : Set \u03b1\nht : t \u2208 (fun i => generateMeasurableRec s i) j\n\u22a2 t \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nt : Set \u03b1\nht :\n  t \u2208\n    let i := j;\n    let S := \u22c3 j, generateMeasurableRec s \u2191j;\n    s \u222a {\u2205} \u222a compl '' S \u222a range fun f => \u22c3 n, \u2191(f n)\n\u22a2 t \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "rcases ht with (((h | (rfl : t = \u2205)) | \u27e8u, \u27e8-, \u27e8\u27e8k, hk\u27e9, rfl\u27e9, hu\u27e9, rfl\u27e9) | \u27e8f, rfl\u27e9)", "annotated_tactic": ["rcases ht with (((h | (rfl : t = \u2205)) | \u27e8u, \u27e8-, \u27e8\u27e8k, hk\u27e9, rfl\u27e9, hu\u27e9, rfl\u27e9) | \u27e8f, rfl\u27e9)", []], "state_before": "case h.refine'_2.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nt : Set \u03b1\nht :\n  t \u2208\n    let i := j;\n    let S := \u22c3 j, generateMeasurableRec s \u2191j;\n    s \u222a {\u2205} \u222a compl '' S \u222a range fun f => \u22c3 n, \u2191(f n)\n\u22a2 t \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro.inl.inl.inl\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nt : Set \u03b1\nh : t \u2208 s\n\u22a2 t \u2208 {t | GenerateMeasurable s t}\n\ncase h.refine'_2.intro.intro.intro.inl.inl.inr\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\n\u22a2 \u2205 \u2208 {t | GenerateMeasurable s t}\n\ncase h.refine'_2.intro.intro.intro.inl.inr.intro.intro.intro.intro.intro.mk\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nu : Set \u03b1\nk : (Quotient.out (ord (aleph 1))).\u03b1\nhk : k \u2208 Iio j\nhu : u \u2208 (fun j_1 => generateMeasurableRec s \u2191j_1) { val := k, property := hk }\n\u22a2 u\u1d9c \u2208 {t | GenerateMeasurable s t}\n\ncase h.refine'_2.intro.intro.intro.inr.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nf : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1)\n\u22a2 (fun f => \u22c3 n, \u2191(f n)) f \u2208 {t | GenerateMeasurable s t}"}, {"tactic": "exact .basic t h", "annotated_tactic": ["exact .basic t h", []], "state_before": "case h.refine'_2.intro.intro.intro.inl.inl.inl\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nt : Set \u03b1\nh : t \u2208 s\n\u22a2 t \u2208 {t | GenerateMeasurable s t}", "state_after": "no goals"}, {"tactic": "exact .empty", "annotated_tactic": ["exact .empty", []], "state_before": "case h.refine'_2.intro.intro.intro.inl.inl.inr\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\n\u22a2 \u2205 \u2208 {t | GenerateMeasurable s t}", "state_after": "no goals"}, {"tactic": "exact .compl u (H k hk u hu)", "annotated_tactic": ["exact .compl u (H k hk u hu)", []], "state_before": "case h.refine'_2.intro.intro.intro.inl.inr.intro.intro.intro.intro.intro.mk\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nu : Set \u03b1\nk : (Quotient.out (ord (aleph 1))).\u03b1\nhk : k \u2208 Iio j\nhu : u \u2208 (fun j_1 => generateMeasurableRec s \u2191j_1) { val := k, property := hk }\n\u22a2 u\u1d9c \u2208 {t | GenerateMeasurable s t}", "state_after": "no goals"}, {"tactic": "refine .iUnion _ @fun n => ?_", "annotated_tactic": ["refine .iUnion _ @fun n => ?_", []], "state_before": "case h.refine'_2.intro.intro.intro.inr.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nf : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1)\n\u22a2 (fun f => \u22c3 n, \u2191(f n)) f \u2208 {t | GenerateMeasurable s t}", "state_after": "case h.refine'_2.intro.intro.intro.inr.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nf : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1)\nn : \u2115\n\u22a2 GenerateMeasurable s \u2191(f n)"}, {"tactic": "obtain \u27e8-, \u27e8\u27e8k, hk\u27e9, rfl\u27e9, hf\u27e9 := (f n).prop", "annotated_tactic": ["obtain \u27e8-, \u27e8\u27e8k, hk\u27e9, rfl\u27e9, hf\u27e9 := (f n).<a>prop</a>", [{"full_name": "Subtype.prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [38, 9], "def_end_pos": [38, 13]}]], "state_before": "case h.refine'_2.intro.intro.intro.inr.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nf : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1)\nn : \u2115\n\u22a2 GenerateMeasurable s \u2191(f n)", "state_after": "case h.refine'_2.intro.intro.intro.inr.intro.intro.intro.intro.mk\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nf : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1)\nn : \u2115\nk : (Quotient.out (ord (aleph 1))).\u03b1\nhk : k \u2208 Iio j\nhf : \u2191(f n) \u2208 (fun j_1 => generateMeasurableRec s \u2191j_1) { val := k, property := hk }\n\u22a2 GenerateMeasurable s \u2191(f n)"}, {"tactic": "exact H k hk _ hf", "annotated_tactic": ["exact H k hk _ hf", []], "state_before": "case h.refine'_2.intro.intro.intro.inr.intro.intro.intro.intro.mk\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni j : (Quotient.out (ord (aleph 1))).\u03b1\nH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192\n      \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}\nf : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1)\nn : \u2115\nk : (Quotient.out (ord (aleph 1))).\u03b1\nhk : k \u2208 Iio j\nhf : \u2191(f n) \u2208 (fun j_1 => generateMeasurableRec s \u2191j_1) { val := k, property := hk }\n\u22a2 GenerateMeasurable s \u2191(f n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Set.range_IicExtend", "start": [224, 1], "end": [225, 76], "traced_tactics": [{"tactic": "simp only [IicExtend, range_comp f, range_projIic, range_id', image_univ]", "annotated_tactic": ["simp only [<a>IicExtend</a>, <a>range_comp</a> f, <a>range_projIic</a>, <a>range_id'</a>, <a>image_univ</a>]", [{"full_name": "Set.IicExtend", "def_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "def_pos": [197, 5], "def_end_pos": [197, 14]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "Set.range_projIic", "def_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "def_pos": [161, 9], "def_end_pos": [161, 22]}, {"full_name": "Set.range_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [877, 9], "def_end_pos": [877, 18]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u2191(Iic b) \u2192 \u03b2\n\u22a2 range (IicExtend f) = range f", "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_Icc_of_neg", "start": [680, 1], "end": [682, 66], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Icc_of_neg a b h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Icc_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_Icc_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [593, 9], "def_end_pos": [593, 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' Icc a b = Icc (b / c) (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/SurjOn.lean", "full_name": "surjOn_Ici_of_monotone_surjective", "start": [75, 1], "end": [80, 52], "traced_tactics": [{"tactic": "rw [\u2190 Ioi_union_left, \u2190 Ioi_union_left]", "annotated_tactic": ["rw [\u2190 <a>Ioi_union_left</a>, \u2190 <a>Ioi_union_left</a>]", [{"full_name": "Set.Ioi_union_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [855, 9], "def_end_pos": [855, 23]}, {"full_name": "Set.Ioi_union_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [855, 9], "def_end_pos": [855, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\nh_surj : Surjective f\na : \u03b1\n\u22a2 SurjOn f (Ici a) (Ici (f a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\nh_surj : Surjective f\na : \u03b1\n\u22a2 SurjOn f (Ioi a \u222a {a}) (Ioi (f a) \u222a {f a})"}, {"tactic": "exact\n  (surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union\n    (@image_singleton _ _ f a \u25b8 surjOn_image _ _)", "annotated_tactic": ["exact\n    (<a>surjOn_Ioi_of_monotone_surjective</a> h_mono h_surj a).<a>union_union</a>\n      (@<a>image_singleton</a> _ _ f a \u25b8 <a>surjOn_image</a> _ _)", [{"full_name": "surjOn_Ioi_of_monotone_surjective", "def_path": "Mathlib/Data/Set/Intervals/SurjOn.lean", "def_pos": [63, 9], "def_end_pos": [63, 42]}, {"full_name": "Set.SurjOn.union_union", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [815, 9], "def_end_pos": [815, 27]}, {"full_name": "Set.image_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [363, 9], "def_end_pos": [363, 24]}, {"full_name": "Set.surjOn_image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [791, 9], "def_end_pos": [791, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\nh_surj : Surjective f\na : \u03b1\n\u22a2 SurjOn f (Ioi a \u222a {a}) (Ioi (f a) \u222a {f a})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.cast_pos", "start": [681, 1], "end": [682, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.predictablePart_bdd_difference", "start": [167, 1], "end": [171, 78], "traced_tactics": [{"tactic": "simp_rw [predictablePart, Finset.sum_apply, Finset.sum_range_succ_sub_sum]", "annotated_tactic": ["simp_rw [<a>predictablePart</a>, <a>Finset.sum_apply</a>, <a>Finset.sum_range_succ_sub_sum</a>]", [{"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"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_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\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), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R", "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\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), |(\u03bc[f (i + 1) - f i|\u2191\u2131 i]) \u03c9| \u2264 \u2191R"}, {"tactic": "exact ae_all_iff.2 fun i => ae_bdd_condexp_of_ae_bdd <| ae_all_iff.1 hbdd i", "annotated_tactic": ["exact <a>ae_all_iff</a>.2 fun i => <a>ae_bdd_condexp_of_ae_bdd</a> <| <a>ae_all_iff</a>.1 hbdd i", [{"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_bdd_condexp_of_ae_bdd", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [146, 9], "def_end_pos": [146, 33]}, {"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "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), |(\u03bc[f (i + 1) - f i|\u2191\u2131 i]) \u03c9| \u2264 \u2191R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.integral_truncation_eq_intervalIntegral", "start": [180, 1], "end": [182, 70], "traced_tactics": [{"tactic": "simpa using moment_truncation_eq_intervalIntegral hf hA one_ne_zero", "annotated_tactic": ["simpa using <a>moment_truncation_eq_intervalIntegral</a> hf hA <a>one_ne_zero</a>", [{"full_name": "ProbabilityTheory.moment_truncation_eq_intervalIntegral", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [142, 9], "def_end_pos": [142, 46]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\n\u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc = \u222b (y : \u211d) in -A..A, y \u2202Measure.map f \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.integrable_kernel_prod_mk_left", "start": [64, 1], "end": [68, 47], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "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)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\n\u22a2 Integrable fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))", "state_after": "case left\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)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\n\u22a2 AEStronglyMeasurable (fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) (\u2191\u03ba a)\n\ncase right\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)\nhs : MeasurableSet s\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))"}, {"tactic": "exact (measurable_kernel_prod_mk_left' hs a).ennreal_toReal.aestronglyMeasurable", "annotated_tactic": ["exact (<a>measurable_kernel_prod_mk_left'</a> hs a).ennreal_toReal.aestronglyMeasurable", [{"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left'", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [111, 9], "def_end_pos": [111, 40]}]], "state_before": "case left\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)\nhs : MeasurableSet s\nh2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4\n\u22a2 AEStronglyMeasurable (fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) (\u2191\u03ba a)", "state_after": "no goals"}, {"tactic": "exact hasFiniteIntegral_prod_mk_left a h2s", "annotated_tactic": ["exact <a>hasFiniteIntegral_prod_mk_left</a> a h2s", [{"full_name": "ProbabilityTheory.hasFiniteIntegral_prod_mk_left", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [48, 9], "def_end_pos": [48, 39]}]], "state_before": "case right\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)\nhs : MeasurableSet s\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": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.negOfNat_mul_negSucc", "start": [392, 1], "end": [393, 56], "traced_tactics": [{"tactic": "rw [Int.mul_comm, negSucc_mul_negOfNat, Nat.mul_comm]", "annotated_tactic": ["rw [<a>Int.mul_comm</a>, <a>negSucc_mul_negOfNat</a>, <a>Nat.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]}, {"full_name": "Int.negSucc_mul_negOfNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [389, 9], "def_end_pos": [389, 29]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "m n : Nat\n\u22a2 negOfNat n * -[m+1] = ofNat (n * succ 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_aux4", "start": [562, 1], "end": [570, 7], "traced_tactics": [{"tactic": "filter_upwards [strong_law_aux2 X hint hindep hident hnonneg c_one] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [<a>strong_law_aux2</a> X hint hindep hident hnonneg c_one] with \u03c9 h\u03c9", [{"full_name": "ProbabilityTheory.strong_law_aux2", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [528, 9], "def_end_pos": [528, 24]}]], "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\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n =>\n      \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\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n  tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)", "annotated_tactic": ["have A : <a>Tendsto</a> (fun n : \u2115 => \u230ac ^ n\u230b\u208a) <a>atTop</a> <a>atTop</a> :=\n    tendsto_nat_floor_atTop.comp (<a>tendsto_pow_atTop_atTop_of_one_lt</a> c_one)", [{"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": "tendsto_pow_atTop_atTop_of_one_lt", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 42]}]], "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\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \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\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "convert h\u03c9.add ((strong_law_aux3 X hint hident).comp_tendsto A) using 1", "annotated_tactic": ["convert h\u03c9.add ((<a>strong_law_aux3</a> X hint hident).<a>comp_tendsto</a> A) using 1", [{"full_name": "ProbabilityTheory.strong_law_aux3", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [545, 9], "def_end_pos": [545, 24]}, {"full_name": "Asymptotics.IsLittleO.comp_tendsto", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [460, 9], "def_end_pos": [460, 31]}]], "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\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a", "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\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>\n    (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n        \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a) +\n      ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>\n          \u230ac ^ n\u230b\u208a)\n        x"}, {"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\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>\n    (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n        \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a) +\n      ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>\n          \u230ac ^ n\u230b\u208a)\n        x", "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\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\nn : \u2115\n\u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =\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      ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>\n          \u230ac ^ n\u230b\u208a)\n        n"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\nn : \u2115\n\u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =\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      ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>\n          \u230ac ^ n\u230b\u208a)\n        n", "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": [802, 1], "end": [811, 37], "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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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\nhp1 : 1 \u2264 p\nhp0 : p = 0\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g 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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc"}, {"tactic": "have hp1_real : 1 \u2264 p.toReal := by\n  rwa [\u2190 ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]", "annotated_tactic": ["have hp1_real : 1 \u2264 p.toReal := by\n    rwa [\u2190 <a>ENNReal.one_toReal</a>, <a>ENNReal.toReal_le_toReal</a> <a>ENNReal.one_ne_top</a> hp_top]", [{"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_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"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\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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc"}, {"tactic": "repeat rw [snorm_eq_snorm' hp0 hp_top]", "annotated_tactic": ["repeat 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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc"}, {"tactic": "exact snorm'_add_le hf hg hp1_real", "annotated_tactic": ["exact <a>snorm'_add_le</a> hf hg hp1_real", [{"full_name": "MeasureTheory.snorm'_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [772, 9], "def_end_pos": [772, 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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm' g (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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : p = 0\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc", "state_after": "no goals"}, {"tactic": "simp [hp_top, snormEssSup_add_le]", "annotated_tactic": ["simp [hp_top, <a>snormEssSup_add_le</a>]", [{"full_name": "MeasureTheory.snormEssSup_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [795, 9], "def_end_pos": [795, 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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc", "state_after": "no goals"}, {"tactic": "rwa [\u2190 ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]", "annotated_tactic": ["rwa [\u2190 <a>ENNReal.one_toReal</a>, <a>ENNReal.toReal_le_toReal</a> <a>ENNReal.one_ne_top</a> hp_top]", [{"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_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"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\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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 1 \u2264 ENNReal.toReal p", "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 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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \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 : \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\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_inj", "start": [1204, 1], "end": [1205, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.comp_quasiMeasurePreserving", "start": [169, 1], "end": [171, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.natSize_to_nat", "start": [760, 1], "end": [760, 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": "Num.size_eq_natSize", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [755, 9], "def_end_pos": [755, 24]}, {"full_name": "Num.size_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [750, 9], "def_end_pos": [750, 20]}]], "state_before": "\u03b1 : Type u_1\nn : Num\n\u22a2 natSize n = Nat.size \u2191n", "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_of_mem_toList", "start": [632, 1], "end": [633, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEmbedding.measurable_extend", "start": [1231, 1], "end": [1237, 42], "traced_tactics": [{"tactic": "refine' measurable_of_restrict_of_restrict_compl hf.measurableSet_range _ _", "annotated_tactic": ["refine' <a>measurable_of_restrict_of_restrict_compl</a> hf.measurableSet_range _ _", [{"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": "\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\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (extend f g g')", "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (restrict (range f) (extend f g g'))\n\ncase refine'_2\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\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (restrict (range f)\u1d9c (extend f g g'))"}, {"tactic": "rw [restrict_extend_range]", "annotated_tactic": ["rw [<a>restrict_extend_range</a>]", [{"full_name": "Set.restrict_extend_range", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [117, 9], "def_end_pos": [117, 30]}]], "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (restrict (range f) (extend f g g'))", "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable fun x => g (Exists.choose (_ : \u2191x \u2208 range f))"}, {"tactic": "simpa only [rangeSplitting] using hg.comp hf.measurable_rangeSplitting", "annotated_tactic": ["simpa only [<a>rangeSplitting</a>] using hg.comp hf.measurable_rangeSplitting", [{"full_name": "Set.rangeSplitting", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1160, 19], "def_end_pos": [1160, 33]}]], "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable fun x => g (Exists.choose (_ : \u2191x \u2208 range f))", "state_after": "no goals"}, {"tactic": "rw [restrict_extend_compl_range]", "annotated_tactic": ["rw [<a>restrict_extend_compl_range</a>]", [{"full_name": "Set.restrict_extend_compl_range", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [124, 9], "def_end_pos": [124, 36]}]], "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (restrict (range f)\u1d9c (extend f g g'))", "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (g' \u2218 Subtype.val)"}, {"tactic": "exact hg'.comp measurable_subtype_coe", "annotated_tactic": ["exact hg'.comp <a>measurable_subtype_coe</a>", [{"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}]], "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (g' \u2218 Subtype.val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.coe_eq_pair", "start": [1155, 1], "end": [1156, 27], "traced_tactics": [{"tactic": "rw [\u2190 coe_pair, coe_inj]", "annotated_tactic": ["rw [\u2190 <a>coe_pair</a>, <a>coe_inj</a>]", [{"full_name": "Finset.coe_pair", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1149, 9], "def_end_pos": [1149, 17]}, {"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}]], "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 : \u03b1\ns : Finset \u03b1\na b : \u03b1\n\u22a2 \u2191s = {a, b} \u2194 s = {a, b}", "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_Iic_rat", "start": [1881, 1], "end": [1887, 76], "traced_tactics": [{"tactic": "rw [borel_eq_generateFrom_Ioi_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]", "annotated_tactic": ["rw [<a>borel_eq_generateFrom_Ioi_rat</a>, <a>iUnion_singleton_eq_range</a>, <a>iUnion_singleton_eq_range</a>]", [{"full_name": "Real.borel_eq_generateFrom_Ioi_rat", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 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, {Iic \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 => Ioi \u2191a) = MeasurableSpace.generateFrom (range fun a => Iic \u2191a)"}, {"tactic": "refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>\nrintro _ \u27e8q, rfl\u27e9 <;>\ndsimp only <;>\n[rw [\u2190 compl_Iic]; rw [\u2190 compl_Ioi]] <;>\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_Iic</a>]; rw [\u2190 <a>compl_Ioi</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_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1074, 9], "def_end_pos": [1074, 18]}, {"full_name": "Set.compl_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1089, 9], "def_end_pos": [1089, 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 => Ioi \u2191a) = MeasurableSpace.generateFrom (range fun a => Iic \u2191a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.nat_cast_val", "start": [260, 1], "end": [261, 36], "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.insert_size", "start": [174, 1], "end": [187, 50], "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 : m.size = Buckets.size m.buckets\n\u22a2 (insert m k v).size = Buckets.size (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 : 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,\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))).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,\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 : 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,\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))).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,\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 : 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,\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) }.size =\n    Buckets.size\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 : 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 (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))).size =\n    Buckets.size\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": "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\nv : \u03b2\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 (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))).size =\n    Buckets.size\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_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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) }.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.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\n\ncase h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\n\u22a2 (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))).size =\n    Buckets.size\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": "unfold Buckets.size", "annotated_tactic": ["unfold <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 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 : 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,\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) }.size =\n    Buckets.size\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 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 : 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,\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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\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\n                          (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.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": "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 : 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,\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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\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\n                          (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.val.data)", "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 : 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\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.replace k v\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 { 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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (w\u271d\u00b9 ++\n          AssocList.replace k v\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": "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 : 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\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.replace k v\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\n\u22a2 { 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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (w\u271d\u00b9 ++\n          AssocList.replace k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n            w\u271d))", "state_after": "no goals"}, {"tactic": "unfold Buckets.size", "annotated_tactic": ["unfold <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 h_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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) }.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.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_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\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\n                          (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.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": "case h_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\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\n                          (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.val.data)", "state_after": "case h_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\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.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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (w\u271d\u00b9 ++\n          AssocList.cons k v\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, Nat.succ_add]", "annotated_tactic": ["simp [h, h\u2081, <a>Buckets.size_eq</a>, <a>Nat.succ_add</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]}, {"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 h_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\n          m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n        w\u271d\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.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) }.size =\n    Nat.sum\n      (List.map (fun x => List.length (AssocList.toList x))\n        (w\u271d\u00b9 ++\n          AssocList.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n            w\u271d))", "state_after": "case h_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\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      Nat.succ\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))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h_2.inl\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 : 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\nh\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\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      Nat.succ\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))", "state_after": "no goals"}, {"tactic": "rw [expand_size]", "annotated_tactic": ["rw [<a>expand_size</a>]", [{"full_name": "Std.HashMap.Imp.expand_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [79, 9], "def_end_pos": [79, 20]}]], "state_before": "case h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\n\u22a2 (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))).size =\n    Buckets.size\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_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\n\u22a2 (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))).size =\n    Buckets.size\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))"}, {"tactic": "simp [h, expand, Buckets.size]", "annotated_tactic": ["simp [h, <a>expand</a>, <a>Buckets.size</a>]", [{"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]}, {"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 h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\n\u22a2 (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))).size =\n    Buckets.size\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))", "state_after": "case h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).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": "case h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data)", "state_after": "case h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\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.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n            w\u271d))"}, {"tactic": "simp [h\u2081, Buckets.size_eq, Nat.succ_add]", "annotated_tactic": ["simp [h\u2081, <a>Buckets.size_eq</a>, <a>Nat.succ_add</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]}, {"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 h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\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.cons k v\n              m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n            w\u271d))", "state_after": "case h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\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      Nat.succ\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))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h_2.inr\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 : 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\nh\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val\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.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)).val.data =\n    w\u271d\u00b9 ++\n      AssocList.cons k v\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      Nat.succ\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))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.drop_append1'", "start": [118, 1], "end": [119, 34], "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": [200, 1], "end": [209, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.fmod_nonneg", "start": [373, 1], "end": [374, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.tendsto_measure_Ioc_atBot", "start": [2768, 1], "end": [2784, 79], "traced_tactics": [{"tactic": "haveI : Nonempty \u03b1 := \u27e8a\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> \u03b1 := \u27e8a\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": "\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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))", "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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))"}, {"tactic": "have h_mono : Antitone fun x => \u03bc (Ioc x a) := fun i j hij =>\n  measure_mono (Ioc_subset_Ioc_left hij)", "annotated_tactic": ["have h_mono : <a>Antitone</a> fun x => \u03bc (<a>Ioc</a> x a) := fun i j hij =>\n    <a>measure_mono</a> (<a>Ioc_subset_Ioc_left</a> hij)", [{"full_name": "Antitone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [82, 5], "def_end_pos": [82, 13]}, {"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_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.Ioc_subset_Ioc_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [484, 9], "def_end_pos": [484, 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\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))", "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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))"}, {"tactic": "convert tendsto_atBot_iSup h_mono", "annotated_tactic": ["convert <a>tendsto_atBot_iSup</a> h_mono", [{"full_name": "tendsto_atBot_iSup", "def_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "def_pos": [160, 9], "def_end_pos": [160, 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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))", "state_after": "case h.e'_5.h.e'_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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a)"}, {"tactic": "obtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := exists_seq_antitone_tendsto_atTop_atBot \u03b1", "annotated_tactic": ["obtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := <a>exists_seq_antitone_tendsto_atTop_atBot</a> \u03b1", [{"full_name": "Filter.exists_seq_antitone_tendsto_atTop_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 48]}]], "state_before": "case h.e'_5.h.e'_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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a)", "state_after": "case h.e'_5.h.e'_3.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a)"}, {"tactic": "have h_Iic : Iic a = \u22c3 n, Ioc (xs n) a := by\n  ext1 x\n  simp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self]\n  intro\n  obtain \u27e8y, hxy\u27e9 := NoMinOrder.exists_lt x\n  obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y\n  exact \u27e8n, (hn n le_rfl).trans_lt hxy\u27e9", "annotated_tactic": ["have h_Iic : <a>Iic</a> a = \u22c3 n, <a>Ioc</a> (xs n) a := by\n    ext1 x\n    simp only [<a>mem_Iic</a>, <a>mem_iUnion</a>, <a>mem_Ioc</a>, <a>exists_and_right</a>, <a>iff_and_self</a>]\n    intro\n    obtain \u27e8y, hxy\u27e9 := <a>NoMinOrder.exists_lt</a> x\n    obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y\n    exact \u27e8n, (hn n <a>le_rfl</a>).<a>trans_lt</a> hxy\u27e9", [{"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 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.mem_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 16]}, {"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_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "iff_and_self", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [217, 17], "def_end_pos": [217, 29]}, {"full_name": "NoMinOrder.exists_lt", "def_path": "Mathlib/Order/Max.lean", "def_pos": [55, 3], "def_end_pos": [55, 12]}, {"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": "case h.e'_5.h.e'_3.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a)", "state_after": "case h.e'_5.h.e'_3.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nh_Iic : Iic a = \u22c3 n, Ioc (xs n) a\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a)"}, {"tactic": "rw [h_Iic, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_antitone h_mono hxs_tendsto]", "annotated_tactic": ["rw [h_Iic, <a>measure_iUnion_eq_iSup</a>, <a>iSup_eq_iSup_subseq_of_antitone</a> h_mono hxs_tendsto]", [{"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": "iSup_eq_iSup_subseq_of_antitone", "def_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "def_pos": [322, 9], "def_end_pos": [322, 40]}]], "state_before": "case h.e'_5.h.e'_3.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nh_Iic : Iic a = \u22c3 n, Ioc (xs n) a\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a)", "state_after": "case h.e'_5.h.e'_3.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nh_Iic : Iic a = \u22c3 n, Ioc (xs n) a\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Ioc (xs n) a"}, {"tactic": "exact Monotone.directed_le fun i j hij => Ioc_subset_Ioc_left (hxs_mono hij)", "annotated_tactic": ["exact <a>Monotone.directed_le</a> fun i j hij => <a>Ioc_subset_Ioc_left</a> (hxs_mono hij)", [{"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [112, 9], "def_end_pos": [112, 29]}, {"full_name": "Set.Ioc_subset_Ioc_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [484, 9], "def_end_pos": [484, 28]}]], "state_before": "case h.e'_5.h.e'_3.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nh_Iic : Iic a = \u22c3 n, Ioc (xs n) a\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Ioc (xs n) a", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\n\u22a2 Iic a = \u22c3 n, Ioc (xs n) 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\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\n\u22a2 x \u2208 Iic a \u2194 x \u2208 \u22c3 n, Ioc (xs n) a"}, {"tactic": "simp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self]", "annotated_tactic": ["simp only [<a>mem_Iic</a>, <a>mem_iUnion</a>, <a>mem_Ioc</a>, <a>exists_and_right</a>, <a>iff_and_self</a>]", [{"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_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"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 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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\n\u22a2 x \u2208 Iic a \u2194 x \u2208 \u22c3 n, Ioc (xs n) 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\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\n\u22a2 x \u2264 a \u2192 \u2203 x_1, xs x_1 < x"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\n\u22a2 x \u2264 a \u2192 \u2203 x_1, xs x_1 < 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\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\n\u22a2 \u2203 x_1, xs x_1 < x"}, {"tactic": "obtain \u27e8y, hxy\u27e9 := NoMinOrder.exists_lt x", "annotated_tactic": ["obtain \u27e8y, hxy\u27e9 := <a>NoMinOrder.exists_lt</a> x", [{"full_name": "NoMinOrder.exists_lt", "def_path": "Mathlib/Order/Max.lean", "def_pos": [55, 3], "def_end_pos": [55, 12]}]], "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\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\n\u22a2 \u2203 x_1, xs x_1 < x", "state_after": "case h.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\ny : \u03b1\nhxy : y < x\n\u22a2 \u2203 x_1, xs x_1 < x"}, {"tactic": "obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y", []], "state_before": "case h.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\ny : \u03b1\nhxy : y < x\n\u22a2 \u2203 x_1, xs x_1 < x", "state_after": "case h.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\ny : \u03b1\nhxy : y < x\nn : \u2115\nhn : \u2200 (a : \u2115), n \u2264 a \u2192 xs a \u2264 y\n\u22a2 \u2203 x_1, xs x_1 < x"}, {"tactic": "exact \u27e8n, (hn n le_rfl).trans_lt hxy\u27e9", "annotated_tactic": ["exact \u27e8n, (hn n <a>le_rfl</a>).<a>trans_lt</a> hxy\u27e9", [{"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": "case h.intro.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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\ny : \u03b1\nhxy : y < x\nn : \u2115\nhn : \u2200 (a : \u2115), n \u2264 a \u2192 xs a \u2264 y\n\u22a2 \u2203 x_1, xs x_1 < x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quot.liftOn\u2082_mk", "start": [139, 1], "end": [142, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.pi_eval_preimage_null", "start": [427, 1], "end": [437, 13], "traced_tactics": [{"tactic": "rcases exists_measurable_superset_of_null hs with \u27e8t, hst, _, h\u03bct\u27e9", "annotated_tactic": ["rcases <a>exists_measurable_superset_of_null</a> hs with \u27e8t, hst, _, h\u03bct\u27e9", [{"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": "\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0", "state_after": "case intro.intro.intro\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0"}, {"tactic": "suffices : Measure.pi \u03bc (eval i \u207b\u00b9' t) = 0", "annotated_tactic": ["suffices : <a>Measure.pi</a> \u03bc (<a>eval</a> i \u207b\u00b9' t) = 0", [{"full_name": "MeasureTheory.Measure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [303, 27], "def_end_pos": [303, 29]}, {"full_name": "Function.eval", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [29, 24], "def_end_pos": [29, 28]}]], "state_before": "case intro.intro.intro\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0", "state_after": "case intro.intro.intro\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\nthis : \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0\n\ncase this\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0"}, {"tactic": "exact measure_mono_null (preimage_mono hst) this", "annotated_tactic": ["exact <a>measure_mono_null</a> (<a>preimage_mono</a> hst) this", [{"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.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}]], "state_before": "case intro.intro.intro\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\nthis : \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0\n\ncase this\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0", "state_after": "case this\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0"}, {"tactic": "clear! s", "annotated_tactic": ["clear! s", []], "state_before": "case this\n\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)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : \u2191\u2191(\u03bc i) s = 0\nt : Set (\u03b1 i)\nhst : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0", "state_after": "case this\n\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)\ni : \u03b9\nt : Set (\u03b1 i)\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0"}, {"tactic": "rw [\u2190 univ_pi_update_univ, pi_pi]", "annotated_tactic": ["rw [\u2190 <a>univ_pi_update_univ</a>, <a>pi_pi</a>]", [{"full_name": "Set.univ_pi_update_univ", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [847, 9], "def_end_pos": [847, 28]}, {"full_name": "MeasureTheory.Measure.pi_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [394, 9], "def_end_pos": [394, 14]}]], "state_before": "case this\n\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)\ni : \u03b9\nt : Set (\u03b1 i)\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0", "state_after": "case this\n\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)\ni : \u03b9\nt : Set (\u03b1 i)\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u220f i_1 : \u03b9, \u2191\u2191(\u03bc i_1) (update (fun j => univ) i t i_1) = 0"}, {"tactic": "apply Finset.prod_eq_zero (Finset.mem_univ i)", "annotated_tactic": ["apply <a>Finset.prod_eq_zero</a> (<a>Finset.mem_univ</a> i)", [{"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\n\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)\ni : \u03b9\nt : Set (\u03b1 i)\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u220f i_1 : \u03b9, \u2191\u2191(\u03bc i_1) (update (fun j => univ) i t i_1) = 0", "state_after": "case this\n\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)\ni : \u03b9\nt : Set (\u03b1 i)\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(\u03bc i) (update (fun j => univ) i t i) = 0"}, {"tactic": "simp [h\u03bct]", "annotated_tactic": ["simp [h\u03bct]", []], "state_before": "case this\n\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)\ni : \u03b9\nt : Set (\u03b1 i)\nleft\u271d : MeasurableSet t\nh\u03bct : \u2191\u2191(\u03bc i) t = 0\n\u22a2 \u2191\u2191(\u03bc i) (update (fun j => univ) i t i) = 0", "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.replicate_sublist_replicate", "start": [484, 9], "end": [490, 48], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, fun h => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, fun h => ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\nm n : Nat\na : \u03b1\n\u22a2 replicate m a <+ replicate n a \u2194 m \u2264 n", "state_after": "case refine_1\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : replicate m a <+ replicate n a\n\u22a2 m \u2264 n\n\ncase refine_2\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : m \u2264 n\n\u22a2 replicate m a <+ replicate n a"}, {"tactic": "have := h.length_le", "annotated_tactic": ["have := h.length_le", []], "state_before": "case refine_1\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : replicate m a <+ replicate n a\n\u22a2 m \u2264 n", "state_after": "case refine_1\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : replicate m a <+ replicate n a\nthis : length (replicate m a) \u2264 length (replicate n a)\n\u22a2 m \u2264 n"}, {"tactic": "simp only [length_replicate] at this \u22a2", "annotated_tactic": ["simp only [<a>length_replicate</a>] at this \u22a2", [{"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]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : replicate m a <+ replicate n a\nthis : length (replicate m a) \u2264 length (replicate n a)\n\u22a2 m \u2264 n", "state_after": "case refine_1\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : replicate m a <+ replicate n a\nthis : m \u2264 n\n\u22a2 m \u2264 n"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "case refine_1\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : replicate m a <+ replicate n a\nthis : m \u2264 n\n\u22a2 m \u2264 n", "state_after": "no goals"}, {"tactic": "induction h with\n| refl => apply Sublist.refl\n| step => simp [*, replicate, Sublist.cons]", "annotated_tactic": ["induction h with\n    | <a>refl</a> => apply <a>Sublist.refl</a>\n    | <a>step</a> => simp [*, <a>replicate</a>, <a>Sublist.cons</a>]", [{"full_name": "Nat.le.refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1557, 5], "def_end_pos": [1557, 9]}, {"full_name": "List.Sublist.refl", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [369, 17], "def_end_pos": [369, 29]}, {"full_name": "Nat.le.step", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1559, 5], "def_end_pos": [1559, 9]}, {"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.Sublist.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [442, 5], "def_end_pos": [442, 9]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nh : m \u2264 n\n\u22a2 replicate m a <+ replicate n a", "state_after": "no goals"}, {"tactic": "apply Sublist.refl", "annotated_tactic": ["apply <a>Sublist.refl</a>", [{"full_name": "List.Sublist.refl", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [369, 17], "def_end_pos": [369, 29]}]], "state_before": "case refine_2.refl\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\n\u22a2 replicate m a <+ replicate m a", "state_after": "no goals"}, {"tactic": "simp [*, replicate, Sublist.cons]", "annotated_tactic": ["simp [*, <a>replicate</a>, <a>Sublist.cons</a>]", [{"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.Sublist.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [442, 5], "def_end_pos": [442, 9]}]], "state_before": "case refine_2.step\n\u03b1 : Type u_1\nm n : Nat\na : \u03b1\nm\u271d : Nat\na\u271d : Nat.le m m\u271d\na_ih\u271d : replicate m a <+ replicate m\u271d a\n\u22a2 replicate m a <+ replicate (succ m\u271d) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.halting_problem", "start": [236, 1], "end": [237, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image_uncurry_product", "start": [343, 1], "end": [344, 87], "traced_tactics": [{"tactic": "rw [\u2190 image\u2082_curry, curry_uncurry]", "annotated_tactic": ["rw [\u2190 <a>image\u2082_curry</a>, <a>curry_uncurry</a>]", [{"full_name": "Finset.image\u2082_curry", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [337, 9], "def_end_pos": [337, 21]}, {"full_name": "Function.curry_uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [221, 9], "def_end_pos": [221, 22]}]], "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\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 image (uncurry f) (s \u00d7\u02e2 t) = image\u2082 f s t", "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.RBSet.ModifyWF.of_eq", "start": [417, 1], "end": [421, 64], "traced_tactics": [{"tactic": "refine \u27e8.modify ?_ t.2\u27e9", "annotated_tactic": ["refine \u27e8.modify ?_ t.2\u27e9", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\nH : \u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x\n\u22a2 ModifyWF t cut f", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\nH : \u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x\n\u22a2 OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst"}, {"tactic": "revert H", "annotated_tactic": ["revert H", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\nH : \u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x\n\u22a2 OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\n\u22a2 (\u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x) \u2192\n    OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst"}, {"tactic": "rw [find?_eq_zoom]", "annotated_tactic": ["rw [<a>find?_eq_zoom</a>]", [{"full_name": "Std.RBNode.find?_eq_zoom", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [405, 9], "def_end_pos": [405, 22]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\n\u22a2 (\u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x) \u2192\n    OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\n\u22a2 (\u2200 {x : \u03b1}, root? (zoom cut t.val Path.root).fst = some x \u2192 cmpEq cmp (f x) x) \u2192\n    OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst"}, {"tactic": "cases (t.1.zoom cut).1 <;> intro H <;> [trivial; exact H rfl]", "annotated_tactic": ["cases (t.1.<a>zoom</a> cut).1 <;> intro H <;> [trivial; exact H <a>rfl</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]}, {"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\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncut : \u03b1 \u2192 Ordering\nf : \u03b1 \u2192 \u03b1\nt : RBSet \u03b1 cmp\n\u22a2 (\u2200 {x : \u03b1}, root? (zoom cut t.val Path.root).fst = some x \u2192 cmpEq cmp (f x) x) \u2192\n    OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "stronglyMeasurable_const_smul_iff\u2080", "start": [531, 1], "end": [533, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.castNum_or", "start": [925, 1], "end": [929, 45], "traced_tactics": [{"tactic": "apply castNum_eq_bitwise fun x y => pos (PosNum.lor x y) <;>\n intros <;> (try cases_type* Bool) <;> rfl", "annotated_tactic": ["apply <a>castNum_eq_bitwise</a> fun x y => <a>pos</a> (<a>PosNum.lor</a> x y) <;>\n   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": "Num.pos", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 8]}, {"full_name": "PosNum.lor", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [26, 5], "def_end_pos": [26, 8]}, {"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(m ||| n) = \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 pos (lor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d)) = bit (a\u271d || b\u271d) (pos (lor m\u271d n\u271d))", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit false m\u271d) (PosNum.bit false n\u271d)) = bit (false || false) (pos (lor m\u271d n\u271d))\n\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit false m\u271d) (PosNum.bit true n\u271d)) = bit (false || true) (pos (lor m\u271d n\u271d))\n\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit true m\u271d) (PosNum.bit false n\u271d)) = bit (true || false) (pos (lor m\u271d n\u271d))\n\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit true m\u271d) (PosNum.bit true n\u271d)) = bit (true || true) (pos (lor m\u271d n\u271d))"}, {"tactic": "cases_type* Bool", "annotated_tactic": ["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 pos (lor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d)) = bit (a\u271d || b\u271d) (pos (lor m\u271d n\u271d))", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit false m\u271d) (PosNum.bit false n\u271d)) = bit (false || false) (pos (lor m\u271d n\u271d))\n\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit false m\u271d) (PosNum.bit true n\u271d)) = bit (false || true) (pos (lor m\u271d n\u271d))\n\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit true m\u271d) (PosNum.bit false n\u271d)) = bit (true || false) (pos (lor m\u271d n\u271d))\n\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (lor (PosNum.bit true m\u271d) (PosNum.bit true n\u271d)) = bit (true || true) (pos (lor m\u271d n\u271d))"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Prod/Basic.lean", "full_name": "Prod.map_id", "start": [142, 1], "end": [143, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.tendsto_set_lintegral_zero", "start": [499, 1], "end": [506, 37], "traced_tactics": [{"tactic": "simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,\n  \u2190 pos_iff_ne_zero] at hl \u22a2", "annotated_tactic": ["simp only [<a>ENNReal.nhds_zero</a>, <a>tendsto_iInf</a>, <a>tendsto_principal</a>, <a>mem_Iio</a>,\n    \u2190 <a>pos_iff_ne_zero</a>] at hl \u22a2", [{"full_name": "ENNReal.nhds_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [193, 9], "def_end_pos": [193, 18]}, {"full_name": "Filter.tendsto_iInf", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3121, 9], "def_end_pos": [3121, 21]}, {"full_name": "Filter.tendsto_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3156, 17], "def_end_pos": [3156, 34]}, {"full_name": "Set.mem_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 16]}, {"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\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1) in s i, f x \u2202\u03bc) l (\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\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u22a2 \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < i"}, {"tactic": "intro \u03b5 \u03b50", "annotated_tactic": ["intro \u03b5 \u03b50", []], "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 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u22a2 \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5"}, {"tactic": "rcases exists_pos_set_lintegral_lt_of_measure_lt h \u03b50.ne' with \u27e8\u03b4, \u03b40, h\u03b4\u27e9", "annotated_tactic": ["rcases <a>exists_pos_set_lintegral_lt_of_measure_lt</a> h \u03b50.ne' with \u27e8\u03b4, \u03b40, h\u03b4\u27e9", [{"full_name": "MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [467, 9], "def_end_pos": [467, 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\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, 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\u271d : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5"}, {"tactic": "exact (hl \u03b4 \u03b40).mono fun i => h\u03b4 _", "annotated_tactic": ["exact (hl \u03b4 \u03b40).<a>mono</a> fun i => h\u03b4 _", [{"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\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_div", "start": [813, 1], "end": [829, 45], "traced_tactics": [{"tactic": "refine of_graph \u27e8_, fst, fun p => Nat.div_le_self _ _\u27e9 ?_", "annotated_tactic": ["refine <a>of_graph</a> \u27e8_, <a>fst</a>, fun p => <a>Nat.div_le_self</a> _ _\u27e9 ?_", [{"full_name": "Primrec.of_graph", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [806, 9], "def_end_pos": [806, 17]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"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]}]], "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\n\u22a2 Primrec\u2082 fun x x_1 => x / x_1", "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\n\u22a2 PrimrecRel fun a b => (fun x x_1 => x / x_1) a.1 a.2 = b"}, {"tactic": "have : PrimrecRel fun (a : \u2115 \u00d7 \u2115) (b : \u2115) => (a.2 = 0 \u2227 b = 0) \u2228\n    (0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2) :=\n  PrimrecPred.or\n    (.and (const 0 |> Primrec.eq.comp (fst |> snd.comp)) (const 0 |> Primrec.eq.comp snd))\n    (.and (nat_lt.comp (const 0) (fst |> snd.comp)) <|\n        .and (nat_le.comp (nat_mul.comp snd (fst |> snd.comp)) (fst |> fst.comp))\n        (nat_lt.comp (fst.comp fst) (nat_mul.comp (Primrec.succ.comp snd) (snd.comp fst))))", "annotated_tactic": ["have : <a>PrimrecRel</a> fun (a : \u2115 \u00d7 \u2115) (b : \u2115) => (a.2 = 0 \u2227 b = 0) \u2228\n      (0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2) :=\n    <a>PrimrecPred.or</a>\n      (.and (<a>const</a> 0 |> Primrec.eq.comp (<a>fst</a> |> snd.comp)) (<a>const</a> 0 |> Primrec.eq.comp <a>snd</a>))\n      (.and (nat_lt.comp (<a>const</a> 0) (<a>fst</a> |> snd.comp)) <|\n          .and (nat_le.comp (nat_mul.comp <a>snd</a> (<a>fst</a> |> snd.comp)) (<a>fst</a> |> fst.comp))\n          (nat_lt.comp (fst.comp <a>fst</a>) (nat_mul.comp (Primrec.succ.comp <a>snd</a>) (snd.comp <a>fst</a>))))", [{"full_name": "PrimrecRel", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [403, 5], "def_end_pos": [403, 15]}, {"full_name": "PrimrecPred.or", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [735, 9], "def_end_pos": [735, 30]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}, {"full_name": "Primrec.snd", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [351, 9], "def_end_pos": [351, 12]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}, {"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.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}, {"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.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "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\n\u22a2 PrimrecRel fun a b => (fun x x_1 => x / x_1) a.1 a.2 = b", "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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\n\u22a2 PrimrecRel fun a b => (fun x x_1 => x / x_1) a.1 a.2 = b"}, {"tactic": "refine this.of_eq ?_", "annotated_tactic": ["refine this.of_eq ?_", []], "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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\n\u22a2 PrimrecRel fun a b => (fun x x_1 => x / x_1) a.1 a.2 = b", "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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\n\u22a2 \u2200 (a : \u2115 \u00d7 \u2115) (b : \u2115),\n    a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 \u2194 (fun x x_1 => x / x_1) a.1 a.2 = b"}, {"tactic": "rintro \u27e8a, k\u27e9 q", "annotated_tactic": ["rintro \u27e8a, k\u27e9 q", []], "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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\n\u22a2 \u2200 (a : \u2115 \u00d7 \u2115) (b : \u2115),\n    a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 \u2194 (fun x x_1 => x / x_1) a.1 a.2 = b", "state_after": "case mk\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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\n\u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194\n    (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q"}, {"tactic": "if H : k = 0 then simp [H, eq_comm]\nelse\n  have : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k := by\n    rw [le_antisymm_iff, \u2190 (@Nat.lt_succ _ q), Nat.le_div_iff_mul_le' (Nat.pos_of_ne_zero H),\n        Nat.div_lt_iff_lt_mul' (Nat.pos_of_ne_zero H)]\n  simpa [H, zero_lt_iff, eq_comm (b := q)]", "annotated_tactic": ["if H : k = 0 then simp [H, <a>eq_comm</a>]\n  else\n    have : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k := by\n      rw [<a>le_antisymm_iff</a>, \u2190 (@<a>Nat.lt_succ</a> _ q), <a>Nat.le_div_iff_mul_le'</a> (<a>Nat.pos_of_ne_zero</a> H),\n          <a>Nat.div_lt_iff_lt_mul'</a> (<a>Nat.pos_of_ne_zero</a> H)]\n    simpa [H, <a>zero_lt_iff</a>, <a>eq_comm</a> (b := q)]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "Nat.lt_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 16]}, {"full_name": "Nat.le_div_iff_mul_le'", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [634, 9], "def_end_pos": [634, 27]}, {"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_lt_iff_lt_mul'", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 27]}, {"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": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case mk\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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\n\u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194\n    (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q", "state_after": "no goals"}, {"tactic": "simp [H, eq_comm]", "annotated_tactic": ["simp [H, <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\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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\nH : k = 0\n\u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194\n    (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q", "state_after": "no goals"}, {"tactic": "have : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k := by\n  rw [le_antisymm_iff, \u2190 (@Nat.lt_succ _ q), Nat.le_div_iff_mul_le' (Nat.pos_of_ne_zero H),\n      Nat.div_lt_iff_lt_mul' (Nat.pos_of_ne_zero H)]", "annotated_tactic": ["have : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k := by\n      rw [<a>le_antisymm_iff</a>, \u2190 (@<a>Nat.lt_succ</a> _ q), <a>Nat.le_div_iff_mul_le'</a> (<a>Nat.pos_of_ne_zero</a> H),\n          <a>Nat.div_lt_iff_lt_mul'</a> (<a>Nat.pos_of_ne_zero</a> H)]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "Nat.lt_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 16]}, {"full_name": "Nat.le_div_iff_mul_le'", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [634, 9], "def_end_pos": [634, 27]}, {"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_lt_iff_lt_mul'", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 27]}, {"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": "\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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\nH : \u00ack = 0\n\u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194\n    (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q", "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\nthis\u271d : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\nH : \u00ack = 0\nthis : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k\n\u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194\n    (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q"}, {"tactic": "simpa [H, zero_lt_iff, eq_comm (b := q)]", "annotated_tactic": ["simpa [H, <a>zero_lt_iff</a>, <a>eq_comm</a> (b := q)]", [{"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}, {"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\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\nthis\u271d : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\nH : \u00ack = 0\nthis : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k\n\u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194\n    (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q", "state_after": "no goals"}, {"tactic": "rw [le_antisymm_iff, \u2190 (@Nat.lt_succ _ q), Nat.le_div_iff_mul_le' (Nat.pos_of_ne_zero H),\n    Nat.div_lt_iff_lt_mul' (Nat.pos_of_ne_zero H)]", "annotated_tactic": ["rw [<a>le_antisymm_iff</a>, \u2190 (@<a>Nat.lt_succ</a> _ q), <a>Nat.le_div_iff_mul_le'</a> (<a>Nat.pos_of_ne_zero</a> H),\n          <a>Nat.div_lt_iff_lt_mul'</a> (<a>Nat.pos_of_ne_zero</a> H)]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "Nat.lt_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 16]}, {"full_name": "Nat.le_div_iff_mul_le'", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [634, 9], "def_end_pos": [634, 27]}, {"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_lt_iff_lt_mul'", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 27]}, {"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": "\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\nthis : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2\na k q : \u2115\nH : \u00ack = 0\n\u22a2 q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k", "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.isOrdered_iff'", "start": [113, 1], "end": [133, 67], "traced_tactics": [{"tactic": "induction t generalizing L R with\n| nil =>\n  simp [isOrdered]; split <;> simp [cmpLT_iff]\n  next h => intro _ ha _ hb; cases h _ _ ha hb\n| node _ l v r =>\n  simp [isOrdered, *]\n  exact \u27e8\n    fun \u27e8\u27e8Ll, lv, Lv, ol\u27e9, \u27e8vr, rR, vR, or\u27e9\u27e9 => \u27e8\n      fun _ h => \u27e8Lv _ h, Ll _ h, (Lv _ h).trans_l vr\u27e9,\n      fun _ h => \u27e8vR _ h, (vR _ h).trans_r lv, rR _ h\u27e9,\n      fun _ hL _ hR => (Lv _ hL).trans (vR _ hR),\n      lv, vr, ol, or\u27e9,\n    fun \u27e8hL, hR, _, lv, vr, ol, or\u27e9 => \u27e8\n      \u27e8fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol\u27e9,\n      \u27e8vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or\u27e9\u27e9\u27e9", "annotated_tactic": ["induction t generalizing L R with\n  | <a>nil</a> =>\n    simp [<a>isOrdered</a>]; split <;> simp [<a>cmpLT_iff</a>]\n    next h => intro _ ha _ hb; cases h _ _ ha hb\n  | <a>node</a> _ l v r =>\n    simp [<a>isOrdered</a>, *]\n    exact \u27e8\n      fun \u27e8\u27e8Ll, lv, Lv, ol\u27e9, \u27e8vr, rR, vR, or\u27e9\u27e9 => \u27e8\n        fun _ h => \u27e8Lv _ h, Ll _ h, (Lv _ h).<a>trans_l</a> vr\u27e9,\n        fun _ h => \u27e8vR _ h, (vR _ h).<a>trans_r</a> lv, rR _ h\u27e9,\n        fun _ hL _ hR => (Lv _ hL).<a>trans</a> (vR _ hR),\n        lv, vr, ol, or\u27e9,\n      fun \u27e8hL, hR, _, lv, vr, ol, or\u27e9 => \u27e8\n        \u27e8fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol\u27e9,\n        \u27e8vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or\u27e9\u27e9\u27e9", [{"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.isOrdered", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [250, 5], "def_end_pos": [250, 14]}, {"full_name": "Std.RBNode.cmpLT_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [238, 9], "def_end_pos": [238, 18]}, {"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.isOrdered", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [250, 5], "def_end_pos": [250, 14]}, {"full_name": "Std.RBNode.cmpLT.trans_l", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [33, 9], "def_end_pos": [33, 22]}, {"full_name": "Std.RBNode.cmpLT.trans_r", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [36, 9], "def_end_pos": [36, 22]}, {"full_name": "Std.RBNode.cmpLT.trans", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [30, 9], "def_end_pos": [30, 20]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nL R : Option \u03b1\ninst\u271d : TransCmp cmp\nt : RBNode \u03b1\n\u22a2 isOrdered cmp t L R = true \u2194\n    (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) t) \u2227\n      (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) t) \u2227\n        (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp t", "state_after": "no goals"}, {"tactic": "simp [isOrdered]", "annotated_tactic": ["simp [<a>isOrdered</a>]", [{"full_name": "Std.RBNode.isOrdered", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [250, 5], "def_end_pos": [250, 14]}]], "state_before": "case nil\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\n\u22a2 isOrdered cmp nil L R = true \u2194\n    (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) nil) \u2227\n      (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) nil) \u2227\n        (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp nil", "state_after": "case nil\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\n\u22a2 (match L, R with\n      | some l, some r => decide (cmp l r = Ordering.lt)\n      | x, x_1 => true) =\n      true \u2194\n    \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp a b"}, {"tactic": "split <;> simp [cmpLT_iff]", "annotated_tactic": ["split <;> simp [<a>cmpLT_iff</a>]", [{"full_name": "Std.RBNode.cmpLT_iff", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [238, 9], "def_end_pos": [238, 18]}]], "state_before": "case nil\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\n\u22a2 (match L, R with\n      | some l, some r => decide (cmp l r = Ordering.lt)\n      | x, x_1 => true) =\n      true \u2194\n    \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp a b", "state_after": "case nil.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\nl\u271d r\u271d : optParam (Option \u03b1) none\nx\u271d : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False\n\u22a2 \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmp a b = Ordering.lt"}, {"tactic": "next h => intro _ ha _ hb; cases h _ _ ha hb", "annotated_tactic": ["next h => intro _ ha _ hb; cases h _ _ ha hb", []], "state_before": "case nil.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\nl\u271d r\u271d : optParam (Option \u03b1) none\nx\u271d : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False\n\u22a2 \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmp a b = Ordering.lt", "state_after": "no goals"}, {"tactic": "intro _ ha _ hb", "annotated_tactic": ["intro _ ha _ hb", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\nl\u271d r\u271d : optParam (Option \u03b1) none\nh : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False\n\u22a2 \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmp a b = Ordering.lt", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\nl\u271d r\u271d : optParam (Option \u03b1) none\nh : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False\na\u271d : \u03b1\nha : L = some a\u271d\nb\u271d : \u03b1\nhb : R = some b\u271d\n\u22a2 cmp a\u271d b\u271d = Ordering.lt"}, {"tactic": "cases h _ _ ha hb", "annotated_tactic": ["cases h _ _ ha hb", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nL R : Option \u03b1\nl\u271d r\u271d : optParam (Option \u03b1) none\nh : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False\na\u271d : \u03b1\nha : L = some a\u271d\nb\u271d : \u03b1\nhb : R = some b\u271d\n\u22a2 cmp a\u271d b\u271d = Ordering.lt", "state_after": "no goals"}, {"tactic": "simp [isOrdered, *]", "annotated_tactic": ["simp [<a>isOrdered</a>, *]", [{"full_name": "Std.RBNode.isOrdered", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [250, 5], "def_end_pos": [250, 14]}]], "state_before": "case node\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nc\u271d : RBColor\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nl_ih\u271d :\n  \u2200 {L R : Option \u03b1},\n    isOrdered cmp l L R = true \u2194\n      (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) l) \u2227\n        (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) l) \u2227\n          (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp l\nr_ih\u271d :\n  \u2200 {L R : Option \u03b1},\n    isOrdered cmp r L R = true \u2194\n      (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) r) \u2227\n        (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) r) \u2227\n          (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp r\nL R : Option \u03b1\n\u22a2 isOrdered cmp (node c\u271d l v r) L R = true \u2194\n    (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) (node c\u271d l v r)) \u2227\n      (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) (node c\u271d l v r)) \u2227\n        (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp (node c\u271d l v r)", "state_after": "case node\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nc\u271d : RBColor\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nl_ih\u271d :\n  \u2200 {L R : Option \u03b1},\n    isOrdered cmp l L R = true \u2194\n      (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) l) \u2227\n        (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) l) \u2227\n          (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp l\nr_ih\u271d :\n  \u2200 {L R : Option \u03b1},\n    isOrdered cmp r L R = true \u2194\n      (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) r) \u2227\n        (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) r) \u2227\n          (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp r\nL R : Option \u03b1\n\u22a2 ((\u2200 (a : \u03b1), L = some a \u2192 All (fun x => cmpLT cmp a x) l) \u2227\n        All (fun x => cmpLT cmp x v) l \u2227 (\u2200 (a : \u03b1), L = some a \u2192 cmpLT cmp a v) \u2227 Ordered cmp l) \u2227\n      All (fun x => cmpLT cmp v x) r \u2227\n        (\u2200 (a : \u03b1), R = some a \u2192 All (fun x => cmpLT cmp x a) r) \u2227\n          (\u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp v b) \u2227 Ordered cmp r \u2194\n    (\u2200 (a : \u03b1), L = some a \u2192 cmpLT cmp a v \u2227 All (fun x => cmpLT cmp a x) l \u2227 All (fun x => cmpLT cmp a x) r) \u2227\n      (\u2200 (a : \u03b1), R = some a \u2192 cmpLT cmp v a \u2227 All (fun x => cmpLT cmp x a) l \u2227 All (fun x => cmpLT cmp x a) r) \u2227\n        (\u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp a b) \u2227\n          All (fun x => cmpLT cmp x v) l \u2227 All (fun x => cmpLT cmp v x) r \u2227 Ordered cmp l \u2227 Ordered cmp r"}, {"tactic": "exact \u27e8\n  fun \u27e8\u27e8Ll, lv, Lv, ol\u27e9, \u27e8vr, rR, vR, or\u27e9\u27e9 => \u27e8\n    fun _ h => \u27e8Lv _ h, Ll _ h, (Lv _ h).trans_l vr\u27e9,\n    fun _ h => \u27e8vR _ h, (vR _ h).trans_r lv, rR _ h\u27e9,\n    fun _ hL _ hR => (Lv _ hL).trans (vR _ hR),\n    lv, vr, ol, or\u27e9,\n  fun \u27e8hL, hR, _, lv, vr, ol, or\u27e9 => \u27e8\n    \u27e8fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol\u27e9,\n    \u27e8vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or\u27e9\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\n      fun \u27e8\u27e8Ll, lv, Lv, ol\u27e9, \u27e8vr, rR, vR, or\u27e9\u27e9 => \u27e8\n        fun _ h => \u27e8Lv _ h, Ll _ h, (Lv _ h).<a>trans_l</a> vr\u27e9,\n        fun _ h => \u27e8vR _ h, (vR _ h).<a>trans_r</a> lv, rR _ h\u27e9,\n        fun _ hL _ hR => (Lv _ hL).<a>trans</a> (vR _ hR),\n        lv, vr, ol, or\u27e9,\n      fun \u27e8hL, hR, _, lv, vr, ol, or\u27e9 => \u27e8\n        \u27e8fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol\u27e9,\n        \u27e8vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or\u27e9\u27e9\u27e9", [{"full_name": "Std.RBNode.cmpLT.trans_l", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [33, 9], "def_end_pos": [33, 22]}, {"full_name": "Std.RBNode.cmpLT.trans_r", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [36, 9], "def_end_pos": [36, 22]}, {"full_name": "Std.RBNode.cmpLT.trans", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [30, 9], "def_end_pos": [30, 20]}]], "state_before": "case node\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nc\u271d : RBColor\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nl_ih\u271d :\n  \u2200 {L R : Option \u03b1},\n    isOrdered cmp l L R = true \u2194\n      (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) l) \u2227\n        (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) l) \u2227\n          (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp l\nr_ih\u271d :\n  \u2200 {L R : Option \u03b1},\n    isOrdered cmp r L R = true \u2194\n      (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) r) \u2227\n        (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) r) \u2227\n          (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp r\nL R : Option \u03b1\n\u22a2 ((\u2200 (a : \u03b1), L = some a \u2192 All (fun x => cmpLT cmp a x) l) \u2227\n        All (fun x => cmpLT cmp x v) l \u2227 (\u2200 (a : \u03b1), L = some a \u2192 cmpLT cmp a v) \u2227 Ordered cmp l) \u2227\n      All (fun x => cmpLT cmp v x) r \u2227\n        (\u2200 (a : \u03b1), R = some a \u2192 All (fun x => cmpLT cmp x a) r) \u2227\n          (\u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp v b) \u2227 Ordered cmp r \u2194\n    (\u2200 (a : \u03b1), L = some a \u2192 cmpLT cmp a v \u2227 All (fun x => cmpLT cmp a x) l \u2227 All (fun x => cmpLT cmp a x) r) \u2227\n      (\u2200 (a : \u03b1), R = some a \u2192 cmpLT cmp v a \u2227 All (fun x => cmpLT cmp x a) l \u2227 All (fun x => cmpLT cmp x a) r) \u2227\n        (\u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp a b) \u2227\n          All (fun x => cmpLT cmp x v) l \u2227 All (fun x => cmpLT cmp v x) r \u2227 Ordered cmp l \u2227 Ordered cmp r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.Submartingale.stoppedValue_leastGE_snorm_le", "start": [129, 1], "end": [136, 85], "traced_tactics": [{"tactic": "refine' snorm_one_le_of_le' ((hf.stoppedValue_leastGE r).integrable _) _\n  (norm_stoppedValue_leastGE_le hr hf0 hbdd i)", "annotated_tactic": ["refine' <a>snorm_one_le_of_le'</a> ((hf.stoppedValue_leastGE r).<a>integrable</a> _) _\n    (<a>norm_stoppedValue_leastGE_le</a> hr hf0 hbdd i)", [{"full_name": "MeasureTheory.snorm_one_le_of_le'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1987, 9], "def_end_pos": [1987, 28]}, {"full_name": "MeasureTheory.Submartingale.integrable", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [219, 19], "def_end_pos": [219, 29]}, {"full_name": "MeasureTheory.norm_stoppedValue_leastGE_le", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [114, 9], "def_end_pos": [114, 37]}]], "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\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 snorm (stoppedValue f (leastGE f r i)) 1 \u03bc \u2264 2 * \u2191\u2191\u03bc Set.univ * ENNReal.ofReal (r + \u2191R)", "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\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 0 \u2264 \u222b (x : \u03a9), stoppedValue f (leastGE f r i) x \u2202\u03bc"}, {"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\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\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 0 \u2264 \u222b (x : \u03a9), stoppedValue f (leastGE f r i) x \u2202\u03bc", "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\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 0 \u2264 \u222b (x : \u03a9) in Set.univ, stoppedValue f (leastGE f r i) x \u2202\u03bc"}, {"tactic": "refine' le_trans _ ((hf.stoppedValue_leastGE r).set_integral_le (zero_le _) MeasurableSet.univ)", "annotated_tactic": ["refine' <a>le_trans</a> _ ((hf.stoppedValue_leastGE r).<a>set_integral_le</a> (<a>zero_le</a> _) <a>MeasurableSet.univ</a>)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.Submartingale.set_integral_le", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [250, 9], "def_end_pos": [250, 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": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "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\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 0 \u2264 \u222b (x : \u03a9) in Set.univ, stoppedValue f (leastGE f r i) x \u2202\u03bc", "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\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 0 \u2264 \u222b (\u03c9 : \u03a9) in Set.univ, stoppedValue f (leastGE f r 0) \u03c9 \u2202\u03bc"}, {"tactic": "simp_rw [stoppedValue, leastGE, hitting_of_le le_rfl, hf0, integral_zero', le_rfl]", "annotated_tactic": ["simp_rw [<a>stoppedValue</a>, <a>leastGE</a>, <a>hitting_of_le</a> <a>le_rfl</a>, hf0, <a>integral_zero'</a>, <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": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.hitting_of_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [98, 9], "def_end_pos": [98, 22]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.integral_zero'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [858, 9], "def_end_pos": [858, 23]}, {"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\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\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 0 \u2264 \u222b (\u03c9 : \u03a9) in Set.univ, stoppedValue f (leastGE f r 0) \u03c9 \u2202\u03bc", "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_ncard_iff", "start": [1059, 1], "end": [1062, 43], "traced_tactics": [{"tactic": "rw [one_lt_ncard hs]", "annotated_tactic": ["rw [<a>one_lt_ncard</a> hs]", [{"full_name": "Set.one_lt_ncard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [1054, 9], "def_end_pos": [1054, 21]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 1 < ncard 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\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 a \u2260 b) \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b"}, {"tactic": "simp only [exists_prop, exists_and_left]", "annotated_tactic": ["simp only [<a>exists_prop</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": "exists_and_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [465, 17], "def_end_pos": [465, 32]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 a \u2260 b) \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Integration.lean", "full_name": "ProbabilityTheory.lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator", "start": [45, 1], "end": [73, 68], "traced_tactics": [{"tactic": "revert f", "annotated_tactic": ["revert f", []], "state_before": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_meas_f : Measurable f\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\n\u22a2 \u2200 {f : \u03a9 \u2192 \u211d\u22650\u221e},\n    Measurable f \u2192\n      \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "have h_mul_indicator : \u2200 g, Measurable g \u2192 Measurable fun a => g a * T.indicator (fun _ => c) a :=\n  fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T)", "annotated_tactic": ["have h_mul_indicator : \u2200 g, <a>Measurable</a> g \u2192 <a>Measurable</a> fun a => g a * T.indicator (fun _ => c) a :=\n    fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"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\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\n\u22a2 \u2200 {f : \u03a9 \u2192 \u211d\u22650\u221e},\n    Measurable f \u2192\n      \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 {f : \u03a9 \u2192 \u211d\u22650\u221e},\n    Measurable f \u2192\n      \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "apply @Measurable.ennreal_induction _ Mf", "annotated_tactic": ["apply @<a>Measurable.ennreal_induction</a> _ Mf", [{"full_name": "Measurable.ennreal_induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 37]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 {f : \u03a9 \u2192 \u211d\u22650\u221e},\n    Measurable f \u2192\n      \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 (c_1 : \u211d\u22650\u221e) \u2983s : Set \u03a9\u2984,\n    MeasurableSet s \u2192\n      \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c_1) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n        (\u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c_1) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\ncase h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 \u2983f g : \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Measurable f \u2192\n        Measurable g \u2192\n          \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n              (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u2192\n            \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n                (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u2192\n              \u222b\u207b (\u03c9 : \u03a9), (f + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n                (\u222b\u207b (\u03c9 : \u03a9), (f + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\ncase h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    (\u2200 (n : \u2115), Measurable (f n)) \u2192\n      Monotone f \u2192\n        (\u2200 (n : \u2115),\n            \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n              (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc) \u2192\n          \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n            (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "intro c' s' h_meas_s'", "annotated_tactic": ["intro c' s' h_meas_s'", []], "state_before": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 (c_1 : \u211d\u22650\u221e) \u2983s : Set \u03a9\u2984,\n    MeasurableSet s \u2192\n      \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c_1) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n        (\u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c_1) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "simp_rw [\u2190 inter_indicator_mul]", "annotated_tactic": ["simp_rw [\u2190 <a>inter_indicator_mul</a>]", [{"full_name": "Set.inter_indicator_mul", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [730, 9], "def_end_pos": [730, 28]}]], "state_before": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), indicator (s' \u2229 T) (fun x => c' * c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "rw [lintegral_indicator _ (MeasurableSet.inter (hMf _ h_meas_s') h_meas_T),\n  lintegral_indicator _ (hMf _ h_meas_s'), lintegral_indicator _ h_meas_T]", "annotated_tactic": ["rw [<a>lintegral_indicator</a> _ (<a>MeasurableSet.inter</a> (hMf _ h_meas_s') h_meas_T),\n      <a>lintegral_indicator</a> _ (hMf _ h_meas_s'), <a>lintegral_indicator</a> _ h_meas_T]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"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_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), indicator (s' \u2229 T) (fun x => c' * c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 \u222b\u207b (a : \u03a9) in s' \u2229 T, c' * c \u2202\u03bc = (\u222b\u207b (a : \u03a9) in s', c' \u2202\u03bc) * \u222b\u207b (a : \u03a9) in T, c \u2202\u03bc"}, {"tactic": "simp only [measurable_const, lintegral_const, univ_inter, lintegral_const_mul,\n  MeasurableSet.univ, Measure.restrict_apply]", "annotated_tactic": ["simp only [<a>measurable_const</a>, <a>lintegral_const</a>, <a>univ_inter</a>, <a>lintegral_const_mul</a>,\n      <a>MeasurableSet.univ</a>, <a>Measure.restrict_apply</a>]", [{"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": "MeasureTheory.lintegral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [671, 9], "def_end_pos": [671, 28]}, {"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": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 \u222b\u207b (a : \u03a9) in s' \u2229 T, c' * c \u2202\u03bc = (\u222b\u207b (a : \u03a9) in s', c' \u2202\u03bc) * \u222b\u207b (a : \u03a9) in T, c \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 c' * c * \u2191\u2191\u03bc (s' \u2229 T) = c' * \u2191\u2191\u03bc s' * (c * \u2191\u2191\u03bc T)"}, {"tactic": "rw [IndepSets_iff] at h_ind", "annotated_tactic": ["rw [<a>IndepSets_iff</a>] at h_ind", [{"full_name": "ProbabilityTheory.IndepSets_iff", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [154, 7], "def_end_pos": [154, 20]}]], "state_before": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 c' * c * \u2191\u2191\u03bc (s' \u2229 T) = c' * \u2191\u2191\u03bc s' * (c * \u2191\u2191\u03bc T)", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : \u2200 (t1 t2 : Set \u03a9), t1 \u2208 {s | MeasurableSet s} \u2192 t2 \u2208 {T} \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 c' * c * \u2191\u2191\u03bc (s' \u2229 T) = c' * \u2191\u2191\u03bc s' * (c * \u2191\u2191\u03bc T)"}, {"tactic": "rw [mul_mul_mul_comm, h_ind s' T h_meas_s' (Set.mem_singleton _)]", "annotated_tactic": ["rw [<a>mul_mul_mul_comm</a>, h_ind s' T h_meas_s' (<a>Set.mem_singleton</a> _)]", [{"full_name": "mul_mul_mul_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [108, 9], "def_end_pos": [108, 25]}, {"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_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : \u2200 (t1 t2 : Set \u03a9), t1 \u2208 {s | MeasurableSet s} \u2192 t2 \u2208 {T} \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nc' : \u211d\u22650\u221e\ns' : Set \u03a9\nh_meas_s' : MeasurableSet s'\n\u22a2 c' * c * \u2191\u2191\u03bc (s' \u2229 T) = c' * \u2191\u2191\u03bc s' * (c * \u2191\u2191\u03bc T)", "state_after": "no goals"}, {"tactic": "intro f' g _ h_meas_f' _ h_ind_f' h_ind_g", "annotated_tactic": ["intro f' g _ h_meas_f' _ h_ind_f' h_ind_g", []], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 \u2983f g : \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Measurable f \u2192\n        Measurable g \u2192\n          \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n              (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u2192\n            \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n                (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u2192\n              \u222b\u207b (\u03c9 : \u03a9), (f + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n                (\u222b\u207b (\u03c9 : \u03a9), (f + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_meas_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' :\n  \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_ind_g :\n  \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "have h_measM_f' : Measurable f' := h_meas_f'.mono hMf le_rfl", "annotated_tactic": ["have h_measM_f' : <a>Measurable</a> f' := h_meas_f'.mono hMf <a>le_rfl</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_meas_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' :\n  \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_ind_g :\n  \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_meas_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' :\n  \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_ind_g :\n  \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "simp_rw [Pi.add_apply, right_distrib]", "annotated_tactic": ["simp_rw [<a>Pi.add_apply</a>, <a>right_distrib</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}]], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_meas_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' :\n  \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_ind_g :\n  \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_meas_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' :\n  \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_ind_g :\n  \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 + g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 + g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "rw [lintegral_add_left (h_mul_indicator _ h_measM_f'), lintegral_add_left h_measM_f',\n  right_distrib, h_ind_f', h_ind_g]", "annotated_tactic": ["rw [<a>lintegral_add_left</a> (h_mul_indicator _ h_measM_f'), <a>lintegral_add_left</a> h_measM_f',\n      <a>right_distrib</a>, h_ind_f', h_ind_g]", [{"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]}, {"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}]], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_meas_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' :\n  \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_ind_g :\n  \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 + g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 + g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro f h_meas_f h_mono_f h_ind_f", "annotated_tactic": ["intro f h_meas_f h_mono_f h_ind_f", []], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    (\u2200 (n : \u2115), Measurable (f n)) \u2192\n      Monotone f \u2192\n        (\u2200 (n : \u2115),\n            \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n              (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc) \u2192\n          \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n            (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "have h_measM_f : \u2200 n, Measurable (f n) := fun n => (h_meas_f n).mono hMf le_rfl", "annotated_tactic": ["have h_measM_f : \u2200 n, <a>Measurable</a> (f n) := fun n => (h_meas_f n).<a>mono</a> hMf <a>le_rfl</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Measurable.mono", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 24]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "simp_rw [ENNReal.iSup_mul]", "annotated_tactic": ["simp_rw [<a>ENNReal.iSup_mul</a>]", [{"full_name": "ENNReal.iSup_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [652, 9], "def_end_pos": [652, 17]}]], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2a06 i, f i \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), \u2a06 n, f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "rw [lintegral_iSup h_measM_f h_mono_f, lintegral_iSup, ENNReal.iSup_mul]", "annotated_tactic": ["rw [<a>lintegral_iSup</a> h_measM_f h_mono_f, <a>lintegral_iSup</a>, <a>ENNReal.iSup_mul</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.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "ENNReal.iSup_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [652, 9], "def_end_pos": [652, 17]}]], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2a06 i, f i \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n    (\u222b\u207b (\u03c9 : \u03a9), \u2a06 n, f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03a9), f n a * indicator T (fun x => c) a \u2202\u03bc =\n    \u2a06 i, (\u222b\u207b (a : \u03a9), f i a \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\n\ncase h_iSup.hf\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03c9 => f n \u03c9 * indicator T (fun x => c) \u03c9\n\ncase h_iSup.h_mono\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 Monotone fun i \u03c9 => f i \u03c9 * indicator T (fun x => c) \u03c9"}, {"tactic": "simp_rw [\u2190 h_ind_f]", "annotated_tactic": ["simp_rw [\u2190 h_ind_f]", []], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03a9), f n a * indicator T (fun x => c) a \u2202\u03bc =\n    \u2a06 i, (\u222b\u207b (a : \u03a9), f i a \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact fun n => h_mul_indicator _ (h_measM_f n)", "annotated_tactic": ["exact fun n => h_mul_indicator _ (h_measM_f n)", []], "state_before": "case h_iSup.hf\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03c9 => f n \u03c9 * indicator T (fun x => c) \u03c9", "state_after": "no goals"}, {"tactic": "exact fun m n h_le a => mul_le_mul_right' (h_mono_f h_le a) _", "annotated_tactic": ["exact fun m n h_le a => <a>mul_le_mul_right'</a> (h_mono_f h_le a) _", [{"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 h_iSup.h_mono\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ng : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nc : \u211d\u22650\u221e\nT : Set \u03a9\nh_meas_T : MeasurableSet T\nh_ind : IndepSets {s | MeasurableSet s} {T}\nh_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f : \u2200 (n : \u2115), Measurable (f n)\nh_mono_f : Monotone f\nh_ind_f :\n  \u2200 (n : \u2115),\n    \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =\n      (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc\nh_measM_f : \u2200 (n : \u2115), Measurable (f n)\n\u22a2 Monotone fun i \u03c9 => f i \u03c9 * indicator T (fun x => c) \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.lift_comp_coe", "start": [1278, 1], "end": [1279, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "full_name": "MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite'", "start": [119, 1], "end": [158, 99], "traced_tactics": [{"tactic": "rw [\u2190 ae_eq_trim_iff_of_aeStronglyMeasurable' hm hfm hgm]", "annotated_tactic": ["rw [\u2190 <a>ae_eq_trim_iff_of_aeStronglyMeasurable'</a> hm hfm hgm]", [{"full_name": "MeasureTheory.ae_eq_trim_iff_of_aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [144, 9], "def_end_pos": [144, 48]}]], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\n\u22a2 f =\u1d50[\u03bc] g", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm"}, {"tactic": "have hf_mk_int_finite :\n  \u2200 s, MeasurableSet[m] s \u2192 \u03bc.trim hm s < \u221e \u2192 @IntegrableOn _ _ m _ (hfm.mk f) s (\u03bc.trim hm) := by\n  intro s hs h\u03bcs\n  rw [trim_measurableSet_eq hm hs] at h\u03bcs\n  unfold IntegrableOn\n  rw [restrict_trim hm _ hs]\n  refine' Integrable.trim hm _ hfm.stronglyMeasurable_mk\n  exact Integrable.congr (hf_int_finite s hs h\u03bcs) (ae_restrict_of_ae hfm.ae_eq_mk)", "annotated_tactic": ["have hf_mk_int_finite :\n    \u2200 s, MeasurableSet[m] s \u2192 \u03bc.trim hm s < \u221e \u2192 @<a>IntegrableOn</a> _ _ m _ (hfm.mk f) s (\u03bc.trim hm) := by\n    intro s hs h\u03bcs\n    rw [<a>trim_measurableSet_eq</a> hm hs] at h\u03bcs\n    -- Porting note: `rw [IntegrableOn]` fails with\n    -- synthesized type class instance is not definitionally equal to expression inferred by typing\n    -- rules, synthesized m0 inferred m\n    unfold <a>IntegrableOn</a>\n    rw [<a>restrict_trim</a> hm _ hs]\n    refine' <a>Integrable.trim</a> hm _ hfm.stronglyMeasurable_mk\n    exact <a>Integrable.congr</a> (hf_int_finite s hs h\u03bcs) (<a>ae_restrict_of_ae</a> hfm.ae_eq_mk)", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 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.Integrable.trim", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1212, 9], "def_end_pos": [1212, 24]}, {"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 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": "\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm"}, {"tactic": "have hg_mk_int_finite :\n  \u2200 s, MeasurableSet[m] s \u2192 \u03bc.trim hm s < \u221e \u2192 @IntegrableOn _ _ m _ (hgm.mk g) s (\u03bc.trim hm) := by\n  intro s hs h\u03bcs\n  rw [trim_measurableSet_eq hm hs] at h\u03bcs\n  unfold IntegrableOn\n  rw [restrict_trim hm _ hs]\n  refine' Integrable.trim hm _ hgm.stronglyMeasurable_mk\n  exact Integrable.congr (hg_int_finite s hs h\u03bcs) (ae_restrict_of_ae hgm.ae_eq_mk)", "annotated_tactic": ["have hg_mk_int_finite :\n    \u2200 s, MeasurableSet[m] s \u2192 \u03bc.trim hm s < \u221e \u2192 @<a>IntegrableOn</a> _ _ m _ (hgm.mk g) s (\u03bc.trim hm) := by\n    intro s hs h\u03bcs\n    rw [<a>trim_measurableSet_eq</a> hm hs] at h\u03bcs\n    -- Porting note: `rw [IntegrableOn]` fails with\n    -- synthesized type class instance is not definitionally equal to expression inferred by typing\n    -- rules, synthesized m0 inferred m\n    unfold <a>IntegrableOn</a>\n    rw [<a>restrict_trim</a> hm _ hs]\n    refine' <a>Integrable.trim</a> hm _ hgm.stronglyMeasurable_mk\n    exact <a>Integrable.congr</a> (hg_int_finite s hs h\u03bcs) (<a>ae_restrict_of_ae</a> hgm.ae_eq_mk)", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 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.Integrable.trim", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1212, 9], "def_end_pos": [1212, 24]}, {"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 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": "\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm"}, {"tactic": "have hfg_mk_eq :\n  \u2200 s : Set \u03b1,\n    MeasurableSet[m] s \u2192\n      \u03bc.trim hm s < \u221e \u2192 \u222b x in s, hfm.mk f x \u2202\u03bc.trim hm = \u222b x in s, hgm.mk g x \u2202\u03bc.trim hm := by\n  intro s hs h\u03bcs\n  rw [trim_measurableSet_eq hm hs] at h\u03bcs\n  rw [restrict_trim hm _ hs, \u2190 integral_trim hm hfm.stronglyMeasurable_mk, \u2190\n    integral_trim hm hgm.stronglyMeasurable_mk,\n    integral_congr_ae (ae_restrict_of_ae hfm.ae_eq_mk.symm),\n    integral_congr_ae (ae_restrict_of_ae hgm.ae_eq_mk.symm)]\n  exact hfg_eq s hs h\u03bcs", "annotated_tactic": ["have hfg_mk_eq :\n    \u2200 s : <a>Set</a> \u03b1,\n      MeasurableSet[m] s \u2192\n        \u03bc.trim hm s < \u221e \u2192 \u222b x in s, hfm.mk f x \u2202\u03bc.trim hm = \u222b x in s, hgm.mk g x \u2202\u03bc.trim hm := by\n    intro s hs h\u03bcs\n    rw [<a>trim_measurableSet_eq</a> hm hs] at h\u03bcs\n    rw [<a>restrict_trim</a> hm _ hs, \u2190 <a>integral_trim</a> hm hfm.stronglyMeasurable_mk, \u2190\n      <a>integral_trim</a> hm hgm.stronglyMeasurable_mk,\n      <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> hfm.ae_eq_mk.symm),\n      <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> hgm.ae_eq_mk.symm)]\n    exact hfg_eq s hs h\u03bcs", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "MeasureTheory.restrict_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [93, 9], "def_end_pos": [93, 22]}, {"full_name": "MeasureTheory.integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 22]}, {"full_name": "MeasureTheory.integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 22]}, {"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.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]}]], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\nhfg_mk_eq :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n          \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm"}, {"tactic": "exact ae_eq_of_forall_set_integral_eq_of_sigmaFinite hf_mk_int_finite hg_mk_int_finite hfg_mk_eq", "annotated_tactic": ["exact <a>ae_eq_of_forall_set_integral_eq_of_sigmaFinite</a> hf_mk_int_finite hg_mk_int_finite hfg_mk_eq", [{"full_name": "MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [423, 9], "def_end_pos": [423, 55]}]], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\nhfg_mk_eq :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n          \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s"}, {"tactic": "rw [trim_measurableSet_eq hm hs] at h\u03bcs", "annotated_tactic": ["rw [<a>trim_measurableSet_eq</a> hm hs] at h\u03bcs", [{"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\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s"}, {"tactic": "unfold IntegrableOn", "annotated_tactic": ["unfold <a>IntegrableOn</a>", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable (AEStronglyMeasurable'.mk f hfm)"}, {"tactic": "exact Integrable.congr (hf_int_finite s hs h\u03bcs) (ae_restrict_of_ae hfm.ae_eq_mk)", "annotated_tactic": ["exact <a>Integrable.congr</a> (hf_int_finite s hs h\u03bcs) (<a>ae_restrict_of_ae</a> hfm.ae_eq_mk)", [{"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 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": "\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable (AEStronglyMeasurable'.mk f hfm)", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s"}, {"tactic": "rw [trim_measurableSet_eq hm hs] at h\u03bcs", "annotated_tactic": ["rw [<a>trim_measurableSet_eq</a> hm hs] at h\u03bcs", [{"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\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s"}, {"tactic": "unfold IntegrableOn", "annotated_tactic": ["unfold <a>IntegrableOn</a>", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable (AEStronglyMeasurable'.mk g hgm)"}, {"tactic": "exact Integrable.congr (hg_int_finite s hs h\u03bcs) (ae_restrict_of_ae hgm.ae_eq_mk)", "annotated_tactic": ["exact <a>Integrable.congr</a> (hg_int_finite s hs h\u03bcs) (<a>ae_restrict_of_ae</a> hgm.ae_eq_mk)", [{"full_name": "MeasureTheory.Integrable.congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [492, 9], "def_end_pos": [492, 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": "\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable (AEStronglyMeasurable'.mk g hgm)", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n          \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n    \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm"}, {"tactic": "rw [trim_measurableSet_eq hm hs] at h\u03bcs", "annotated_tactic": ["rw [<a>trim_measurableSet_eq</a> hm hs] at h\u03bcs", [{"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\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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n    \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n    \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm"}, {"tactic": "rw [restrict_trim hm _ hs, \u2190 integral_trim hm hfm.stronglyMeasurable_mk, \u2190\n  integral_trim hm hgm.stronglyMeasurable_mk,\n  integral_congr_ae (ae_restrict_of_ae hfm.ae_eq_mk.symm),\n  integral_congr_ae (ae_restrict_of_ae hgm.ae_eq_mk.symm)]", "annotated_tactic": ["rw [<a>restrict_trim</a> hm _ hs, \u2190 <a>integral_trim</a> hm hfm.stronglyMeasurable_mk, \u2190\n      <a>integral_trim</a> hm hgm.stronglyMeasurable_mk,\n      <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> hfm.ae_eq_mk.symm),\n      <a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> hgm.ae_eq_mk.symm)]", [{"full_name": "MeasureTheory.restrict_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [93, 9], "def_end_pos": [93, 22]}, {"full_name": "MeasureTheory.integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 22]}, {"full_name": "MeasureTheory.integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 22]}, {"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.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]}]], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =\n    \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm", "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u222b (a : \u03b1) in s, g a \u2202\u03bc"}, {"tactic": "exact hfg_eq s hs h\u03bcs", "annotated_tactic": ["exact hfg_eq s hs h\u03bcs", []], "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\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 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nf g : \u03b1 \u2192 F'\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nhf_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s\nhg_mk_int_finite :\n  \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u222b (a : \u03b1) in s, g a \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_bindOnSupport_apply", "start": [329, 1], "end": [332, 97], "traced_tactics": [{"tactic": "simp only [toMeasure_apply_eq_toOuterMeasure_apply _ _ hs, toOuterMeasure_bindOnSupport_apply]", "annotated_tactic": ["simp only [<a>toMeasure_apply_eq_toOuterMeasure_apply</a> _ _ hs, <a>toOuterMeasure_bindOnSupport_apply</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_bindOnSupport_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [309, 9], "def_end_pos": [309, 43]}]], "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\ns : Set \u03b2\ninst\u271d : MeasurableSpace \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure (bindOnSupport p f)) s = \u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191\u2191(toMeasure (f a h)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_append", "start": [1098, 1], "end": [1100, 30], "traced_tactics": [{"tactic": "induction l\u2081 <;> simp [*]", "annotated_tactic": ["induction l\u2081 <;> 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\nl\u2081 l\u2082 : List \u03b1\n\u22a2 List.foldr (fun b s => ((l\u2081, l\u2082), b, s).2.1 :: ((l\u2081, l\u2082), b, s).2.2) (l\u2081, l\u2082).2 (l\u2081, l\u2082).1 = l\u2081 ++ l\u2082", "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", "start": [1057, 1], "end": [1096, 46], "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\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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)"}, {"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\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)"}, {"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\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)"}, {"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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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]", "annotated_tactic": ["conv_lhs => 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\nhf : InjOn f s\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 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\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\nhf : InjOn f s\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 \u222b\u207b (x : E) in \u22c3 n, s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in 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\nhf : InjOn f s\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 \u222b\u207b (x : E) in \u22c3 n, s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n    \u2211' (n : \u2115), \u222b\u207b (x : E) in 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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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"}, {"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\nhf : InjOn f s\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 \u2264\n    \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\nhf : InjOn f s\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 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' (s \u2229 u n))"}, {"tactic": "apply\n  lintegral_abs_det_fderiv_le_addHaar_image_aux2 \u03bc (hs.inter (u_meas n)) _\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) (hf.mono (inter_subset_left _ _))", "annotated_tactic": ["apply\n        <a>lintegral_abs_det_fderiv_le_addHaar_image_aux2</a> \u03bc (hs.inter (u_meas n)) _\n          (fun x hx => (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _)) (hf.mono (<a>inter_subset_left</a> _ _))", [{"full_name": "MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1039, 9], "def_end_pos": [1039, 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]}, {"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\nhf : InjOn f s\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 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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\nhf : InjOn f s\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, image_iUnion]", "annotated_tactic": ["conv_rhs => 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\nhf : InjOn f s\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)) = \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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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)) = \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 u i))"}, {"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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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)) = \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 u i))", "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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 i => f '' (s \u2229 u i))\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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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 (f '' (s \u2229 u i))"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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 i => f '' (s \u2229 u i))", "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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => f '' (s \u2229 u i)) i j"}, {"tactic": "apply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)", "annotated_tactic": ["apply <a>Disjoint.image</a> _ hf (<a>inter_subset_left</a> _ _) (<a>inter_subset_left</a> _ _)", [{"full_name": "Disjoint.image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [757, 9], "def_end_pos": [757, 30]}, {"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 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => f '' (s \u2229 u i)) i j", "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\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\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 u i) (s \u2229 u j)"}, {"tactic": "exact\n  Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _)\n    (disjoint_disjointed _ hij)", "annotated_tactic": ["exact\n          <a>Disjoint.mono</a> (<a>inter_subset_right</a> _ _) (<a>inter_subset_right</a> _ _)\n            (<a>disjoint_disjointed</a> _ hij)", [{"full_name": "Disjoint.mono", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}, {"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": "disjoint_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [74, 9], "def_end_pos": [74, 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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 u i) (s \u2229 u j)", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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 (f '' (s \u2229 u 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\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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\ni : \u2115\n\u22a2 MeasurableSet (f '' (s \u2229 u i))"}, {"tactic": "exact\n  measurable_image_of_fderivWithin (hs.inter (u_meas i))\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n    (hf.mono (inter_subset_left _ _))", "annotated_tactic": ["exact\n          <a>measurable_image_of_fderivWithin</a> (hs.inter (u_meas i))\n            (fun x hx => (hf' x hx.1).<a>mono</a> (<a>inter_subset_left</a> _ _))\n            (hf.mono (<a>inter_subset_left</a> _ _))", [{"full_name": "MeasureTheory.measurable_image_of_fderivWithin", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [778, 9], "def_end_pos": [778, 41]}, {"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_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "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\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\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\ni : \u2115\n\u22a2 MeasurableSet (f '' (s \u2229 u 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.mul_lt_mul'", "start": [534, 11], "end": [535, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sizeOf_lt_sizeOf_of_mem", "start": [2444, 1], "end": [2450, 44], "traced_tactics": [{"tactic": "cases s", "annotated_tactic": ["cases s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\ns : Finset \u03b1\nhx : x \u2208 s\n\u22a2 sizeOf x < sizeOf s", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < sizeOf { val := val\u271d, nodup := nodup\u271d }"}, {"tactic": "dsimp [SizeOf.sizeOf, SizeOf.sizeOf, Multiset.sizeOf]", "annotated_tactic": ["dsimp [<a>SizeOf.sizeOf</a>, <a>SizeOf.sizeOf</a>, <a>Multiset.sizeOf</a>]", [{"full_name": "SizeOf.sizeOf", "def_path": "lake-packages/lean4/src/lean/Init/SizeOf.lean", "def_pos": [28, 3], "def_end_pos": [28, 9]}, {"full_name": "SizeOf.sizeOf", "def_path": "lake-packages/lean4/src/lean/Init/SizeOf.lean", "def_pos": [28, 3], "def_end_pos": [28, 9]}, {"full_name": "Multiset.sizeOf", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < sizeOf { val := val\u271d, nodup := nodup\u271d }", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < 1 + Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1)"}, {"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": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < 1 + Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1)", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1) + 1"}, {"tactic": "refine' lt_trans _ (Nat.lt_succ_self _)", "annotated_tactic": ["refine' <a>lt_trans</a> _ (<a>Nat.lt_succ_self</a> _)", [{"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 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": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1) + 1", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1)"}, {"tactic": "exact Multiset.sizeOf_lt_sizeOf_of_mem hx", "annotated_tactic": ["exact <a>Multiset.sizeOf_lt_sizeOf_of_mem</a> hx", [{"full_name": "Multiset.sizeOf_lt_sizeOf_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1510, 9], "def_end_pos": [1510, 32]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "full_name": "MeasureTheory.sigmaFiniteTrim_mono", "start": [107, 1], "end": [123, 47], "traced_tactics": [{"tactic": "have _ := Measure.FiniteSpanningSetsIn (\u03bc.trim (hm\u2082.trans hm)) Set.univ", "annotated_tactic": ["have _ := <a>Measure.FiniteSpanningSetsIn</a> (\u03bc.trim (hm\u2082.trans hm)) <a>Set.univ</a>", [{"full_name": "MeasureTheory.Measure.FiniteSpanningSetsIn", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3272, 11], "def_end_pos": [3272, 31]}, {"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\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\n\u22a2 SigmaFinite (Measure.trim \u03bc hm)", "state_after": "\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 SigmaFinite (Measure.trim \u03bc hm)"}, {"tactic": "refine' Measure.FiniteSpanningSetsIn.sigmaFinite _", "annotated_tactic": ["refine' <a>Measure.FiniteSpanningSetsIn.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": "\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 SigmaFinite (Measure.trim \u03bc hm)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 Set (Set \u03b1)\n\ncase refine'_2\n\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 Measure.FiniteSpanningSetsIn (Measure.trim \u03bc hm) ?refine'_1"}, {"tactic": "exact Set.univ", "annotated_tactic": ["exact <a>Set.univ</a>", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 Set (Set \u03b1)", "state_after": "no goals"}, {"tactic": "refine'\n  { set := spanningSets (\u03bc.trim (hm\u2082.trans hm))\n    set_mem := fun _ => Set.mem_univ _\n    finite := fun i => _ spanning := iUnion_spanningSets _ }", "annotated_tactic": ["refine'\n      { set := <a>spanningSets</a> (\u03bc.trim (hm\u2082.trans hm))\n        set_mem := fun _ => <a>Set.mem_univ</a> _\n        finite := fun i => _ -- This is the only one left to prove\n        spanning := <a>iUnion_spanningSets</a> _ }", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "MeasureTheory.iUnion_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3334, 9], "def_end_pos": [3334, 28]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 Measure.FiniteSpanningSetsIn (Measure.trim \u03bc hm) Set.univ", "state_after": "case refine'_2\n\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) < \u22a4"}, {"tactic": "calc\n  (\u03bc.trim hm) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) =\n      ((\u03bc.trim hm).trim hm\u2082) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) :=\n    by rw [@trim_measurableSet_eq \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (measurable_spanningSets _ _)]\n  _ = (\u03bc.trim (hm\u2082.trans hm)) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) := by\n    rw [@trim_trim _ _ \u03bc _ _ hm\u2082 hm]\n  _ < \u221e := measure_spanningSets_lt_top _ _", "annotated_tactic": ["calc\n      (\u03bc.trim hm) (<a>spanningSets</a> (\u03bc.trim (hm\u2082.trans hm)) i) =\n          ((\u03bc.trim hm).<a>trim</a> hm\u2082) (<a>spanningSets</a> (\u03bc.trim (hm\u2082.trans hm)) i) :=\n        by rw [@<a>trim_measurableSet_eq</a> \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (<a>measurable_spanningSets</a> _ _)]\n      _ = (\u03bc.trim (hm\u2082.trans hm)) (<a>spanningSets</a> (\u03bc.trim (hm\u2082.trans hm)) i) := by\n        rw [@<a>trim_trim</a> _ _ \u03bc _ _ hm\u2082 hm]\n      _ < \u221e := <a>measure_spanningSets_lt_top</a> _ _", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "MeasureTheory.Measure.trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [32, 5], "def_end_pos": [32, 17]}, {"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}, {"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "MeasureTheory.trim_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}, {"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 refine'_2\n\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) < \u22a4", "state_after": "no goals"}, {"tactic": "rw [@trim_measurableSet_eq \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (measurable_spanningSets _ _)]", "annotated_tactic": ["rw [@<a>trim_measurableSet_eq</a> \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (<a>measurable_spanningSets</a> _ _)]", [{"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"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\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) =\n    \u2191\u2191(Measure.trim (Measure.trim \u03bc hm) hm\u2082) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i)", "state_after": "no goals"}, {"tactic": "rw [@trim_trim _ _ \u03bc _ _ hm\u2082 hm]", "annotated_tactic": ["rw [@<a>trim_trim</a> _ _ \u03bc _ _ hm\u2082 hm]", [{"full_name": "MeasureTheory.trim_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}]], "state_before": "\u03b1 : Type u_1\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(Measure.trim (Measure.trim \u03bc hm) hm\u2082) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) =\n    \u2191\u2191(Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.self_mem_range_succ", "start": [3070, 1], "end": [3071, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.Coprime.pow", "start": [390, 1], "end": [391, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.Measure.ofMeasurable_apply", "start": [120, 1], "end": [126, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_sub", "start": [772, 1], "end": [775, 94], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, setToL1S_add T h_zero h_add, setToL1S_neg h_zero h_add, sub_eq_add_neg]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>setToL1S_add</a> T h_zero h_add, <a>setToL1S_neg</a> h_zero h_add, <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]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_add", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [752, 9], "def_end_pos": [752, 21]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_neg", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [763, 9], "def_end_pos": [763, 21]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 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\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\nf g : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 setToL1S T (f - g) = setToL1S T f - setToL1S T g", "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.subNat_addNat", "start": [542, 9], "end": [543, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.trim_eq_trim_iff", "start": [1643, 1], "end": [1645, 60], "traced_tactics": [{"tactic": "simp only [le_antisymm_iff, trim_le_trim_iff, forall_and]", "annotated_tactic": ["simp only [<a>le_antisymm_iff</a>, <a>trim_le_trim_iff</a>, <a>forall_and</a>]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "MeasureTheory.OuterMeasure.trim_le_trim_iff", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1638, 9], "def_end_pos": [1638, 25]}, {"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_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\n\u22a2 trim m\u2081 = trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s", "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.out", "start": [283, 1], "end": [290, 60], "traced_tactics": [{"tactic": "induction h with\n| mk h\u2081 h\u2082 => exact \u27e8h\u2081, h\u2082\u27e9\n| @empty' _ h => exact \u27e8(Buckets.mk_size h).symm, .mk' h\u27e9\n| insert _ ih => exact \u27e8insert_size ih.1, insert_WF ih.2\u27e9\n| erase _ ih => exact \u27e8erase_size ih.1, erase_WF ih.2\u27e9\n| modify _ ih => exact \u27e8modify_size ih.1, modify_WF ih.2\u27e9", "annotated_tactic": ["induction h with\n  | <a>mk</a> h\u2081 h\u2082 => exact \u27e8h\u2081, h\u2082\u27e9\n  | @<a>empty'</a> _ h => exact \u27e8(<a>Buckets.mk_size</a> h).<a>symm</a>, .mk' h\u27e9\n  | <a>insert</a> _ ih => exact \u27e8<a>insert_size</a> ih.1, <a>insert_WF</a> ih.2\u27e9\n  | <a>erase</a> _ ih => exact \u27e8<a>erase_size</a> ih.1, <a>erase_WF</a> ih.2\u27e9\n  | <a>modify</a> _ ih => exact \u27e8<a>modify_size</a> ih.1, <a>modify_WF</a> ih.2\u27e9", [{"full_name": "Std.HashMap.Imp.WF.mk", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [230, 5], "def_end_pos": [230, 7]}, {"full_name": "Std.HashMap.Imp.WF.empty'", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [232, 5], "def_end_pos": [232, 11]}, {"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]}, {"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.HashMap.Imp.WF.insert", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [234, 5], "def_end_pos": [234, 11]}, {"full_name": "Std.HashMap.Imp.insert_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "Std.HashMap.Imp.insert_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [221, 9], "def_end_pos": [221, 18]}, {"full_name": "Std.HashMap.Imp.WF.erase", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [236, 5], "def_end_pos": [236, 10]}, {"full_name": "Std.HashMap.Imp.erase_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}, {"full_name": "Std.HashMap.Imp.erase_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [257, 9], "def_end_pos": [257, 17]}, {"full_name": "Std.HashMap.Imp.WF.modify", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [238, 5], "def_end_pos": [238, 11]}, {"full_name": "Std.HashMap.Imp.modify_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [265, 9], "def_end_pos": [265, 20]}, {"full_name": "Std.HashMap.Imp.modify_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [273, 9], "def_end_pos": [273, 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\nh : WF m\n\u22a2 m.size = Buckets.size m.buckets \u2227 Buckets.WF m.buckets", "state_after": "no goals"}, {"tactic": "exact \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["exact \u27e8h\u2081, h\u2082\u27e9", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm m\u271d : Imp \u03b1 \u03b2\nh\u2081 : m\u271d.size = Buckets.size m\u271d.buckets\nh\u2082 : Buckets.WF m\u271d.buckets\n\u22a2 m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets", "state_after": "no goals"}, {"tactic": "exact \u27e8(Buckets.mk_size h).symm, .mk' h\u27e9", "annotated_tactic": ["exact \u27e8(<a>Buckets.mk_size</a> h).<a>symm</a>, .mk' h\u27e9", [{"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]}, {"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 empty'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm : Imp \u03b1 \u03b2\nn\u271d : Nat\nh : 0 < n\u271d\n\u22a2 (Imp.empty' n\u271d).size = Buckets.size (Imp.empty' n\u271d).buckets \u2227 Buckets.WF (Imp.empty' n\u271d).buckets", "state_after": "no goals"}, {"tactic": "exact \u27e8insert_size ih.1, insert_WF ih.2\u27e9", "annotated_tactic": ["exact \u27e8<a>insert_size</a> ih.1, <a>insert_WF</a> ih.2\u27e9", [{"full_name": "Std.HashMap.Imp.insert_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}, {"full_name": "Std.HashMap.Imp.insert_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [221, 9], "def_end_pos": [221, 18]}]], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm m\u271d : Imp \u03b1 \u03b2\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\na\u271d : WF m\u271d\nih : m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets\n\u22a2 (Imp.insert m\u271d a\u271d\u00b9 b\u271d).size = Buckets.size (Imp.insert m\u271d a\u271d\u00b9 b\u271d).buckets \u2227 Buckets.WF (Imp.insert m\u271d a\u271d\u00b9 b\u271d).buckets", "state_after": "no goals"}, {"tactic": "exact \u27e8erase_size ih.1, erase_WF ih.2\u27e9", "annotated_tactic": ["exact \u27e8<a>erase_size</a> ih.1, <a>erase_WF</a> ih.2\u27e9", [{"full_name": "Std.HashMap.Imp.erase_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}, {"full_name": "Std.HashMap.Imp.erase_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [257, 9], "def_end_pos": [257, 17]}]], "state_before": "case erase\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm m\u271d : Imp \u03b1 \u03b2\na\u271d\u00b9 : \u03b1\na\u271d : WF m\u271d\nih : m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets\n\u22a2 (Imp.erase m\u271d a\u271d\u00b9).size = Buckets.size (Imp.erase m\u271d a\u271d\u00b9).buckets \u2227 Buckets.WF (Imp.erase m\u271d a\u271d\u00b9).buckets", "state_after": "no goals"}, {"tactic": "exact \u27e8modify_size ih.1, modify_WF ih.2\u27e9", "annotated_tactic": ["exact \u27e8<a>modify_size</a> ih.1, <a>modify_WF</a> ih.2\u27e9", [{"full_name": "Std.HashMap.Imp.modify_size", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [265, 9], "def_end_pos": [265, 20]}, {"full_name": "Std.HashMap.Imp.modify_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [273, 9], "def_end_pos": [273, 18]}]], "state_before": "case modify\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nm m\u271d : Imp \u03b1 \u03b2\na\u271d\u00b9 : \u03b1\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\na\u271d : WF m\u271d\nih : m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets\n\u22a2 (Imp.modify m\u271d a\u271d\u00b9 f\u271d).size = Buckets.size (Imp.modify m\u271d a\u271d\u00b9 f\u271d).buckets \u2227 Buckets.WF (Imp.modify m\u271d a\u271d\u00b9 f\u271d).buckets", "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_assoc'", "start": [723, 11], "end": [725, 57], "traced_tactics": [{"tactic": "rw [Int.mul_comm, Int.mul_div_assoc _ h, Int.mul_comm]", "annotated_tactic": ["rw [<a>Int.mul_comm</a>, <a>Int.mul_div_assoc</a> _ h, <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]}, {"full_name": "Int.mul_div_assoc", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [713, 19], "def_end_pos": [713, 32]}, {"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": "b a c : Int\nh : c \u2223 a\n\u22a2 div (a * b) c = div a c * b", "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.add", "start": [1293, 1], "end": [1296, 20], "traced_tactics": [{"tactic": "refine' induction_on\u2082 f g fun f hf g hg hfi hgi => _", "annotated_tactic": ["refine' <a>induction_on\u2082</a> f g fun f hf g hg hfi hgi => _", [{"full_name": "MeasureTheory.AEEqFun.induction_on\u2082", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [193, 9], "def_end_pos": [193, 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 g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 Integrable f \u2192 Integrable g \u2192 Integrable (f + g)", "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 g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : Integrable (mk f hf)\nhgi : Integrable (mk g hg)\n\u22a2 Integrable (mk f hf + mk g hg)"}, {"tactic": "simp only [integrable_mk, mk_add_mk] at hfi hgi \u22a2", "annotated_tactic": ["simp only [<a>integrable_mk</a>, <a>mk_add_mk</a>] at hfi hgi \u22a2", [{"full_name": "MeasureTheory.AEEqFun.integrable_mk", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1263, 9], "def_end_pos": [1263, 22]}, {"full_name": "MeasureTheory.AEEqFun.mk_add_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [701, 3], "def_end_pos": [701, 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 g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : Integrable (mk f hf)\nhgi : Integrable (mk g hg)\n\u22a2 Integrable (mk f hf + mk g hg)", "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 g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : MeasureTheory.Integrable f\nhgi : MeasureTheory.Integrable g\n\u22a2 MeasureTheory.Integrable (f + g)"}, {"tactic": "exact hfi.add hgi", "annotated_tactic": ["exact hfi.add hgi", []], "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 g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : MeasureTheory.Integrable f\nhgi : MeasureTheory.Integrable g\n\u22a2 MeasureTheory.Integrable (f + g)", "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", "start": [1775, 1], "end": [1788, 43], "traced_tactics": [{"tactic": "have A : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n  rw [closure_eq_iInter_cthickening]\n  exact\n    tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)\n      (fun i j _ ij => cthickening_mono ij _) hs", "annotated_tactic": ["have A : <a>Tendsto</a> (fun r => \u03bc (<a>cthickening</a> r s)) (\ud835\udcdd[<a>Ioi</a> 0] 0) (\ud835\udcdd (\u03bc (<a>closure</a> s))) := by\n    rw [<a>closure_eq_iInter_cthickening</a>]\n    exact\n      <a>tendsto_measure_biInter_gt</a> (fun r _ => isClosed_cthickening.measurableSet)\n        (fun i j _ ij => <a>cthickening_mono</a> ij _) hs", [{"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": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Metric.closure_eq_iInter_cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1385, 9], "def_end_pos": [1385, 38]}, {"full_name": "MeasureTheory.tendsto_measure_biInter_gt", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [547, 9], "def_end_pos": [547, 35]}, {"full_name": "Metric.cthickening_mono", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 25]}]], "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\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))", "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 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))"}, {"tactic": "have B : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n  apply Tendsto.congr' _ tendsto_const_nhds\n  filter_upwards [self_mem_nhdsWithin (\u03b1 := \u211d)] with _ hr\n  rw [cthickening_of_nonpos hr]", "annotated_tactic": ["have B : <a>Tendsto</a> (fun r => \u03bc (<a>cthickening</a> r s)) (\ud835\udcdd[<a>Iic</a> 0] 0) (\ud835\udcdd (\u03bc (<a>closure</a> s))) := by\n    apply <a>Tendsto.congr'</a> _ <a>tendsto_const_nhds</a>\n    filter_upwards [<a>self_mem_nhdsWithin</a> (\u03b1 := \u211d)] with _ hr\n    rw [<a>cthickening_of_nonpos</a> hr]", [{"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": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 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 : 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))", "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 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\nB : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))"}, {"tactic": "convert B.sup A", "annotated_tactic": ["convert B.sup A", []], "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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\nB : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))", "state_after": "case h.e'_4\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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\nB : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 \ud835\udcdd 0 = \ud835\udcdd[Iic 0] 0 \u2294 \ud835\udcdd[Ioi 0] 0"}, {"tactic": "exact (nhds_left_sup_nhds_right' 0).symm", "annotated_tactic": ["exact (<a>nhds_left_sup_nhds_right'</a> 0).<a>symm</a>", [{"full_name": "nhds_left_sup_nhds_right'", "def_path": "Mathlib/Topology/Algebra/Order/LeftRight.lean", "def_pos": [66, 9], "def_end_pos": [66, 34]}, {"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'_4\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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\nB : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 \ud835\udcdd 0 = \ud835\udcdd[Iic 0] 0 \u2294 \ud835\udcdd[Ioi 0] 0", "state_after": "no goals"}, {"tactic": "rw [closure_eq_iInter_cthickening]", "annotated_tactic": ["rw [<a>closure_eq_iInter_cthickening</a>]", [{"full_name": "Metric.closure_eq_iInter_cthickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1385, 9], "def_end_pos": [1385, 38]}]], "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\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))", "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 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\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 \u03b4, \u22c2 (_ : 0 < \u03b4), cthickening \u03b4 s)))"}, {"tactic": "exact\n  tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)\n    (fun i j _ ij => cthickening_mono ij _) hs", "annotated_tactic": ["exact\n      <a>tendsto_measure_biInter_gt</a> (fun r _ => isClosed_cthickening.measurableSet)\n        (fun i j _ ij => <a>cthickening_mono</a> ij _) hs", [{"full_name": "MeasureTheory.tendsto_measure_biInter_gt", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [547, 9], "def_end_pos": [547, 35]}, {"full_name": "Metric.cthickening_mono", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 25]}]], "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\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 \u03b4, \u22c2 (_ : 0 < \u03b4), cthickening \u03b4 s)))", "state_after": "no goals"}, {"tactic": "apply Tendsto.congr' _ tendsto_const_nhds", "annotated_tactic": ["apply <a>Tendsto.congr'</a> _ <a>tendsto_const_nhds</a>", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}, {"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\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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))", "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 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 (fun x => \u2191\u2191\u03bc (closure s)) =\u1da0[\ud835\udcdd[Iic 0] 0] fun r => \u2191\u2191\u03bc (cthickening r s)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin (\u03b1 := \u211d)] with _ hr", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a> (\u03b1 := \u211d)] with _ hr", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 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 : 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\n\u22a2 (fun x => \u2191\u2191\u03bc (closure s)) =\u1da0[\ud835\udcdd[Iic 0] 0] fun r => \u2191\u2191\u03bc (cthickening r s)", "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 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\na\u271d : \u211d\nhr : a\u271d \u2208 Iic 0\n\u22a2 \u2191\u2191\u03bc (closure s) = \u2191\u2191\u03bc (cthickening a\u271d s)"}, {"tactic": "rw [cthickening_of_nonpos hr]", "annotated_tactic": ["rw [<a>cthickening_of_nonpos</a> hr]", [{"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 30]}]], "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 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\nA : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s)))\na\u271d : \u211d\nhr : a\u271d \u2208 Iic 0\n\u22a2 \u2191\u2191\u03bc (closure s) = \u2191\u2191\u03bc (cthickening a\u271d s)", "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_left_Ioc", "start": [1147, 1], "end": [1148, 78], "traced_tactics": [{"tactic": "rw [\u2190 map_add_left_Ioc, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_left_Ioc</a>, <a>map_eq_image</a>, <a>addLeftEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_left_Ioc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 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) (Ioc a b) = Ioc (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "torusIntegral_sub", "start": [177, 1], "end": [179, 85], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg, \u2190 torusIntegral_neg] using torusIntegral_add hf hg.neg", "annotated_tactic": ["simpa only [<a>sub_eq_add_neg</a>, \u2190 <a>torusIntegral_neg</a>] using <a>torusIntegral_add</a> hf hg.neg", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "torusIntegral_neg", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [167, 9], "def_end_pos": [167, 26]}, {"full_name": "torusIntegral_add", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [171, 9], "def_end_pos": [171, 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 g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n\u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x - g x) = (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x) - \u222f (x : Fin n \u2192 \u2102) in T(c, R), g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "full_name": "MvQPF.liftpPreservation_iff_uniform", "start": [282, 1], "end": [283, 80], "traced_tactics": [{"tactic": "rw [\u2190 suppPreservation_iff_liftpPreservation, suppPreservation_iff_isUniform]", "annotated_tactic": ["rw [\u2190 <a>suppPreservation_iff_liftpPreservation</a>, <a>suppPreservation_iff_isUniform</a>]", [{"full_name": "MvQPF.suppPreservation_iff_liftpPreservation", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [267, 9], "def_end_pos": [267, 47]}, {"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": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u22a2 LiftPPreservation \u2194 IsUniform", "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", 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39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.cons_subset_cons", "start": [920, 1], "end": [921, 83], "traced_tactics": [{"tactic": "rwa [\u2190 coe_subset, coe_cons, coe_cons, Set.insert_subset_insert_iff, coe_subset]", "annotated_tactic": ["rwa [\u2190 <a>coe_subset</a>, <a>coe_cons</a>, <a>coe_cons</a>, <a>Set.insert_subset_insert_iff</a>, <a>coe_subset</a>]", [{"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "Finset.coe_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 17]}, {"full_name": "Finset.coe_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 17]}, {"full_name": "Set.insert_subset_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1179, 17], "def_end_pos": [1179, 41]}, {"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nhs : \u00aca \u2208 s\nht : \u00aca \u2208 t\n\u22a2 cons a s hs \u2286 cons a t ht \u2194 s \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.vars_add_of_disjoint", "start": [331, 1], "end": [337, 11], "traced_tactics": [{"tactic": "apply Finset.Subset.antisymm (vars_add_subset p q)", "annotated_tactic": ["apply <a>Finset.Subset.antisymm</a> (<a>vars_add_subset</a> p q)", [{"full_name": "Finset.Subset.antisymm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [367, 9], "def_end_pos": [367, 24]}, {"full_name": "MvPolynomial.vars_add_subset", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [324, 9], "def_end_pos": [324, 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\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nh : Disjoint (vars p) (vars q)\n\u22a2 vars (p + q) = vars p \u222a vars q", "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 q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nh : Disjoint (vars p) (vars q)\n\u22a2 vars p \u222a vars q \u2286 vars (p + q)"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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 : DecidableEq \u03c3\nh : Disjoint (vars p) (vars q)\n\u22a2 vars p \u222a vars q \u2286 vars (p + q)", "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 q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nh : Disjoint (vars p) (vars q)\nx : \u03c3\nhx : x \u2208 vars p \u222a vars q\n\u22a2 x \u2208 vars (p + q)"}, {"tactic": "simp only [vars_def, Multiset.disjoint_toFinset] at h hx \u22a2", "annotated_tactic": ["simp only [<a>vars_def</a>, <a>Multiset.disjoint_toFinset</a>] at h hx \u22a2", [{"full_name": "MvPolynomial.vars_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}, {"full_name": "Multiset.disjoint_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3859, 9], "def_end_pos": [3859, 26]}]], "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 : DecidableEq \u03c3\nh : Disjoint (vars p) (vars q)\nx : \u03c3\nhx : x \u2208 vars p \u222a vars q\n\u22a2 x \u2208 vars (p + q)", "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 q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nx : \u03c3\nh : Multiset.Disjoint (degrees p) (degrees q)\nhx : x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q)\n\u22a2 x \u2208 Multiset.toFinset (degrees (p + q))"}, {"tactic": "rw [degrees_add_of_disjoint h, Multiset.toFinset_union]", "annotated_tactic": ["rw [<a>degrees_add_of_disjoint</a> h, <a>Multiset.toFinset_union</a>]", [{"full_name": "MvPolynomial.degrees_add_of_disjoint", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [216, 9], "def_end_pos": [216, 32]}, {"full_name": "Multiset.toFinset_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3239, 9], "def_end_pos": [3239, 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 q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nx : \u03c3\nh : Multiset.Disjoint (degrees p) (degrees q)\nhx : x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q)\n\u22a2 x \u2208 Multiset.toFinset (degrees (p + q))", "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 q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nx : \u03c3\nh : Multiset.Disjoint (degrees p) (degrees q)\nhx : x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q)\n\u22a2 x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q)"}, {"tactic": "exact hx", "annotated_tactic": ["exact hx", []], "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 : DecidableEq \u03c3\nx : \u03c3\nh : Multiset.Disjoint (degrees p) (degrees q)\nhx : x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q)\n\u22a2 x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.le_total_of_le_of_le", "start": [407, 1], "end": [417, 11], "traced_tactics": [{"tactic": "rcases Part.eq_none_or_eq_some x with (h | \u27e8b, h\u2080\u27e9)", "annotated_tactic": ["rcases <a>Part.eq_none_or_eq_some</a> x with (h | \u27e8b, h\u2080\u27e9)", [{"full_name": "Part.eq_none_or_eq_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [210, 9], "def_end_pos": [210, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\n\u22a2 x \u2264 y \u2228 y \u2264 x", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nh : x = none\n\u22a2 x \u2264 y \u2228 y \u2264 x\n\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : x = some b\n\u22a2 x \u2264 y \u2228 y \u2264 x"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : x = some b\n\u22a2 x \u2264 y \u2228 y \u2264 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : x = some b\n\u22a2 y \u2264 x"}, {"tactic": "intro b' h\u2081", "annotated_tactic": ["intro b' h\u2081", []], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : x = some b\n\u22a2 y \u2264 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : x = some b\nb' : \u03b1\nh\u2081 : b' \u2208 y\n\u22a2 b' \u2208 x"}, {"tactic": "rw [Part.eq_some_iff] at h\u2080", "annotated_tactic": ["rw [<a>Part.eq_some_iff</a>] at h\u2080", [{"full_name": "Part.eq_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [174, 9], "def_end_pos": [174, 20]}]], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : x = some b\nb' : \u03b1\nh\u2081 : b' \u2208 y\n\u22a2 b' \u2208 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\n\u22a2 b' \u2208 x"}, {"tactic": "have hx := hx _ h\u2080", "annotated_tactic": ["have hx := hx _ h\u2080", []], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\n\u22a2 b' \u2208 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\nhx : b \u2208 z\n\u22a2 b' \u2208 x"}, {"tactic": "have hy := hy _ h\u2081", "annotated_tactic": ["have hy := hy _ h\u2081", []], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d : x \u2264 z\nhy : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\nhx : b \u2208 z\n\u22a2 b' \u2208 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d : x \u2264 z\nhy\u271d : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\nhx : b \u2208 z\nhy : b' \u2208 z\n\u22a2 b' \u2208 x"}, {"tactic": "have hx := Part.mem_unique hx hy", "annotated_tactic": ["have hx := <a>Part.mem_unique</a> hx hy", [{"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}]], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d : x \u2264 z\nhy\u271d : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\nhx : b \u2208 z\nhy : b' \u2208 z\n\u22a2 b' \u2208 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d\u00b9 : x \u2264 z\nhy\u271d : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\nhx\u271d : b \u2208 z\nhy : b' \u2208 z\nhx : b = b'\n\u22a2 b' \u2208 x"}, {"tactic": "subst hx", "annotated_tactic": ["subst hx", []], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d\u00b9 : x \u2264 z\nhy\u271d : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nb' : \u03b1\nh\u2081 : b' \u2208 y\nhx\u271d : b \u2208 z\nhy : b' \u2208 z\nhx : b = b'\n\u22a2 b' \u2208 x", "state_after": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d : x \u2264 z\nhy\u271d : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nhx : b \u2208 z\nh\u2081 : b \u2208 y\nhy : b \u2208 z\n\u22a2 b \u2208 x"}, {"tactic": "exact h\u2080", "annotated_tactic": ["exact h\u2080", []], "state_before": "case inr.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx\u271d : x \u2264 z\nhy\u271d : y \u2264 z\nb : \u03b1\nh\u2080 : b \u2208 x\nhx : b \u2208 z\nh\u2081 : b \u2208 y\nhy : b \u2208 z\n\u22a2 b \u2208 x", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nh : x = none\n\u22a2 x \u2264 y \u2228 y \u2264 x", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nh : x = none\n\u22a2 none \u2264 y \u2228 y \u2264 none"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nh : x = none\n\u22a2 none \u2264 y \u2228 y \u2264 none", "state_after": "case inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nh : x = none\n\u22a2 none \u2264 y"}, {"tactic": "apply OrderBot.bot_le _", "annotated_tactic": ["apply <a>OrderBot.bot_le</a> _", [{"full_name": "OrderBot.bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [236, 3], "def_end_pos": [236, 9]}]], "state_before": "case inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : Part \u03b1\nhx : x \u2264 z\nhy : y \u2264 z\nh : x = none\n\u22a2 none \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnected.lean", "full_name": "Set.dual_ordConnected", "start": [222, 1], "end": [223, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.kernel.integral_integral_sub", "start": [201, 1], "end": [206, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.integrable_iff_integrable_mul_pdf", "start": [154, 1], "end": [160, 95], "traced_tactics": [{"tactic": "erw [\u2190 integrable_map_measure hf.aestronglyMeasurable (HasPDF.measurable X \u2119 \u03bc).aemeasurable,\n  map_eq_withDensity_pdf X \u2119 \u03bc, integrable_withDensity_iff (measurable_pdf _ _ _) ae_lt_top]", "annotated_tactic": ["erw [\u2190 <a>integrable_map_measure</a> hf.aestronglyMeasurable (<a>HasPDF.measurable</a> X \u2119 \u03bc).<a>aemeasurable</a>,\n    <a>map_eq_withDensity_pdf</a> X \u2119 \u03bc, <a>integrable_withDensity_iff</a> (<a>measurable_pdf</a> _ _ _) <a>ae_lt_top</a>]", [{"full_name": "MeasureTheory.integrable_map_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [610, 9], "def_end_pos": [610, 31]}, {"full_name": "MeasureTheory.HasPDF.measurable", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [74, 9], "def_end_pos": [74, 26]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "MeasureTheory.map_eq_withDensity_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [116, 9], "def_end_pos": [116, 31]}, {"full_name": "MeasureTheory.integrable_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [957, 9], "def_end_pos": [957, 35]}, {"full_name": "MeasureTheory.measurable_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [109, 9], "def_end_pos": [109, 23]}, {"full_name": "MeasureTheory.pdf.ae_lt_top", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [139, 16], "def_end_pos": [139, 25]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\n\u22a2 (Integrable fun x => f (X x)) \u2194 Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)", "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.mapKey_toList", "start": [85, 9], "end": [87, 27], "traced_tactics": [{"tactic": "induction l <;> simp [*]", "annotated_tactic": ["induction l <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\n\u03b4 : Type u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b4\nl : AssocList \u03b1 \u03b2\n\u22a2 toList (mapKey f l) =\n    List.map\n      (fun x =>\n        match x with\n        | (a, b) => (f a, b))\n      (toList l)", "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_inter_support", "start": [215, 1], "end": [217, 77], "traced_tactics": [{"tactic": "simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support]", "annotated_tactic": ["simp only [<a>toOuterMeasure_apply</a>, <a>PMF.support</a>, <a>Set.indicator_inter_support</a>]", [{"full_name": "PMF.toOuterMeasure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 29]}, {"full_name": "PMF.support", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [89, 5], "def_end_pos": [89, 12]}, {"full_name": "Set.indicator_inter_support", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [230, 3], "def_end_pos": [230, 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 \u2191(toOuterMeasure p) (s \u2229 support p) = \u2191(toOuterMeasure p) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.mk_preimage_prod_right_fn_eq_if", "start": [286, 1], "end": [288, 78], "traced_tactics": [{"tactic": "rw [\u2190 mk_preimage_prod_right_eq_if, prod_preimage_right, preimage_preimage]", "annotated_tactic": ["rw [\u2190 <a>mk_preimage_prod_right_eq_if</a>, <a>prod_preimage_right</a>, <a>preimage_preimage</a>]", [{"full_name": "Set.mk_preimage_prod_right_eq_if", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [277, 9], "def_end_pos": [277, 37]}, {"full_name": "Set.prod_preimage_right", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [234, 9], "def_end_pos": [234, 28]}, {"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [176, 9], "def_end_pos": [176, 26]}]], "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\ninst\u271d : DecidablePred fun x => x \u2208 s\ng : \u03b4 \u2192 \u03b2\n\u22a2 (fun b => (a, g b)) \u207b\u00b9' s \u00d7\u02e2 t = if a \u2208 s then g \u207b\u00b9' t else \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_pi_closedBall", "start": [687, 1], "end": [690, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.stoppedValue_sub_eq_sum'", "start": [1071, 1], "end": [1080, 81], "traced_tactics": [{"tactic": "rw [stoppedValue_sub_eq_sum hle]", "annotated_tactic": ["rw [<a>stoppedValue_sub_eq_sum</a> hle]", [{"full_name": "MeasureTheory.stoppedValue_sub_eq_sum", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 32]}]], "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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 stoppedValue u \u03c0 - stoppedValue u \u03c4 = fun \u03c9 =>\n    Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - u i)) \u03c9", "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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 (fun \u03c9 => Finset.sum (Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9)) (fun i => u (i + 1) - u i) \u03c9) = fun \u03c9 =>\n    Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - u i)) \u03c9"}, {"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\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u22a2 (fun \u03c9 => Finset.sum (Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9)) (fun i => u (i + 1) - u i) \u03c9) = fun \u03c9 =>\n    Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - 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\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\n\u22a2 Finset.sum (Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9)) (fun i => u (i + 1) - u i) \u03c9 =\n    Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - u i)) \u03c9"}, {"tactic": "simp only [Finset.sum_apply, Finset.sum_indicator_eq_sum_filter]", "annotated_tactic": ["simp only [<a>Finset.sum_apply</a>, <a>Finset.sum_indicator_eq_sum_filter</a>]", [{"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_indicator_eq_sum_filter", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [648, 3], "def_end_pos": [648, 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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\n\u22a2 Finset.sum (Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9)) (fun i => u (i + 1) - u i) \u03c9 =\n    Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - 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\nf : Filtration \u2115 m\nu : \u2115 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\n\u22a2 \u2211 c in Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9), (u (c + 1) - u c) \u03c9 =\n    \u2211 c in Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1)), (u (c + 1) - u c) \u03c9"}, {"tactic": "refine' Finset.sum_congr _ fun _ _ => rfl", "annotated_tactic": ["refine' <a>Finset.sum_congr</a> _ fun _ _ => <a>rfl</a>", [{"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": "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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\n\u22a2 \u2211 c in Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9), (u (c + 1) - u c) \u03c9 =\n    \u2211 c in Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1)), (u (c + 1) - u c) \u03c9", "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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\n\u22a2 Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9) = Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1))"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\n\u22a2 Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9) = Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1))", "state_after": "case h.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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\ni : \u2115\n\u22a2 i \u2208 Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9) \u2194 i \u2208 Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1))"}, {"tactic": "simp only [Finset.mem_filter, Set.mem_setOf_eq, Finset.mem_range, Finset.mem_Ico]", "annotated_tactic": ["simp only [<a>Finset.mem_filter</a>, <a>Set.mem_setOf_eq</a>, <a>Finset.mem_range</a>, <a>Finset.mem_Ico</a>]", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 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": "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]}]], "state_before": "case h.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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\ni : \u2115\n\u22a2 i \u2208 Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9) \u2194 i \u2208 Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1))", "state_after": "case h.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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\ni : \u2115\n\u22a2 \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9 \u2194 i < N + 1 \u2227 \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9"}, {"tactic": "exact \u27e8fun h => \u27e8lt_trans h.2 (Nat.lt_succ_iff.2 <| hbdd _), h\u27e9, fun h => h.2\u27e9", "annotated_tactic": ["exact \u27e8fun h => \u27e8<a>lt_trans</a> h.2 (<a>Nat.lt_succ_iff</a>.2 <| hbdd _), h\u27e9, fun h => h.2\u27e9", [{"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}, {"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 h.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 : AddCommGroup \u03b2\nhle : \u03c4 \u2264 \u03c0\nN : \u2115\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N\n\u03c9 : \u03a9\ni : \u2115\n\u22a2 \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9 \u2194 i < N + 1 \u2227 \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.eval\u2082Hom_bind\u2082", "start": [323, 1], "end": [325, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.StronglyMeasurable.integral_prod_right", "start": [77, 1], "end": [121, 49], "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : \u00acCompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd"}, {"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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : \u00acCompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : \u00acCompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd"}, {"tactic": "borelize E", "annotated_tactic": ["borelize E", []], "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd"}, {"tactic": "haveI : SeparableSpace (range (uncurry f) \u222a {0} : Set E) :=\n  hf.separableSpace_range_union_singleton", "annotated_tactic": ["haveI : <a>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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd"}, {"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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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\u03bd"}, {"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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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\u03bd"}, {"tactic": "let f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => {x | Integrable (f x) \u03bd}.indicator fun x => (s' n x).integral \u03bd", "annotated_tactic": ["let f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => {x | <a>Integrable</a> (f x) \u03bd}.<a>indicator</a> fun x => (s' n x).<a>integral</a> \u03bd", [{"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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd"}, {"tactic": "have hf' : \u2200 n, StronglyMeasurable (f' n) := by\n  intro n; refine' StronglyMeasurable.indicator _ (measurableSet_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 measurable_measure_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_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>measurable_measure_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": "measurableSet_integrable", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [64, 9], "def_end_pos": [64, 33]}, {"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": "measurable_measure_prod_mk_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 40]}, {"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 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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd"}, {"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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "state_after": "no goals"}, {"tactic": "simp [integral, hE, stronglyMeasurable_const]", "annotated_tactic": ["simp [<a>integral</a>, hE, <a>stronglyMeasurable_const</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : \u00acCompleteSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\n\u22a2 \u2200 (n : \u2115), StronglyMeasurable (f' n)", "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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable (f' n)"}, {"tactic": "refine' StronglyMeasurable.indicator _ (measurableSet_integrable hf)", "annotated_tactic": ["refine' <a>StronglyMeasurable.indicator</a> _ (<a>measurableSet_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": "measurableSet_integrable", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [64, 9], "def_end_pos": [64, 33]}]], "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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable (f' n)", "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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral \u03bd (s' n 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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 \u03bd (s' n 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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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 measurable_measure_prod_mk_left", "annotated_tactic": ["apply <a>measurable_measure_prod_mk_left</a>", [{"full_name": "measurable_measure_prod_mk_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 40]}]], "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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd\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 hs\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 hs\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))"}, {"tactic": "by_cases hfx : Integrable (f x) \u03bd", "annotated_tactic": ["by_cases hfx : <a>Integrable</a> (f x) \u03bd", [{"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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))"}, {"tactic": "have : \u2200 n, Integrable (s' n x) \u03bd := 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) \u03bd := 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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd))"}, {"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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd))", "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd)\n    atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))"}, {"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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd)\n    atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))", "state_after": "case pos.refine'_1\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd,\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd,\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd, \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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd, \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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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": "refine' fun n => eventually_of_forall fun y => SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n", "annotated_tactic": ["refine' fun n => <a>eventually_of_forall</a> fun y => <a>SimpleFunc.norm_approxOn_zero_le</a> _ _ (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd,\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": "case pos.refine'_1.refine'_1\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 Measurable (uncurry f)\n\ncase pos.refine'_1.refine'_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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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}"}, {"tactic": "exact hf.measurable", "annotated_tactic": ["exact hf.measurable", []], "state_before": "case pos.refine'_1.refine'_1\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 Measurable (uncurry f)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos.refine'_1.refine'_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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 _ _ _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun y => <a>SimpleFunc.tendsto_approxOn</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": "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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\u03bd,\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.refine'_1\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 Measurable fun x_1 => uncurry f (x, x_1)\n\ncase pos.refine'_2.refine'_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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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}\n\ncase pos.refine'_2.refine'_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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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.refine'_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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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.refine'_3.a\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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.refine'_3.a\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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": "exact hf.measurable.of_uncurry_left", "annotated_tactic": ["exact hf.measurable.of_uncurry_left", []], "state_before": "case pos.refine'_2.refine'_1\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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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 Measurable fun x_1 => uncurry f (x, x_1)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos.refine'_2.refine'_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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : 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 \u03bd (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\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 : NormedSpace \u211d E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : 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 \u03bd (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\u03bd))", "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_mul_cancel", "start": [701, 11], "end": [702, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UnionFind.lt_rankMax", "start": [194, 1], "end": [195, 62], "traced_tactics": [{"tactic": "simp [rank]", "annotated_tactic": ["simp [<a>rank</a>]", [{"full_name": "UnionFind.rank", "def_path": "Mathlib/Data/UnionFind.lean", "def_pos": [175, 5], "def_end_pos": [175, 9]}]], "state_before": "\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\n\u22a2 rank self i < rankMax self", "state_after": "\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\n\u22a2 (if h : i < size self then self.arr[i].rank else 0) < rankMax self"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\n\u22a2 (if h : i < size self then self.arr[i].rank else 0) < rankMax self", "state_after": "case inl\n\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\nh\u271d : i < size self\n\u22a2 self.arr[i].rank < rankMax self\n\ncase inr\n\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\nh\u271d : \u00aci < size self\n\u22a2 0 < rankMax self"}, {"tactic": "{apply lt_rankMax'}", "annotated_tactic": ["{apply <a>lt_rankMax'</a>}", [{"full_name": "UnionFind.lt_rankMax'", "def_path": "Mathlib/Data/UnionFind.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}]], "state_before": "case inl\n\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\nh\u271d : i < size self\n\u22a2 self.arr[i].rank < rankMax self\n\ncase inr\n\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\nh\u271d : \u00aci < size self\n\u22a2 0 < rankMax self", "state_after": "case inr\n\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\nh\u271d : \u00aci < size self\n\u22a2 0 < rankMax self"}, {"tactic": "apply Nat.succ_pos", "annotated_tactic": ["apply <a>Nat.succ_pos</a>", [{"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 inr\n\u03b1 : Type u_1\nself : UnionFind \u03b1\ni : \u2115\nh\u271d : \u00aci < size self\n\u22a2 0 < rankMax self", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.mem_spanningSetsIndex", "start": [3368, 1], "end": [3370, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.Integrable.lintegral_lt_top", "start": [373, 1], "end": [377, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Cardinal.lean", "full_name": "WType.cardinal_mk_eq_sum", "start": [41, 1], "end": [43, 32], "traced_tactics": [{"tactic": "simp only [Cardinal.power_def, \u2190 Cardinal.mk_sigma]", "annotated_tactic": ["simp only [<a>Cardinal.power_def</a>, \u2190 <a>Cardinal.mk_sigma</a>]", [{"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]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u22a2 #(WType \u03b2) = sum fun a => #(WType \u03b2) ^ #(\u03b2 a)", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u22a2 #(WType \u03b2) = #((i : \u03b1) \u00d7 (\u03b2 i \u2192 WType \u03b2))"}, {"tactic": "exact mk_congr (equivSigma \u03b2)", "annotated_tactic": ["exact <a>mk_congr</a> (<a>equivSigma</a> \u03b2)", [{"full_name": "Cardinal.mk_congr", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}, {"full_name": "WType.equivSigma", "def_path": "Mathlib/Data/W/Basic.lean", "def_pos": [75, 5], "def_end_pos": [75, 15]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u22a2 #(WType \u03b2) = #((i : \u03b1) \u00d7 (\u03b2 i \u2192 WType \u03b2))", "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_iff_tendsto", "start": [486, 1], "end": [495, 71], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "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\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2194\n    Tendsto \u03bcs F (\ud835\udcdd \u03bc)", "state_after": "case mp\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\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2192\n    Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n\ncase mpr\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\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192\n    Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))"}, {"tactic": "rintro \u27e8normalized_lim, mass_lim\u27e9", "annotated_tactic": ["rintro \u27e8normalized_lim, mass_lim\u27e9", []], "state_before": "case mp\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\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2192\n    Tendsto \u03bcs F (\ud835\udcdd \u03bc)", "state_after": "case mp.intro\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\nnonzero : \u03bc \u2260 0\nnormalized_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))\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc)"}, {"tactic": "exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim", "annotated_tactic": ["exact <a>tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass</a> normalized_lim mass_lim", [{"full_name": 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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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x"}, {"tactic": "rw [ae_const_le_iff_forall_lt_measure_zero]", "annotated_tactic": ["rw [<a>ae_const_le_iff_forall_lt_measure_zero</a>]", [{"full_name": "MeasureTheory.ae_const_le_iff_forall_lt_measure_zero", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [126, 9], "def_end_pos": [126, 47]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x", "state_after": "\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200 (b : \u211d), b < 0 \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "intro b hb_neg", "annotated_tactic": ["intro b hb_neg", []], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200 (b : \u211d), b < 0 \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "let s := {x | f x \u2264 b}", "annotated_tactic": ["let s := {x | f x \u2264 b}", []], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const", "annotated_tactic": ["have hs : <a>MeasurableSet</a> s := hfm.measurableSet_le <a>stronglyMeasurable_const</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"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\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "have mus : \u03bc s < \u221e := by\n  let c : \u211d\u22650 := \u27e8|b|, abs_nonneg _\u27e9\n  have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne\n  calc\n    \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n      apply measure_mono\n      intro x hx\n      simp only [Set.mem_setOf_eq] at hx\n      simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,\n        Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n    _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c :=\n      (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)\n    _ < \u221e := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos", "annotated_tactic": ["have mus : \u03bc s < \u221e := by\n    let c : \u211d\u22650 := \u27e8|b|, <a>abs_nonneg</a> _\u27e9\n    have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 <a>NNReal.coe_eq_zero</a>] using hb_neg.ne\n    calc\n      \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n        apply <a>measure_mono</a>\n        intro x hx\n        simp only [<a>Set.mem_setOf_eq</a>] at hx\n        simpa only [<a>nnnorm</a>, <a>abs_of_neg</a> hb_neg, <a>abs_of_neg</a> (hx.trans_lt hb_neg), <a>Real.norm_eq_abs</a>,\n          <a>Subtype.mk_le_mk</a>, <a>neg_le_neg_iff</a>, <a>Set.mem_setOf_eq</a>, <a>ENNReal.coe_le_coe</a>, <a>NNReal</a>] using hx\n      _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c :=\n        (<a>meas_ge_le_lintegral_div</a> hfm.aemeasurable.ennnorm c_pos <a>ENNReal.coe_ne_top</a>)\n      _ < \u221e := <a>ENNReal.div_lt_top</a> (<a>ne_of_lt</a> hf.2) c_pos", [{"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "NNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [212, 19], "def_end_pos": [212, 30]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 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": "NNNorm.nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [66, 3], "def_end_pos": [66, 9]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"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": "Subtype.mk_le_mk", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1150, 9], "def_end_pos": [1150, 17]}, {"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": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "NNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [59, 5], "def_end_pos": [59, 11]}, {"full_name": "MeasureTheory.meas_ge_le_lintegral_div", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [864, 9], "def_end_pos": [864, 33]}, {"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.div_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1460, 9], "def_end_pos": [1460, 19]}, {"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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "have h_int_gt : (\u222b x in s, f x \u2202\u03bc) \u2264 b * (\u03bc s).toReal := by\n  have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc := by\n    refine'\n      set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n    rw [EventuallyLE, ae_restrict_iff hs]\n    exact eventually_of_forall fun x hxs => hxs\n  rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le", "annotated_tactic": ["have h_int_gt : (\u222b x in s, f x \u2202\u03bc) \u2264 b * (\u03bc s).<a>toReal</a> := by\n    have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc := by\n      refine'\n        <a>set_integral_mono_ae_restrict</a> hf.integrableOn (integrableOn_const.mpr (<a>Or.inr</a> mus)) _\n      rw [<a>EventuallyLE</a>, <a>ae_restrict_iff</a> hs]\n      exact <a>eventually_of_forall</a> fun x hxs => hxs\n    rwa [<a>set_integral_const</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>] at h_const_le", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.set_integral_mono_ae_restrict", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [712, 9], "def_end_pos": [712, 38]}, {"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": "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": [2560, 9], "def_end_pos": [2560, 24]}, {"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.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}, {"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]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 False"}, {"tactic": "refine' (lt_self_iff_false (\u222b x in s, f x \u2202\u03bc)).mp (h_int_gt.trans_lt _)", "annotated_tactic": ["refine' (<a>lt_self_iff_false</a> (\u222b x in s, f x \u2202\u03bc)).<a>mp</a> (h_int_gt.trans_lt _)", [{"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]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 False", "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 b * ENNReal.toReal (\u2191\u2191\u03bc s) < \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "refine' (mul_neg_iff.mpr (Or.inr \u27e8hb_neg, _\u27e9)).trans_le _", "annotated_tactic": ["refine' (mul_neg_iff.mpr (<a>Or.inr</a> \u27e8hb_neg, _\u27e9)).<a>trans_le</a> _", [{"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.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 b * ENNReal.toReal (\u2191\u2191\u03bc s) < \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "case refine'_1\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n\ncase refine'_2\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine'_1\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n\ncase refine'_2\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "case refine'_2\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\ncase refine'_1\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _", "annotated_tactic": ["refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _", []], "state_before": "case refine'_1\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)", "state_after": "case refine'_1\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with h\u03bcs_eq_zero h\u03bcs_eq_top", "annotated_tactic": ["cases' (<a>ENNReal.toReal_eq_zero_iff</a> _).<a>mp</a> h_eq.symm with h\u03bcs_eq_zero h\u03bcs_eq_top", [{"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": "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'_1\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "case refine'_1.inl\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\nh\u03bcs_eq_zero : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\ncase refine'_1.inr\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\nh\u03bcs_eq_top : \u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "let c : \u211d\u22650 := \u27e8|b|, abs_nonneg _\u27e9", "annotated_tactic": ["let c : \u211d\u22650 := \u27e8|b|, <a>abs_nonneg</a> _\u27e9", [{"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s < \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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\n\u22a2 \u2191\u2191\u03bc s < \u22a4"}, {"tactic": "have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne", "annotated_tactic": ["have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 <a>NNReal.coe_eq_zero</a>] using hb_neg.ne", [{"full_name": "NNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [212, 19], "def_end_pos": [212, 30]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\n\u22a2 \u2191\u2191\u03bc s < \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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4"}, {"tactic": "calc\n  \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n    apply measure_mono\n    intro x hx\n    simp only [Set.mem_setOf_eq] at hx\n    simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,\n      Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n  _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c :=\n    (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)\n  _ < \u221e := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos", "annotated_tactic": ["calc\n      \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n        apply <a>measure_mono</a>\n        intro x hx\n        simp only [<a>Set.mem_setOf_eq</a>] at hx\n        simpa only [<a>nnnorm</a>, <a>abs_of_neg</a> hb_neg, <a>abs_of_neg</a> (hx.trans_lt hb_neg), <a>Real.norm_eq_abs</a>,\n          <a>Subtype.mk_le_mk</a>, <a>neg_le_neg_iff</a>, <a>Set.mem_setOf_eq</a>, <a>ENNReal.coe_le_coe</a>, <a>NNReal</a>] using hx\n      _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c :=\n        (<a>meas_ge_le_lintegral_div</a> hfm.aemeasurable.ennnorm c_pos <a>ENNReal.coe_ne_top</a>)\n      _ < \u221e := <a>ENNReal.div_lt_top</a> (<a>ne_of_lt</a> hf.2) c_pos", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 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": "NNNorm.nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [66, 3], "def_end_pos": [66, 9]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"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": "Subtype.mk_le_mk", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1150, 9], "def_end_pos": [1150, 17]}, {"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": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "NNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [59, 5], "def_end_pos": [59, 11]}, {"full_name": "MeasureTheory.meas_ge_le_lintegral_div", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [864, 9], "def_end_pos": [864, 33]}, {"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.div_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1460, 9], "def_end_pos": [1460, 19]}, {"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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne", "annotated_tactic": ["simpa [\u2190 <a>NNReal.coe_eq_zero</a>] using hb_neg.ne", [{"full_name": "NNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [212, 19], "def_end_pos": [212, 30]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\n\u22a2 \u2191c \u2260 0", "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\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 s \u2286 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 s \u2286 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}"}, {"tactic": "simp only [Set.mem_setOf_eq] at hx", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>] at hx", [{"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\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}", "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\nx : \u03b1\nhx : f x \u2264 b\n\u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}"}, {"tactic": "simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,\n  Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx", "annotated_tactic": ["simpa only [<a>nnnorm</a>, <a>abs_of_neg</a> hb_neg, <a>abs_of_neg</a> (hx.trans_lt hb_neg), <a>Real.norm_eq_abs</a>,\n          <a>Subtype.mk_le_mk</a>, <a>neg_le_neg_iff</a>, <a>Set.mem_setOf_eq</a>, <a>ENNReal.coe_le_coe</a>, <a>NNReal</a>] using hx", [{"full_name": "NNNorm.nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [66, 3], "def_end_pos": [66, 9]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"full_name": "abs_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [119, 9], "def_end_pos": [119, 19]}, {"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": "Subtype.mk_le_mk", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1150, 9], "def_end_pos": [1150, 17]}, {"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": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}, {"full_name": "NNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [59, 5], "def_end_pos": [59, 11]}]], "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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\nx : \u03b1\nhx : f x \u2264 b\n\u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}", "state_after": "no goals"}, {"tactic": "have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc := by\n  refine'\n    set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n  rw [EventuallyLE, ae_restrict_iff hs]\n  exact eventually_of_forall fun x hxs => hxs", "annotated_tactic": ["have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc := by\n      refine'\n        <a>set_integral_mono_ae_restrict</a> hf.integrableOn (integrableOn_const.mpr (<a>Or.inr</a> mus)) _\n      rw [<a>EventuallyLE</a>, <a>ae_restrict_iff</a> hs]\n      exact <a>eventually_of_forall</a> fun x hxs => hxs", [{"full_name": "MeasureTheory.set_integral_mono_ae_restrict", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [712, 9], "def_end_pos": [712, 38]}, {"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": "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": [2560, 9], "def_end_pos": [2560, 24]}, {"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\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_const_le : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le", "annotated_tactic": ["rwa [<a>set_integral_const</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>] at h_const_le", [{"full_name": "MeasureTheory.set_integral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 27]}, {"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]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_const_le : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)", "state_after": "no goals"}, {"tactic": "refine'\n  set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _", "annotated_tactic": ["refine'\n        <a>set_integral_mono_ae_restrict</a> hf.integrableOn (integrableOn_const.mpr (<a>Or.inr</a> mus)) _", [{"full_name": "MeasureTheory.set_integral_mono_ae_restrict", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [712, 9], "def_end_pos": [712, 38]}, {"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\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 (fun x => f x) \u2264\u1d50[Measure.restrict \u03bc s] fun x => b"}, {"tactic": "rw [EventuallyLE, ae_restrict_iff hs]", "annotated_tactic": ["rw [<a>EventuallyLE</a>, <a>ae_restrict_iff</a> hs]", [{"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": [2560, 9], "def_end_pos": [2560, 24]}]], "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 (fun x => f x) \u2264\u1d50[Measure.restrict \u03bc s] fun x => b", "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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 b"}, {"tactic": "exact eventually_of_forall fun x hxs => hxs", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x hxs => hxs", [{"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\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 b", "state_after": "no goals"}, {"tactic": "exact hf_zero s hs mus", "annotated_tactic": ["exact hf_zero s hs mus", []], "state_before": "case refine'_2\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact h\u03bcs_eq_zero", "annotated_tactic": ["exact h\u03bcs_eq_zero", []], "state_before": "case refine'_1.inl\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\nh\u03bcs_eq_zero : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "no goals"}, {"tactic": "exact absurd h\u03bcs_eq_top mus.ne", "annotated_tactic": ["exact <a>absurd</a> h\u03bcs_eq_top mus.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 refine'_1.inr\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 : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\nh\u03bcs_eq_top : \u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "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_norm_ge_le", "start": [259, 1], "end": [293, 19], "traced_tactics": [{"tactic": "by_cases hp_ne_zero : p = 0", "annotated_tactic": ["by_cases hp_ne_zero : p = 0", []], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : p = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "by_cases hp_ne_top : p = \u221e", "annotated_tactic": ["by_cases hp_ne_top : p = \u221e", []], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : p = \u22a4\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8M, hM', hM\u27e9 := Mem\u2112p.integral_indicator_norm_ge_nonneg_le\n  (\u03bc := \u03bc) (hf.norm_rpow hp_ne_zero hp_ne_top) (Real.rpow_pos_of_pos h\u03b5 p.toReal)", "annotated_tactic": ["obtain \u27e8M, hM', hM\u27e9 := <a>Mem\u2112p.integral_indicator_norm_ge_nonneg_le</a>\n    (\u03bc := \u03bc) (hf.norm_rpow hp_ne_zero hp_ne_top) (<a>Real.rpow_pos_of_pos</a> h\u03b5 p.toReal)", [{"full_name": "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_nonneg_le", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [220, 9], "def_end_pos": [220, 51]}, {"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 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8M ^ (1 / p.toReal), _\u27e9", "annotated_tactic": ["refine' \u27e8M ^ (1 / p.toReal), _\u27e9", []], "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 snorm (Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "rw [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top, \u2190 ENNReal.rpow_one (ENNReal.ofReal \u03b5)]", "annotated_tactic": ["rw [<a>snorm_eq_lintegral_rpow_nnnorm</a> hp_ne_zero hp_ne_top, \u2190 <a>ENNReal.rpow_one</a> (<a>ENNReal.ofReal</a> \u03b5)]", [{"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": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 snorm (Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal \u03b5 ^ 1"}, {"tactic": "conv_rhs => rw [\u2190 mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>mul_one_div_cancel</a> (<a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top).ne.symm]", [{"full_name": "mul_one_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [75, 9], "def_end_pos": [75, 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.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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal \u03b5 ^ 1", "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal \u03b5 ^ (ENNReal.toReal p * (1 / ENNReal.toReal p))"}, {"tactic": "rw [ENNReal.rpow_mul,\n  ENNReal.rpow_le_rpow_iff (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top),\n  ENNReal.ofReal_rpow_of_pos h\u03b5]", "annotated_tactic": ["rw [<a>ENNReal.rpow_mul</a>,\n    <a>ENNReal.rpow_le_rpow_iff</a> (<a>one_div_pos</a>.2 <| <a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top),\n    <a>ENNReal.ofReal_rpow_of_pos</a> h\u03b5]", [{"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [532, 9], "def_end_pos": [532, 17]}, {"full_name": "ENNReal.rpow_le_rpow_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [650, 9], "def_end_pos": [650, 25]}, {"full_name": "one_div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [73, 9], "def_end_pos": [73, 20]}, {"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]}]], "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n      (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal \u03b5 ^ (ENNReal.toReal p * (1 / ENNReal.toReal p))", "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)"}, {"tactic": "convert hM", "annotated_tactic": ["convert hM", []], "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)", "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx\u271d : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u271d\u2016\u208a ^ ENNReal.toReal p =\n    \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u271d\u2016\u208a"}, {"tactic": "rename_i x", "annotated_tactic": ["rename_i x", []], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx\u271d : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u271d\u2016\u208a ^ ENNReal.toReal p =\n    \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u271d\u2016\u208a", "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p =\n    \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a"}, {"tactic": "rw [ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg, nnnorm_indicator_eq_indicator_nnnorm,\n  nnnorm_indicator_eq_indicator_nnnorm]", "annotated_tactic": ["rw [<a>ENNReal.coe_rpow_of_nonneg</a> _ <a>ENNReal.toReal_nonneg</a>, <a>nnnorm_indicator_eq_indicator_nnnorm</a>,\n    <a>nnnorm_indicator_eq_indicator_nnnorm</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.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}, {"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": "nnnorm_indicator_eq_indicator_nnnorm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [29, 9], "def_end_pos": [29, 45]}]], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\n\u22a2 \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p =\n    \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a", "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x)"}, {"tactic": "have hiff : M ^ (1 / p.toReal) \u2264 \u2016f x\u2016\u208a \u2194 M \u2264 \u2016\u2016f x\u2016 ^ p.toReal\u2016\u208a := by\n  rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,\n    \u2190 Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _)\n      (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top), \u2190 Real.rpow_mul (norm_nonneg _),\n    mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one]", "annotated_tactic": ["have hiff : M ^ (1 / p.toReal) \u2264 \u2016f x\u2016\u208a \u2194 M \u2264 \u2016\u2016f x\u2016 ^ p.toReal\u2016\u208a := by\n    rw [<a>coe_nnnorm</a>, <a>coe_nnnorm</a>, <a>Real.norm_rpow_of_nonneg</a> (<a>norm_nonneg</a> _), <a>norm_norm</a>,\n      \u2190 <a>Real.rpow_le_rpow_iff</a> hM' (<a>Real.rpow_nonneg_of_nonneg</a> (<a>norm_nonneg</a> _) _)\n        (<a>one_div_pos</a>.2 <| <a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top), \u2190 <a>Real.rpow_mul</a> (<a>norm_nonneg</a> _),\n      <a>mul_one_div_cancel</a> (<a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top).ne.symm, <a>Real.rpow_one</a>]", [{"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"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_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]}, {"full_name": "norm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [837, 9], "def_end_pos": [837, 18]}, {"full_name": "Real.rpow_le_rpow_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [447, 9], "def_end_pos": [447, 25]}, {"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": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "one_div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [73, 9], "def_end_pos": [73, 20]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "mul_one_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [75, 9], "def_end_pos": [75, 27]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"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.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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x)", "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x)"}, {"tactic": "by_cases hx : x \u2208 { x : \u03b1 | M ^ (1 / p.toReal) \u2264 \u2016f x\u2016\u208a }", "annotated_tactic": ["by_cases hx : x \u2208 { x : \u03b1 | M ^ (1 / p.toReal) \u2264 \u2016f x\u2016\u208a }", []], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x)"}, {"tactic": "refine' \u27e81, hp_ne_zero.symm \u25b8 _\u27e9", "annotated_tactic": ["refine' \u27e81, hp_ne_zero.symm \u25b8 _\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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : p = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : p = 0\n\u22a2 snorm (Set.indicator {x | 1 \u2264 \u2191\u2016f x\u2016\u208a} f) 0 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp [snorm_exponent_zero]", "annotated_tactic": ["simp [<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\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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : p = 0\n\u22a2 snorm (Set.indicator {x | 1 \u2264 \u2191\u2016f x\u2016\u208a} f) 0 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "subst hp_ne_top", "annotated_tactic": ["subst hp_ne_top", []], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : p = \u22a4\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) 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\nf : \u03b1 \u2192 \u03b2\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhf : Mem\u2112p f \u22a4\nhp_ne_zero : \u00ac\u22a4 = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8M, hM\u27e9 := hf.snormEssSup_indicator_norm_ge_eq_zero \u03bc hmeas", "annotated_tactic": ["obtain \u27e8M, hM\u27e9 := hf.snormEssSup_indicator_norm_ge_eq_zero \u03bc hmeas", []], "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\nf : \u03b1 \u2192 \u03b2\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhf : Mem\u2112p f \u22a4\nhp_ne_zero : \u00ac\u22a4 = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos.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\nf : \u03b1 \u2192 \u03b2\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhf : Mem\u2112p f \u22a4\nhp_ne_zero : \u00ac\u22a4 = 0\nM : \u211d\nhM : snormEssSup (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8M, _\u27e9", "annotated_tactic": ["refine' \u27e8M, _\u27e9", []], "state_before": "case pos.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\nf : \u03b1 \u2192 \u03b2\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhf : Mem\u2112p f \u22a4\nhp_ne_zero : \u00ac\u22a4 = 0\nM : \u211d\nhM : snormEssSup (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc = 0\n\u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos.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\nf : \u03b1 \u2192 \u03b2\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhf : Mem\u2112p f \u22a4\nhp_ne_zero : \u00ac\u22a4 = 0\nM : \u211d\nhM : snormEssSup (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc = 0\n\u22a2 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp only [snorm_exponent_top, hM, zero_le]", "annotated_tactic": ["simp only [<a>snorm_exponent_top</a>, hM, <a>zero_le</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": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos.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\nf : \u03b1 \u2192 \u03b2\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhf : Mem\u2112p f \u22a4\nhp_ne_zero : \u00ac\u22a4 = 0\nM : \u211d\nhM : snormEssSup (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc = 0\n\u22a2 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,\n  \u2190 Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _)\n    (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top), \u2190 Real.rpow_mul (norm_nonneg _),\n  mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one]", "annotated_tactic": ["rw [<a>coe_nnnorm</a>, <a>coe_nnnorm</a>, <a>Real.norm_rpow_of_nonneg</a> (<a>norm_nonneg</a> _), <a>norm_norm</a>,\n      \u2190 <a>Real.rpow_le_rpow_iff</a> hM' (<a>Real.rpow_nonneg_of_nonneg</a> (<a>norm_nonneg</a> _) _)\n        (<a>one_div_pos</a>.2 <| <a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top), \u2190 <a>Real.rpow_mul</a> (<a>norm_nonneg</a> _),\n      <a>mul_one_div_cancel</a> (<a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top).ne.symm, <a>Real.rpow_one</a>]", [{"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [905, 41], "def_end_pos": [905, 51]}, {"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_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]}, {"full_name": "norm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [837, 9], "def_end_pos": [837, 18]}, {"full_name": "Real.rpow_le_rpow_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [447, 9], "def_end_pos": [447, 25]}, {"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": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "one_div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [73, 9], "def_end_pos": [73, 20]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "mul_one_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [75, 9], "def_end_pos": [75, 27]}, {"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2131, 9], "def_end_pos": [2131, 19]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\n\u22a2 M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [Set.indicator_of_mem hx, Set.indicator_of_mem, Real.nnnorm_of_nonneg]", "annotated_tactic": ["rw [<a>Set.indicator_of_mem</a> hx, <a>Set.indicator_of_mem</a>, <a>Real.nnnorm_of_nonneg</a>]", [{"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]}, {"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\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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(\u2016f x\u2016\u208a ^ ENNReal.toReal p) = \u2191{ val := \u2016f x\u2016 ^ ENNReal.toReal p, property := ?pos\u271d }\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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 0 \u2264 \u2016f x\u2016 ^ ENNReal.toReal p\n\ncase 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a}"}, {"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 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(\u2016f x\u2016\u208a ^ ENNReal.toReal p) = \u2191{ val := \u2016f x\u2016 ^ ENNReal.toReal p, property := ?pos\u271d }\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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 0 \u2264 \u2016f x\u2016 ^ ENNReal.toReal p\n\ncase 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a}", "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a}"}, {"tactic": "rw [Set.mem_setOf_eq]", "annotated_tactic": ["rw [<a>Set.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 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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a}", "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a"}, {"tactic": "rwa [\u2190 hiff]", "annotated_tactic": ["rwa [\u2190 hiff]", []], "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [Set.indicator_of_not_mem hx, Set.indicator_of_not_mem]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a> hx, <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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =\n    \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(0 ^ ENNReal.toReal p) = \u21910\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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u00acx \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a}"}, {"tactic": "simp [(ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]", "annotated_tactic": ["simp [(<a>ENNReal.toReal_pos</a> hp_ne_zero hp_ne_top).ne.symm]", [{"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\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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u2191(0 ^ ENNReal.toReal p) = \u21910", "state_after": "no goals"}, {"tactic": "rw [Set.mem_setOf_eq]", "annotated_tactic": ["rw [<a>Set.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 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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u00acx \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a}", "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 : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u00acM \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a"}, {"tactic": "rwa [\u2190 hiff]", "annotated_tactic": ["rwa [\u2190 hiff]", []], "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 p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhp_ne_zero : \u00acp = 0\nhp_ne_top : \u00acp = \u22a4\nM : \u211d\nhM' : 0 \u2264 M\nhM :\n  \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264\n    ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p)\nx : \u03b1\nhiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a\nhx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 \u00acM \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a", "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_nnreal'", "start": [49, 1], "end": [55, 69], "traced_tactics": [{"tactic": "simp_rw [\u2190 measurable_coe_nnreal_ennreal_iff] at hf \u22a2", "annotated_tactic": ["simp_rw [\u2190 <a>measurable_coe_nnreal_ennreal_iff</a>] at hf \u22a2", [{"full_name": "measurable_coe_nnreal_ennreal_iff", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2111, 9], "def_end_pos": [2111, 42]}]], "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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : Tendsto f u (\ud835\udcdd g)\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 Measurable fun x => \u2191(g x)"}, {"tactic": "refine' measurable_of_tendsto_ennreal' u hf _", "annotated_tactic": ["refine' <a>measurable_of_tendsto_ennreal'</a> u hf _", [{"full_name": "measurable_of_tendsto_ennreal'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [29, 9], "def_end_pos": [29, 39]}]], "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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : Tendsto f u (\ud835\udcdd g)\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 Measurable fun x => \u2191(g x)", "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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : Tendsto f u (\ud835\udcdd g)\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 Tendsto (fun i x => \u2191(f i x)) u (\ud835\udcdd fun x => \u2191(g x))"}, {"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": "\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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : Tendsto f u (\ud835\udcdd g)\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 Tendsto (fun i x => \u2191(f i x)) u (\ud835\udcdd fun x => \u2191(g x))", "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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => \u2191(f i x)) u (\ud835\udcdd \u2191(g x))"}, {"tactic": "exact fun x => (ENNReal.continuous_coe.tendsto (g x)).comp (lim x)", "annotated_tactic": ["exact fun x => (ENNReal.continuous_coe.tendsto (g x)).<a>comp</a> (lim x)", [{"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\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 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => \u2191(f i x)) u (\ud835\udcdd \u2191(g x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.coe_mk", "start": [64, 1], "end": [64, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst", "start": [806, 1], "end": [814, 48], "traced_tactics": [{"tactic": "have h_empty : T \u2205 = 0 := h_zero _ MeasurableSet.empty measure_empty", "annotated_tactic": ["have h_empty : T \u2205 = 0 := h_zero _ <a>MeasurableSet.empty</a> <a>measure_empty</a>", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"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\nE : Type u_2\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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\n\u22a2 setToL1S T (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x) = \u2191(T s) x", "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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 setToL1S T (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x) = \u2191(T s) x"}, {"tactic": "rw [setToL1S_eq_setToSimpleFunc]", "annotated_tactic": ["rw [<a>setToL1S_eq_setToSimpleFunc</a>]", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_eq_setToSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [689, 9], "def_end_pos": [689, 36]}]], "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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 setToL1S T (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x) = \u2191(T s) x", "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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) = \u2191(T s) x"}, {"tactic": "refine' Eq.trans _ (SimpleFunc.setToSimpleFunc_indicator T h_empty hs x)", "annotated_tactic": ["refine' <a>Eq.trans</a> _ (<a>SimpleFunc.setToSimpleFunc_indicator</a> T h_empty hs x)", [{"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.SimpleFunc.setToSimpleFunc_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [604, 9], "def_end_pos": [604, 34]}]], "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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) = \u2191(T s) x", "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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) =\n    SimpleFunc.setToSimpleFunc T (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 x) (SimpleFunc.const \u03b1 0))"}, {"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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) =\n    SimpleFunc.setToSimpleFunc T (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 x) (SimpleFunc.const \u03b1 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\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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 \u2191(toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) =\u1d50[\u03bc]\n    \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 x) (SimpleFunc.const \u03b1 0))"}, {"tactic": "exact toSimpleFunc_indicatorConst hs h\u03bcs.ne x", "annotated_tactic": ["exact <a>toSimpleFunc_indicatorConst</a> hs h\u03bcs.ne x", [{"full_name": "MeasureTheory.Lp.simpleFunc.toSimpleFunc_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [705, 9], "def_end_pos": [705, 36]}]], "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\ns : Set \u03b1\nh_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0\nh_add : FinMeasAdditive \u03bc T\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nx : E\nh_empty : T \u2205 = 0\n\u22a2 \u2191(toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) =\u1d50[\u03bc]\n    \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 x) (SimpleFunc.const \u03b1 0))", "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_iff_forall_integral_tendsto", "start": [293, 1], "end": [300, 6], "traced_tactics": [{"tactic": "rw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]", "annotated_tactic": ["rw [<a>tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds</a>]", [{"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": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto (toFiniteMeasure \u2218 \u03bcs) F (\ud835\udcdd (toFiniteMeasure \u03bc)) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))"}, {"tactic": "rw [FiniteMeasure.tendsto_iff_forall_integral_tendsto]", "annotated_tactic": ["rw [<a>FiniteMeasure.tendsto_iff_forall_integral_tendsto</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [680, 9], "def_end_pos": [680, 44]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto (toFiniteMeasure \u2218 \u03bcs) F (\ud835\udcdd (toFiniteMeasure \u03bc)) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d),\n      Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191((toFiniteMeasure \u2218 \u03bcs) i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191(toFiniteMeasure \u03bc)))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d),\n      Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191((toFiniteMeasure \u2218 \u03bcs) i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191(toFiniteMeasure \u03bc)))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\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.foldr_eq", "start": [720, 1], "end": [721, 46], "traced_tactics": [{"tactic": "simpa using foldrAux_of_valid f [] s.1 [] a", "annotated_tactic": ["simpa using <a>foldrAux_of_valid</a> f [] s.1 [] a", [{"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\nf : Char \u2192 \u03b1 \u2192 \u03b1\ns : String\na : \u03b1\n\u22a2 foldr f a s = List.foldr f a s.data", "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.inv_divInt", "start": [305, 9], "end": [308, 38], "traced_tactics": [{"tactic": "if z : d = 0 then simp [z] else\ncases e : n /. d; rcases divInt_num_den z e with \u27e8g, zg, rfl, rfl\u27e9\nsimp [inv_def, divInt_mul_right zg]", "annotated_tactic": ["if z : d = 0 then simp [z] else\n  cases e : n /. d; rcases <a>divInt_num_den</a> z e with \u27e8g, zg, rfl, rfl\u27e9\n  simp [<a>inv_def</a>, <a>divInt_mul_right</a> zg]", [{"full_name": "Rat.divInt_num_den", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [155, 9], "def_end_pos": [155, 23]}, {"full_name": "Rat.inv_def", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [291, 9], "def_end_pos": [291, 16]}, {"full_name": "Rat.divInt_mul_right", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 25]}]], "state_before": "n d : Int\n\u22a2 Rat.inv (n /. d) = d /. n", "state_after": "no goals"}, {"tactic": "simp [z]", "annotated_tactic": ["simp [z]", []], "state_before": "n d : Int\nz : d = 0\n\u22a2 Rat.inv (n /. d) = d /. n", "state_after": "no goals"}, {"tactic": "cases e : n /. d", "annotated_tactic": ["cases e : n /. d", []], "state_before": "n d : Int\nz : \u00acd = 0\n\u22a2 Rat.inv (n /. d) = d /. n", "state_after": "case mk'\nn d : Int\nz : \u00acd = 0\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d\ne : n /. d = mk' num\u271d den\u271d\n\u22a2 Rat.inv (mk' num\u271d den\u271d) = d /. n"}, {"tactic": "rcases divInt_num_den z e with \u27e8g, zg, rfl, rfl\u27e9", "annotated_tactic": ["rcases <a>divInt_num_den</a> z e with \u27e8g, zg, rfl, rfl\u27e9", [{"full_name": "Rat.divInt_num_den", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [155, 9], "def_end_pos": [155, 23]}]], "state_before": "case mk'\nn d : Int\nz : \u00acd = 0\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d\ne : n /. d = mk' num\u271d den\u271d\n\u22a2 Rat.inv (mk' num\u271d den\u271d) = d /. n", "state_after": "case mk'.intro.intro.intro\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d\ng : Int\nzg : g \u2260 0\nz : \u00ac\u2191den\u271d * g = 0\ne : num\u271d * g /. (\u2191den\u271d * g) = mk' num\u271d den\u271d\n\u22a2 Rat.inv (mk' num\u271d den\u271d) = \u2191den\u271d * g /. (num\u271d * g)"}, {"tactic": "simp [inv_def, divInt_mul_right zg]", "annotated_tactic": ["simp [<a>inv_def</a>, <a>divInt_mul_right</a> zg]", [{"full_name": "Rat.inv_def", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [291, 9], "def_end_pos": [291, 16]}, {"full_name": "Rat.divInt_mul_right", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 25]}]], "state_before": "case mk'.intro.intro.intro\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d\ng : Int\nzg : g \u2260 0\nz : \u00ac\u2191den\u271d * g = 0\ne : num\u271d * g /. (\u2191den\u271d * g) = mk' num\u271d den\u271d\n\u22a2 Rat.inv (mk' num\u271d den\u271d) = \u2191den\u271d * g /. (num\u271d * g)", "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_inter", "start": [2131, 1], "end": [2133, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.erase_append_left", "start": [1189, 1], "end": [1191, 86], "traced_tactics": [{"tactic": "simp [erase_eq_eraseP]", "annotated_tactic": ["simp [<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\u2081 l\u2082 : List \u03b1\nh : a \u2208 l\u2081\n\u22a2 List.erase (l\u2081 ++ l\u2082) a = List.erase l\u2081 a ++ l\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : a \u2208 l\u2081\n\u22a2 eraseP (fun b => decide (a = b)) (l\u2081 ++ l\u2082) = eraseP (fun b => decide (a = b)) l\u2081 ++ l\u2082"}, {"tactic": "exact eraseP_append_left (by exact decide_eq_true rfl) l\u2082 h", "annotated_tactic": ["exact <a>eraseP_append_left</a> (by exact <a>decide_eq_true</a> <a>rfl</a>) l\u2082 h", [{"full_name": "List.eraseP_append_left", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1093, 9], "def_end_pos": [1093, 27]}, {"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\u2081 l\u2082 : List \u03b1\nh : a \u2208 l\u2081\n\u22a2 eraseP (fun b => decide (a = b)) (l\u2081 ++ l\u2082) = eraseP (fun b => decide (a = b)) l\u2081 ++ l\u2082", "state_after": "no goals"}, {"tactic": "exact decide_eq_true rfl", "annotated_tactic": ["exact <a>decide_eq_true</a> <a>rfl</a>", [{"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\u2081 l\u2082 : List \u03b1\nh : a \u2208 l\u2081\n\u22a2 decide (a = a) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "Function.Periodic.tendsto_atBot_intervalIntegral_of_pos", "start": [350, 1], "end": [355, 79], "traced_tactics": [{"tactic": "apply tendsto_atBot_mono (hg.integral_le_sSup_add_zsmul_of_pos h_int hT)", "annotated_tactic": ["apply <a>tendsto_atBot_mono</a> (hg.integral_le_sSup_add_zsmul_of_pos h_int hT)", [{"full_name": "Filter.tendsto_atBot_mono", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [414, 9], "def_end_pos": [414, 27]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun t => \u222b (x : \u211d) in 0 ..t, g x) atBot atBot", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun n => sSup ((fun t => \u222b (x : \u211d) in 0 ..t, g x) '' Icc 0 T) + \u230an / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atBot\n    atBot"}, {"tactic": "apply atBot.tendsto_atBot_add_const_left (sSup <| (fun t => \u222b x in (0)..t, g x) '' Icc 0 T)", "annotated_tactic": ["apply atBot.tendsto_atBot_add_const_left (<a>sSup</a> <| (fun t => \u222b x in (0)..t, g x) '' <a>Icc</a> 0 T)", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun n => sSup ((fun t => \u222b (x : \u211d) in 0 ..t, g x) '' Icc 0 T) + \u230an / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atBot\n    atBot", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atBot atBot"}, {"tactic": "apply Tendsto.atBot_zsmul_const h\u2080", "annotated_tactic": ["apply <a>Tendsto.atBot_zsmul_const</a> h\u2080", [{"full_name": "Filter.Tendsto.atBot_zsmul_const", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [260, 9], "def_end_pos": [260, 34]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atBot atBot", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b) atBot atBot"}, {"tactic": "exact tendsto_floor_atBot.comp (tendsto_id.atBot_mul_const (inv_pos.mpr hT))", "annotated_tactic": ["exact tendsto_floor_atBot.comp (tendsto_id.atBot_mul_const (inv_pos.mpr hT))", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b) atBot atBot", "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.empty_eq", "start": [619, 9], "end": [619, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasurableSet.image_of_continuousOn_injOn", "start": [823, 1], "end": [832, 34], "traced_tactics": [{"tactic": "obtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @PolishSpace \u03b3 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> \u03b3, t' \u2264 t\u03b3 \u2227 @<a>PolishSpace</a> \u03b3 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 : 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\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : PolishSpace \u03b3\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : BorelSpace \u03b3\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (f '' s)", "state_after": "case intro.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\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : PolishSpace \u03b3\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : BorelSpace \u03b3\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 MeasurableSet (f '' s)"}, {"tactic": "exact\n  @IsClosed.measurableSet_image_of_continuousOn_injOn \u03b3 t' t'_polish \u03b2 _ _ _ _ s s_closed f\n    (f_cont.mono_dom t't) f_inj", "annotated_tactic": ["exact\n    @<a>IsClosed.measurableSet_image_of_continuousOn_injOn</a> \u03b3 t' t'_polish \u03b2 _ _ _ _ s s_closed f\n      (f_cont.mono_dom t't) f_inj", [{"full_name": "IsClosed.measurableSet_image_of_continuousOn_injOn", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [807, 9], "def_end_pos": [807, 66]}]], "state_before": "case intro.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\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : PolishSpace \u03b3\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : BorelSpace \u03b3\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 MeasurableSet (f '' 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.restrict_le_restrict_iff", "start": [861, 1], "end": [866, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.ediv_le_self", "start": [576, 1], "end": [578, 36], "traced_tactics": [{"tactic": "have := Int.le_trans le_natAbs (ofNat_le.2 <| natAbs_div_le_natAbs a b)", "annotated_tactic": ["have := <a>Int.le_trans</a> <a>le_natAbs</a> (<a>ofNat_le</a>.2 <| <a>natAbs_div_le_natAbs</a> a b)", [{"full_name": "Int.le_trans", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [612, 19], "def_end_pos": [612, 27]}, {"full_name": "Int.le_natAbs", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [733, 9], "def_end_pos": [733, 18]}, {"full_name": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}, {"full_name": "Int.natAbs_div_le_natAbs", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [566, 9], "def_end_pos": [566, 29]}]], "state_before": "a b : Int\nHa : 0 \u2264 a\n\u22a2 a / b \u2264 a", "state_after": "a b : Int\nHa : 0 \u2264 a\nthis : a / b \u2264 \u2191(natAbs a)\n\u22a2 a / b \u2264 a"}, {"tactic": "rwa [natAbs_of_nonneg Ha] at this", "annotated_tactic": ["rwa [<a>natAbs_of_nonneg</a> Ha] at this", [{"full_name": "Int.natAbs_of_nonneg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 25]}]], "state_before": "a b : Int\nHa : 0 \u2264 a\nthis : a / b \u2264 \u2191(natAbs a)\n\u22a2 a / b \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/NFA.lean", "full_name": "DFA.toNFA_correct", "start": [162, 1], "end": [168, 38], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\n\u22a2 NFA.accepts (toNFA M) = accepts M", "state_after": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 x \u2208 NFA.accepts (toNFA M) \u2194 x \u2208 accepts M"}, {"tactic": "rw [NFA.mem_accepts, toNFA_start, toNFA_evalFrom_match]", "annotated_tactic": ["rw [<a>NFA.mem_accepts</a>, <a>toNFA_start</a>, <a>toNFA_evalFrom_match</a>]", [{"full_name": "NFA.mem_accepts", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [111, 9], "def_end_pos": [111, 20]}, {"full_name": "DFA.toNFA_start", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [143, 3], "def_end_pos": [143, 8]}, {"full_name": "DFA.toNFA_evalFrom_match", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [151, 9], "def_end_pos": [151, 29]}]], "state_before": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 x \u2208 NFA.accepts (toNFA M) \u2194 x \u2208 accepts M", "state_after": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 (\u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}) \u2194 x \u2208 accepts M"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 (\u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}) \u2194 x \u2208 accepts M", "state_after": "case h.mp\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 (\u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}) \u2192 x \u2208 accepts M\n\ncase h.mpr\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 x \u2208 accepts M \u2192 \u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}"}, {"tactic": "rintro \u27e8S, hS\u2081, hS\u2082\u27e9", "annotated_tactic": ["rintro \u27e8S, hS\u2081, hS\u2082\u27e9", []], "state_before": "case h.mp\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 (\u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}) \u2192 x \u2208 accepts M", "state_after": "case h.mp.intro.intro\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\nS : \u03c3\nhS\u2081 : S \u2208 (toNFA M).accept\nhS\u2082 : S \u2208 {evalFrom M M.start x}\n\u22a2 x \u2208 accepts M"}, {"tactic": "rwa [Set.mem_singleton_iff.mp hS\u2082] at hS\u2081", "annotated_tactic": ["rwa [Set.mem_singleton_iff.mp hS\u2082] at hS\u2081", []], "state_before": "case h.mp.intro.intro\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\nS : \u03c3\nhS\u2081 : S \u2208 (toNFA M).accept\nhS\u2082 : S \u2208 {evalFrom M M.start x}\n\u22a2 x \u2208 accepts M", "state_after": "no goals"}, {"tactic": "exact fun h => \u27e8M.eval x, h, rfl\u27e9", "annotated_tactic": ["exact fun h => \u27e8M.eval x, h, <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 h.mpr\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : NFA \u03b1 \u03c3\nM : DFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 x \u2208 accepts M \u2192 \u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_surjective", "start": [1565, 1], "end": [1569, 16], "traced_tactics": [{"tactic": "refine' \u27e8fun h y => _, Surjective.image_surjective\u27e9", "annotated_tactic": ["refine' \u27e8fun h y => _, <a>Surjective.image_surjective</a>\u27e9", [{"full_name": "Function.Surjective.image_surjective", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1319, 9], "def_end_pos": [1319, 36]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\n\u22a2 Surjective (image f) \u2194 Surjective f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\n\u22a2 \u2203 a, f a = y"}, {"tactic": "cases' h {y} with s hs", "annotated_tactic": ["cases' h {y} with s hs", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\n\u22a2 \u2203 a, f a = y"}, {"tactic": "have := mem_singleton y", "annotated_tactic": ["have := <a>mem_singleton</a> y", [{"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 (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 {y}\n\u22a2 \u2203 a, f a = y"}, {"tactic": "rw [\u2190 hs] at this", "annotated_tactic": ["rw [\u2190 hs] at this", []], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 {y}\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 f '' s\n\u22a2 \u2203 a, f a = y"}, {"tactic": "rcases this with \u27e8x, _, hx\u27e9", "annotated_tactic": ["rcases this with \u27e8x, _, hx\u27e9", []], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 f '' s\n\u22a2 \u2203 a, f a = y", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nx : \u03b1\nleft\u271d : x \u2208 s\nhx : f x = y\n\u22a2 \u2203 a, f a = y"}, {"tactic": "exact \u27e8x, hx\u27e9", "annotated_tactic": ["exact \u27e8x, hx\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nx : \u03b1\nleft\u271d : x \u2208 s\nhx : f x = y\n\u22a2 \u2203 a, f a = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.succ_ofInt'", "start": [1244, 1], "end": [1257, 60], "traced_tactics": [{"tactic": "change ZNum.ofInt' (n + 1 : \u2115) = ZNum.ofInt' (n : \u2115) + 1", "annotated_tactic": ["change <a>ZNum.ofInt'</a> (n + 1 : \u2115) = <a>ZNum.ofInt'</a> (n : \u2115) + 1", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' (\u2191n + 1) = ZNum.ofInt' \u2191n + 1", "state_after": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' \u2191(n + 1) = ZNum.ofInt' \u2191n + 1"}, {"tactic": "dsimp only [ZNum.ofInt', ZNum.ofInt']", "annotated_tactic": ["dsimp only [<a>ZNum.ofInt'</a>, <a>ZNum.ofInt'</a>]", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' \u2191(n + 1) = ZNum.ofInt' \u2191n + 1", "state_after": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNum (ofNat' (n + 1)) = toZNum (ofNat' n) + 1"}, {"tactic": "rw [Num.ofNat'_succ, Num.add_one, toZNum_succ, ZNum.add_one]", "annotated_tactic": ["rw [<a>Num.ofNat'_succ</a>, <a>Num.add_one</a>, <a>toZNum_succ</a>, <a>ZNum.add_one</a>]", [{"full_name": "Num.ofNat'_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [252, 9], "def_end_pos": [252, 20]}, {"full_name": "Num.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 16]}, {"full_name": "Num.toZNum_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 20]}, {"full_name": "ZNum.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 16]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNum (ofNat' (n + 1)) = toZNum (ofNat' n) + 1", "state_after": "no goals"}, {"tactic": "change ZNum.ofInt' 0 = ZNum.ofInt' (-[0+1]) + 1", "annotated_tactic": ["change <a>ZNum.ofInt'</a> 0 = <a>ZNum.ofInt'</a> (-[0+1]) + 1", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}]], "state_before": "\u03b1 : Type u_1\n\u22a2 ZNum.ofInt' (-[0+1] + 1) = ZNum.ofInt' -[0+1] + 1", "state_after": "\u03b1 : Type u_1\n\u22a2 ZNum.ofInt' 0 = ZNum.ofInt' -[0+1] + 1"}, {"tactic": "dsimp only [ZNum.ofInt', ZNum.ofInt']", "annotated_tactic": ["dsimp only [<a>ZNum.ofInt'</a>, <a>ZNum.ofInt'</a>]", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}]], "state_before": "\u03b1 : Type u_1\n\u22a2 ZNum.ofInt' 0 = ZNum.ofInt' -[0+1] + 1", "state_after": "\u03b1 : Type u_1\n\u22a2 toZNum (ofNat' 0) = toZNumNeg (ofNat' (0 + 1)) + 1"}, {"tactic": "rw [ofNat'_succ, ofNat'_zero]", "annotated_tactic": ["rw [<a>ofNat'_succ</a>, <a>ofNat'_zero</a>]", [{"full_name": "Num.ofNat'_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [252, 9], "def_end_pos": [252, 20]}, {"full_name": "Num.ofNat'_zero", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [236, 9], "def_end_pos": [236, 20]}]], "state_before": "\u03b1 : Type u_1\n\u22a2 toZNum (ofNat' 0) = toZNumNeg (ofNat' (0 + 1)) + 1", "state_after": "\u03b1 : Type u_1\n\u22a2 toZNum 0 = toZNumNeg (0 + 1) + 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u22a2 toZNum 0 = toZNumNeg (0 + 1) + 1", "state_after": "no goals"}, {"tactic": "change ZNum.ofInt' -[n+1] = ZNum.ofInt' -[(n + 1)+1] + 1", "annotated_tactic": ["change <a>ZNum.ofInt'</a> -[n+1] = <a>ZNum.ofInt'</a> -[(n + 1)+1] + 1", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' (-[n + 1+1] + 1) = ZNum.ofInt' -[n + 1+1] + 1", "state_after": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' -[n+1] = ZNum.ofInt' -[n + 1+1] + 1"}, {"tactic": "dsimp only [ZNum.ofInt', ZNum.ofInt']", "annotated_tactic": ["dsimp only [<a>ZNum.ofInt'</a>, <a>ZNum.ofInt'</a>]", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' -[n+1] = ZNum.ofInt' -[n + 1+1] + 1", "state_after": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNumNeg (ofNat' (n + 1)) = toZNumNeg (ofNat' (n + 1 + 1)) + 1"}, {"tactic": "rw [@Num.ofNat'_succ (n + 1), Num.add_one, toZNumNeg_succ,\n  @ofNat'_succ n, Num.add_one, ZNum.add_one, pred_succ]", "annotated_tactic": ["rw [@<a>Num.ofNat'_succ</a> (n + 1), <a>Num.add_one</a>, <a>toZNumNeg_succ</a>,\n      @<a>ofNat'_succ</a> n, <a>Num.add_one</a>, <a>ZNum.add_one</a>, <a>pred_succ</a>]", [{"full_name": "Num.ofNat'_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [252, 9], "def_end_pos": [252, 20]}, {"full_name": "Num.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 16]}, {"full_name": "Num.toZNumNeg_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 23]}, {"full_name": "Num.ofNat'_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [252, 9], "def_end_pos": [252, 20]}, {"full_name": "Num.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 16]}, {"full_name": "ZNum.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 16]}, {"full_name": "Num.pred_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNumNeg (ofNat' (n + 1)) = toZNumNeg (ofNat' (n + 1 + 1)) + 1", "state_after": "no goals"}]}, {"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_aux", "start": [85, 1], "end": [155, 55], "traced_tactics": [{"tactic": "refine' Metric.tendsto_nhds.2 fun \u03b5 \u03b5pos => _", "annotated_tactic": ["refine' <a>Metric.tendsto_nhds</a>.2 fun \u03b5 \u03b5pos => _", [{"full_name": "Metric.tendsto_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\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 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4, \u03b4pos\u27e9 : \u2203 \u03b4, (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 < \u03b5 \u2227 0 < \u03b4 := by\n  have A :\n    Tendsto (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) := by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id\n  rw [zero_mul, zero_add] at A\n  exact (((tendsto_order.1 A).2 \u03b5 \u03b5pos).and self_mem_nhdsWithin).exists", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4, \u03b4pos\u27e9 : \u2203 \u03b4, (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 < \u03b5 \u2227 0 < \u03b4 := by\n    have A :\n      <a>Tendsto</a> (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) := by\n      apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n      exact (tendsto_id.mul <a>tendsto_const_nhds</a>).<a>add</a> <a>tendsto_id</a>\n    rw [<a>zero_mul</a>, <a>zero_add</a>] at A\n    exact (((<a>tendsto_order</a>.1 A).2 \u03b5 \u03b5pos).<a>and</a> <a>self_mem_nhdsWithin</a>).<a>exists</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": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"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_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"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": "\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5", "state_after": "case 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5"}, {"tactic": "suffices \u2200\u1da0 i in l, \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 by\n  filter_upwards [this] with i hi\n  simp only [dist_zero_right]\n  exact hi.trans_lt h\u03b4", "annotated_tactic": ["suffices \u2200\u1da0 i in l, \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 by\n    filter_upwards [this] with i hi\n    simp only [<a>dist_zero_right</a>]\n    exact hi.trans_lt h\u03b4", [{"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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5", "state_after": "case 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4"}, {"tactic": "obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) \u03b4", "annotated_tactic": ["obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, <a>IsOpen</a> u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 <a>ball</a> (g x\u2080) \u03b4", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}]], "state_before": "case 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\ncase 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4"}, {"tactic": "exact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ \u03b4pos))", "annotated_tactic": ["exact <a>mem_nhdsWithin</a>.1 (hcg (<a>ball_mem_nhds</a> _ \u03b4pos))", [{"full_name": "mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [93, 9], "def_end_pos": [93, 23]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}]], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\ncase 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4", "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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4"}, {"tactic": "filter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) \u03b4 \u03b4pos, hi\u03c6, hn\u03c6,\n  integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hl\u03c6 hi\u03c6 hmg hcg] with i hi h'i\n  h\u03c6pos h''i", "annotated_tactic": ["filter_upwards [<a>tendstoUniformlyOn_iff</a>.1 (hl\u03c6 u u_open x\u2080u) \u03b4 \u03b4pos, hi\u03c6, hn\u03c6,\n    <a>integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt</a> hs hl\u03c6 hi\u03c6 hmg hcg] with i hi h'i\n    h\u03c6pos h''i", [{"full_name": "Metric.tendstoUniformlyOn_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [922, 9], "def_end_pos": [922, 31]}, {"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": "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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4"}, {"tactic": "calc\n  \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =\n    \u2016(\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc) + \u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := by\n    conv_lhs => rw [\u2190 diff_union_inter s u]\n    rw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet)\n        (h''i.mono_set (diff_subset _ _)) (h''i.mono_set (inter_subset_left _ _))]\n  _ \u2264 \u2016\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 + \u2016\u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := (norm_add_le _ _)\n  _ \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 := add_le_add C B", "annotated_tactic": ["calc\n    \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =\n      \u2016(\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc) + \u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := by\n      conv_lhs => rw [\u2190 <a>diff_union_inter</a> s u]\n      rw [<a>integral_union</a> (<a>disjoint_sdiff_inter</a> _ _) (hs.inter u_open.measurableSet)\n          (h''i.mono_set (<a>diff_subset</a> _ _)) (h''i.mono_set (<a>inter_subset_left</a> _ _))]\n    _ \u2264 \u2016\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 + \u2016\u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := (<a>norm_add_le</a> _ _)\n    _ \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 := <a>add_le_add</a> C B", [{"full_name": "Set.diff_union_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1894, 9], "def_end_pos": [1894, 25]}, {"full_name": "MeasureTheory.integral_union", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}, {"full_name": "Set.disjoint_sdiff_inter", "def_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "def_pos": [42, 9], "def_end_pos": [42, 33]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "norm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [482, 15], "def_end_pos": [482, 26]}, {"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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4", "state_after": "no goals"}, {"tactic": "have A :\n  Tendsto (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id", "annotated_tactic": ["have A :\n      <a>Tendsto</a> (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) := by\n      apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n      exact (tendsto_id.mul <a>tendsto_const_nhds</a>).<a>add</a> <a>tendsto_id</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": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"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_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nA : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4"}, {"tactic": "rw [zero_mul, zero_add] at A", "annotated_tactic": ["rw [<a>zero_mul</a>, <a>zero_add</a>] at 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_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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nA : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nA : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4"}, {"tactic": "exact (((tendsto_order.1 A).2 \u03b5 \u03b5pos).and self_mem_nhdsWithin).exists", "annotated_tactic": ["exact (((<a>tendsto_order</a>.1 A).2 \u03b5 \u03b5pos).<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": "\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nA : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4", "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": "\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))"}, {"tactic": "exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id", "annotated_tactic": ["exact (tendsto_id.mul <a>tendsto_const_nhds</a>).<a>add</a> <a>tendsto_id</a>", [{"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"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_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))", "state_after": "no goals"}, {"tactic": "filter_upwards [this] with i hi", "annotated_tactic": ["filter_upwards [this] with i hi", []], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\ni : \u03b9\nhi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 dist (\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) 0 < \u03b5"}, {"tactic": "simp only [dist_zero_right]", "annotated_tactic": ["simp only [<a>dist_zero_right</a>]", [{"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 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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\ni : \u03b9\nhi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 dist (\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) 0 < \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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\ni : \u03b9\nhi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 < \u03b5"}, {"tactic": "exact hi.trans_lt h\u03b4", "annotated_tactic": ["exact hi.trans_lt h\u03b4", []], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\ni : \u03b9\nhi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 < \u03b5", "state_after": "no goals"}, {"tactic": "refine' set_integral_mono_on _ _ (hs.inter u_open.measurableSet) fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_mono_on</a> _ _ (hs.inter u_open.measurableSet) 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]}]], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc", "state_after": "case refine'_1\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \u2229 u)\n\ncase refine'_2\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) (s \u2229 u)\n\ncase refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4"}, {"tactic": "rw [norm_smul]", "annotated_tactic": ["rw [<a>norm_smul</a>]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}]], "state_before": "case refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4", "state_after": "case refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4"}, {"tactic": "apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_left</a> _ (<a>norm_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": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u03b4"}, {"tactic": "rw [inter_comm, h'g] at hu", "annotated_tactic": ["rw [<a>inter_comm</a>, h'g] at hu", [{"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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u03b4", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 s \u2229 u \u2192 g x \u2208 ball 0 \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u03b4"}, {"tactic": "exact (mem_ball_zero_iff.1 (hu x hx)).le", "annotated_tactic": ["exact (<a>mem_ball_zero_iff</a>.1 (hu x hx)).<a>le</a>", [{"full_name": "mem_ball_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [641, 3], "def_end_pos": [641, 14]}, {"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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 s \u2229 u \u2192 g x \u2208 ball 0 \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u03b4", "state_after": "no goals"}, {"tactic": "exact IntegrableOn.mono_set h''i.norm (inter_subset_left _ _)", "annotated_tactic": ["exact <a>IntegrableOn.mono_set</a> h''i.norm (<a>inter_subset_left</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.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}]], "state_before": "case refine'_1\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \u2229 u)", "state_after": "no goals"}, {"tactic": "exact\n  IntegrableOn.mono_set ((integrable_of_integral_eq_one h'i).norm.mul_const _)\n    (inter_subset_left _ _)", "annotated_tactic": ["exact\n            <a>IntegrableOn.mono_set</a> ((<a>integrable_of_integral_eq_one</a> h'i).norm.mul_const _)\n              (<a>inter_subset_left</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": "MeasureTheory.integrable_of_integral_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [864, 9], "def_end_pos": [864, 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": "case refine'_2\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) (s \u2229 u)", "state_after": "no goals"}, {"tactic": "apply set_integral_mono_set", "annotated_tactic": ["apply <a>set_integral_mono_set</a>", [{"full_name": "MeasureTheory.set_integral_mono_set", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [736, 9], "def_end_pos": [736, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc", "state_after": "case hfi\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) s\n\ncase hf\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2016\u03c6 i x\u2016 * \u03b4\n\ncase hst\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 s \u2229 u \u2264\u1da0[ae \u03bc] s"}, {"tactic": "exact (integrable_of_integral_eq_one h'i).norm.mul_const _", "annotated_tactic": ["exact (<a>integrable_of_integral_eq_one</a> h'i).norm.mul_const _", [{"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 hfi\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) s", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) \u03b4pos.le", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => <a>mul_nonneg</a> (<a>norm_nonneg</a> _) \u03b4pos.le", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"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]}]], "state_before": "case hf\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2016\u03c6 i x\u2016 * \u03b4", "state_after": "no goals"}, {"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 hst\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 s \u2229 u \u2264\u1da0[ae \u03bc] s", "state_after": "case hst.hp\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u2200 (x : \u03b1), (s \u2229 u) x \u2264 s x"}, {"tactic": "exact inter_subset_left s u", "annotated_tactic": ["exact <a>inter_subset_left</a> s u", [{"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 hst.hp\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u2200 (x : \u03b1), (s \u2229 u) x \u2264 s x", "state_after": "no goals"}, {"tactic": "apply set_integral_congr hs fun x hx => ?_", "annotated_tactic": ["apply <a>set_integral_congr</a> hs fun x hx => ?_", [{"full_name": "MeasureTheory.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 i x * \u03b4 \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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2016\u03c6 i x\u2016 * \u03b4 = \u03c6 i x * \u03b4"}, {"tactic": "rw [Real.norm_of_nonneg (h\u03c6pos _ hx)]", "annotated_tactic": ["rw [<a>Real.norm_of_nonneg</a> (h\u03c6pos _ hx)]", [{"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": "\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2016\u03c6 i x\u2016 * \u03b4 = \u03c6 i x * \u03b4", "state_after": "no goals"}, {"tactic": "rw [integral_mul_right, h'i, one_mul]", "annotated_tactic": ["rw [<a>integral_mul_right</a>, h'i, <a>one_mul</a>]", [{"full_name": "MeasureTheory.integral_mul_right", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [928, 9], "def_end_pos": [928, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x * \u03b4 \u2202\u03bc = \u03b4", "state_after": "no goals"}, {"tactic": "refine' set_integral_mono_on _ _ (hs.diff u_open.measurableSet) fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_mono_on</a> _ _ (hs.diff u_open.measurableSet) 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]}]], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u222b (x : \u03b1) in s \\ u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc", "state_after": "case refine'_1\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \\ u)\n\ncase refine'_2\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 IntegrableOn (fun x => \u03b4 * \u2016g x\u2016) (s \\ u)\n\ncase refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u03b4 * \u2016g x\u2016"}, {"tactic": "rw [norm_smul]", "annotated_tactic": ["rw [<a>norm_smul</a>]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}]], "state_before": "case refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u03b4 * \u2016g x\u2016", "state_after": "case refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u03b4 * \u2016g x\u2016"}, {"tactic": "apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_right</a> _ (<a>norm_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": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case refine'_3\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u03b4 * \u2016g x\u2016", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 \u03b4"}, {"tactic": "simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le", "annotated_tactic": ["simpa only [<a>Pi.zero_apply</a>, <a>dist_zero_left</a>] using (hi x hx).<a>le</a>", [{"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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 \u03b4", "state_after": "no goals"}, {"tactic": "exact IntegrableOn.mono_set h''i.norm (diff_subset _ _)", "annotated_tactic": ["exact <a>IntegrableOn.mono_set</a> h''i.norm (<a>diff_subset</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.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case refine'_1\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \\ u)", "state_after": "no goals"}, {"tactic": "exact IntegrableOn.mono_set (hmg.norm.const_mul _) (diff_subset _ _)", "annotated_tactic": ["exact <a>IntegrableOn.mono_set</a> (hmg.norm.const_mul _) (<a>diff_subset</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.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case refine'_2\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 IntegrableOn (fun x => \u03b4 * \u2016g x\u2016) (s \\ u)", "state_after": "no goals"}, {"tactic": "rw [integral_mul_left]", "annotated_tactic": ["rw [<a>integral_mul_left</a>]", [{"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}]], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u222b (x : \u03b1) in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u03b4 * \u222b (a : \u03b1) in s \\ u, \u2016g a\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc"}, {"tactic": "apply mul_le_mul_of_nonneg_left (set_integral_mono_set hmg.norm _ _) \u03b4pos.le", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_left</a> (<a>set_integral_mono_set</a> hmg.norm _ _) \u03b4pos.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": "MeasureTheory.set_integral_mono_set", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [736, 9], "def_end_pos": [736, 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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u03b4 * \u222b (a : \u03b1) in s \\ u, \u2016g a\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2016g a\u2016\n\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 s \\ u \u2264\u1da0[ae \u03bc] s"}, {"tactic": "exact eventually_of_forall fun x => norm_nonneg _", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => <a>norm_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": "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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2016g a\u2016", "state_after": "no goals"}, {"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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 s \\ u \u2264\u1da0[ae \u03bc] s", "state_after": "case hp\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u2200 (x : \u03b1), (s \\ u) x \u2264 s x"}, {"tactic": "exact diff_subset s u", "annotated_tactic": ["exact <a>diff_subset</a> s u", [{"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "case hp\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u2200 (x : \u03b1), (s \\ u) x \u2264 s x", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [\u2190 diff_union_inter s u]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>diff_union_inter</a> s u]", [{"full_name": "Set.diff_union_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1894, 9], "def_end_pos": [1894, 25]}]], "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 = \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016", "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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s \\ u \u222a s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =\n    \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016"}, {"tactic": "rw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet)\n    (h''i.mono_set (diff_subset _ _)) (h''i.mono_set (inter_subset_left _ _))]", "annotated_tactic": ["rw [<a>integral_union</a> (<a>disjoint_sdiff_inter</a> _ _) (hs.inter u_open.measurableSet)\n          (h''i.mono_set (<a>diff_subset</a> _ _)) (h''i.mono_set (<a>inter_subset_left</a> _ _))]", [{"full_name": "MeasureTheory.integral_union", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}, {"full_name": "Set.disjoint_sdiff_inter", "def_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "def_pos": [42, 9], "def_end_pos": [42, 33]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"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\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\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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s \\ u \u222a s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =\n    \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016", "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_Icc_of_le", "start": [175, 1], "end": [176, 43], "traced_tactics": [{"tactic": "rw [card_fintype_Icc, toNat_sub_of_le h]", "annotated_tactic": ["rw [<a>card_fintype_Icc</a>, <a>toNat_sub_of_le</a> h]", [{"full_name": "Int.card_fintype_Icc", "def_path": "Mathlib/Data/Int/Interval.lean", "def_pos": [152, 9], "def_end_pos": [152, 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 + 1\n\u22a2 \u2191(Fintype.card \u2191(Set.Icc a b)) = b + 1 - 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_bot'", "start": [248, 1], "end": [266, 49], "traced_tactics": [{"tactic": "by_cases h\u03bc_finite : IsFiniteMeasure \u03bc", "annotated_tactic": ["by_cases h\u03bc_finite : <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\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"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 : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc\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 : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \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\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : \u00acIntegrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"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 : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\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\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : \u00acIntegrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : \u00acIntegrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc\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 : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "have h_meas : StronglyMeasurable[\u22a5] (\u03bc[f|\u22a5]) := stronglyMeasurable_condexp", "annotated_tactic": ["have h_meas : StronglyMeasurable[\u22a5] (\u03bc[f|\u22a5]) := <a>stronglyMeasurable_condexp</a>", [{"full_name": "MeasureTheory.stronglyMeasurable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 35]}]], "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "obtain \u27e8c, h_eq\u27e9 := stronglyMeasurable_bot_iff.mp h_meas", "annotated_tactic": ["obtain \u27e8c, h_eq\u27e9 := stronglyMeasurable_bot_iff.mp h_meas", []], "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [h_eq]", "annotated_tactic": ["rw [h_eq]", []], "state_before": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\n\u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "have h_integral : \u222b x, (\u03bc[f|\u22a5]) x \u2202\u03bc = \u222b x, f x \u2202\u03bc := integral_condexp bot_le hf", "annotated_tactic": ["have h_integral : \u222b x, (\u03bc[f|\u22a5]) x \u2202\u03bc = \u222b x, f x \u2202\u03bc := <a>integral_condexp</a> <a>bot_le</a> hf", [{"full_name": "MeasureTheory.integral_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 25]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\n\u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : \u222b (x : \u03b1), (\u03bc[f|\u22a5]) x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "simp_rw [h_eq, integral_const] at h_integral", "annotated_tactic": ["simp_rw [h_eq, <a>integral_const</a>] at h_integral", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}]], "state_before": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : \u222b (x : \u03b1), (\u03bc[f|\u22a5]) x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [\u2190 h_integral, \u2190 smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]", "annotated_tactic": ["rw [\u2190 h_integral, \u2190 <a>smul_assoc</a>, <a>smul_eq_mul</a>, <a>inv_mul_cancel</a>, <a>one_smul</a>]", [{"full_name": "smul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"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 pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \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": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2260 0", "state_after": "case pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u00ac\u2191\u2191\u03bc Set.univ = 0 \u2227 \u00ac\u2191\u2191\u03bc Set.univ = \u22a4"}, {"tactic": "exact \u27e8NeZero.ne _, measure_ne_top \u03bc Set.univ\u27e9", "annotated_tactic": ["exact \u27e8<a>NeZero.ne</a> _, <a>measure_ne_top</a> \u03bc <a>Set.univ</a>\u27e9", [{"full_name": "NeZero.ne", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [30, 9], "def_end_pos": [30, 18]}, {"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 pos.intro\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : Integrable f\nh_meas : StronglyMeasurable (\u03bc[f|\u22a5])\nc : F'\nh_eq : \u03bc[f|\u22a5] = fun x => c\nh_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u00ac\u2191\u2191\u03bc Set.univ = 0 \u2227 \u00ac\u2191\u2191\u03bc Set.univ = \u22a4", "state_after": "no goals"}, {"tactic": "have h : \u00acSigmaFinite (\u03bc.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]", "annotated_tactic": ["have h : \u00ac<a>SigmaFinite</a> (\u03bc.trim <a>bot_le</a>) := by rwa [<a>sigmaFinite_trim_bot_iff</a>]", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}, {"full_name": "MeasureTheory.sigmaFinite_trim_bot_iff", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [126, 9], "def_end_pos": [126, 33]}]], "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [not_isFiniteMeasure_iff] at h\u03bc_finite", "annotated_tactic": ["rw [<a>not_isFiniteMeasure_iff</a>] at h\u03bc_finite", [{"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\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [condexp_of_not_sigmaFinite bot_le h]", "annotated_tactic": ["rw [<a>condexp_of_not_sigmaFinite</a> <a>bot_le</a> h]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 0 = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "simp only [h\u03bc_finite, ENNReal.top_toReal, inv_zero, zero_smul]", "annotated_tactic": ["simp only [h\u03bc_finite, <a>ENNReal.top_toReal</a>, <a>inv_zero</a>, <a>zero_smul</a>]", [{"full_name": "ENNReal.top_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [227, 17], "def_end_pos": [227, 27]}, {"full_name": "GroupWithZero.inv_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [186, 3], "def_end_pos": [186, 11]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 0 = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 0 = fun x => 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 : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 0 = fun x => 0", "state_after": "no goals"}, {"tactic": "rwa [sigmaFinite_trim_bot_iff]", "annotated_tactic": ["rwa [<a>sigmaFinite_trim_bot_iff</a>]", [{"full_name": "MeasureTheory.sigmaFinite_trim_bot_iff", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [126, 9], "def_end_pos": [126, 33]}]], "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))", "state_after": "no goals"}, {"tactic": "rw [integral_undef hf, smul_zero, condexp_undef hf]", "annotated_tactic": ["rw [<a>integral_undef</a> hf, <a>smul_zero</a>, <a>condexp_undef</a> hf]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"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\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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : \u00acIntegrable f\n\u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : \u00acIntegrable f\n\u22a2 0 = fun x => 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 : \u03b1 \u2192 F'\ns : Set \u03b1\nh\u03bc : NeZero \u03bc\nf : \u03b1 \u2192 F'\nh\u03bc_finite : IsFiniteMeasure \u03bc\nhf : \u00acIntegrable f\n\u22a2 0 = fun x => 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Partrec.sum_casesOn", "start": [125, 8], "end": [131, 76], "traced_tactics": [{"tactic": "cases f a <;> simp only [Bool.cond_true, Bool.cond_false]", "annotated_tactic": ["cases f a <;> simp only [<a>Bool.cond_true</a>, <a>Bool.cond_false</a>]", [{"full_name": "Bool.cond_true", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [40, 15], "def_end_pos": [40, 24]}, {"full_name": "Bool.cond_false", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [43, 15], "def_end_pos": [43, 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 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Partrec\u2082 h\na : \u03b1\n\u22a2 (bif Sum.casesOn (f a) (fun b => true) fun b => false then\n      Sum.casesOn (f a) (fun b => Part.map Option.some (g (a, b).1 (a, b).2)) fun c => Part.some Option.none\n    else Sum.casesOn (f a) (fun b => Part.some Option.none) fun b => Part.map Option.some (h (a, b).1 (a, b).2)) =\n    Part.map Option.some (Sum.casesOn (f a) (g a) (h a))", "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.countP_cons_of_pos", "start": [36, 9], "end": [39, 49], "traced_tactics": [{"tactic": "have : countP.go p (a :: l) 0 = countP.go p l 1 := show cond .. = _ by rw [pa]; rfl", "annotated_tactic": ["have : <a>countP.go</a> p (a :: l) 0 = <a>countP.go</a> p l 1 := show <a>cond</a> .. = _ by rw [pa]; rfl", [{"full_name": "List.countP.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [799, 17], "def_end_pos": [799, 19]}, {"full_name": "List.countP.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [799, 17], "def_end_pos": [799, 19]}, {"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\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\n\u22a2 countP p (a :: l) = countP p l + 1", "state_after": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\nthis : countP.go p (a :: l) 0 = countP.go p l 1\n\u22a2 countP p (a :: l) = countP p l + 1"}, {"tactic": "unfold countP", "annotated_tactic": ["unfold <a>countP</a>", [{"full_name": "List.countP", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [797, 15], "def_end_pos": [797, 21]}]], "state_before": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\nthis : countP.go p (a :: l) 0 = countP.go p l 1\n\u22a2 countP p (a :: l) = countP p l + 1", "state_after": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\nthis : countP.go p (a :: l) 0 = countP.go p l 1\n\u22a2 countP.go p (a :: l) 0 = countP.go p l 0 + 1"}, {"tactic": "rw [this, Nat.add_comm, List.countP_go_eq_add]", "annotated_tactic": ["rw [this, <a>Nat.add_comm</a>, <a>List.countP_go_eq_add</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]}, {"full_name": "List.countP_go_eq_add", "def_path": "lake-packages/std/Std/Data/List/Count.lean", "def_pos": [28, 19], "def_end_pos": [28, 35]}]], "state_before": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\nthis : countP.go p (a :: l) 0 = countP.go p l 1\n\u22a2 countP.go p (a :: l) 0 = countP.go p l 0 + 1", "state_after": "no goals"}, {"tactic": "rw [pa]", "annotated_tactic": ["rw [pa]", []], "state_before": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\n\u22a2 (bif p a then countP.go p l (0 + 1) else countP.go p l 0) = countP.go p l 1", "state_after": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\n\u22a2 (bif true then countP.go p l (0 + 1) else countP.go p l 0) = countP.go p l 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\npa : p a = true\n\u22a2 (bif true then countP.go p l (0 + 1) else countP.go p l 0) = countP.go p l 1", "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_get", "start": [244, 9], "end": [244, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.unary_decode_encode_nat", "start": [198, 1], "end": [199, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.mem\u2112p_finset_sum", "start": [1243, 1], "end": [1251, 95], "traced_tactics": [{"tactic": "haveI : DecidableEq \u03b9 := Classical.decEq _", "annotated_tactic": ["haveI : <a>DecidableEq</a> \u03b9 := <a>Classical.decEq</a> _", [{"full_name": "DecidableEq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [839, 8], "def_end_pos": [839, 19]}, {"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [983, 19], "def_end_pos": [983, 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\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\n\u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\nthis : DecidableEq \u03b9\n\u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) p"}, {"tactic": "revert hf", "annotated_tactic": ["revert hf", []], "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\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\nthis : DecidableEq \u03b9\n\u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p"}, {"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 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\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) 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 : \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\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in \u2205, f i a) p\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 : \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\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n    \u00aca \u2208 s \u2192\n      ((\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p) \u2192\n        (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a_3 => \u2211 i in insert a s, f i a_3) p"}, {"tactic": "simp only [zero_mem_\u2112p', Finset.sum_empty, imp_true_iff]", "annotated_tactic": ["simp only [<a>zero_mem_\u2112p'</a>, <a>Finset.sum_empty</a>, <a>imp_true_iff</a>]", [{"full_name": "MeasureTheory.zero_mem_\u2112p'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [220, 9], "def_end_pos": [220, 21]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 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 refine'_1\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\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in \u2205, f i a) p", "state_after": "no goals"}, {"tactic": "intro i s his ih hf", "annotated_tactic": ["intro i s his ih hf", []], "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 : \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\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n    \u00aca \u2208 s \u2192\n      ((\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p) \u2192\n        (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a_3 => \u2211 i in insert a s, f i a_3) 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 : \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\u03b9 : Type u_5\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p\n\u22a2 Mem\u2112p (fun a => \u2211 i in insert i s, f i a) p"}, {"tactic": "simp only [his, Finset.sum_insert, not_false_iff]", "annotated_tactic": ["simp only [his, <a>Finset.sum_insert</a>, <a>not_false_iff</a>]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 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'_2\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\u03b9 : Type u_5\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p\n\u22a2 Mem\u2112p (fun a => \u2211 i in insert i s, f i a) 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 : \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\u03b9 : Type u_5\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p\n\u22a2 Mem\u2112p (fun a => f i a + \u2211 i in s, f i a) p"}, {"tactic": "exact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_mem hj))", "annotated_tactic": ["exact (hf i (s.mem_insert_self i)).<a>add</a> (ih fun j hj => hf j (<a>Finset.mem_insert_of_mem</a> hj))", [{"full_name": "MeasureTheory.Mem\u2112p.add", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1234, 9], "def_end_pos": [1234, 18]}, {"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 refine'_2\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\u03b9 : Type u_5\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p\n\u22a2 Mem\u2112p (fun a => f i a + \u2211 i in s, f i a) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.tendsto_Lp_of_tendsto_\u2112p", "start": [1402, 1], "end": [1406, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_subtype_coe", "start": [577, 1], "end": [578, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "generateFrom_piiUnionInter_measurableSet", "start": [526, 1], "end": [533, 70], "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\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) = \u2a06 i \u2208 S, m i", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) \u2264 \u2a06 i \u2208 S, m i\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 \u2a06 i \u2208 S, m i \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)"}, {"tactic": "rw [\u2190 @generateFrom_measurableSet \u03b1 (\u2a06 i \u2208 S, m i)]", "annotated_tactic": ["rw [\u2190 @<a>generateFrom_measurableSet</a> \u03b1 (\u2a06 i \u2208 S, m i)]", [{"full_name": "MeasurableSpace.generateFrom_measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [396, 9], "def_end_pos": [396, 35]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) \u2264 \u2a06 i \u2208 S, m i", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) \u2264 generateFrom {s | MeasurableSet s}"}, {"tactic": "exact generateFrom_mono (measurableSet_iSup_of_mem_piiUnionInter m S)", "annotated_tactic": ["exact <a>generateFrom_mono</a> (<a>measurableSet_iSup_of_mem_piiUnionInter</a> m S)", [{"full_name": "MeasurableSpace.generateFrom_mono", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [438, 9], "def_end_pos": [438, 26]}, {"full_name": "measurableSet_iSup_of_mem_piiUnionInter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [516, 9], "def_end_pos": [516, 48]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) \u2264 generateFrom {s | MeasurableSet s}", "state_after": "no goals"}, {"tactic": "refine' iSup\u2082_le fun i hi => _", "annotated_tactic": ["refine' <a>iSup\u2082_le</a> fun i hi => _", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\n\u22a2 \u2a06 i \u2208 S, m i \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\ni : \u03b9\nhi : i \u2208 S\n\u22a2 m i \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)"}, {"tactic": "rw [\u2190 @generateFrom_measurableSet \u03b1 (m i)]", "annotated_tactic": ["rw [\u2190 @<a>generateFrom_measurableSet</a> \u03b1 (m i)]", [{"full_name": "MeasurableSpace.generateFrom_measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [396, 9], "def_end_pos": [396, 35]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\ni : \u03b9\nhi : i \u2208 S\n\u22a2 m i \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\ni : \u03b9\nhi : i \u2208 S\n\u22a2 generateFrom {s | MeasurableSet s} \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)"}, {"tactic": "exact generateFrom_mono (mem_piiUnionInter_of_measurableSet m hi)", "annotated_tactic": ["exact <a>generateFrom_mono</a> (<a>mem_piiUnionInter_of_measurableSet</a> m hi)", [{"full_name": "MeasurableSpace.generateFrom_mono", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [438, 9], "def_end_pos": [438, 26]}, {"full_name": "mem_piiUnionInter_of_measurableSet", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [505, 9], "def_end_pos": [505, 43]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 MeasurableSpace \u03b1\nS : Set \u03b9\ni : \u03b9\nhi : i \u2208 S\n\u22a2 generateFrom {s | MeasurableSet s} \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) 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_zero", "start": [981, 1], "end": [986, 33], "traced_tactics": [{"tactic": "simp only [Pi.zero_apply, SimpleFunc.coe_zero, support_zero', measure_empty,\n  WithTop.zero_lt_top, forall_const]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>SimpleFunc.coe_zero</a>, <a>support_zero'</a>, <a>measure_empty</a>,\n      <a>WithTop.zero_lt_top</a>, <a>forall_const</a>]", [{"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.coe_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Function.support_zero'", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [127, 3], "def_end_pos": [127, 14]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"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": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(OfNat.ofNat 0 n)) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.iff_comp_neg", "start": [338, 1], "end": [340, 71], "traced_tactics": [{"tactic": "rw [\u2190 comp_mul_left_iff (neg_ne_zero.2 one_ne_zero)]", "annotated_tactic": ["rw [\u2190 <a>comp_mul_left_iff</a> (<a>neg_ne_zero</a>.2 <a>one_ne_zero</a>)]", [{"full_name": "IntervalIntegrable.comp_mul_left_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [306, 9], "def_end_pos": [306, 26]}, {"full_name": "neg_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [434, 3], "def_end_pos": [434, 14]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "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\n\u22a2 IntervalIntegrable f volume a b \u2194 IntervalIntegrable (fun x => f (-x)) volume (-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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na b : \u211d\n\u03bc : Measure \u211d\n\u22a2 IntervalIntegrable (fun x => f (-1 * x)) volume (a / -1) (b / -1) \u2194\n    IntervalIntegrable (fun x => f (-x)) volume (-a) (-b)"}, {"tactic": "simp [div_neg]", "annotated_tactic": ["simp [<a>div_neg</a>]", [{"full_name": "div_neg", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedRing A\nf g : \u211d \u2192 E\na b : \u211d\n\u03bc : Measure \u211d\n\u22a2 IntervalIntegrable (fun x => f (-1 * x)) volume (a / -1) (b / -1) \u2194\n    IntervalIntegrable (fun x => f (-x)) volume (-a) (-b)", "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.getRight_inr", "start": [84, 9], "end": [84, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurableSet_min_const_iff", "start": [617, 1], "end": [620, 77], "traced_tactics": [{"tactic": "apply MeasurableSpace.measurableSet_inf", "annotated_tactic": ["apply <a>MeasurableSpace.measurableSet_inf</a>", [{"full_name": "MeasurableSpace.measurableSet_inf", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 26]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : LinearOrder \u03b9\nf : Filtration \u03b9 m\n\u03c4 \u03c0 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\ns : Set \u03a9\ni : \u03b9\n\u22a2 MeasurableSet s \u2194 MeasurableSet s \u2227 MeasurableSet 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.le_sup_lintegral", "start": [1090, 1], "end": [1100, 66], "traced_tactics": [{"tactic": "rw [map_lintegral, map_lintegral]", "annotated_tactic": ["rw [<a>map_lintegral</a>, <a>map_lintegral</a>]", [{"full_name": "MeasureTheory.SimpleFunc.map_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [976, 9], "def_end_pos": [976, 22]}, {"full_name": "MeasureTheory.SimpleFunc.map_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [976, 9], "def_end_pos": [976, 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\nf g : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 lintegral (map Prod.fst (pair f g)) \u03bc \u2294 lintegral (map Prod.snd (pair f g)) \u03bc \u2264\n    \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 (\u2211 x in SimpleFunc.range (pair f g), x.1 * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x})) \u2294\n      \u2211 x in SimpleFunc.range (pair f g), x.2 * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) \u2264\n    \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x})"}, {"tactic": "refine' sup_le _ _ <;> refine' Finset.sum_le_sum fun a _ => mul_le_mul_right' _ _", "annotated_tactic": ["refine' <a>sup_le</a> _ _ <;> refine' <a>Finset.sum_le_sum</a> fun a _ => <a>mul_le_mul_right'</a> _ _", [{"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [167, 9], "def_end_pos": [167, 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_right'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [67, 9], "def_end_pos": [67, 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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 (\u2211 x in SimpleFunc.range (pair f g), x.1 * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x})) \u2294\n      \u2211 x in SimpleFunc.range (pair f g), x.2 * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) \u2264\n    \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x})", "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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\na : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nx\u271d : a \u2208 SimpleFunc.range (pair f g)\n\u22a2 a.1 \u2264 a.1 \u2294 a.2\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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\na : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nx\u271d : a \u2208 SimpleFunc.range (pair f g)\n\u22a2 a.2 \u2264 a.1 \u2294 a.2"}, {"tactic": "exact le_sup_left", "annotated_tactic": ["exact <a>le_sup_left</a>", [{"full_name": "le_sup_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}]], "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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\na : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nx\u271d : a \u2208 SimpleFunc.range (pair f g)\n\u22a2 a.1 \u2264 a.1 \u2294 a.2", "state_after": "no goals"}, {"tactic": "exact le_sup_right", "annotated_tactic": ["exact <a>le_sup_right</a>", [{"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [141, 9], "def_end_pos": [141, 21]}]], "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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e\na : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nx\u271d : a \u2208 SimpleFunc.range (pair f g)\n\u22a2 a.2 \u2264 a.1 \u2294 a.2", "state_after": "no goals"}, {"tactic": "rw [sup_eq_map\u2082, map_lintegral]", "annotated_tactic": ["rw [<a>sup_eq_map\u2082</a>, <a>map_lintegral</a>]", [{"full_name": "MeasureTheory.SimpleFunc.sup_eq_map\u2082", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "MeasureTheory.SimpleFunc.map_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [976, 9], "def_end_pos": [976, 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\nf g : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) = lintegral (f \u2294 g) \u03bc", "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", "start": [874, 1], "end": [917, 48], "traced_tactics": [{"tactic": "by_cases h\u03c1_zero : \u03c1.fst.restrict s = 0", "annotated_tactic": ["by_cases h\u03c1_zero : \u03c1.fst.restrict s = 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\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": "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\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)\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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\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": "have h :\n  \u222b\u207b a in s, ENNReal.ofReal (condCdf \u03c1 a x) \u2202\u03c1.fst =\n    \u222b\u207b a in s, ENNReal.ofReal (\u2a05 r : { r' : \u211a // x < r' }, condCdf \u03c1 a r) \u2202\u03c1.fst := by\n  congr with a : 1\n  rw [\u2190 (condCdf \u03c1 a).iInf_rat_gt_eq x]", "annotated_tactic": ["have h :\n    \u222b\u207b a in s, <a>ENNReal.ofReal</a> (<a>condCdf</a> \u03c1 a x) \u2202\u03c1.fst =\n      \u222b\u207b a in s, <a>ENNReal.ofReal</a> (\u2a05 r : { r' : \u211a // x < r' }, <a>condCdf</a> \u03c1 a r) \u2202\u03c1.fst := by\n    congr with a : 1\n    rw [\u2190 (<a>condCdf</a> \u03c1 a).<a>iInf_rat_gt_eq</a> x]", [{"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": "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.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"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 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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\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": "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\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": "have h_nonempty : Nonempty { r' : \u211a // x < \u2191r' } := by\n  obtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n  exact \u27e8\u27e8r, hrx\u27e9\u27e9", "annotated_tactic": ["have h_nonempty : <a>Nonempty</a> { r' : \u211a // x < \u2191r' } := by\n    obtain \u27e8r, hrx\u27e9 := <a>exists_rat_gt</a> x\n    exact \u27e8\u27e8r, hrx\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": "exists_rat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\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": "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\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": "rw [h]", "annotated_tactic": ["rw [h]", []], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\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": "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "simp_rw [ENNReal.ofReal_cinfi]", "annotated_tactic": ["simp_rw [<a>ENNReal.ofReal_cinfi</a>]", [{"full_name": "ENNReal.ofReal_cinfi", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [119, 9], "def_end_pos": [119, 29]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 i, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "have h_coe : \u2200 b : { r' : \u211a // x < \u2191r' }, (b : \u211d) = ((b : \u211a) : \u211d) := fun _ => by congr", "annotated_tactic": ["have h_coe : \u2200 b : { r' : \u211a // x < \u2191r' }, (b : \u211d) = ((b : \u211a) : \u211d) := fun _ => by congr", []], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 i, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 i, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "rw [lintegral_iInf_directed_of_measurable h\u03c1_zero fun q : { r' : \u211a // x < \u2191r' } =>\n    (measurable_condCdf \u03c1 q).ennreal_ofReal]", "annotated_tactic": ["rw [<a>lintegral_iInf_directed_of_measurable</a> h\u03c1_zero fun q : { r' : \u211a // x < \u2191r' } =>\n      (<a>measurable_condCdf</a> \u03c1 q).<a>ennreal_ofReal</a>]", [{"full_name": "lintegral_iInf_directed_of_measurable", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [135, 9], "def_end_pos": [135, 46]}, {"full_name": "ProbabilityTheory.measurable_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [854, 9], "def_end_pos": [854, 27]}, {"full_name": "Measurable.ennreal_ofReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2005, 9], "def_end_pos": [2005, 34]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 i, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 b, \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\ncase neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4\n\ncase neg.h_directed\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q)"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 b, \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\ncase neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4\n\ncase neg.h_directed\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q)", "state_after": "case neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4\n\ncase neg.h_directed\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q)\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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 b, \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "simp_rw [set_lintegral_condCdf_rat \u03c1 _ hs]", "annotated_tactic": ["simp_rw [<a>set_lintegral_condCdf_rat</a> \u03c1 _ hs]", [{"full_name": "ProbabilityTheory.set_lintegral_condCdf_rat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [865, 9], "def_end_pos": [865, 34]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 b, \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 b, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "rw [\u2190 measure_iInter_eq_iInf]", "annotated_tactic": ["rw [\u2190 <a>measure_iInter_eq_iInf</a>]", [{"full_name": "MeasureTheory.measure_iInter_eq_iInf", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [507, 9], "def_end_pos": [507, 31]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 b, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\ncase neg.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (i : { r' // x < \u2191r' }), MeasurableSet (s \u00d7\u02e2 Iic \u2191\u2191i)\n\ncase neg.hd\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun b => s \u00d7\u02e2 Iic \u2191\u2191b\n\ncase neg.hfin\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2203 i, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191i) \u2260 \u22a4"}, {"tactic": "rw [h\u03c1_zero, lintegral_zero_measure]", "annotated_tactic": ["rw [h\u03c1_zero, <a>lintegral_zero_measure</a>]", [{"full_name": "MeasureTheory.lintegral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [632, 9], "def_end_pos": [632, 31]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\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": "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 0 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "refine' le_antisymm (zero_le _) _", "annotated_tactic": ["refine' <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 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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 0 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u2264 0"}, {"tactic": "calc\n  \u03c1 (s \u00d7\u02e2 Iic x) \u2264 \u03c1 (Prod.fst \u207b\u00b9' s) := measure_mono (prod_subset_preimage_fst s (Iic x))\n  _ = \u03c1.fst s := by rw [Measure.fst_apply hs]\n  _ = \u03c1.fst.restrict s univ := by rw [Measure.restrict_apply_univ]\n  _ = 0 := by simp only [h\u03c1_zero, Measure.coe_zero, Pi.zero_apply]", "annotated_tactic": ["calc\n      \u03c1 (s \u00d7\u02e2 <a>Iic</a> x) \u2264 \u03c1 (<a>Prod.fst</a> \u207b\u00b9' s) := <a>measure_mono</a> (<a>prod_subset_preimage_fst</a> s (<a>Iic</a> x))\n      _ = \u03c1.fst s := by rw [<a>Measure.fst_apply</a> hs]\n      _ = \u03c1.fst.restrict s <a>univ</a> := by rw [<a>Measure.restrict_apply_univ</a>]\n      _ = 0 := by simp only [h\u03c1_zero, <a>Measure.coe_zero</a>, <a>Pi.zero_apply</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.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "Set.prod_subset_preimage_fst", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [375, 9], "def_end_pos": [375, 33]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"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": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"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.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": "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u2264 0", "state_after": "no goals"}, {"tactic": "rw [Measure.fst_apply hs]", "annotated_tactic": ["rw [<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]}]], "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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s) = \u2191\u2191(Measure.fst \u03c1) s", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_apply_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_apply_univ</a>]", [{"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\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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191(Measure.fst \u03c1) s = \u2191\u2191(Measure.restrict (Measure.fst \u03c1) s) univ", "state_after": "no goals"}, {"tactic": "simp only [h\u03c1_zero, Measure.coe_zero, Pi.zero_apply]", "annotated_tactic": ["simp only [h\u03c1_zero, <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": "\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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191(Measure.restrict (Measure.fst \u03c1) s) univ = 0", "state_after": "no goals"}, {"tactic": "congr with a : 1", "annotated_tactic": ["congr with a : 1", []], "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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1", "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\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\na : \u03b1\n\u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) = ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r)"}, {"tactic": "rw [\u2190 (condCdf \u03c1 a).iInf_rat_gt_eq x]", "annotated_tactic": ["rw [\u2190 (<a>condCdf</a> \u03c1 a).<a>iInf_rat_gt_eq</a> x]", [{"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"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 e_f.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\na : \u03b1\n\u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) = ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r)", "state_after": "no goals"}, {"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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\n\u22a2 Nonempty { r' // x < \u2191r' }", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nr : \u211a\nhrx : x < \u2191r\n\u22a2 Nonempty { r' // x < \u2191r' }"}, {"tactic": "exact \u27e8\u27e8r, hrx\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\u27e8r, hrx\u27e9\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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nr : \u211a\nhrx : x < \u2191r\n\u22a2 Nonempty { r' // x < \u2191r' }", "state_after": "no goals"}, {"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\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nx\u271d : { r' // x < \u2191r' }\n\u22a2 \u2191\u2191x\u271d = \u2191\u2191x\u271d", "state_after": "no goals"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "state_before": "case neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4", "state_after": "case neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\nb : { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4"}, {"tactic": "rw [set_lintegral_condCdf_rat \u03c1 _ hs]", "annotated_tactic": ["rw [<a>set_lintegral_condCdf_rat</a> \u03c1 _ hs]", [{"full_name": "ProbabilityTheory.set_lintegral_condCdf_rat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [865, 9], "def_end_pos": [865, 34]}]], "state_before": "case neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\nb : { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4", "state_after": "case neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\nb : { r' // x < \u2191r' }\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) \u2260 \u22a4"}, {"tactic": "exact measure_ne_top \u03c1 _", "annotated_tactic": ["exact <a>measure_ne_top</a> \u03c1 _", [{"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 neg.hf_int\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\nb : { r' // x < \u2191r' }\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "refine' Monotone.directed_ge fun i j hij a => ENNReal.ofReal_le_ofReal ((condCdf \u03c1 a).mono _)", "annotated_tactic": ["refine' <a>Monotone.directed_ge</a> fun i j hij a => <a>ENNReal.ofReal_le_ofReal</a> ((<a>condCdf</a> \u03c1 a).<a>mono</a> _)", [{"full_name": "Monotone.directed_ge", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [149, 9], "def_end_pos": [149, 29]}, {"full_name": "ENNReal.ofReal_le_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"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]}]], "state_before": "case neg.h_directed\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q)", "state_after": "case neg.h_directed\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\na : \u03b1\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j"}, {"tactic": "exact_mod_cast hij", "annotated_tactic": ["exact_mod_cast hij", []], "state_before": "case neg.h_directed\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\na : \u03b1\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j", "state_after": "no goals"}, {"tactic": "rw [\u2190 prod_iInter]", "annotated_tactic": ["rw [\u2190 <a>prod_iInter</a>]", [{"full_name": "prod_iInter", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [70, 9], "def_end_pos": [70, 20]}]], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c2 i, Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)"}, {"tactic": "congr with y", "annotated_tactic": ["congr with y", []], "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)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c2 i, Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)", "state_after": "case neg.e_a.e_a.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ny : \u211d\n\u22a2 y \u2208 \u22c2 i, Iic \u2191\u2191i \u2194 y \u2208 Iic x"}, {"tactic": "simp only [mem_iInter, mem_Iic, Subtype.forall, Subtype.coe_mk]", "annotated_tactic": ["simp only [<a>mem_iInter</a>, <a>mem_Iic</a>, <a>Subtype.forall</a>, <a>Subtype.coe_mk</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": "Subtype.forall", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [43, 19], "def_end_pos": [43, 27]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case neg.e_a.e_a.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ny : \u211d\n\u22a2 y \u2208 \u22c2 i, Iic \u2191\u2191i \u2194 y \u2208 Iic x", "state_after": "case neg.e_a.e_a.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ny : \u211d\n\u22a2 (\u2200 (a : \u211a), x < \u2191a \u2192 y \u2264 \u2191a) \u2194 y \u2264 x"}, {"tactic": "exact \u27e8le_of_forall_lt_rat_imp_le, fun hyx q hq => hyx.trans hq.le\u27e9", "annotated_tactic": ["exact \u27e8<a>le_of_forall_lt_rat_imp_le</a>, fun hyx q hq => hyx.trans hq.le\u27e9", [{"full_name": "le_of_forall_lt_rat_imp_le", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [290, 9], "def_end_pos": [290, 35]}]], "state_before": "case neg.e_a.e_a.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ny : \u211d\n\u22a2 (\u2200 (a : \u211a), x < \u2191a \u2192 y \u2264 \u2191a) \u2194 y \u2264 x", "state_after": "no goals"}, {"tactic": "exact fun i => hs.prod measurableSet_Iic", "annotated_tactic": ["exact fun i => hs.prod <a>measurableSet_Iic</a>", [{"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}]], "state_before": "case neg.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\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (i : { r' // x < \u2191r' }), MeasurableSet (s \u00d7\u02e2 Iic \u2191\u2191i)", "state_after": "no goals"}, {"tactic": "refine' Monotone.directed_ge fun i j hij => _", "annotated_tactic": ["refine' <a>Monotone.directed_ge</a> fun i j hij => _", [{"full_name": "Monotone.directed_ge", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [149, 9], "def_end_pos": [149, 29]}]], "state_before": "case neg.hd\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun b => s \u00d7\u02e2 Iic \u2191\u2191b", "state_after": "case neg.hd\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\n\u22a2 s \u00d7\u02e2 Iic \u2191\u2191i \u2264 s \u00d7\u02e2 Iic \u2191\u2191j"}, {"tactic": "refine' prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, Iic_subset_Iic.mpr _\u27e9)", "annotated_tactic": ["refine' prod_subset_prod_iff.mpr (<a>Or.inl</a> \u27e8<a>subset_rfl</a>, Iic_subset_Iic.mpr _\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 neg.hd\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\n\u22a2 s \u00d7\u02e2 Iic \u2191\u2191i \u2264 s \u00d7\u02e2 Iic \u2191\u2191j", "state_after": "case neg.hd\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j"}, {"tactic": "exact_mod_cast hij", "annotated_tactic": ["exact_mod_cast hij", []], "state_before": "case neg.hd\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j", "state_after": "no goals"}, {"tactic": "exact \u27e8h_nonempty.some, measure_ne_top _ _\u27e9", "annotated_tactic": ["exact \u27e8h_nonempty.some, <a>measure_ne_top</a> _ _\u27e9", [{"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 neg.hfin\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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n  \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2203 i, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191i) \u2260 \u22a4", "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_join", "start": [219, 1], "end": [224, 23], "traced_tactics": [{"tactic": "show bind \u03bc join = join (join \u03bc)", "annotated_tactic": ["show <a>bind</a> \u03bc <a>join</a> = <a>join</a> (<a>join</a> \u03bc)", [{"full_name": "MeasureTheory.Measure.bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [152, 5], "def_end_pos": [152, 9]}, {"full_name": "MeasureTheory.Measure.join", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [99, 5], "def_end_pos": [99, 9]}, {"full_name": "MeasureTheory.Measure.join", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [99, 5], "def_end_pos": [99, 9]}, {"full_name": "MeasureTheory.Measure.join", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [99, 5], "def_end_pos": [99, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 join (map join \u03bc) = join (join \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 bind \u03bc join = join (join \u03bc)"}, {"tactic": "rw [join_eq_bind, join_eq_bind, bind_bind measurable_id measurable_id]", "annotated_tactic": ["rw [<a>join_eq_bind</a>, <a>join_eq_bind</a>, <a>bind_bind</a> <a>measurable_id</a> <a>measurable_id</a>]", [{"full_name": "MeasureTheory.Measure.join_eq_bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [208, 9], "def_end_pos": [208, 21]}, {"full_name": "MeasureTheory.Measure.join_eq_bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [208, 9], "def_end_pos": [208, 21]}, {"full_name": "MeasureTheory.Measure.bind_bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}, {"full_name": "measurable_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}, {"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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 bind \u03bc join = join (join \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 bind \u03bc join = bind \u03bc fun a => bind (id a) id"}, {"tactic": "apply congr_arg (bind \u03bc)", "annotated_tactic": ["apply <a>congr_arg</a> (<a>bind</a> \u03bc)", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "MeasureTheory.Measure.bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [152, 5], "def_end_pos": [152, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 bind \u03bc join = bind \u03bc fun a => bind (id a) id", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 join = fun a => bind (id a) id"}, {"tactic": "funext \u03bd", "annotated_tactic": ["funext \u03bd", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u22a2 join = fun a => bind (id a) id", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u03bd : Measure (Measure \u03b1)\n\u22a2 join \u03bd = bind (id \u03bd) id"}, {"tactic": "exact join_eq_bind \u03bd", "annotated_tactic": ["exact <a>join_eq_bind</a> \u03bd", [{"full_name": "MeasureTheory.Measure.join_eq_bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [208, 9], "def_end_pos": [208, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure (Measure (Measure \u03b1))\n\u03bd : Measure (Measure \u03b1)\n\u22a2 join \u03bd = bind (id \u03bd) id", "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_zero", "start": [1058, 1], "end": [1060, 35], "traced_tactics": [{"tactic": "refine' (eq_singularPart 0 0 VectorMeasure.MutuallySingular.zero_left _).symm", "annotated_tactic": ["refine' (<a>eq_singularPart</a> 0 0 <a>VectorMeasure.MutuallySingular.zero_left</a> _).<a>symm</a>", [{"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.VectorMeasure.MutuallySingular.zero_left", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1196, 9], "def_end_pos": [1196, 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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 singularPart 0 \u03bc = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 0 = 0 + withDensity\u1d65 \u03bc 0"}, {"tactic": "rw [zero_add, withDensity\u1d65_zero]", "annotated_tactic": ["rw [<a>zero_add</a>, <a>withDensity\u1d65_zero</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "MeasureTheory.withDensity\u1d65_zero", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [65, 9], "def_end_pos": [65, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 0 = 0 + withDensity\u1d65 \u03bc 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integrable_of_summable_norm_restrict", "start": [855, 1], "end": [858, 92], "traced_tactics": [{"tactic": "simpa only [hs, integrableOn_univ] using integrableOn_iUnion_of_summable_norm_restrict hf", "annotated_tactic": ["simpa only [hs, <a>integrableOn_univ</a>] using <a>integrableOn_iUnion_of_summable_norm_restrict</a> hf", [{"full_name": "MeasureTheory.integrableOn_univ", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [110, 9], "def_end_pos": [110, 26]}, {"full_name": "MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [839, 9], "def_end_pos": [839, 54]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : Countable \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nf : C(\u03b1, E)\ns : \u03b2 \u2192 Compacts \u03b1\nhf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i))\nhs : \u22c3 i, \u2191(s i) = univ\n\u22a2 Integrable \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.disjSups_inter_subset_left", "start": [549, 1], "end": [550, 92], "traced_tactics": [{"tactic": "simpa only [disjSups, inter_product, filter_inter_distrib] using image_inter_subset _ _ _", "annotated_tactic": ["simpa only [<a>disjSups</a>, <a>inter_product</a>, <a>filter_inter_distrib</a>] using <a>image_inter_subset</a> _ _ _", [{"full_name": "Finset.disjSups", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [465, 5], "def_end_pos": [465, 13]}, {"full_name": "Finset.inter_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [249, 9], "def_end_pos": [249, 22]}, {"full_name": "Finset.filter_inter_distrib", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2881, 9], "def_end_pos": [2881, 29]}, {"full_name": "Finset.image_inter_subset", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 27]}]], "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\u2081 \u2229 s\u2082) \u25cb t \u2286 s\u2081 \u25cb t \u2229 s\u2082 \u25cb t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.restrict_piecewise", "start": [106, 1], "end": [108, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto", "start": [532, 1], "end": [539, 26], "traced_tactics": [{"tactic": "rw [tendsto_iff_forall_toWeakDualBCNN_tendsto]", "annotated_tactic": ["rw [<a>tendsto_iff_forall_toWeakDualBCNN_tendsto</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [487, 9], "def_end_pos": [487, 50]}]], "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 => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))", "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 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))"}, {"tactic": "simp_rw [toWeakDualBCNN_apply _ _, \u2190 testAgainstNN_coe_eq, ENNReal.tendsto_coe,\n  ENNReal.toNNReal_coe]", "annotated_tactic": ["simp_rw [<a>toWeakDualBCNN_apply</a> _ _, \u2190 <a>testAgainstNN_coe_eq</a>, <a>ENNReal.tendsto_coe</a>,\n    <a>ENNReal.toNNReal_coe</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.toWeakDualBCNN_apply", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [443, 9], "def_end_pos": [443, 29]}, {"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_coe_eq", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [326, 9], "def_end_pos": [326, 29]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "ENNReal.toNNReal_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [176, 9], "def_end_pos": [176, 21]}]], "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 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_to_bool", "start": [465, 1], "end": [470, 10], "traced_tactics": [{"tactic": "apply measurable_to_countable'", "annotated_tactic": ["apply <a>measurable_to_countable'</a>", [{"full_name": "measurable_to_countable'", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 33]}]], "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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 Measurable f", "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\u03b9\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 \u2200 (x : Bool), MeasurableSet (f \u207b\u00b9' {x})"}, {"tactic": "rintro (- | -)", "annotated_tactic": ["rintro (- | -)", []], "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\u03b9\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 \u2200 (x : Bool), MeasurableSet (f \u207b\u00b9' {x})", "state_after": "case h.false\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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 MeasurableSet (f \u207b\u00b9' {false})\n\ncase h.true\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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 MeasurableSet (f \u207b\u00b9' {true})"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case h.true\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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 MeasurableSet (f \u207b\u00b9' {true})", "state_after": "no goals"}, {"tactic": "convert h.compl", "annotated_tactic": ["convert h.compl", []], "state_before": "case h.false\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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 MeasurableSet (f \u207b\u00b9' {false})", "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 f \u207b\u00b9' {false} = (f \u207b\u00b9' {true})\u1d9c"}, {"tactic": "rw [\u2190 preimage_compl, Bool.compl_singleton, Bool.not_true]", "annotated_tactic": ["rw [\u2190 <a>preimage_compl</a>, <a>Bool.compl_singleton</a>, <a>Bool.not_true</a>]", [{"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Bool.compl_singleton", "def_path": "Mathlib/Data/Bool/Set.lean", "def_pos": [32, 17], "def_end_pos": [32, 32]}, {"full_name": "Bool.not_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [124, 17], "def_end_pos": [124, 30]}]], "state_before": "case h.e'_3\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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Bool\nh : MeasurableSet (f \u207b\u00b9' {true})\n\u22a2 f \u207b\u00b9' {false} = (f \u207b\u00b9' {true})\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.indicator\u2080", "start": [374, 1], "end": [376, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_indexOf", "start": [1136, 1], "end": [1137, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.aestronglyMeasurable'_condexpInd", "start": [285, 1], "end": [288, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.AEEqFun.snorm_compMeasurePreserving", "start": [945, 1], "end": [949, 63], "traced_tactics": [{"tactic": "rw [snorm_congr_ae (g.coeFn_compMeasurePreserving _)]", "annotated_tactic": ["rw [<a>snorm_congr_ae</a> (g.coeFn_compMeasurePreserving _)]", [{"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\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\u271d : 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\u271d : \u03b2 \u2192 E\n\u03bd : Measure \u03b2\ng : \u03b2 \u2192\u2098[\u03bd] E\nhf : MeasurePreserving f\n\u22a2 snorm (\u2191(compMeasurePreserving g f hf)) p \u03bc = snorm (\u2191g) p \u03bd", "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\u271d : 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\u271d : \u03b2 \u2192 E\n\u03bd : Measure \u03b2\ng : \u03b2 \u2192\u2098[\u03bd] E\nhf : MeasurePreserving f\n\u22a2 snorm (\u2191g \u2218 f) p \u03bc = snorm (\u2191g) p \u03bd"}, {"tactic": "exact snorm_comp_measurePreserving g.aestronglyMeasurable hf", "annotated_tactic": ["exact <a>snorm_comp_measurePreserving</a> g.aestronglyMeasurable hf", [{"full_name": "MeasureTheory.snorm_comp_measurePreserving", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [941, 9], "def_end_pos": [941, 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\u271d : 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\u271d : \u03b2 \u2192 E\n\u03bd : Measure \u03b2\ng : \u03b2 \u2192\u2098[\u03bd] E\nhf : MeasurePreserving f\n\u22a2 snorm (\u2191g \u2218 f) p \u03bc = snorm (\u2191g) p \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "full_name": "MeasureTheory.Measure.lintegral_bind", "start": [183, 1], "end": [185, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.dvd_of_dvd_mul_right_of_gcd_one", "start": [424, 1], "end": [427, 48], "traced_tactics": [{"tactic": "rw [mul_comm] at habc", "annotated_tactic": ["rw [<a>mul_comm</a>] at habc", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "a b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a b = 1\n\u22a2 a \u2223 c", "state_after": "a b c : \u2124\nhabc : a \u2223 c * b\nhab : gcd a b = 1\n\u22a2 a \u2223 c"}, {"tactic": "exact dvd_of_dvd_mul_left_of_gcd_one habc hab", "annotated_tactic": ["exact <a>dvd_of_dvd_mul_left_of_gcd_one</a> habc hab", [{"full_name": "Int.dvd_of_dvd_mul_left_of_gcd_one", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [412, 9], "def_end_pos": [412, 39]}]], "state_before": "a b c : \u2124\nhabc : a \u2223 c * b\nhab : gcd a b = 1\n\u22a2 a \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.union_comm_of_disjoint", "start": [679, 1], "end": [682, 82], "traced_tactics": [{"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081\u271d s\u2082\u271d : Finmap \u03b2\ns\u2081 s\u2082 : AList \u03b2\n\u22a2 Disjoint \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 \u2192 \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2082\u27e7 \u222a \u27e6s\u2081\u27e7", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081\u271d s\u2082\u271d : Finmap \u03b2\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7\n\u22a2 \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2082\u27e7 \u222a \u27e6s\u2081\u27e7"}, {"tactic": "simp only [AList.toFinmap_eq, union_toFinmap, AList.union_comm_of_disjoint h]", "annotated_tactic": ["simp only [<a>AList.toFinmap_eq</a>, <a>union_toFinmap</a>, <a>AList.union_comm_of_disjoint</a> h]", [{"full_name": "AList.toFinmap_eq", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}, {"full_name": "Finmap.union_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [574, 9], "def_end_pos": [574, 23]}, {"full_name": "AList.union_comm_of_disjoint", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [509, 9], "def_end_pos": [509, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081\u271d s\u2082\u271d : Finmap \u03b2\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7\n\u22a2 \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2082\u27e7 \u222a \u27e6s\u2081\u27e7", "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.re", "start": [1610, 1], "end": [1616, 60], "traced_tactics": [{"tactic": "have : \u2200 x, \u2016IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n  intro x\n  rw [one_mul]\n  exact IsROrC.norm_re_le_norm (f x)", "annotated_tactic": ["have : \u2200 x, \u2016<a>IsROrC.re</a> (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n    intro x\n    rw [<a>one_mul</a>]\n    exact <a>IsROrC.norm_re_le_norm</a> (f x)", [{"full_name": "IsROrC.re", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 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_re_le_norm", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [731, 9], "def_end_pos": [731, 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.re (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.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (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.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (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.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.re (f x)) \u03bc"}, {"tactic": "exact IsROrC.continuous_re.comp_aestronglyMeasurable hf.1", "annotated_tactic": ["exact IsROrC.continuous_re.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.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.re (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.re (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.re (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.re (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.re (f x)\u2016 \u2264 \u2016f x\u2016"}, {"tactic": "exact IsROrC.norm_re_le_norm (f x)", "annotated_tactic": ["exact <a>IsROrC.norm_re_le_norm</a> (f x)", [{"full_name": "IsROrC.norm_re_le_norm", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [731, 9], "def_end_pos": [731, 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.re (f x)\u2016 \u2264 \u2016f x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.measurable_condCdf", "start": [854, 1], "end": [861, 61], "traced_tactics": [{"tactic": "have : (fun a => condCdf \u03c1 a x) = fun a => \u2a05 r : { r' : \u211a // x < r' }, condCdfRat \u03c1 a \u2191r := by\n  ext1 a\n  rw [\u2190 StieltjesFunction.iInf_rat_gt_eq]\n  congr with q\n  rw [condCdf_eq_condCdfRat]", "annotated_tactic": ["have : (fun a => <a>condCdf</a> \u03c1 a x) = fun a => \u2a05 r : { r' : \u211a // x < r' }, <a>condCdfRat</a> \u03c1 a \u2191r := by\n    ext1 a\n    rw [\u2190 <a>StieltjesFunction.iInf_rat_gt_eq</a>]\n    congr with q\n    rw [<a>condCdf_eq_condCdfRat</a>]", [{"full_name": "ProbabilityTheory.condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [777, 19], "def_end_pos": [777, 26]}, {"full_name": "ProbabilityTheory.condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [570, 19], "def_end_pos": [570, 29]}, {"full_name": "StieltjesFunction.iInf_rat_gt_eq", "def_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "def_pos": [82, 9], "def_end_pos": [82, 23]}, {"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)\nx : \u211d\n\u22a2 Measurable fun a => \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)\nx : \u211d\nthis : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 Measurable fun a => \u2191(condCdf \u03c1 a) x"}, {"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\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\nthis : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 Measurable fun a => \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)\nx : \u211d\nthis : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 Measurable fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r"}, {"tactic": "exact measurable_iInf (fun q => measurable_condCdfRat \u03c1 q)", "annotated_tactic": ["exact <a>measurable_iInf</a> (fun q => <a>measurable_condCdfRat</a> \u03c1 q)", [{"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_condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [598, 9], "def_end_pos": [598, 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)\nx : \u211d\nthis : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r\n\u22a2 Measurable fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r", "state_after": "no goals"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "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)\nx : \u211d\n\u22a2 (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \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)\nx : \u211d\na : \u03b1\n\u22a2 \u2191(condCdf \u03c1 a) x = \u2a05 r, condCdfRat \u03c1 a \u2191r"}, {"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 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)\nx : \u211d\na : \u03b1\n\u22a2 \u2191(condCdf \u03c1 a) x = \u2a05 r, condCdfRat \u03c1 a \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)\nx : \u211d\na : \u03b1\n\u22a2 \u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r"}, {"tactic": "congr with q", "annotated_tactic": ["congr with q", []], "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)\nx : \u211d\na : \u03b1\n\u22a2 \u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r", "state_after": "case h.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)\nx : \u211d\na : \u03b1\nq : { r' // x < \u2191r' }\n\u22a2 \u2191(condCdf \u03c1 a) \u2191\u2191q = condCdfRat \u03c1 a \u2191q"}, {"tactic": "rw [condCdf_eq_condCdfRat]", "annotated_tactic": ["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 h.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)\nx : \u211d\na : \u03b1\nq : { r' // x < \u2191r' }\n\u22a2 \u2191(condCdf \u03c1 a) \u2191\u2191q = condCdfRat \u03c1 a \u2191q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.fst_apply'", "start": [782, 1], "end": [783, 97], "traced_tactics": [{"tactic": "rw [fst_apply, Measure.map_apply measurable_fst hs]", "annotated_tactic": ["rw [<a>fst_apply</a>, <a>Measure.map_apply</a> <a>measurable_fst</a> hs]", [{"full_name": "ProbabilityTheory.kernel.fst_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [778, 9], "def_end_pos": [778, 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": "\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 \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(fst \u03ba) a) s = \u2191\u2191(\u2191\u03ba a) {p | p.1 \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 \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.fst \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | p.1 \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 \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.fst \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | p.1 \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indepSet_iff_measure_inter_eq_mul", "start": [497, 1], "end": [500, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top", "start": [105, 1], "end": [125, 87], "traced_tactics": [{"tactic": "let npf := (\u222b\u207b c : \u03b1, f c ^ p \u2202\u03bc) ^ (1 / p)", "annotated_tactic": ["let npf := (\u222b\u207b c : \u03b1, f c ^ 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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "let nqg := (\u222b\u207b c : \u03b1, g c ^ q \u2202\u03bc) ^ (1 / q)", "annotated_tactic": ["let nqg := (\u222b\u207b c : \u03b1, g c ^ q \u2202\u03bc) ^ (1 / q)", []], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "calc\n  (\u222b\u207b a : \u03b1, (f * g) a \u2202\u03bc) =\n      \u222b\u207b a : \u03b1, (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc := by\n    refine' lintegral_congr fun a => _\n    rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f hf_nonzero hf_nontop,\n      fun_eq_funMulInvSnorm_mul_snorm g hg_nonzero hg_nontop, Pi.mul_apply]\n    ring\n  _ \u2264 npf * nqg := by\n    rw [lintegral_mul_const' (npf * nqg) _\n        (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])]\n    refine' mul_le_of_le_one_left' _\n    have hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop\n    have hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop\n    exact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1", "annotated_tactic": ["calc\n    (\u222b\u207b a : \u03b1, (f * g) a \u2202\u03bc) =\n        \u222b\u207b a : \u03b1, (<a>funMulInvSnorm</a> f p \u03bc * <a>funMulInvSnorm</a> g q \u03bc) a * (npf * nqg) \u2202\u03bc := by\n      refine' <a>lintegral_congr</a> fun a => _\n      rw [<a>Pi.mul_apply</a>, <a>fun_eq_funMulInvSnorm_mul_snorm</a> f hf_nonzero hf_nontop,\n        <a>fun_eq_funMulInvSnorm_mul_snorm</a> g hg_nonzero hg_nontop, <a>Pi.mul_apply</a>]\n      ring\n    _ \u2264 npf * nqg := by\n      rw [<a>lintegral_mul_const'</a> (npf * nqg) _\n          (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, <a>ENNReal.mul_eq_top</a>])]\n      refine' <a>mul_le_of_le_one_left'</a> _\n      have hf1 := <a>lintegral_rpow_funMulInvSnorm_eq_one</a> hpq.pos hf_nonzero hf_nontop\n      have hg1 := <a>lintegral_rpow_funMulInvSnorm_eq_one</a> hpq.symm.pos hg_nonzero hg_nontop\n      exact <a>lintegral_mul_le_one_of_lintegral_rpow_eq_one</a> hpq (hf.mul_const _) hf1 hg1", [{"full_name": "ENNReal.funMulInvSnorm", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [78, 5], "def_end_pos": [78, 19]}, {"full_name": "ENNReal.funMulInvSnorm", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [78, 5], "def_end_pos": [78, 19]}, {"full_name": "MeasureTheory.lintegral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [308, 9], "def_end_pos": [308, 24]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "ENNReal.fun_eq_funMulInvSnorm_mul_snorm", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [82, 9], "def_end_pos": [82, 40]}, {"full_name": "ENNReal.fun_eq_funMulInvSnorm_mul_snorm", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [82, 9], "def_end_pos": [82, 40]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "MeasureTheory.lintegral_mul_const'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [737, 9], "def_end_pos": [737, 29]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"full_name": "mul_le_of_le_one_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [419, 9], "def_end_pos": [419, 31]}, {"full_name": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [96, 9], "def_end_pos": [96, 45]}, {"full_name": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [96, 9], "def_end_pos": [96, 45]}, {"full_name": "ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [61, 9], "def_end_pos": [61, 54]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}, {"tactic": "refine' lintegral_congr fun a => _", "annotated_tactic": ["refine' <a>lintegral_congr</a> fun a => _", [{"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 : 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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc", "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\na : \u03b1\n\u22a2 (f * g) a = (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg)"}, {"tactic": "rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f hf_nonzero hf_nontop,\n  fun_eq_funMulInvSnorm_mul_snorm g hg_nonzero hg_nontop, Pi.mul_apply]", "annotated_tactic": ["rw [<a>Pi.mul_apply</a>, <a>fun_eq_funMulInvSnorm_mul_snorm</a> f hf_nonzero hf_nontop,\n        <a>fun_eq_funMulInvSnorm_mul_snorm</a> g hg_nonzero hg_nontop, <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]}, {"full_name": "ENNReal.fun_eq_funMulInvSnorm_mul_snorm", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [82, 9], "def_end_pos": [82, 40]}, {"full_name": "ENNReal.fun_eq_funMulInvSnorm_mul_snorm", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [82, 9], "def_end_pos": [82, 40]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\na : \u03b1\n\u22a2 (f * g) a = (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg)", "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\na : \u03b1\n\u22a2 funMulInvSnorm f p \u03bc a * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) *\n      (funMulInvSnorm g q \u03bc a * (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)) =\n    funMulInvSnorm f p \u03bc a * funMulInvSnorm g q \u03bc a * (npf * nqg)"}, {"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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\na : \u03b1\n\u22a2 funMulInvSnorm f p \u03bc a * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) *\n      (funMulInvSnorm g q \u03bc a * (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)) =\n    funMulInvSnorm f p \u03bc a * funMulInvSnorm g q \u03bc a * (npf * nqg)", "state_after": "no goals"}, {"tactic": "rw [lintegral_mul_const' (npf * nqg) _\n    (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])]", "annotated_tactic": ["rw [<a>lintegral_mul_const'</a> (npf * nqg) _\n          (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, <a>ENNReal.mul_eq_top</a>])]", [{"full_name": "MeasureTheory.lintegral_mul_const'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [737, 9], "def_end_pos": [737, 29]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc \u2264 npf * nqg", "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 (\u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc) * (npf * nqg) \u2264 npf * nqg"}, {"tactic": "refine' mul_le_of_le_one_left' _", "annotated_tactic": ["refine' <a>mul_le_of_le_one_left'</a> _", [{"full_name": "mul_le_of_le_one_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [419, 9], "def_end_pos": [419, 31]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 (\u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc) * (npf * nqg) \u2264 npf * nqg", "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1"}, {"tactic": "have hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop", "annotated_tactic": ["have hf1 := <a>lintegral_rpow_funMulInvSnorm_eq_one</a> hpq.pos hf_nonzero hf_nontop", [{"full_name": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [96, 9], "def_end_pos": [96, 45]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\nhf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1"}, {"tactic": "have hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop", "annotated_tactic": ["have hg1 := <a>lintegral_rpow_funMulInvSnorm_eq_one</a> hpq.symm.pos hg_nonzero hg_nontop", [{"full_name": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [96, 9], "def_end_pos": [96, 45]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\nhf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\nhf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1\nhg1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => g a) q \u03bc c ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1"}, {"tactic": "exact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1", "annotated_tactic": ["exact <a>lintegral_mul_le_one_of_lintegral_rpow_eq_one</a> hpq (hf.mul_const _) hf1 hg1", [{"full_name": "ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [61, 9], "def_end_pos": [61, 54]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\nhf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1\nhg1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => g a) q \u03bc c ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1", "state_after": "no goals"}, {"tactic": "simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top]", "annotated_tactic": ["simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, <a>ENNReal.mul_eq_top</a>]", [{"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}]], "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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 npf * nqg \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.self_trans_symm", "start": [270, 1], "end": [278, 44], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "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) = ofSet {a | isSome (\u2191f a) = true}", "state_after": "case h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (PEquiv.symm f)) x\u271d \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d"}, {"tactic": "dsimp [PEquiv.trans]", "annotated_tactic": ["dsimp [<a>PEquiv.trans</a>]", [{"full_name": "PEquiv.trans", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [117, 15], "def_end_pos": [117, 20]}]], "state_before": "case h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (PEquiv.symm f)) x\u271d \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d", "state_after": "case h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 Option.bind (\u2191f x\u271d) \u2191(PEquiv.symm f) \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d"}, {"tactic": "simp only [eq_some_iff f, Option.isSome_iff_exists, Option.mem_def, bind_eq_some',\n  ofSet_eq_some_iff]", "annotated_tactic": ["simp only [<a>eq_some_iff</a> f, <a>Option.isSome_iff_exists</a>, <a>Option.mem_def</a>, <a>bind_eq_some'</a>,\n    <a>ofSet_eq_some_iff</a>]", [{"full_name": "PEquiv.eq_some_iff", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [111, 9], "def_end_pos": [111, 20]}, {"full_name": "Option.isSome_iff_exists", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [55, 9], "def_end_pos": [55, 26]}, {"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.bind_eq_some'", "def_path": "Mathlib/Data/Option/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 22]}, {"full_name": "PEquiv.ofSet_eq_some_iff", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [234, 9], "def_end_pos": [234, 26]}]], "state_before": "case h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 Option.bind (\u2191f x\u271d) \u2191(PEquiv.symm f) \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d", "state_after": "case h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2194 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2194 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}", "state_after": "case h.a.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2192 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}\n\ncase h.a.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} \u2192 \u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a"}, {"tactic": "rintro \u27e8b, hb\u2081, hb\u2082\u27e9", "annotated_tactic": ["rintro \u27e8b, hb\u2081, hb\u2082\u27e9", []], "state_before": "case h.a.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2192 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}", "state_after": "case h.a.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\nb : \u03b2\nhb\u2081 : \u2191f x\u271d = some b\nhb\u2082 : \u2191f a\u271d = some b\n\u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}"}, {"tactic": "exact \u27e8PEquiv.inj _ hb\u2082 hb\u2081, b, hb\u2082\u27e9", "annotated_tactic": ["exact \u27e8<a>PEquiv.inj</a> _ hb\u2082 hb\u2081, b, hb\u2082\u27e9", [{"full_name": "PEquiv.inj", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [174, 19], "def_end_pos": [174, 22]}]], "state_before": "case h.a.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\nb : \u03b2\nhb\u2081 : \u2191f x\u271d = some b\nhb\u2082 : \u2191f a\u271d = some b\n\u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}", "state_after": "no goals"}, {"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 h.a.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} \u2192 \u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_inj_on", "start": [1208, 1], "end": [1209, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.degrees_map_of_injective", "start": [252, 1], "end": [254, 66], "traced_tactics": [{"tactic": "simp only [degrees, MvPolynomial.support_map_of_injective _ hf]", "annotated_tactic": ["simp only [<a>degrees</a>, <a>MvPolynomial.support_map_of_injective</a> _ hf]", [{"full_name": "MvPolynomial.degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [87, 5], "def_end_pos": [87, 12]}, {"full_name": "MvPolynomial.support_map_of_injective", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1390, 9], "def_end_pos": [1390, 33]}]], "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 : R \u2192+* S\nhf : Injective \u2191f\n\u22a2 degrees (\u2191(map f) p) = degrees p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasurableSet.nhdsWithin_isMeasurablyGenerated", "start": [382, 1], "end": [385, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.setLintegral_setLaverage", "start": [228, 1], "end": [230, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DList/Defs.lean", "full_name": "Std.DList.toList_cons", "start": [80, 1], "end": [81, 16], "traced_tactics": [{"tactic": "cases l", "annotated_tactic": ["cases l", []], "state_before": "\u03b1 : Type u\nx : \u03b1\nl : DList \u03b1\n\u22a2 toList (cons x l) = x :: toList l", "state_after": "case mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (cons x { apply := apply\u271d, invariant := invariant\u271d }) =\n    x :: toList { apply := apply\u271d, invariant := invariant\u271d }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (cons x { apply := apply\u271d, invariant := invariant\u271d }) =\n    x :: toList { apply := apply\u271d, invariant := invariant\u271d }", "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.toArray_data", "start": [37, 9], "end": [38, 33], "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", "start": [891, 1], "end": [907, 57], "traced_tactics": [{"tactic": "rcases hf.exists_nat_integrableOn with \u27e8u, u_open, u_univ, hu\u27e9", "annotated_tactic": ["rcases hf.exists_nat_integrableOn with \u27e8u, u_open, u_univ, hu\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 : LocallyIntegrable 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": "case intro.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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\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": "have : \u2200 n, \u2200\u1d50 x \u2202\u03bc,\n    Tendsto (fun a => (\u222b\u207b y in a, \u2016(u n).indicator f y - (u n).indicator f x\u2016\u208a \u2202\u03bc) / \u03bc a)\n    (v.filterAt x) (\ud835\udcdd 0) := by\n  intro n\n  apply ae_tendsto_lintegral_nnnorm_sub_div_of_integrable\n  exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n)", "annotated_tactic": ["have : \u2200 n, \u2200\u1d50 x \u2202\u03bc,\n      <a>Tendsto</a> (fun a => (\u222b\u207b y in a, \u2016(u n).<a>indicator</a> f y - (u n).<a>indicator</a> f x\u2016\u208a \u2202\u03bc) / \u03bc a)\n      (v.filterAt x) (\ud835\udcdd 0) := by\n    intro n\n    apply <a>ae_tendsto_lintegral_nnnorm_sub_div_of_integrable</a>\n    exact (<a>integrable_indicator_iff</a> (u_open n).<a>measurableSet</a>).2 (hu n)", [{"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": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div_of_integrable", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [877, 9], "def_end_pos": [877, 58]}, {"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]}]], "state_before": "case intro.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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\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.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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\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 [ae_all_iff.2 this] with x hx", "annotated_tactic": ["filter_upwards [<a>ae_all_iff</a>.2 this] with x hx", [{"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\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\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": "obtain \u27e8n, hn\u27e9 : \u2203 n, x \u2208 u n := by simpa only [\u2190 u_univ, mem_iUnion] using mem_univ x", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n, x \u2208 u n := by simpa only [\u2190 u_univ, <a>mem_iUnion</a>] using <a>mem_univ</a> x", [{"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_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\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.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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\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 Tendsto.congr' _ (hx n)", "annotated_tactic": ["apply <a>Tendsto.congr'</a> _ (hx n)", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}]], "state_before": "case h.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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\n\u22a2 (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) =\u1da0[filterAt v x] fun a =>\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a"}, {"tactic": "filter_upwards [v.eventually_filterAt_subset_of_nhds ((u_open n).mem_nhds hn),\n  v.eventually_filterAt_measurableSet x] with a ha h'a", "annotated_tactic": ["filter_upwards [v.eventually_filterAt_subset_of_nhds ((u_open n).<a>mem_nhds</a> hn),\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\n\u22a2 (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) =\u1da0[filterAt v x] fun a =>\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\na : Set \u03b1\nha : a \u2286 u n\nh'a : MeasurableSet a\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a =\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\na : Set \u03b1\nha : a \u2286 u n\nh'a : MeasurableSet a\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a =\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\na : Set \u03b1\nha : a \u2286 u n\nh'a : MeasurableSet a\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc = \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc"}, {"tactic": "refine' set_lintegral_congr_fun h'a (eventually_of_forall (fun y hy \u21a6 _))", "annotated_tactic": ["refine' <a>set_lintegral_congr_fun</a> h'a (<a>eventually_of_forall</a> (fun y hy \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": "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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\na : Set \u03b1\nha : a \u2286 u n\nh'a : MeasurableSet a\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc = \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\na : Set \u03b1\nha : a \u2286 u n\nh'a : MeasurableSet a\ny : \u03b1\nhy : y \u2208 a\n\u22a2 \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a = \u2191\u2016f y - f x\u2016\u208a"}, {"tactic": "rw [indicator_of_mem (ha hy) f, indicator_of_mem hn f]", "annotated_tactic": ["rw [<a>indicator_of_mem</a> (ha hy) f, <a>indicator_of_mem</a> hn f]", [{"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 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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\nn : \u2115\nhn : x \u2208 u n\na : Set \u03b1\nha : a \u2286 u n\nh'a : MeasurableSet a\ny : \u03b1\nhy : y \u2208 a\n\u22a2 \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a = \u2191\u2016f y - f x\u2016\u208a", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\n\u22a2 \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)"}, {"tactic": "apply ae_tendsto_lintegral_nnnorm_sub_div_of_integrable", "annotated_tactic": ["apply <a>ae_tendsto_lintegral_nnnorm_sub_div_of_integrable</a>", [{"full_name": "VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div_of_integrable", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [877, 9], "def_end_pos": [877, 58]}]], "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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)", "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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nn : \u2115\n\u22a2 Integrable fun y => indicator (u n) f y"}, {"tactic": "exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n)", "annotated_tactic": ["exact (<a>integrable_indicator_iff</a> (u_open n).<a>measurableSet</a>).2 (hu n)", [{"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]}]], "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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nn : \u2115\n\u22a2 Integrable fun y => indicator (u n) f y", "state_after": "no goals"}, {"tactic": "simpa only [\u2190 u_univ, mem_iUnion] using mem_univ x", "annotated_tactic": ["simpa only [\u2190 u_univ, <a>mem_iUnion</a>] using <a>mem_univ</a> x", [{"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_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 : LocallyIntegrable f\nu : \u2115 \u2192 Set \u03b1\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_univ : \u22c3 n, u n = univ\nhu : \u2200 (n : \u2115), IntegrableOn f (u n)\nthis :\n  \u2200 (n : \u2115),\n    \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)\n        (\ud835\udcdd 0)\nx : \u03b1\nhx :\n  \u2200 (i : \u2115),\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0)\n\u22a2 \u2203 n, x \u2208 u n", "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_Ioo", "start": [104, 1], "end": [105, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.dvd_iff_mod_eq_zero", "start": [1765, 1], "end": [1766, 77], "traced_tactics": [{"tactic": "rw [\u2190 dvd_to_int, Int.dvd_iff_emod_eq_zero, \u2190 to_int_inj, mod_to_int]", "annotated_tactic": ["rw [\u2190 <a>dvd_to_int</a>, <a>Int.dvd_iff_emod_eq_zero</a>, \u2190 <a>to_int_inj</a>, <a>mod_to_int</a>]", [{"full_name": "ZNum.dvd_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1562, 9], "def_end_pos": [1562, 19]}, {"full_name": "Int.dvd_iff_emod_eq_zero", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [688, 9], "def_end_pos": [688, 29]}, {"full_name": "ZNum.to_int_inj", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1359, 9], "def_end_pos": [1359, 19]}, {"full_name": "ZNum.mod_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1747, 9], "def_end_pos": [1747, 19]}]], "state_before": "m n : ZNum\n\u22a2 m \u2223 n \u2194 n % m = 0", "state_after": "m n : ZNum\n\u22a2 \u2191n % \u2191m = 0 \u2194 \u2191n % \u2191m = \u21910"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "m n : ZNum\n\u22a2 \u2191n % \u2191m = 0 \u2194 \u2191n % \u2191m = \u21910", "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_eq", "start": [25, 1], "end": [31, 73], "traced_tactics": [{"tactic": "simp only [normalize, maybeNormalize_eq,\n  Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]", "annotated_tactic": ["simp only [<a>normalize</a>, <a>maybeNormalize_eq</a>,\n    <a>Int.div_eq_ediv_of_dvd</a> (<a>Int.ofNat_dvd_left</a>.2 (<a>Nat.gcd_dvd_left</a> ..))]", [{"full_name": "Rat.normalize", "def_path": "lake-packages/std/Std/Data/Rat/Basic.lean", "def_pos": [72, 15], "def_end_pos": [72, 28]}, {"full_name": "Rat.maybeNormalize_eq", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [13, 17], "def_end_pos": [13, 34]}, {"full_name": "Int.div_eq_ediv_of_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [794, 9], "def_end_pos": [794, 27]}, {"full_name": "Int.ofNat_dvd_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [662, 9], "def_end_pos": [662, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}]], "state_before": "num : Int\nden : Nat\nden_nz : den \u2260 0\n\u22a2 normalize num den = mk' (num / \u2191(Nat.gcd (Int.natAbs num) den)) (den / Nat.gcd (Int.natAbs num) den)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.meas_ge_le_variance_div_sq", "start": [285, 1], "end": [291, 8], "traced_tactics": [{"tactic": "rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.ofReal_variance_eq]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_div_of_pos</a> (<a>sq_pos_of_ne_zero</a> _ hc.ne.symm), hX.ofReal_variance_eq]", [{"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": "sq_pos_of_ne_zero", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [656, 9], "def_end_pos": [656, 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 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} \u2264 ENNReal.ofReal (variance X \u2119 / c ^ 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 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} \u2264 evariance X \u2119 / ENNReal.ofReal (c ^ 2)"}, {"tactic": "convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1", "annotated_tactic": ["convert @<a>meas_ge_le_evariance_div_sq</a> _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1", [{"full_name": "ProbabilityTheory.meas_ge_le_evariance_div_sq", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [266, 9], "def_end_pos": [266, 36]}]], "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 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} \u2264 evariance X \u2119 / ENNReal.ofReal (c ^ 2)", "state_after": "case h.e'_3\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} = \u2191\u2191\u2119 {\u03c9 | \u2191(Real.toNNReal c) \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|}\n\ncase h.e'_4\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 evariance X \u2119 / ENNReal.ofReal (c ^ 2) = evariance X \u2119 / \u2191(Real.toNNReal c ^ 2)"}, {"tactic": "simp [hc]", "annotated_tactic": ["simp [hc]", []], "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 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 Real.toNNReal c \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]", "annotated_tactic": ["simp only [<a>Real.coe_toNNReal'</a>, <a>max_le_iff</a>, <a>abs_nonneg</a>, <a>and_true_iff</a>]", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}]], "state_before": "case h.e'_3\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} = \u2191\u2191\u2119 {\u03c9 | \u2191(Real.toNNReal c) \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|}", "state_after": "no goals"}, {"tactic": "rw [ENNReal.ofReal_pow hc.le, ENNReal.coe_pow]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_pow</a> hc.le, <a>ENNReal.coe_pow</a>]", [{"full_name": "ENNReal.ofReal_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2235, 9], "def_end_pos": [2235, 19]}, {"full_name": "ENNReal.coe_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [554, 9], "def_end_pos": [554, 16]}]], "state_before": "case h.e'_4\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 evariance X \u2119 / ENNReal.ofReal (c ^ 2) = evariance X \u2119 / \u2191(Real.toNNReal c ^ 2)", "state_after": "case h.e'_4\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 evariance X \u2119 / ENNReal.ofReal c ^ 2 = evariance X \u2119 / \u2191(Real.toNNReal c) ^ 2"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_4\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhX : Mem\u2112p X 2\nc : \u211d\nhc : 0 < c\n\u22a2 evariance X \u2119 / ENNReal.ofReal c ^ 2 = evariance X \u2119 / \u2191(Real.toNNReal c) ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Integration.lean", "full_name": "ProbabilityTheory.indepFun_iff_integral_comp_mul", "start": [305, 1], "end": [320, 69], "traced_tactics": [{"tactic": "refine' \u27e8fun hfg _ _ h\u03c6 h\u03c8 => IndepFun.integral_mul_of_integrable (hfg.comp h\u03c6 h\u03c8), _\u27e9", "annotated_tactic": ["refine' \u27e8fun hfg _ _ h\u03c6 h\u03c8 => <a>IndepFun.integral_mul_of_integrable</a> (hfg.comp h\u03c6 h\u03c8), _\u27e9", [{"full_name": "ProbabilityTheory.IndepFun.integral_mul_of_integrable", "def_path": "Mathlib/Probability/Integration.lean", "def_pos": [225, 9], "def_end_pos": [225, 44]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\n\u22a2 IndepFun f g \u2194\n    \u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n      Measurable \u03c6 \u2192\n        Measurable \u03c8 \u2192\n          Integrable (\u03c6 \u2218 f) \u2192 Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)", "state_after": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\n\u22a2 (\u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n      Measurable \u03c6 \u2192\n        Measurable \u03c8 \u2192\n          Integrable (\u03c6 \u2218 f) \u2192\n            Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)) \u2192\n    IndepFun f g"}, {"tactic": "rw [IndepFun_iff]", "annotated_tactic": ["rw [<a>IndepFun_iff</a>]", [{"full_name": "ProbabilityTheory.IndepFun_iff", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [212, 7], "def_end_pos": [212, 19]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\n\u22a2 (\u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n      Measurable \u03c6 \u2192\n        Measurable \u03c8 \u2192\n          Integrable (\u03c6 \u2218 f) \u2192\n            Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)) \u2192\n    IndepFun f g", "state_after": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\n\u22a2 (\u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n      Measurable \u03c6 \u2192\n        Measurable \u03c8 \u2192\n          Integrable (\u03c6 \u2218 f) \u2192\n            Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)) \u2192\n    \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2"}, {"tactic": "rintro h _ _ \u27e8A, hA, rfl\u27e9 \u27e8B, hB, rfl\u27e9", "annotated_tactic": ["rintro h _ _ \u27e8A, hA, rfl\u27e9 \u27e8B, hB, rfl\u27e9", []], "state_before": "\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\n\u22a2 (\u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n      Measurable \u03c6 \u2192\n        Measurable \u03c8 \u2192\n          Integrable (\u03c6 \u2218 f) \u2192\n            Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)) \u2192\n    \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2", "state_after": "case intro.intro.intro.intro\n\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\nh :\n  \u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n    Measurable \u03c6 \u2192\n      Measurable \u03c8 \u2192\n        Integrable (\u03c6 \u2218 f) \u2192 Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)\nA : Set \u03b2\nhA : MeasurableSet A\nB : Set \u03b2'\nhB : MeasurableSet B\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A \u2229 g \u207b\u00b9' B) = \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B)"}, {"tactic": "specialize\n  h (measurable_one.indicator hA) (measurable_one.indicator hB)\n    ((integrable_const 1).indicator (hfm.comp measurable_id hA))\n    ((integrable_const 1).indicator (hgm.comp measurable_id hB))", "annotated_tactic": ["specialize\n    h (measurable_one.indicator hA) (measurable_one.indicator hB)\n      ((<a>integrable_const</a> 1).<a>indicator</a> (hfm.comp <a>measurable_id</a> hA))\n      ((<a>integrable_const</a> 1).<a>indicator</a> (hgm.comp <a>measurable_id</a> hB))", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}, {"full_name": "MeasureTheory.Integrable.indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}, {"full_name": "measurable_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}, {"full_name": "MeasureTheory.Integrable.indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 29]}, {"full_name": "measurable_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}]], "state_before": "case intro.intro.intro.intro\n\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\nh :\n  \u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},\n    Measurable \u03c6 \u2192\n      Measurable \u03c8 \u2192\n        Integrable (\u03c6 \u2218 f) \u2192 Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)\nA : Set \u03b2\nhA : MeasurableSet A\nB : Set \u03b2'\nhB : MeasurableSet B\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A \u2229 g \u207b\u00b9' B) = \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B)", "state_after": "case intro.intro.intro.intro\n\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\nA : Set \u03b2\nhA : MeasurableSet A\nB : Set \u03b2'\nhB : MeasurableSet B\nh : integral \u03bc (indicator A 1 \u2218 f * indicator B 1 \u2218 g) = integral \u03bc (indicator A 1 \u2218 f) * integral \u03bc (indicator B 1 \u2218 g)\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A \u2229 g \u207b\u00b9' B) = \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B)"}, {"tactic": "rwa [\u2190 ENNReal.toReal_eq_toReal (measure_ne_top \u03bc _), ENNReal.toReal_mul, \u2190\n  integral_indicator_one ((hfm hA).inter (hgm hB)), \u2190 integral_indicator_one (hfm hA), \u2190\n  integral_indicator_one (hgm hB), Set.inter_indicator_one]", "annotated_tactic": ["rwa [\u2190 <a>ENNReal.toReal_eq_toReal</a> (<a>measure_ne_top</a> \u03bc _), <a>ENNReal.toReal_mul</a>, \u2190\n    <a>integral_indicator_one</a> ((hfm hA).<a>inter</a> (hgm hB)), \u2190 <a>integral_indicator_one</a> (hfm hA), \u2190\n    <a>integral_indicator_one</a> (hgm hB), <a>Set.inter_indicator_one</a>]", [{"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": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "MeasureTheory.integral_indicator_one", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [485, 9], "def_end_pos": [485, 31]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "MeasureTheory.integral_indicator_one", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [485, 9], "def_end_pos": [485, 31]}, {"full_name": "MeasureTheory.integral_indicator_one", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [485, 9], "def_end_pos": [485, 31]}, {"full_name": "Set.inter_indicator_one", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [743, 9], "def_end_pos": [743, 28]}]], "state_before": "case intro.intro.intro.intro\n\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\nA : Set \u03b2\nhA : MeasurableSet A\nB : Set \u03b2'\nhB : MeasurableSet B\nh : integral \u03bc (indicator A 1 \u2218 f * indicator B 1 \u2218 g) = integral \u03bc (indicator A 1 \u2218 f) * integral \u03bc (indicator B 1 \u2218 g)\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A \u2229 g \u207b\u00b9' B) = \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B)", "state_after": "case intro.intro.intro.intro\n\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\nA : Set \u03b2\nhA : MeasurableSet A\nB : Set \u03b2'\nhB : MeasurableSet B\nh : integral \u03bc (indicator A 1 \u2218 f * indicator B 1 \u2218 g) = integral \u03bc (indicator A 1 \u2218 f) * integral \u03bc (indicator B 1 \u2218 g)\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B) \u2260 \u22a4"}, {"tactic": "exact ENNReal.mul_ne_top (measure_ne_top \u03bc _) (measure_ne_top \u03bc _)", "annotated_tactic": ["exact <a>ENNReal.mul_ne_top</a> (<a>measure_ne_top</a> \u03bc _) (<a>measure_ne_top</a> \u03bc _)", [{"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]}, {"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\n\u03a9 : Type u_1\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b2 : Type u_2\n\u03b2' : Type u_3\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\nf : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nhfm : Measurable f\nhgm : Measurable g\nA : Set \u03b2\nhA : MeasurableSet A\nB : Set \u03b2'\nhB : MeasurableSet B\nh : integral \u03bc (indicator A 1 \u2218 f * indicator B 1 \u2218 g) = integral \u03bc (indicator A 1 \u2218 f) * integral \u03bc (indicator B 1 \u2218 g)\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "eq_borel_upgradeStandardBorel", "start": [100, 1], "end": [103, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.castSucc_ne_zero_iff", "start": [388, 1], "end": [389, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.classes_inj", "start": [108, 1], "end": [109, 101], "traced_tactics": [{"tactic": "simp only [rel_iff_exists_classes, exists_prop, h]", "annotated_tactic": ["simp only [<a>rel_iff_exists_classes</a>, <a>exists_prop</a>, h]", [{"full_name": "Setoid.rel_iff_exists_classes", "def_path": "Mathlib/Data/Setoid/Partition.lean", "def_pos": [101, 9], "def_end_pos": [101, 31]}, {"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\nr\u2081 r\u2082 : Setoid \u03b1\nh : classes r\u2081 = classes r\u2082\na b : \u03b1\n\u22a2 Rel r\u2081 a b \u2194 Rel r\u2082 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.inter_left_idem", "start": [1762, 1], "end": [1763, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.ringHom_ext", "start": [456, 1], "end": [466, 15], "traced_tactics": [{"tactic": "refine AddMonoidAlgebra.ringHom_ext' ?_ ?_", "annotated_tactic": ["refine <a>AddMonoidAlgebra.ringHom_ext'</a> ?_ ?_", [{"full_name": "AddMonoidAlgebra.ringHom_ext'", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1853, 9], "def_end_pos": [1853, 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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 f = g", "state_after": "case refine_1\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 RingHom.comp f singleZeroRingHom = RingHom.comp g singleZeroRingHom\n\ncase refine_2\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115)) = MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case refine_1\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 RingHom.comp f singleZeroRingHom = RingHom.comp g singleZeroRingHom", "state_after": "case refine_1.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\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\nx : R\n\u22a2 \u2191(RingHom.comp f singleZeroRingHom) x = \u2191(RingHom.comp g singleZeroRingHom) x"}, {"tactic": "exact hC _", "annotated_tactic": ["exact hC _", []], "state_before": "case refine_1.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\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\nx : R\n\u22a2 \u2191(RingHom.comp f singleZeroRingHom) x = \u2191(RingHom.comp g singleZeroRingHom) x", "state_after": "no goals"}, {"tactic": "apply Finsupp.mulHom_ext'", "annotated_tactic": ["apply <a>Finsupp.mulHom_ext'</a>", [{"full_name": "Finsupp.mulHom_ext'", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [1168, 9], "def_end_pos": [1168, 20]}]], "state_before": "case refine_2\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115)) = MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))", "state_after": "case refine_2.H\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 \u2200 (x : \u03c3),\n    MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) =\n      MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x))"}, {"tactic": "intros x", "annotated_tactic": ["intros x", []], "state_before": "case refine_2.H\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\n\u22a2 \u2200 (x : \u03c3),\n    MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) =\n      MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x))", "state_after": "case refine_2.H\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\nx : \u03c3\n\u22a2 MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) =\n    MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x))"}, {"tactic": "apply MonoidHom.ext_mnat", "annotated_tactic": ["apply <a>MonoidHom.ext_mnat</a>", [{"full_name": "MonoidHom.ext_mnat", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [899, 9], "def_end_pos": [899, 27]}]], "state_before": "case refine_2.H\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\nx : \u03c3\n\u22a2 MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) =\n    MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x))", "state_after": "case refine_2.H.h\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\nx : \u03c3\n\u22a2 \u2191(MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)))\n      (\u2191Multiplicative.ofAdd 1) =\n    \u2191(MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)))\n      (\u2191Multiplicative.ofAdd 1)"}, {"tactic": "exact hX _", "annotated_tactic": ["exact hX _", []], "state_before": "case refine_2.H.h\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\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nA : Type u_2\ninst\u271d : Semiring A\nf g : MvPolynomial \u03c3 R \u2192+* A\nhC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r)\nhX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i)\nx : \u03c3\n\u22a2 \u2191(MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)))\n      (\u2191Multiplicative.ofAdd 1) =\n    \u2191(MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)))\n      (\u2191Multiplicative.ofAdd 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.finset_sum_apply", "start": [885, 1], "end": [886, 84], "traced_tactics": [{"tactic": "rw [coe_finset_sum, Finset.sum_apply]", "annotated_tactic": ["rw [<a>coe_finset_sum</a>, <a>Finset.sum_apply</a>]", [{"full_name": "MeasureTheory.Measure.coe_finset_sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [881, 9], "def_end_pos": [881, 23]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 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\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : MeasurableSpace \u03b1\nI : Finset \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u2191(\u2211 i in I, \u03bc i) s = \u2211 i in I, \u2191\u2191(\u03bc i) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "torusMap_zero_radius", "start": [95, 1], "end": [96, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/EpsilonNFA.lean", "full_name": "\u03b5NFA.evalFrom_empty", "start": [111, 1], "end": [114, 54], "traced_tactics": [{"tactic": "induction' x using List.reverseRecOn with x a ih", "annotated_tactic": ["induction' x using <a>List.reverseRecOn</a> with x a ih", [{"full_name": "List.reverseRecOn", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [954, 5], "def_end_pos": [954, 17]}]], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx\u271d : List \u03b1\ns : \u03c3\na : \u03b1\nx : List \u03b1\n\u22a2 evalFrom M \u2205 x = \u2205", "state_after": "case H0\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\n\u22a2 evalFrom M \u2205 [] = \u2205\n\ncase H1\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx\u271d : List \u03b1\ns : \u03c3\na\u271d : \u03b1\nx : List \u03b1\na : \u03b1\nih : evalFrom M \u2205 x = \u2205\n\u22a2 evalFrom M \u2205 (x ++ [a]) = \u2205"}, {"tactic": "rw [evalFrom_nil, \u03b5Closure_empty]", "annotated_tactic": ["rw [<a>evalFrom_nil</a>, <a>\u03b5Closure_empty</a>]", [{"full_name": "\u03b5NFA.evalFrom_nil", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [95, 9], "def_end_pos": [95, 21]}, {"full_name": "\u03b5NFA.\u03b5Closure_empty", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [60, 9], "def_end_pos": [60, 23]}]], "state_before": "case H0\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\n\u22a2 evalFrom M \u2205 [] = \u2205", "state_after": "no goals"}, {"tactic": "rw [evalFrom_append_singleton, ih, stepSet_empty]", "annotated_tactic": ["rw [<a>evalFrom_append_singleton</a>, ih, <a>stepSet_empty</a>]", [{"full_name": "\u03b5NFA.evalFrom_append_singleton", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [105, 9], "def_end_pos": [105, 34]}, {"full_name": "\u03b5NFA.stepSet_empty", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case H1\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx\u271d : List \u03b1\ns : \u03c3\na\u271d : \u03b1\nx : List \u03b1\na : \u03b1\nih : evalFrom M \u2205 x = \u2205\n\u22a2 evalFrom M \u2205 (x ++ [a]) = \u2205", "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.tendsto_approxBounded_ae", "start": [233, 1], "end": [237, 85], "traced_tactics": [{"tactic": "filter_upwards [hf_bound] with x hfx using tendsto_approxBounded_of_norm_le hf hfx", "annotated_tactic": ["filter_upwards [hf_bound] with x hfx using <a>tendsto_approxBounded_of_norm_le</a> hf hfx", [{"full_name": "MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [201, 9], "def_end_pos": [201, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedSpace \u211d \u03b2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhf : StronglyMeasurable f\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(approxBounded hf c n) x) atTop (\ud835\udcdd (f x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "meas_lt_essInf", "start": [131, 1], "end": [135, 24], "traced_tactics": [{"tactic": "simp_rw [\u2190 not_le]", "annotated_tactic": ["simp_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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | f y < essInf f \u03bc} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | \u00acessInf f \u03bc \u2264 f y} = 0"}, {"tactic": "exact ae_essInf_le hf", "annotated_tactic": ["exact <a>ae_essInf_le</a> hf", [{"full_name": "ae_essInf_le", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | \u00acessInf f \u03bc \u2264 f y} = 0", "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_smul", "start": [65, 1], "end": [82, 100], "traced_tactics": [{"tactic": "rcases eq_or_ne R 0 with (rfl | hR)", "annotated_tactic": ["rcases <a>eq_or_ne</a> R 0 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": "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\nf : E \u2192 F\nR : \u211d\n\u22a2 \u222b (x : E), f (R \u2022 x) \u2202\u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "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 : Set E\nf : E \u2192 F\n\u22a2 \u222b (x : E), f (0 \u2022 x) \u2202\u03bc = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc\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\nf : E \u2192 F\nR : \u211d\nhR : R \u2260 0\n\u22a2 \u222b (x : E), f (R \u2022 x) \u2202\u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc"}, {"tactic": "simp only [zero_smul, integral_const]", "annotated_tactic": ["simp only [<a>zero_smul</a>, <a>integral_const</a>]", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}]], "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 : Set E\nf : E \u2192 F\n\u22a2 \u222b (x : E), f (0 \u2022 x) \u2202\u03bc = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "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 : Set E\nf : E \u2192 F\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc"}, {"tactic": "rcases Nat.eq_zero_or_pos (finrank \u211d E) with (hE | hE)", "annotated_tactic": ["rcases <a>Nat.eq_zero_or_pos</a> (<a>finrank</a> \u211d E) with (hE | hE)", [{"full_name": "Nat.eq_zero_or_pos", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}]], "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 : Set E\nf : E \u2192 F\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc\n\ncase inl.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\nf : E \u2192 F\nhE : finrank \u211d E > 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc"}, {"tactic": "have : Subsingleton E := finrank_zero_iff.1 hE", "annotated_tactic": ["have : <a>Subsingleton</a> E := <a>finrank_zero_iff</a>.1 hE", [{"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 inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis : Subsingleton E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc"}, {"tactic": "have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim x 0]", "annotated_tactic": ["have : f = fun _ => f 0 := by ext x; rw [<a>Subsingleton.elim</a> x 0]", [{"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.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis : Subsingleton E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis\u271d : Subsingleton E\nthis : f = fun x => f 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc"}, {"tactic": "conv_rhs => rw [this]", "annotated_tactic": ["conv_rhs => rw [this]", []], "state_before": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis\u271d : Subsingleton E\nthis : f = fun x => f 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis\u271d : Subsingleton E\nthis : f = fun x => f 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), (fun x => f 0) x \u2202\u03bc"}, {"tactic": "simp only [hE, pow_zero, inv_one, abs_one, one_smul, integral_const]", "annotated_tactic": ["simp only [hE, <a>pow_zero</a>, <a>inv_one</a>, <a>abs_one</a>, <a>one_smul</a>, <a>integral_const</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 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]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}]], "state_before": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis\u271d : Subsingleton E\nthis : f = fun x => f 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), (fun x => f 0) x \u2202\u03bc", "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\ns : Set E\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis : Subsingleton E\n\u22a2 f = fun x => f 0", "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 : Set E\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis : Subsingleton E\nx : E\n\u22a2 f x = f 0"}, {"tactic": "rw [Subsingleton.elim x 0]", "annotated_tactic": ["rw [<a>Subsingleton.elim</a> x 0]", [{"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\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\nf : E \u2192 F\nhE : finrank \u211d E = 0\nthis : Subsingleton E\nx : E\n\u22a2 f x = f 0", "state_after": "no goals"}, {"tactic": "have : Nontrivial E := finrank_pos_iff.1 hE", "annotated_tactic": ["have : <a>Nontrivial</a> E := <a>finrank_pos_iff</a>.1 hE", [{"full_name": "Nontrivial", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [29, 7], "def_end_pos": [29, 17]}, {"full_name": "FiniteDimensional.finrank_pos_iff", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [325, 9], "def_end_pos": [325, 24]}]], "state_before": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E > 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E > 0\nthis : Nontrivial E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc"}, {"tactic": "simp only [zero_pow hE, measure_univ_of_isAddLeftInvariant, ENNReal.top_toReal, zero_smul,\n  inv_zero, abs_zero]", "annotated_tactic": ["simp only [<a>zero_pow</a> hE, <a>measure_univ_of_isAddLeftInvariant</a>, <a>ENNReal.top_toReal</a>, <a>zero_smul</a>,\n        <a>inv_zero</a>, <a>abs_zero</a>]", [{"full_name": "zero_pow", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [32, 9], "def_end_pos": [32, 17]}, {"full_name": "MeasureTheory.measure_univ_of_isAddLeftInvariant", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [650, 3], "def_end_pos": [650, 14]}, {"full_name": "ENNReal.top_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [227, 17], "def_end_pos": [227, 27]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "GroupWithZero.inv_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [186, 3], "def_end_pos": [186, 11]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}]], "state_before": "case inl.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\nf : E \u2192 F\nhE : finrank \u211d E > 0\nthis : Nontrivial E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "calc\n  (\u222b x, f (R \u2022 x) \u2202\u03bc) = \u222b y, f y \u2202Measure.map (fun x => R \u2022 x) \u03bc :=\n    (integral_map_equiv (Homeomorph.smul (isUnit_iff_ne_zero.2 hR).unit).toMeasurableEquiv\n        f).symm\n  _ = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b x, f x \u2202\u03bc := by\n    simp only [map_addHaar_smul \u03bc hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]", "annotated_tactic": ["calc\n      (\u222b x, f (R \u2022 x) \u2202\u03bc) = \u222b y, f y \u2202<a>Measure.map</a> (fun x => R \u2022 x) \u03bc :=\n        (<a>integral_map_equiv</a> (<a>Homeomorph.smul</a> (<a>isUnit_iff_ne_zero</a>.2 hR).<a>unit</a>).<a>toMeasurableEquiv</a>\n            f).<a>symm</a>\n      _ = |(R ^ <a>finrank</a> \u211d E)\u207b\u00b9| \u2022 \u222b x, f x \u2202\u03bc := by\n        simp only [<a>map_addHaar_smul</a> \u03bc hR, <a>integral_smul_measure</a>, <a>ENNReal.toReal_ofReal</a>, <a>abs_nonneg</a>]", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1163, 17], "def_end_pos": [1163, 20]}, {"full_name": "MeasureTheory.integral_map_equiv", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1637, 9], "def_end_pos": [1637, 27]}, {"full_name": "Homeomorph.smul", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [236, 5], "def_end_pos": [236, 20]}, {"full_name": "isUnit_iff_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 27]}, {"full_name": "IsUnit.unit", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [718, 29], "def_end_pos": [718, 33]}, {"full_name": "Homeomorph.toMeasurableEquiv", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [983, 5], "def_end_pos": [983, 33]}, {"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": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "MeasureTheory.Measure.map_addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [347, 9], "def_end_pos": [347, 25]}, {"full_name": "MeasureTheory.integral_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1571, 9], "def_end_pos": [1571, 30]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "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\nf : E \u2192 F\nR : \u211d\nhR : R \u2260 0\n\u22a2 \u222b (x : E), f (R \u2022 x) \u2202\u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [map_addHaar_smul \u03bc hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]", "annotated_tactic": ["simp only [<a>map_addHaar_smul</a> \u03bc hR, <a>integral_smul_measure</a>, <a>ENNReal.toReal_ofReal</a>, <a>abs_nonneg</a>]", [{"full_name": "MeasureTheory.Measure.map_addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [347, 9], "def_end_pos": [347, 25]}, {"full_name": "MeasureTheory.integral_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1571, 9], "def_end_pos": [1571, 30]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "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\nf : E \u2192 F\nR : \u211d\nhR : R \u2260 0\n\u22a2 \u222b (y : E), f y \u2202map (fun x => R \u2022 x) \u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.sum_measure", "start": [75, 1], "end": [111, 41], "traced_tactics": [{"tactic": "nontriviality \u03b2", "annotated_tactic": ["nontriviality \u03b2", []], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u22a2 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\n\u22a2 AEMeasurable f"}, {"tactic": "inhabit \u03b2", "annotated_tactic": ["inhabit \u03b2", []], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\n\u22a2 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\n\u22a2 AEMeasurable f"}, {"tactic": "set s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) { x | f x \u2260 (h i).mk f x }", "annotated_tactic": ["set s : \u03b9 \u2192 <a>Set</a> \u03b1 := fun i => <a>toMeasurable</a> (\u03bc i) { x | f x \u2260 (h i).<a>mk</a> f x }", [{"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]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\n\u22a2 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 AEMeasurable f"}, {"tactic": "have hs\u03bc : \u2200 i, \u03bc i (s i) = 0 := by\n  intro i\n  rw [measure_toMeasurable]\n  exact (h i).ae_eq_mk", "annotated_tactic": ["have hs\u03bc : \u2200 i, \u03bc i (s i) = 0 := by\n    intro i\n    rw [<a>measure_toMeasurable</a>]\n    exact (h i).<a>ae_eq_mk</a>", [{"full_name": "MeasureTheory.measure_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}, {"full_name": "AEMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [731, 9], "def_end_pos": [731, 17]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\n\u22a2 AEMeasurable f"}, {"tactic": "have hsm : MeasurableSet (\u22c2 i, s i) :=\n  MeasurableSet.iInter fun i => measurableSet_toMeasurable _ _", "annotated_tactic": ["have hsm : <a>MeasurableSet</a> (\u22c2 i, s i) :=\n    <a>MeasurableSet.iInter</a> fun i => <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.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\n\u22a2 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\n\u22a2 AEMeasurable f"}, {"tactic": "have hs : \u2200 i x, x \u2209 s i \u2192 f x = (h i).mk f x := by\n  intro i x hx\n  contrapose! hx\n  exact subset_toMeasurable _ _ hx", "annotated_tactic": ["have hs : \u2200 i x, x \u2209 s i \u2192 f x = (h i).<a>mk</a> f x := by\n    intro i x hx\n    contrapose! hx\n    exact <a>subset_toMeasurable</a> _ _ hx", [{"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\n\u22a2 AEMeasurable f", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\n\u22a2 AEMeasurable f"}, {"tactic": "set g : \u03b1 \u2192 \u03b2 := (\u22c2 i, s i).piecewise (const \u03b1 default) f", "annotated_tactic": ["set g : \u03b1 \u2192 \u03b2 := (\u22c2 i, s i).<a>piecewise</a> (<a>const</a> \u03b1 <a>default</a>) f", [{"full_name": "Set.piecewise", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [993, 5], "def_end_pos": [993, 18]}, {"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": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\n\u22a2 AEMeasurable f", "state_after": "\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 AEMeasurable f"}, {"tactic": "refine' \u27e8g, measurable_of_restrict_of_restrict_compl hsm _ _, ae_sum_iff.mpr fun i => _\u27e9", "annotated_tactic": ["refine' \u27e8g, <a>measurable_of_restrict_of_restrict_compl</a> hsm _ _, ae_sum_iff.mpr fun i => _\u27e9", [{"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": "\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 AEMeasurable f", "state_after": "case refine'_1\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c2 i, s i) g)\n\ncase refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c2 i, s i)\u1d9c g)\n\ncase refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc i, f x = g x"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bc i) (s i) = 0"}, {"tactic": "rw [measure_toMeasurable]", "annotated_tactic": ["rw [<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": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bc i) (s i) = 0", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} = 0"}, {"tactic": "exact (h i).ae_eq_mk", "annotated_tactic": ["exact (h i).<a>ae_eq_mk</a>", [{"full_name": "AEMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [731, 9], "def_end_pos": [731, 17]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} = 0", "state_after": "no goals"}, {"tactic": "intro i x hx", "annotated_tactic": ["intro i x hx", []], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\n\u22a2 \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 s i\n\u22a2 f x = mk f (_ : AEMeasurable f) x"}, {"tactic": "contrapose! hx", "annotated_tactic": ["contrapose! hx", []], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 s i\n\u22a2 f x = mk f (_ : AEMeasurable f) x", "state_after": "\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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\ni : \u03b9\nx : \u03b1\nhx : f x \u2260 mk f (_ : AEMeasurable f) x\n\u22a2 x \u2208 s i"}, {"tactic": "exact subset_toMeasurable _ _ hx", "annotated_tactic": ["exact <a>subset_toMeasurable</a> _ _ hx", [{"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "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 g : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\ni : \u03b9\nx : \u03b1\nhx : f x \u2260 mk f (_ : AEMeasurable f) x\n\u22a2 x \u2208 s i", "state_after": "no goals"}, {"tactic": "rw [restrict_piecewise]", "annotated_tactic": ["rw [<a>restrict_piecewise</a>]", [{"full_name": "Set.restrict_piecewise", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [106, 9], "def_end_pos": [106, 27]}]], "state_before": "case refine'_1\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c2 i, s i) g)", "state_after": "case refine'_1\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c2 i, s i) (const \u03b1 default))"}, {"tactic": "simp only [Set.restrict, const]", "annotated_tactic": ["simp only [<a>Set.restrict</a>, <a>const</a>]", [{"full_name": "Set.restrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}, {"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": "case refine'_1\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c2 i, s i) (const \u03b1 default))", "state_after": "case refine'_1\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable fun x => default"}, {"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 refine'_1\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable fun x => default", "state_after": "no goals"}, {"tactic": "rw [restrict_piecewise_compl, compl_iInter]", "annotated_tactic": ["rw [<a>restrict_piecewise_compl</a>, <a>compl_iInter</a>]", [{"full_name": "Set.restrict_piecewise_compl", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [112, 9], "def_end_pos": [112, 33]}, {"full_name": "Set.compl_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [615, 9], "def_end_pos": [615, 21]}]], "state_before": "case refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c2 i, s i)\u1d9c g)", "state_after": "case refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c3 i, (s i)\u1d9c) f)"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "case refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\n\u22a2 Measurable (Set.restrict (\u22c3 i, (s i)\u1d9c) f)", "state_after": "case refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 MeasurableSet (Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t)"}, {"tactic": "refine \u27e8\u22c3 i, (h i).mk f \u207b\u00b9' t \u2229 (s i)\u1d9c, MeasurableSet.iUnion fun i \u21a6\n  (measurable_mk _ ht).inter (measurableSet_toMeasurable _ _).compl, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u22c3 i, (h i).<a>mk</a> f \u207b\u00b9' t \u2229 (s i)\u1d9c, <a>MeasurableSet.iUnion</a> fun i \u21a6\n      (<a>measurable_mk</a> _ ht).<a>inter</a> (<a>measurableSet_toMeasurable</a> _ _).<a>compl</a>, ?_\u27e9", [{"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}, {"full_name": "AEMeasurable.measurable_mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [727, 9], "def_end_pos": [727, 22]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"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 refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 MeasurableSet (Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t)", "state_after": "case refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 Subtype.val \u207b\u00b9' \u22c3 i, mk f (_ : AEMeasurable f) \u207b\u00b9' t \u2229 (s i)\u1d9c = Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t"}, {"tactic": "ext \u27e8x, hx\u27e9", "annotated_tactic": ["ext \u27e8x, hx\u27e9", []], "state_before": "case refine'_2\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 Subtype.val \u207b\u00b9' \u22c3 i, mk f (_ : AEMeasurable f) \u207b\u00b9' t \u2229 (s i)\u1d9c = Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t", "state_after": "case refine'_2.h.mk\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : x \u2208 \u22c3 i, (s i)\u1d9c\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' \u22c3 i, mk f (_ : AEMeasurable f) \u207b\u00b9' t \u2229 (s i)\u1d9c \u2194\n    { val := x, property := hx } \u2208 Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t"}, {"tactic": "simp only [mem_preimage, mem_iUnion, Subtype.coe_mk, Set.restrict, mem_inter_iff,\n  mem_compl_iff] at hx \u22a2", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>mem_iUnion</a>, <a>Subtype.coe_mk</a>, <a>Set.restrict</a>, <a>mem_inter_iff</a>,\n      <a>mem_compl_iff</a>] at hx \u22a2", [{"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_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "Set.restrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [47, 5], "def_end_pos": [47, 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": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}]], "state_before": "case refine'_2.h.mk\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : x \u2208 \u22c3 i, (s i)\u1d9c\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' \u22c3 i, mk f (_ : AEMeasurable f) \u207b\u00b9' t \u2229 (s i)\u1d9c \u2194\n    { val := x, property := hx } \u2208 Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t", "state_after": "case refine'_2.h.mk\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 (\u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}) \u2194 f x \u2208 t"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case refine'_2.h.mk\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 (\u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}) \u2194 f x \u2208 t", "state_after": "case refine'_2.h.mk.mp\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 (\u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}) \u2192 f x \u2208 t\n\ncase refine'_2.h.mk.mpr\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 f x \u2208 t \u2192 \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}"}, {"tactic": "rintro \u27e8i, hxt, hxs\u27e9", "annotated_tactic": ["rintro \u27e8i, hxt, hxs\u27e9", []], "state_before": "case refine'_2.h.mk.mp\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 (\u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}) \u2192 f x \u2208 t", "state_after": "case refine'_2.h.mk.mp.intro.intro\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\ni : \u03b9\nhxt : mk f (_ : AEMeasurable f) x \u2208 t\nhxs : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 f x \u2208 t"}, {"tactic": "rwa [hs _ _ hxs]", "annotated_tactic": ["rwa [hs _ _ hxs]", []], "state_before": "case refine'_2.h.mk.mp.intro.intro\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\ni : \u03b9\nhxt : mk f (_ : AEMeasurable f) x \u2208 t\nhxs : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 f x \u2208 t", "state_after": "no goals"}, {"tactic": "rcases hx with \u27e8i, hi\u27e9", "annotated_tactic": ["rcases hx with \u27e8i, hi\u27e9", []], "state_before": "case refine'_2.h.mk.mpr\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\nhx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 f x \u2208 t \u2192 \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}", "state_after": "case refine'_2.h.mk.mpr.intro\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\ni : \u03b9\nhi : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 f x \u2208 t \u2192 \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}"}, {"tactic": "rw [hs _ _ hi]", "annotated_tactic": ["rw [hs _ _ hi]", []], "state_before": "case refine'_2.h.mk.mpr.intro\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\ni : \u03b9\nhi : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 f x \u2208 t \u2192 \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}", "state_after": "case refine'_2.h.mk.mpr.intro\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\ni : \u03b9\nhi : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 mk f (_ : AEMeasurable f) x \u2208 t \u2192\n    \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}"}, {"tactic": "exact fun h => \u27e8i, h, hi\u27e9", "annotated_tactic": ["exact fun h => \u27e8i, h, hi\u27e9", []], "state_before": "case refine'_2.h.mk.mpr.intro\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\nt : Set \u03b2\nht : MeasurableSet t\nx : \u03b1\ni : \u03b9\nhi : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\n\u22a2 mk f (_ : AEMeasurable f) x \u2208 t \u2192\n    \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}", "state_after": "no goals"}, {"tactic": "refine' measure_mono_null (fun x (hx : f x \u2260 g x) => _) (hs\u03bc i)", "annotated_tactic": ["refine' <a>measure_mono_null</a> (fun x (hx : f x \u2260 g x) => _) (hs\u03bc i)", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}]], "state_before": "case refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc i, f x = g x", "state_after": "case refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\nx : \u03b1\nhx : f x \u2260 g x\n\u22a2 x \u2208 s i"}, {"tactic": "contrapose! hx", "annotated_tactic": ["contrapose! hx", []], "state_before": "case refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\nx : \u03b1\nhx : f x \u2260 g x\n\u22a2 x \u2208 s i", "state_after": "case refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 s i\n\u22a2 f x = g x"}, {"tactic": "refine' (piecewise_eq_of_not_mem _ _ _ _).symm", "annotated_tactic": ["refine' (<a>piecewise_eq_of_not_mem</a> _ _ _ _).<a>symm</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": "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'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 s i\n\u22a2 f x = g x", "state_after": "case refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 s i\n\u22a2 \u00acx \u2208 \u22c2 i, s i"}, {"tactic": "exact fun h => hx (mem_iInter.1 h i)", "annotated_tactic": ["exact fun h => hx (<a>mem_iInter</a>.1 h i)", [{"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}]], "state_before": "case refine'_3\n\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 g\u271d : \u03b1 \u2192 \u03b2\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nh : \u2200 (i : \u03b9), AEMeasurable f\n\u271d : Nontrivial \u03b2\ninhabited_h : Inhabited \u03b2\ns : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhsm : MeasurableSet (\u22c2 i, s i)\nhs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x\ng : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f\ni : \u03b9\nx : \u03b1\nhx : \u00acx \u2208 s i\n\u22a2 \u00acx \u2208 \u22c2 i, s i", "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", "start": [555, 1], "end": [571, 24], "traced_tactics": [{"tactic": "have : \u2200 n, snorm (f n - g) p \u03bc = snorm ((hf n).mk (f n) - hg.1.mk g) p \u03bc :=\n  fun n => snorm_congr_ae ((hf n).ae_eq_mk.sub hg.1.ae_eq_mk)", "annotated_tactic": ["have : \u2200 n, <a>snorm</a> (f n - g) p \u03bc = <a>snorm</a> ((hf n).<a>mk</a> (f n) - hg.1.<a>mk</a> g) p \u03bc :=\n    fun n => <a>snorm_congr_ae</a> ((hf n).ae_eq_mk.sub hg.1.<a>ae_eq_mk</a>)", [{"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.AEStronglyMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1202, 29], "def_end_pos": [1202, 31]}, {"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_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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), AEStronglyMeasurable (f n) \u03bc\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "simp_rw [this]", "annotated_tactic": ["simp_rw [this]", []], "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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 Tendsto\n    (fun n =>\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc)\n    atTop (\ud835\udcdd 0)"}, {"tactic": "refine' tendsto_Lp_of_tendsto_ae_of_meas \u03bc hp hp' (fun n => (hf n).stronglyMeasurable_mk)\n  hg.1.stronglyMeasurable_mk (hg.ae_eq hg.1.ae_eq_mk) (hui.ae_eq fun n => (hf n).ae_eq_mk) _", "annotated_tactic": ["refine' <a>tendsto_Lp_of_tendsto_ae_of_meas</a> \u03bc hp hp' (fun n => (hf n).<a>stronglyMeasurable_mk</a>)\n    hg.1.<a>stronglyMeasurable_mk</a> (hg.ae_eq hg.1.<a>ae_eq_mk</a>) (hui.ae_eq fun n => (hf n).<a>ae_eq_mk</a>) _", [{"full_name": "MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [484, 9], "def_end_pos": [484, 41]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 Tendsto\n    (fun n =>\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc)\n    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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x))"}, {"tactic": "have h_ae_forall_eq : \u2200\u1d50 x \u2202\u03bc, \u2200 n, f n x = (hf n).mk (f n) x := by\n  rw [ae_all_iff]\n  exact fun n => (hf n).ae_eq_mk", "annotated_tactic": ["have h_ae_forall_eq : \u2200\u1d50 x \u2202\u03bc, \u2200 n, f n x = (hf n).<a>mk</a> (f n) x := by\n    rw [<a>ae_all_iff</a>]\n    exact fun n => (hf n).<a>ae_eq_mk</a>", [{"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.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x))", "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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x))"}, {"tactic": "filter_upwards [hfg, h_ae_forall_eq, hg.1.ae_eq_mk] with x hx_tendsto hxf_eq hxg_eq", "annotated_tactic": ["filter_upwards [hfg, h_ae_forall_eq, hg.1.<a>ae_eq_mk</a>] with x hx_tendsto hxf_eq hxg_eq", [{"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x))", "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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\n\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n    (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x))"}, {"tactic": "rw [\u2190 hxg_eq]", "annotated_tactic": ["rw [\u2190 hxg_eq]", []], "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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\n\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n    (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x))", "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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\n\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "convert hx_tendsto using 1", "annotated_tactic": ["convert hx_tendsto using 1", []], "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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\n\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop (\ud835\udcdd (g x))", "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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\n\u22a2 (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) = fun n => f n x"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\n\u22a2 (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) = fun n => f n x", "state_after": "case h.e'_3.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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\nn : \u2115\n\u22a2 AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x"}, {"tactic": "exact (hxf_eq n).symm", "annotated_tactic": ["exact (hxf_eq n).<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'_3.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\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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\nh_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nx : \u03b1\nhx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x\nhxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x\nn : \u2115\n\u22a2 AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x", "state_after": "no goals"}, {"tactic": "rw [ae_all_iff]", "annotated_tactic": ["rw [<a>ae_all_iff</a>]", [{"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 : 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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x", "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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f i a = AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc) a"}, {"tactic": "exact fun n => (hf n).ae_eq_mk", "annotated_tactic": ["exact fun n => (hf n).<a>ae_eq_mk</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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), AEStronglyMeasurable (f n) \u03bc\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))\nthis :\n  \u2200 (n : \u2115),\n    snorm (f n - g) p \u03bc =\n      snorm\n        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -\n          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))\n        p \u03bc\n\u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f i a = AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc) a", "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_Ioo", "start": [1169, 1], "end": [1170, 80], "traced_tactics": [{"tactic": "rw [\u2190 map_add_right_Ioo, map_eq_image, addRightEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_right_Ioo</a>, <a>map_eq_image</a>, <a>addRightEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_right_Ioo", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 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) (Ioo a b) = Ioo (a + c) (b + c)", "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.succ_succ_ne_one", "start": [270, 1], "end": [271, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.div'_to_nat", "start": [1617, 1], "end": [1618, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.coeff_C_mul", "start": [702, 1], "end": [706, 7], "traced_tactics": [{"tactic": "rw [mul_def, sum_C]", "annotated_tactic": ["rw [<a>mul_def</a>, <a>sum_C</a>]", [{"full_name": "MvPolynomial.mul_def", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 16]}, {"full_name": "MvPolynomial.sum_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [369, 9], "def_end_pos": [369, 14]}]], "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 m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nm : \u03c3 \u2192\u2080 \u2115\na : R\np : MvPolynomial \u03c3 R\n\u22a2 coeff m (\u2191C a * p) = a * coeff m p", "state_after": "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 m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nm : \u03c3 \u2192\u2080 \u2115\na : R\np : MvPolynomial \u03c3 R\n\u22a2 coeff m (sum p fun n b => \u2191(monomial (0 + n)) (a * b)) = a * coeff m p\n\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 m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nm : \u03c3 \u2192\u2080 \u2115\na : R\np : MvPolynomial \u03c3 R\n\u22a2 (sum p fun n b => \u2191(monomial (0 + n)) (0 * b)) = 0"}, {"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\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nm : \u03c3 \u2192\u2080 \u2115\na : R\np : MvPolynomial \u03c3 R\n\u22a2 (sum p fun n b => \u2191(monomial (0 + n)) (0 * b)) = 0", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) [sum_def, coeff_sum]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) [<a>sum_def</a>, <a>coeff_sum</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.sum_def", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [595, 9], "def_end_pos": [595, 16]}, {"full_name": "MvPolynomial.coeff_sum", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [643, 9], "def_end_pos": [643, 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 m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np\u271d q : MvPolynomial \u03c3 R\nm : \u03c3 \u2192\u2080 \u2115\na : R\np : MvPolynomial \u03c3 R\n\u22a2 coeff m (sum p fun n b => \u2191(monomial (0 + n)) (a * b)) = a * coeff m p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.comp\u2082Measurable_eq_mk", "start": [420, 1], "end": [425, 70], "traced_tactics": [{"tactic": "rw [comp\u2082Measurable_eq_pair, pair_eq_mk, compMeasurable_mk]", "annotated_tactic": ["rw [<a>comp\u2082Measurable_eq_pair</a>, <a>pair_eq_mk</a>, <a>compMeasurable_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.comp\u2082Measurable_eq_pair", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [415, 9], "def_end_pos": [415, 32]}, {"full_name": "MeasureTheory.AEEqFun.pair_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [344, 9], "def_end_pos": [344, 19]}, {"full_name": "MeasureTheory.AEEqFun.compMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [312, 9], "def_end_pos": [312, 26]}]], "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 comp\u2082Measurable g hg f\u2081 f\u2082 =\n    mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)", "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 mk (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) =\n    mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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 mk (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) =\n    mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.mod_to_nat", "start": [1655, 1], "end": [1659, 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/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_of_ssubset_of_subset", "start": [428, 1], "end": [430, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.injective_of_forall_isSome", "start": [193, 1], "end": [196, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpIndL1Fin_smul'", "start": [116, 1], "end": [125, 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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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": [412, 9], "def_end_pos": [412, 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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\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-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.comp_inf_eq_inf_comp_of_is_total", "start": [707, 1], "end": [709, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.eq_succ_of_ne_zero", "start": [122, 1], "end": [123, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpIndL1_add", "start": [202, 1], "end": [209, 41], "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 y : G\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y", "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 y : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y\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 y : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y"}, {"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 y : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y\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 y : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y", "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 y : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y\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 y : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y"}, {"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 y : G\nhs : MeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y", "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 y : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y\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 y : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y"}, {"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 y : G\nhs : \u00acMeasurableSet s\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y", "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 y : G\nhs : \u00acMeasurableSet s\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\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 y : G\nhs : \u00acMeasurableSet s\n\u22a2 0 = 0 + 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)\nx y : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y", "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 y : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\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 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 y : G\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = \u22a4\n\u22a2 0 = 0 + 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)\nx y : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y", "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 y : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1Fin hm hs h\u03bcs (x + y) = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y"}, {"tactic": "exact condexpIndL1Fin_add hs h\u03bcs x y", "annotated_tactic": ["exact <a>condexpIndL1Fin_add</a> hs h\u03bcs x y", [{"full_name": "MeasureTheory.condexpIndL1Fin_add", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [92, 9], "def_end_pos": [92, 28]}]], "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 y : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4\n\u22a2 condexpIndL1Fin hm hs h\u03bcs (x + y) = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y", "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_strict_mono", "start": [975, 1], "end": [978, 90], "traced_tactics": [{"tactic": "simp [hs]", "annotated_tactic": ["simp [hs]", []], "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\ns : Set \u03b1\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 0\nhg : Measurable g\nhfi : \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2260 \u22a4\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x < g x\n\u22a2 Measure.restrict \u03bc s \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.mem_pure'", "start": [76, 1], "end": [77, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.val_inj", "start": [36, 1], "end": [36, 90], "traced_tactics": []}, {"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_aux2", "start": [680, 1], "end": [709, 12], "traced_tactics": [{"tactic": "set t' := R\u207b\u00b9 \u2022 t with ht'", "annotated_tactic": ["set t' := R\u207b\u00b9 \u2022 t with ht'", []], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\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": "set u' := R\u207b\u00b9 \u2022 u with hu'", "annotated_tactic": ["set u' := R\u207b\u00b9 \u2022 u with hu'", []], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 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": "rw [mul_zero] at B", "annotated_tactic": ["rw [<a>mul_zero</a>] at B", [{"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 u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi (R * 0)] (R * 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 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 (A.comp B).congr' _", "annotated_tactic": ["apply (A.comp B).<a>congr'</a> _", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r =>\n    \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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r =>\n    \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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) a =\n        \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u)"}, {"tactic": "rintro r -", "annotated_tactic": ["rintro 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) a =\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "have T : (R * r) \u2022 t' = r \u2022 t := by\n  rw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]", "annotated_tactic": ["have T : (R * r) \u2022 t' = r \u2022 t := by\n    rw [<a>mul_comm</a>, ht', <a>smul_smul</a>, <a>mul_assoc</a>, <a>mul_inv_cancel</a> Rpos.ne', <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =\n    \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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "have U : (R * r) \u2022 u' = r \u2022 u := by\n  rw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]", "annotated_tactic": ["have U : (R * r) \u2022 u' = r \u2022 u := by\n    rw [<a>mul_comm</a>, hu', <a>smul_smul</a>, <a>mul_assoc</a>, <a>mul_inv_cancel</a> Rpos.ne', <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =\n    \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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\nU : (R * r) \u2022 u' = r \u2022 u\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\nU : (R * r) \u2022 u' = r \u2022 u\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =\n    \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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\nU : (R * r) \u2022 u' = r \u2022 u\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 u) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)"}, {"tactic": "rw [T, U]", "annotated_tactic": ["rw [T, U]", []], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\nU : (R * r) \u2022 u' = r \u2022 u\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 u) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)", "state_after": "no goals"}, {"tactic": "apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 \u03bc s x h t' u'", "annotated_tactic": ["apply <a>tendsto_addHaar_inter_smul_zero_of_density_zero_aux1</a> \u03bc s x h t' u'", [{"full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux1", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [628, 9], "def_end_pos": [628, 61]}]], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 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": "case h'u\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 \u2191\u2191\u03bc u' \u2260 0\n\ncase t_bound\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 t' \u2286 closedBall 0 1"}, {"tactic": "simp only [h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff,\n  ENNReal.ofReal_eq_zero, inv_eq_zero, inv_pow, Ne.def, or_self_iff, mul_eq_zero]", "annotated_tactic": ["simp only [h'u, (<a>pow_pos</a> Rpos _).<a>ne'</a>, <a>abs_nonpos_iff</a>, <a>addHaar_smul</a>, <a>not_false_iff</a>,\n        <a>ENNReal.ofReal_eq_zero</a>, <a>inv_eq_zero</a>, <a>inv_pow</a>, <a>Ne.def</a>, <a>or_self_iff</a>, <a>mul_eq_zero</a>]", [{"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "abs_nonpos_iff", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [185, 9], "def_end_pos": [185, 23]}, {"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": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 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": "inv_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 20]}, {"full_name": "inv_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}]], "state_before": "case h'u\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 \u2191\u2191\u03bc u' \u2260 0", "state_after": "no goals"}, {"tactic": "refine (smul_set_mono t_bound).trans_eq ?_", "annotated_tactic": ["refine (<a>smul_set_mono</a> t_bound).<a>trans_eq</a> ?_", [{"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": "LE.le.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [211, 7], "def_end_pos": [211, 21]}]], "state_before": "case t_bound\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 t' \u2286 closedBall 0 1", "state_after": "case t_bound\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 R\u207b\u00b9 \u2022 closedBall 0 R = closedBall 0 1"}, {"tactic": "rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),\n  inv_mul_cancel Rpos.ne']", "annotated_tactic": ["rw [<a>smul_closedBall</a> _ _ Rpos.le, <a>smul_zero</a>, <a>Real.norm_of_nonneg</a> (<a>inv_nonneg</a>.2 Rpos.le),\n        <a>inv_mul_cancel</a> Rpos.ne']", [{"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": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 19]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}]], "state_before": "case t_bound\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 R\u207b\u00b9 \u2022 closedBall 0 R = closedBall 0 1", "state_after": "no goals"}, {"tactic": "apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within", "annotated_tactic": ["apply <a>tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within</a>", [{"full_name": "tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [453, 9], "def_end_pos": [453, 64]}]], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi (R * 0)] (R * 0))", "state_after": "case h1\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (R * 0))\n\ncase h2\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi 0] 0, R * x \u2208 Ioi (R * 0)"}, {"tactic": "exact (tendsto_const_nhds.mul tendsto_id).mono_left nhdsWithin_le_nhds", "annotated_tactic": ["exact (tendsto_const_nhds.mul <a>tendsto_id</a>).<a>mono_left</a> <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"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 h1\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (R * 0))", "state_after": "no goals"}, {"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": "case h2\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi 0] 0, R * x \u2208 Ioi (R * 0)", "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 R * a \u2208 Ioi (R * 0)"}, {"tactic": "intro r rpos", "annotated_tactic": ["intro r rpos", []], "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 R * a \u2208 Ioi (R * 0)", "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nr : \u211d\nrpos : r \u2208 Ioi 0\n\u22a2 R * r \u2208 Ioi (R * 0)"}, {"tactic": "rw [mul_zero]", "annotated_tactic": ["rw [<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 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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nr : \u211d\nrpos : r \u2208 Ioi 0\n\u22a2 R * r \u2208 Ioi (R * 0)", "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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nr : \u211d\nrpos : r \u2208 Ioi 0\n\u22a2 R * r \u2208 Ioi 0"}, {"tactic": "exact mul_pos Rpos rpos", "annotated_tactic": ["exact <a>mul_pos</a> Rpos rpos", [{"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\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nr : \u211d\nrpos : r \u2208 Ioi 0\n\u22a2 R * r \u2208 Ioi 0", "state_after": "no goals"}, {"tactic": "rw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]", "annotated_tactic": ["rw [<a>mul_comm</a>, ht', <a>smul_smul</a>, <a>mul_assoc</a>, <a>mul_inv_cancel</a> Rpos.ne', <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"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\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\n\u22a2 (R * r) \u2022 t' = r \u2022 t", "state_after": "no goals"}, {"tactic": "rw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]", "annotated_tactic": ["rw [<a>mul_comm</a>, hu', <a>smul_smul</a>, <a>mul_assoc</a>, <a>mul_inv_cancel</a> Rpos.ne', <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [194, 15], "def_end_pos": [194, 29]}, {"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\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\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : 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)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\n\u22a2 (R * r) \u2022 u' = r \u2022 u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.map_inl_disjUnion_map_inr", "start": [60, 1], "end": [63, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.DominatedFinMeasAdditive.smul", "start": [224, 1], "end": [229, 74], "traced_tactics": [{"tactic": "refine' \u27e8hT.1.smul c, fun s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8hT.1.<a>smul</a> c, fun s hs h\u03bcs => _\u27e9", [{"full_name": "MeasureTheory.FinMeasAdditive.smul", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [113, 9], "def_end_pos": [113, 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\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\n\u22a2 DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016(fun s => c \u2022 T s) s\u2016 \u2264 \u2016c\u2016 * C * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "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\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016(fun s => c \u2022 T s) s\u2016 \u2264 \u2016c\u2016 * 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\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016c \u2022 T s\u2016 \u2264 \u2016c\u2016 * C * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "rw [norm_smul, mul_assoc]", "annotated_tactic": ["rw [<a>norm_smul</a>, <a>mul_assoc</a>]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"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\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016c \u2022 T s\u2016 \u2264 \u2016c\u2016 * 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\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016c\u2016 * \u2016T s\u2016 \u2264 \u2016c\u2016 * (C * ENNReal.toReal (\u2191\u2191\u03bc s))"}, {"tactic": "exact mul_le_mul le_rfl (hT.2 s hs h\u03bcs) (norm_nonneg _) (norm_nonneg _)", "annotated_tactic": ["exact <a>mul_le_mul</a> <a>le_rfl</a> (hT.2 s hs h\u03bcs) (<a>norm_nonneg</a> _) (<a>norm_nonneg</a> _)", [{"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_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"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\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\ninst\u271d\u00b3 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c \u03b2\nhT : DominatedFinMeasAdditive \u03bc T C\nc : \ud835\udd5c\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016c\u2016 * \u2016T s\u2016 \u2264 \u2016c\u2016 * (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/Prod.lean", "full_name": "MeasureTheory.measure_mul_lintegral_eq", "start": [244, 1], "end": [259, 23], "traced_tactics": [{"tactic": "rw [\u2190 set_lintegral_one, \u2190 lintegral_indicator _ sm,\n  \u2190 lintegral_lintegral_mul (measurable_const.indicator sm).aemeasurable hf.aemeasurable,\n  \u2190 lintegral_lintegral_mul_inv \u03bc \u03bd]", "annotated_tactic": ["rw [\u2190 <a>set_lintegral_one</a>, \u2190 <a>lintegral_indicator</a> _ sm,\n    \u2190 <a>lintegral_lintegral_mul</a> (measurable_const.indicator sm).<a>aemeasurable</a> hf.aemeasurable,\n    \u2190 <a>lintegral_lintegral_mul_inv</a> \u03bc \u03bd]", [{"full_name": "MeasureTheory.set_lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [156, 9], "def_end_pos": [156, 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.lintegral_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [743, 9], "def_end_pos": [743, 32]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "MeasureTheory.lintegral_lintegral_mul_inv", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [195, 9], "def_end_pos": [195, 36]}]], "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc s * \u222b\u207b (y : G), f y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\n\ncase hf\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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\n\ncase hf\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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y)", "state_after": "case hf\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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y)\n\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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc"}, {"tactic": "have ms :\n  \u2200 x : G, Measurable fun y => ((fun z => z * x) \u207b\u00b9' s).indicator (fun _ => (1 : \u211d\u22650\u221e)) y :=\n  fun x => measurable_const.indicator (measurable_mul_const _ sm)", "annotated_tactic": ["have ms :\n    \u2200 x : G, <a>Measurable</a> fun y => ((fun z => z * x) \u207b\u00b9' s).<a>indicator</a> (fun _ => (1 : \u211d\u22650\u221e)) y :=\n    fun x => measurable_const.indicator (<a>measurable_mul_const</a> _ sm)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasurableMul.measurable_mul_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [83, 3], "def_end_pos": [83, 23]}]], "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc"}, {"tactic": "have : \u2200 x y, s.indicator (fun _ : G => (1 : \u211d\u22650\u221e)) (y * x) =\n    ((fun z => z * x) \u207b\u00b9' s).indicator (fun b : G => 1) y := by\n  intro x y; symm; convert indicator_comp_right (M := \u211d\u22650\u221e) fun y => y * x using 2; ext1; rfl", "annotated_tactic": ["have : \u2200 x y, s.indicator (fun _ : G => (1 : \u211d\u22650\u221e)) (y * x) =\n      ((fun z => z * x) \u207b\u00b9' s).<a>indicator</a> (fun b : G => 1) y := by\n    intro x y; symm; convert <a>indicator_comp_right</a> (M := \u211d\u22650\u221e) fun y => y * x using 2; ext1; rfl", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.indicator_comp_right", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [245, 3], "def_end_pos": [245, 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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nthis : \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc"}, {"tactic": "simp_rw [this, lintegral_mul_const _ (ms _), lintegral_indicator _ (measurable_mul_const _ sm),\n  set_lintegral_one]", "annotated_tactic": ["simp_rw [this, <a>lintegral_mul_const</a> _ (ms _), <a>lintegral_indicator</a> _ (<a>measurable_mul_const</a> _ sm),\n    <a>set_lintegral_one</a>]", [{"full_name": "MeasureTheory.lintegral_mul_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [725, 9], "def_end_pos": [725, 28]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MeasurableMul.measurable_mul_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [83, 3], "def_end_pos": [83, 23]}, {"full_name": "MeasureTheory.set_lintegral_one", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [156, 9], "def_end_pos": [156, 26]}]], "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nthis : \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n    \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact (((measurable_const.indicator sm).comp measurable_fst).mul\n  (hf.comp measurable_snd)).aemeasurable", "annotated_tactic": ["exact (((measurable_const.indicator sm).<a>comp</a> <a>measurable_fst</a>).<a>mul</a>\n      (hf.comp <a>measurable_snd</a>)).<a>aemeasurable</a>", [{"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.mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [141, 9], "def_end_pos": [141, 23]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}]], "state_before": "case hf\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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y)", "state_after": "no goals"}, {"tactic": "intro x y", "annotated_tactic": ["intro x y", []], "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\n\u22a2 \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y", "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y", "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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y = indicator s (fun x => 1) (y * x)"}, {"tactic": "convert indicator_comp_right (M := \u211d\u22650\u221e) fun y => y * x using 2", "annotated_tactic": ["convert <a>indicator_comp_right</a> (M := \u211d\u22650\u221e) fun y => y * x using 2", [{"full_name": "Set.indicator_comp_right", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [245, 3], "def_end_pos": [245, 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\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y = indicator s (fun x => 1) (y * x)", "state_after": "case h.e'_2.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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 (fun b => 1) = (fun x => 1) \u2218 fun y => y * x"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case h.e'_2.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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 (fun b => 1) = (fun x => 1) \u2218 fun y => y * x", "state_after": "case h.e'_2.h.e'_5.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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y x\u271d : G\n\u22a2 1 = ((fun x => 1) \u2218 fun y => y * x) x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_2.h.e'_5.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\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y x\u271d : G\n\u22a2 1 = ((fun x => 1) \u2218 fun y => y * x) x\u271d", "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_left", "start": [929, 1], "end": [932, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_top_smul_measure", "start": [1666, 1], "end": [1672, 67], "traced_tactics": [{"tactic": "refine' setToFun_measure_zero' hT fun s _ h\u03bcs => _", "annotated_tactic": ["refine' <a>setToFun_measure_zero'</a> hT fun s _ h\u03bcs => _", [{"full_name": "MeasureTheory.setToFun_measure_zero'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1433, 9], "def_end_pos": [1433, 31]}]], "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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\n\u22a2 setToFun (\u22a4 \u2022 \u03bc) T hT f = 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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191(\u22a4 \u2022 \u03bc) s < \u22a4\n\u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 0"}, {"tactic": "rw [lt_top_iff_ne_top] at h\u03bcs", "annotated_tactic": ["rw [<a>lt_top_iff_ne_top</a>] at h\u03bcs", [{"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\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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191(\u22a4 \u2022 \u03bc) s < \u22a4\n\u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191(\u22a4 \u2022 \u03bc) s \u2260 \u22a4\n\u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 0"}, {"tactic": "simp only [true_and_iff, Measure.smul_apply, ENNReal.mul_eq_top, eq_self_iff_true,\n  top_ne_zero, Ne.def, not_false_iff, not_or, Classical.not_not, smul_eq_mul] at h\u03bcs", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>Measure.smul_apply</a>, <a>ENNReal.mul_eq_top</a>, <a>eq_self_iff_true</a>,\n    <a>top_ne_zero</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>not_or</a>, <a>Classical.not_not</a>, <a>smul_eq_mul</a>] at h\u03bcs", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 19]}, {"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": "ENNReal.top_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [337, 17], "def_end_pos": [337, 28]}, {"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": "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": "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\nE : Type u_2\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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191(\u22a4 \u2022 \u03bc) s \u2260 \u22a4\n\u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u2227 \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 0"}, {"tactic": "simp only [h\u03bcs.right, Measure.smul_apply, mul_zero, smul_eq_mul]", "annotated_tactic": ["simp only [h\u03bcs.right, <a>Measure.smul_apply</a>, <a>mul_zero</a>, <a>smul_eq_mul</a>]", [{"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"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\nE : Type u_2\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 (\u22a4 \u2022 \u03bc) T C\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx\u271d : MeasurableSet s\nh\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u2227 \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 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_add_adjacent_intervals_cancel", "start": [886, 1], "end": [897, 76], "traced_tactics": [{"tactic": "have hac := hab.trans hbc", "annotated_tactic": ["have hac := hab.trans hbc", []], "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 c..a, f x \u2202\u03bc = 0", "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\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\nhac : IntervalIntegrable f \u03bc a 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 c..a, f x \u2202\u03bc = 0"}, {"tactic": "simp only [intervalIntegral, sub_add_sub_comm, sub_eq_zero]", "annotated_tactic": ["simp only [<a>intervalIntegral</a>, <a>sub_add_sub_comm</a>, <a>sub_eq_zero</a>]", [{"full_name": "intervalIntegral", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}, {"full_name": "sub_add_sub_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [581, 3], "def_end_pos": [581, 14]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 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\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a 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 c..a, f x \u2202\u03bc = 0", "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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc b c, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc c a, f x \u2202\u03bc =\n    \u222b (x : \u211d) in Ioc b a, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc c b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc a c, f x \u2202\u03bc"}, {"tactic": "iterate 4 rw [\u2190 integral_union]", "annotated_tactic": ["iterate 4 rw [\u2190 <a>integral_union</a>]", [{"full_name": "MeasureTheory.integral_union", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [102, 9], "def_end_pos": [102, 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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc b c, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc c a, f x \u2202\u03bc =\n    \u222b (x : \u211d) in Ioc b a, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc c b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc a c, f x \u2202\u03bc", "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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc = \u222b (x : \u211d) in Ioc b a \u222a Ioc c b \u222a Ioc a c, f x \u2202\u03bc\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a \u222a Ioc c b) (Ioc a c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc a c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a \u222a Ioc c b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a c)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a) (Ioc c b)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c b)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c b)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b \u222a Ioc b c) (Ioc c a)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c a)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b \u222a Ioc b c)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c a)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b) (Ioc b c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc b c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b c)"}, {"tactic": "all_goals\n  simp [*, MeasurableSet.union, measurableSet_Ioc, Ioc_disjoint_Ioc_same,\n    Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2]", "annotated_tactic": ["all_goals\n    simp [*, <a>MeasurableSet.union</a>, <a>measurableSet_Ioc</a>, <a>Ioc_disjoint_Ioc_same</a>,\n      Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2]", [{"full_name": "MeasurableSet.union", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [191, 19], "def_end_pos": [191, 38]}, {"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_disjoint_Ioc_same", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [44, 9], "def_end_pos": [44, 30]}]], "state_before": "case hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a \u222a Ioc c b) (Ioc a c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc a c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a \u222a Ioc c b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a c)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a) (Ioc c b)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c b)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c b)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b \u222a Ioc b c) (Ioc c a)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c a)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b \u222a Ioc b c)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c a)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b) (Ioc b c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc b c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b c)", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_union]", "annotated_tactic": ["rw [\u2190 <a>integral_union</a>]", [{"full_name": "MeasureTheory.integral_union", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [102, 9], "def_end_pos": [102, 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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc =\n    \u222b (x : \u211d) in Ioc b a \u222a Ioc c b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc a c, f x \u2202\u03bc\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a) (Ioc c b)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c b)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c b)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b \u222a Ioc b c) (Ioc c a)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c a)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b \u222a Ioc b c)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c a)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b) (Ioc b c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc b c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc 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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc = \u222b (x : \u211d) in Ioc b a \u222a Ioc c b \u222a Ioc a c, f x \u2202\u03bc\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a \u222a Ioc c b) (Ioc a c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc a c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a \u222a Ioc c b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a c)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc b a) (Ioc c b)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c b)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b a)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c b)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b \u222a Ioc b c) (Ioc c a)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc c a)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b \u222a Ioc b c)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc c a)\n\ncase hst\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Disjoint (Ioc a b) (Ioc b c)\n\ncase ht\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 MeasurableSet (Ioc b c)\n\ncase hfs\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc a b)\n\ncase hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b c)"}, {"tactic": "suffices Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c by rw [this]", "annotated_tactic": ["suffices <a>Ioc</a> a b \u222a <a>Ioc</a> b c \u222a <a>Ioc</a> c a = <a>Ioc</a> b a \u222a <a>Ioc</a> c b \u222a <a>Ioc</a> a c by rw [this]", [{"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": "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": "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]}]], "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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc = \u222b (x : \u211d) in Ioc b a \u222a Ioc c b \u222a Ioc a c, f x \u2202\u03bc", "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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c"}, {"tactic": "rw [Ioc_union_Ioc_union_Ioc_cycle, union_right_comm, Ioc_union_Ioc_union_Ioc_cycle,\n  min_left_comm, max_left_comm]", "annotated_tactic": ["rw [<a>Ioc_union_Ioc_union_Ioc_cycle</a>, <a>union_right_comm</a>, <a>Ioc_union_Ioc_union_Ioc_cycle</a>,\n      <a>min_left_comm</a>, <a>max_left_comm</a>]", [{"full_name": "Set.Ioc_union_Ioc_union_Ioc_cycle", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1879, 9], "def_end_pos": [1879, 38]}, {"full_name": "Set.union_right_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 25]}, {"full_name": "Set.Ioc_union_Ioc_union_Ioc_cycle", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1879, 9], "def_end_pos": [1879, 38]}, {"full_name": "min_left_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [92, 9], "def_end_pos": [92, 22]}, {"full_name": "max_left_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [125, 9], "def_end_pos": [125, 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\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\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "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\nhac : IntervalIntegrable f \u03bc a c\nthis : Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c\n\u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc = \u222b (x : \u211d) in Ioc b a \u222a Ioc c b \u222a Ioc a c, f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp [*, MeasurableSet.union, measurableSet_Ioc, Ioc_disjoint_Ioc_same,\n  Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2]", "annotated_tactic": ["simp [*, <a>MeasurableSet.union</a>, <a>measurableSet_Ioc</a>, <a>Ioc_disjoint_Ioc_same</a>,\n      Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2]", [{"full_name": "MeasurableSet.union", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [191, 19], "def_end_pos": [191, 38]}, {"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_disjoint_Ioc_same", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [44, 9], "def_end_pos": [44, 30]}]], "state_before": "case hft\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\nhab : IntervalIntegrable f \u03bc a b\nhbc : IntervalIntegrable f \u03bc b c\nhac : IntervalIntegrable f \u03bc a c\n\u22a2 IntegrableOn (fun x => f x) (Ioc b c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.get_cons_succ", "start": [285, 1], "end": [286, 34], "traced_tactics": [{"tactic": "rw [\u2190 get_tail_succ, tail_cons]", "annotated_tactic": ["rw [\u2190 <a>get_tail_succ</a>, <a>tail_cons</a>]", [{"full_name": "Vector.get_tail_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [177, 9], "def_end_pos": [177, 22]}, {"full_name": "Vector.tail_cons", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\na : \u03b1\nv : Vector \u03b1 n\ni : Fin n\n\u22a2 get (a ::\u1d65 v) (Fin.succ i) = get v 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.sublist_nil", "start": [468, 9], "end": [469, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "full_name": "MeasureTheory.Measure.join_dirac", "start": [231, 1], "end": [232, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.bindOnSupport_bindOnSupport", "start": [275, 1], "end": [291, 15], "traced_tactics": [{"tactic": "refine' PMF.ext fun a => _", "annotated_tactic": ["refine' <a>PMF.ext</a> fun a => _", [{"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\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\n\u22a2 bindOnSupport (bindOnSupport p f) g =\n    bindOnSupport p fun a ha => bindOnSupport (f a ha) fun b hb => g b (_ : b \u2208 support (bindOnSupport p f))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 \u2191(bindOnSupport (bindOnSupport p f) g) a =\n    \u2191(bindOnSupport p fun a ha => bindOnSupport (f a ha) fun b hb => g b (_ : b \u2208 support (bindOnSupport p f))) a"}, {"tactic": "dsimp only [bindOnSupport_apply]", "annotated_tactic": ["dsimp only [<a>bindOnSupport_apply</a>]", [{"full_name": "PMF.bindOnSupport_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [219, 9], "def_end_pos": [219, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 \u2191(bindOnSupport (bindOnSupport p f) g) a =\n    \u2191(bindOnSupport p fun a ha => bindOnSupport (f a ha) fun b hb => g b (_ : b \u2208 support (bindOnSupport p f))) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2),\n      (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) *\n        if h : (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0 then 0 else \u2191(g a_1 h) a) =\n    \u2211' (a_1 : \u03b1),\n      \u2191p a_1 *\n        if h : \u2191p a_1 = 0 then 0\n        else\n          \u2211' (a_2 : \u03b2),\n            \u2191(f a_1 h) a_2 * if h : \u2191(f a_1 h) a_2 = 0 then 0 else \u2191(g a_2 (_ : a_2 \u2208 support (bindOnSupport p f))) a"}, {"tactic": "simp only [\u2190 tsum_dite_right, ENNReal.tsum_mul_left.symm, ENNReal.tsum_mul_right.symm]", "annotated_tactic": ["simp only [\u2190 <a>tsum_dite_right</a>, ENNReal.tsum_mul_left.symm, ENNReal.tsum_mul_right.symm]", [{"full_name": "tsum_dite_right", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [575, 9], "def_end_pos": [575, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2),\n      (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) *\n        if h : (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0 then 0 else \u2191(g a_1 h) a) =\n    \u2211' (a_1 : \u03b1),\n      \u2191p a_1 *\n        if h : \u2191p a_1 = 0 then 0\n        else\n          \u2211' (a_2 : \u03b2),\n            \u2191(f a_1 h) a_2 * if h : \u2191(f a_1 h) a_2 = 0 then 0 else \u2191(g a_2 (_ : a_2 \u2208 support (bindOnSupport p f))) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) *\n        if h : (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0 then 0 else \u2191(g a_1 h) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      \u2191p a_1 *\n        if h : \u2191p a_1 = 0 then 0\n        else \u2191(f a_1 h) i * if h : \u2191(f a_1 h) i = 0 then 0 else \u2191(g i (_ : i \u2208 support (bindOnSupport p f))) a"}, {"tactic": "simp only [ENNReal.tsum_eq_zero, dite_eq_left_iff]", "annotated_tactic": ["simp only [<a>ENNReal.tsum_eq_zero</a>, <a>dite_eq_left_iff</a>]", [{"full_name": "ENNReal.tsum_eq_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [862, 19], "def_end_pos": [862, 31]}, {"full_name": "dite_eq_left_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1151, 17], "def_end_pos": [1151, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) *\n        if h : (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0 then 0 else \u2191(g a_1 h) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      \u2191p a_1 *\n        if h : \u2191p a_1 = 0 then 0\n        else \u2191(f a_1 h) i * if h : \u2191(f a_1 h) i = 0 then 0 else \u2191(g i (_ : i \u2208 support (bindOnSupport p f))) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) *\n        if h : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) = 0 then 0\n        else \u2191(g a_1 (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0)) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      \u2191p a_1 *\n        if h : \u2191p a_1 = 0 then 0\n        else \u2191(f a_1 h) i * if h : \u2191(f a_1 h) i = 0 then 0 else \u2191(g i (_ : i \u2208 support (bindOnSupport p f))) a"}, {"tactic": "refine' ENNReal.tsum_comm.trans (tsum_congr fun a' => tsum_congr fun b => _)", "annotated_tactic": ["refine' ENNReal.tsum_comm.trans (<a>tsum_congr</a> fun a' => <a>tsum_congr</a> fun b => _)", [{"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) *\n        if h : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) = 0 then 0\n        else \u2191(g a_1 (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0)) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      \u2191p a_1 *\n        if h : \u2191p a_1 = 0 then 0\n        else \u2191(f a_1 h) i * if h : \u2191(f a_1 h) i = 0 then 0 else \u2191(g i (_ : i \u2208 support (bindOnSupport p f))) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\n\u22a2 ((\u2191p a' * if h : \u2191p a' = 0 then 0 else \u2191(f a' h) b) *\n      if h : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 then 0\n      else \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a) =\n    \u2191p a' *\n      if h : \u2191p a' = 0 then 0\n      else \u2191(f a' h) b * if h : \u2191(f a' h) b = 0 then 0 else \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a"}, {"tactic": "split_ifs with h _ h_1 _ h_2", "annotated_tactic": ["split_ifs with h _ h_1 _ h_2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\n\u22a2 ((\u2191p a' * if h : \u2191p a' = 0 then 0 else \u2191(f a' h) b) *\n      if h : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 then 0\n      else \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a) =\n    \u2191p a' *\n      if h : \u2191p a' = 0 then 0\n      else \u2191(f a' h) b * if h : \u2191(f a' h) b = 0 then 0 else \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u2191p a' = 0\n_ : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n\u22a2 \u2191p a' * 0 * 0 = \u2191p a' * 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u2191p a' = 0\n_ : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n\u22a2 \u2191p a' * 0 * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a = \u2191p a' * 0\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)"}, {"tactic": "any_goals ring1", "annotated_tactic": ["any_goals ring1", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u2191p a' = 0\n_ : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n\u22a2 \u2191p a' * 0 * 0 = \u2191p a' * 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u2191p a' = 0\n_ : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n\u22a2 \u2191p a' * 0 * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a = \u2191p a' * 0\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * 0)"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)", "state_after": "no goals"}, {"tactic": "have := h_1 a'", "annotated_tactic": ["have := h_1 a'", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\nthis : (\u2191p a' * if h : \u2191p a' = 0 then 0 else \u2191(f a' h) b) = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)"}, {"tactic": "simp [h] at this", "annotated_tactic": ["simp [h] at this", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\nthis : (\u2191p a' * if h : \u2191p a' = 0 then 0 else \u2191(f a' h) b) = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\nthis : \u2191(f a' (_ : \u00ac\u2191p a' = 0)) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\n_ : \u00ac\u2191(f a' h) b = 0\nthis : \u2191(f a' (_ : \u00ac\u2191p a' = 0)) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)", "state_after": "no goals"}, {"tactic": "simp [h_2]", "annotated_tactic": ["simp [h_2]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00ac\u2191p a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0\nh_2 : \u2191(f a' h) b = 0\n\u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =\n    \u2191p a' * (\u2191(f a' h) b * 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_max_add", "start": [263, 1], "end": [266, 89], "traced_tactics": [{"tactic": "classical\n  induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right]", "annotated_tactic": ["classical\n    induction' s using <a>Finset.induction_on</a> with a s _ ih <;> simp [*, <a>max_add_add_right</a>]", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}, {"full_name": "max_add_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/MinMax.lean", "def_pos": [72, 3], "def_end_pos": [72, 14]}]], "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\u271d : Finset \u03b1\na : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\nc : \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nn : WithBot \u03b2\ns : Finset \u03b1\n\u22a2 fold max \u22a5 (fun x => \u2191(f x) + n) s = fold max \u22a5 (WithBot.some \u2218 f) s + n", "state_after": "no goals"}, {"tactic": "induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right]", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with a s _ ih <;> simp [*, <a>max_add_add_right</a>]", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}, {"full_name": "max_add_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/MinMax.lean", "def_pos": [72, 3], "def_end_pos": [72, 14]}]], "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\u271d : Finset \u03b1\na : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\nc : \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nn : WithBot \u03b2\ns : Finset \u03b1\n\u22a2 fold max \u22a5 (fun x => \u2191(f x) + n) s = fold max \u22a5 (WithBot.some \u2218 f) s + n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_pi_iff", "start": [890, 1], "end": [892, 60], "traced_tactics": [{"tactic": "simp_rw [measurable_iff_comap_le, MeasurableSpace.pi, MeasurableSpace.comap_iSup,\n  MeasurableSpace.comap_comp, Function.comp, iSup_le_iff]", "annotated_tactic": ["simp_rw [<a>measurable_iff_comap_le</a>, <a>MeasurableSpace.pi</a>, <a>MeasurableSpace.comap_iSup</a>,\n    <a>MeasurableSpace.comap_comp</a>, <a>Function.comp</a>, <a>iSup_le_iff</a>]", [{"full_name": "measurable_iff_comap_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 32]}, {"full_name": "MeasurableSpace.pi", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [884, 10], "def_end_pos": [884, 28]}, {"full_name": "MeasurableSpace.comap_iSup", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 19]}, {"full_name": "MeasurableSpace.comap_comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 19]}, {"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": "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ng : \u03b1 \u2192 (a : \u03b4) \u2192 \u03c0 a\n\u22a2 Measurable g \u2194 \u2200 (a : \u03b4), Measurable fun x => g x a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.integral_condCdf", "start": [958, 1], "end": [960, 76], "traced_tactics": [{"tactic": "rw [\u2190 set_integral_condCdf \u03c1 _ MeasurableSet.univ, Measure.restrict_univ]", "annotated_tactic": ["rw [\u2190 <a>set_integral_condCdf</a> \u03c1 _ <a>MeasurableSet.univ</a>, <a>Measure.restrict_univ</a>]", [{"full_name": "ProbabilityTheory.set_integral_condCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [948, 9], "def_end_pos": [948, 29]}, {"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\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 \u222b (a : \u03b1), \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic x))", "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.sign_eq_zero_iff_zero", "start": [1296, 1], "end": [1297, 59], "traced_tactics": [{"tactic": "rw [h, sign_zero]", "annotated_tactic": ["rw [h, <a>sign_zero</a>]", [{"full_name": "Int.sign_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [195, 17], "def_end_pos": [195, 26]}]], "state_before": "a : Int\nh : a = 0\n\u22a2 sign a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.liftOn\u2082'_mk", "start": [815, 11], "end": [817, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_set_univ\u2080", "start": [1060, 1], "end": [1061, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.data_take", "start": [1115, 9], "end": [1115, 96], "traced_tactics": [{"tactic": "rw [take_eq]", "annotated_tactic": ["rw [<a>take_eq</a>]", [{"full_name": "String.take_eq", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [1112, 9], "def_end_pos": [1112, 16]}]], "state_before": "s : String\nn : Nat\n\u22a2 (take s n).data = List.take n s.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.monomial_finsupp_sum_index", "start": [384, 1], "end": [386, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bitwise_or", "start": [221, 1], "end": [235, 32], "traced_tactics": [{"tactic": "funext m n", "annotated_tactic": ["funext m n", []], "state_before": "\u22a2 bitwise or = lor", "state_after": "case h.h\nm n : \u2124\n\u22a2 bitwise or m n = lor 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, cond_true, lor, Nat.ldiff,\n    negSucc.injEq, Bool.true_or, Nat.land]", "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>cond_true</a>, <a>lor</a>, <a>Nat.ldiff</a>,\n      negSucc.injEq, <a>Bool.true_or</a>, <a>Nat.land</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": "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.true_or", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 29]}, {"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]}]], "state_before": "case h.h\nm n : \u2124\n\u22a2 bitwise or m n = lor m n", "state_after": "case h.h.ofNat.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !(x || !y)) m n = Nat.bitwise (fun a b => a && !b) n m\n\ncase h.h.negSucc.ofNat\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !(!x || y)) m n = Nat.bitwise (fun a b => a && !b) m n\n\ncase h.h.negSucc.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !(!x || !y)) m n = m &&& n"}, {"tactic": "rw [Nat.bitwise_swap, Function.swap]", "annotated_tactic": ["rw [<a>Nat.bitwise_swap</a>, <a>Function.swap</a>]", [{"full_name": "Nat.bitwise_swap", "def_path": "Mathlib/Data/Nat/Bitwise.lean", "def_pos": [317, 9], "def_end_pos": [317, 21]}, {"full_name": "Function.swap", "def_path": "Mathlib/Init/Function.lean", "def_pos": [66, 5], "def_end_pos": [66, 9]}]], "state_before": "case h.h.ofNat.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !(x || !y)) m n = Nat.bitwise (fun a b => a && !b) n m", "state_after": "case h.h.ofNat.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun y x => !(x || !y)) n m = Nat.bitwise (fun a b => a && !b) n m"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.h.ofNat.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun y x => !(x || !y)) n m = Nat.bitwise (fun a b => a && !b) n m", "state_after": "case h.h.ofNat.negSucc.e_f\nm n : \u2115\n\u22a2 (fun y x => !(x || !y)) = fun a b => a && !b"}, {"tactic": "funext x y", "annotated_tactic": ["funext x y", []], "state_before": "case h.h.ofNat.negSucc.e_f\nm n : \u2115\n\u22a2 (fun y x => !(x || !y)) = fun a b => a && !b", "state_after": "case h.h.ofNat.negSucc.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!(y || !x)) = (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 (!(y || !x)) = (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 => !(!x || y)) m n = Nat.bitwise (fun a b => a && !b) m n", "state_after": "case h.h.negSucc.ofNat.e_f\nm n : \u2115\n\u22a2 (fun x y => !(!x || y)) = fun a b => a && !b"}, {"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 => !(!x || y)) = fun a b => a && !b", "state_after": "case h.h.negSucc.ofNat.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!(!x || y)) = (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 (!(!x || y)) = (x && !y)", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.h.negSucc.negSucc\nm n : \u2115\n\u22a2 Nat.bitwise (fun x y => !(!x || !y)) m n = m &&& n", "state_after": "case h.h.negSucc.negSucc.e_f\nm n : \u2115\n\u22a2 (fun x y => !(!x || !y)) = and"}, {"tactic": "funext x y", "annotated_tactic": ["funext x y", []], "state_before": "case h.h.negSucc.negSucc.e_f\nm n : \u2115\n\u22a2 (fun x y => !(!x || !y)) = and", "state_after": "case h.h.negSucc.negSucc.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!(!x || !y)) = (x && y)"}, {"tactic": "cases x <;> cases y <;> rfl", "annotated_tactic": ["cases x <;> cases y <;> rfl", []], "state_before": "case h.h.negSucc.negSucc.e_f.h.h\nm n : \u2115\nx y : Bool\n\u22a2 (!(!x || !y)) = (x && y)", "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_iff", "start": [1081, 1], "end": [1083, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.cond_eq_inv_mul_cond_mul", "start": [159, 1], "end": [163, 83], "traced_tactics": [{"tactic": "by_cases ht : \u03bc t = 0", "annotated_tactic": ["by_cases ht : \u03bc t = 0", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t", "state_after": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\nht : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t\n\ncase neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\nht : \u00ac\u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t"}, {"tactic": "simp [cond, ht, Measure.restrict_apply hmt, Or.inr (measure_inter_null_of_null_left s ht)]", "annotated_tactic": ["simp [<a>cond</a>, ht, <a>Measure.restrict_apply</a> hmt, <a>Or.inr</a> (<a>measure_inter_null_of_null_left</a> s ht)]", [{"full_name": "ProbabilityTheory.cond", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [72, 5], "def_end_pos": [72, 9]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"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_inter_null_of_null_left", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [357, 9], "def_end_pos": [357, 40]}]], "state_before": "case pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\nht : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t", "state_after": "no goals"}, {"tactic": "rw [mul_assoc, cond_mul_eq_inter \u03bc hmt ht s, Set.inter_comm, cond_apply _ hms]", "annotated_tactic": ["rw [<a>mul_assoc</a>, <a>cond_mul_eq_inter</a> \u03bc hmt ht s, <a>Set.inter_comm</a>, <a>cond_apply</a> _ hms]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ProbabilityTheory.cond_mul_eq_inter", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [145, 9], "def_end_pos": [145, 26]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [102, 9], "def_end_pos": [102, 19]}]], "state_before": "case neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\nht : \u00ac\u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc 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.sub_add_eq_max", "start": [493, 11], "end": [496, 62], "traced_tactics": [{"tactic": "match a.le_total b with\n| .inl hl => rw [Nat.max_eq_right hl, Nat.sub_eq_zero_iff_le.mpr hl, Nat.zero_add]\n| .inr hr => rw [Nat.max_eq_left hr, Nat.sub_add_cancel hr]", "annotated_tactic": ["match a.le_total b with\n  | .inl hl => rw [<a>Nat.max_eq_right</a> hl, Nat.sub_eq_zero_iff_le.mpr hl, <a>Nat.zero_add</a>]\n  | .inr hr => rw [<a>Nat.max_eq_left</a> hr, <a>Nat.sub_add_cancel</a> hr]", [{"full_name": "Nat.max_eq_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [475, 19], "def_end_pos": [475, 31]}, {"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]}, {"full_name": "Nat.max_eq_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [478, 19], "def_end_pos": [478, 30]}, {"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]}]], "state_before": "a b : Nat\n\u22a2 a - b + b = max a b", "state_after": "no goals"}, {"tactic": "rw [Nat.max_eq_right hl, Nat.sub_eq_zero_iff_le.mpr hl, Nat.zero_add]", "annotated_tactic": ["rw [<a>Nat.max_eq_right</a> hl, Nat.sub_eq_zero_iff_le.mpr hl, <a>Nat.zero_add</a>]", [{"full_name": "Nat.max_eq_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [475, 19], "def_end_pos": [475, 31]}, {"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": "a b : Nat\nhl : a \u2264 b\n\u22a2 a - b + b = max a b", "state_after": "no goals"}, {"tactic": "rw [Nat.max_eq_left hr, Nat.sub_add_cancel hr]", "annotated_tactic": ["rw [<a>Nat.max_eq_left</a> hr, <a>Nat.sub_add_cancel</a> hr]", [{"full_name": "Nat.max_eq_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [478, 19], "def_end_pos": [478, 30]}, {"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]}]], "state_before": "a b : Nat\nhr : b \u2264 a\n\u22a2 a - b + b = max a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.natAbs_valMinAbs_add_le", "start": [1159, 1], "end": [1165, 30], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "n\u271d a\u271d n : \u2115\na b : ZMod n\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)", "state_after": "case zero\nn a\u271d : \u2115\na b : ZMod Nat.zero\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)\n\ncase succ\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)"}, {"tactic": "apply natAbs_min_of_le_div_two n.succ", "annotated_tactic": ["apply <a>natAbs_min_of_le_div_two</a> n.succ", [{"full_name": "ZMod.natAbs_min_of_le_div_two", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1143, 9], "def_end_pos": [1143, 33]}]], "state_before": "case succ\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)", "state_after": "case succ.he\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 \u2191(valMinAbs (a + b)) = \u2191(valMinAbs a + valMinAbs b)\n\ncase succ.hl\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Nat.succ n / 2"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\nn a\u271d : \u2115\na b : ZMod Nat.zero\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)", "state_after": "no goals"}, {"tactic": "simp_rw [Int.cast_add, coe_valMinAbs]", "annotated_tactic": ["simp_rw [<a>Int.cast_add</a>, <a>coe_valMinAbs</a>]", [{"full_name": "Int.cast_add", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [108, 9], "def_end_pos": [108, 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 succ.he\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 \u2191(valMinAbs (a + b)) = \u2191(valMinAbs a + valMinAbs b)", "state_after": "no goals"}, {"tactic": "apply natAbs_valMinAbs_le", "annotated_tactic": ["apply <a>natAbs_valMinAbs_le</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": "case succ.hl\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Nat.succ n / 2", "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.measurableSet_mulSupport", "start": [362, 8], "end": [365, 47], "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\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : MetrizableSpace \u03b2\nhf : StronglyMeasurable f\n\u22a2 MeasurableSet (mulSupport f)", "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\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : MetrizableSpace \u03b2\nhf : StronglyMeasurable f\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\n\u22a2 MeasurableSet (mulSupport f)"}, {"tactic": "exact measurableSet_mulSupport hf.measurable", "annotated_tactic": ["exact <a>measurableSet_mulSupport</a> hf.measurable", [{"full_name": "measurableSet_mulSupport", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [369, 9], "def_end_pos": [369, 33]}]], "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\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : MetrizableSpace \u03b2\nhf : StronglyMeasurable f\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\n\u22a2 MeasurableSet (mulSupport f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.aestronglyMeasurable'_condexpL1", "start": [536, 1], "end": [543, 88], "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\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 : \u03b1 \u2192 F'\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc", "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\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 : \u03b1 \u2192 F'\nhf : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc\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\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 : \u03b1 \u2192 F'\nhf : \u00acIntegrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc"}, {"tactic": "rw [condexpL1_eq hf]", "annotated_tactic": ["rw [<a>condexpL1_eq</a> hf]", [{"full_name": "MeasureTheory.condexpL1_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 9], "def_end_pos": [522, 21]}]], "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\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 : \u03b1 \u2192 F'\nhf : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc", "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\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 : \u03b1 \u2192 F'\nhf : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hf))) \u03bc"}, {"tactic": "exact aestronglyMeasurable'_condexpL1Clm _", "annotated_tactic": ["exact <a>aestronglyMeasurable'_condexpL1Clm</a> _", [{"full_name": "MeasureTheory.aestronglyMeasurable'_condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [465, 9], "def_end_pos": [465, 43]}]], "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\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 : \u03b1 \u2192 F'\nhf : Integrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hf))) \u03bc", "state_after": "no goals"}, {"tactic": "rw [condexpL1_undef hf]", "annotated_tactic": ["rw [<a>condexpL1_undef</a> hf]", [{"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\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 : \u03b1 \u2192 F'\nhf : \u00acIntegrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc", "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\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 : \u03b1 \u2192 F'\nhf : \u00acIntegrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u21910) \u03bc"}, {"tactic": "refine AEStronglyMeasurable'.congr ?_ (coeFn_zero _ _ _).symm", "annotated_tactic": ["refine <a>AEStronglyMeasurable'.congr</a> ?_ (<a>coeFn_zero</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_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\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 : \u03b1 \u2192 F'\nhf : \u00acIntegrable f\n\u22a2 AEStronglyMeasurable' m (\u2191\u21910) \u03bc", "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\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 : \u03b1 \u2192 F'\nhf : \u00acIntegrable f\n\u22a2 AEStronglyMeasurable' m 0 \u03bc"}, {"tactic": "exact StronglyMeasurable.aeStronglyMeasurable' (@stronglyMeasurable_zero _ _ m _ _)", "annotated_tactic": ["exact <a>StronglyMeasurable.aeStronglyMeasurable'</a> (@<a>stronglyMeasurable_zero</a> _ _ m _ _)", [{"full_name": "MeasureTheory.StronglyMeasurable.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [138, 9], "def_end_pos": [138, 49]}, {"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\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 : \u03b1 \u2192 F'\nhf : \u00acIntegrable f\n\u22a2 AEStronglyMeasurable' m 0 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSet.compl", "start": [87, 11], "end": [88, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg", "start": [1760, 1], "end": [1777, 49], "traced_tactics": [{"tactic": "have h_left : \u222b a, f a * g a \u2202\u03bc = \u222b a, \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc := by\n  refine' integral_congr_ae _\n  filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg\n  rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg]", "annotated_tactic": ["have h_left : \u222b a, f a * g a \u2202\u03bc = \u222b a, \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc := by\n    refine' <a>integral_congr_ae</a> _\n    filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg\n    rw [<a>Real.norm_of_nonneg</a> hxf, <a>Real.norm_of_nonneg</a> hxg]", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 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": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}]], "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "have h_right_f : \u222b a, f a ^ p \u2202\u03bc = \u222b a, \u2016f a\u2016 ^ p \u2202\u03bc := by\n  refine' integral_congr_ae _\n  filter_upwards [hf_nonneg] with x hxf\n  rw [Real.norm_of_nonneg hxf]", "annotated_tactic": ["have h_right_f : \u222b a, f a ^ p \u2202\u03bc = \u222b a, \u2016f a\u2016 ^ p \u2202\u03bc := by\n    refine' <a>integral_congr_ae</a> _\n    filter_upwards [hf_nonneg] with x hxf\n    rw [<a>Real.norm_of_nonneg</a> hxf]", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"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": "\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "have h_right_g : \u222b a, g a ^ q \u2202\u03bc = \u222b a, \u2016g a\u2016 ^ q \u2202\u03bc := by\n  refine' integral_congr_ae _\n  filter_upwards [hg_nonneg] with x hxg\n  rw [Real.norm_of_nonneg hxg]", "annotated_tactic": ["have h_right_g : \u222b a, g a ^ q \u2202\u03bc = \u222b a, \u2016g a\u2016 ^ q \u2202\u03bc := by\n    refine' <a>integral_congr_ae</a> _\n    filter_upwards [hg_nonneg] with x hxg\n    rw [<a>Real.norm_of_nonneg</a> hxg]", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"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": "\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\nh_right_g : \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "rw [h_left, h_right_f, h_right_g]", "annotated_tactic": ["rw [h_left, h_right_f, h_right_g]", []], "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\nh_right_g : \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\nh_right_g : \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc \u2264 (\u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "exact integral_mul_norm_le_Lp_mul_Lq hpq hf hg", "annotated_tactic": ["exact <a>integral_mul_norm_le_Lp_mul_Lq</a> hpq hf hg", [{"full_name": "MeasureTheory.integral_mul_norm_le_Lp_mul_Lq", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1711, 9], "def_end_pos": [1711, 39]}]], "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\nh_right_g : \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc \u2264 (\u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc) ^ (1 / q)", "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": "\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\n\u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\n\u22a2 (fun a => f a * g a) =\u1d50[\u03bc] fun a => \u2016f a\u2016 * \u2016g a\u2016"}, {"tactic": "filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg", "annotated_tactic": ["filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg", []], "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\n\u22a2 (fun a => f a * g a) =\u1d50[\u03bc] fun a => \u2016f a\u2016 * \u2016g a\u2016", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nx : \u03b1\nhxf : OfNat.ofNat 0 x \u2264 f x\nhxg : OfNat.ofNat 0 x \u2264 g x\n\u22a2 f x * g x = \u2016f x\u2016 * \u2016g x\u2016"}, {"tactic": "rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg]", "annotated_tactic": ["rw [<a>Real.norm_of_nonneg</a> hxf, <a>Real.norm_of_nonneg</a> hxg]", [{"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": "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\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nx : \u03b1\nhxf : OfNat.ofNat 0 x \u2264 f x\nhxg : OfNat.ofNat 0 x \u2264 g x\n\u22a2 f x * g x = \u2016f x\u2016 * \u2016g x\u2016", "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": "\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\n\u22a2 (fun a => f a ^ p) =\u1d50[\u03bc] fun a => \u2016f a\u2016 ^ p"}, {"tactic": "filter_upwards [hf_nonneg] with x hxf", "annotated_tactic": ["filter_upwards [hf_nonneg] with x hxf", []], "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\n\u22a2 (fun a => f a ^ p) =\u1d50[\u03bc] fun a => \u2016f a\u2016 ^ p", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nx : \u03b1\nhxf : OfNat.ofNat 0 x \u2264 f x\n\u22a2 f x ^ p = \u2016f x\u2016 ^ p"}, {"tactic": "rw [Real.norm_of_nonneg hxf]", "annotated_tactic": ["rw [<a>Real.norm_of_nonneg</a> hxf]", [{"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\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nx : \u03b1\nhxf : OfNat.ofNat 0 x \u2264 f x\n\u22a2 f x ^ p = \u2016f x\u2016 ^ p", "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": "\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\n\u22a2 (fun a => g a ^ q) =\u1d50[\u03bc] fun a => \u2016g a\u2016 ^ q"}, {"tactic": "filter_upwards [hg_nonneg] with x hxg", "annotated_tactic": ["filter_upwards [hg_nonneg] with x hxg", []], "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\n\u22a2 (fun a => g a ^ q) =\u1d50[\u03bc] fun a => \u2016g a\u2016 ^ q", "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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\nx : \u03b1\nhxg : OfNat.ofNat 0 x \u2264 g x\n\u22a2 g x ^ q = \u2016g x\u2016 ^ q"}, {"tactic": "rw [Real.norm_of_nonneg hxg]", "annotated_tactic": ["rw [<a>Real.norm_of_nonneg</a> hxg]", [{"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\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\u271d : \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\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\nhf_nonneg : 0 \u2264\u1d50[\u03bc] f\nhg_nonneg : 0 \u2264\u1d50[\u03bc] g\nhf : Mem\u2112p f (ENNReal.ofReal p)\nhg : Mem\u2112p g (ENNReal.ofReal q)\nh_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc\nh_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc\nx : \u03b1\nhxg : OfNat.ofNat 0 x \u2264 g x\n\u22a2 g x ^ q = \u2016g x\u2016 ^ q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.mem_sup", "start": [1869, 1], "end": [1873, 25], "traced_tactics": [{"tactic": "change _ \u2194 \u2203 v \u2208 s, x \u2208 (f v).val", "annotated_tactic": ["change _ \u2194 \u2203 v \u2208 s, x \u2208 (f v).<a>val</a>", [{"full_name": "Finset.val", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [140, 3], "def_end_pos": [140, 6]}]], "state_before": "F : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nx : \u03b2\n\u22a2 x \u2208 sup s f \u2194 \u2203 v, v \u2208 s \u2227 x \u2208 f v", "state_after": "F : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nx : \u03b2\n\u22a2 x \u2208 sup s f \u2194 \u2203 v, v \u2208 s \u2227 x \u2208 (f v).val"}, {"tactic": "rw [\u2190 Multiset.mem_sup, \u2190 Multiset.mem_toFinset, sup_toFinset]", "annotated_tactic": ["rw [\u2190 <a>Multiset.mem_sup</a>, \u2190 <a>Multiset.mem_toFinset</a>, <a>sup_toFinset</a>]", [{"full_name": "Multiset.mem_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 16]}, {"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3200, 9], "def_end_pos": [3200, 21]}, {"full_name": "Finset.sup_toFinset", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [208, 9], "def_end_pos": [208, 21]}]], "state_before": "F : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nx : \u03b2\n\u22a2 x \u2208 sup s f \u2194 \u2203 v, v \u2208 s \u2227 x \u2208 (f v).val", "state_after": "F : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nx : \u03b2\n\u22a2 x \u2208 sup s f \u2194 x \u2208 sup s fun x => Multiset.toFinset (f x).val"}, {"tactic": "simp_rw [val_toFinset]", "annotated_tactic": ["simp_rw [<a>val_toFinset</a>]", [{"full_name": "Finset.val_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3284, 9], "def_end_pos": [3284, 21]}]], "state_before": "F : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nx : \u03b2\n\u22a2 x \u2208 sup s f \u2194 x \u2208 sup s fun x => Multiset.toFinset (f x).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.haarMeasure_unique", "start": [692, 1], "end": [697, 40], "traced_tactics": [{"tactic": "rw [haarMeasure_self, div_one]", "annotated_tactic": ["rw [<a>haarMeasure_self</a>, <a>div_one</a>]", [{"full_name": "MeasureTheory.Measure.haarMeasure_self", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [632, 9], "def_end_pos": [632, 25]}, {"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}]], "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 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK\u2080 : PositiveCompacts G\n\u22a2 (\u2191\u2191\u03bc \u2191K\u2080 / \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080) \u2022 haarMeasure K\u2080 = \u2191\u2191\u03bc \u2191K\u2080 \u2022 haarMeasure K\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.range_orderEmbOfFin", "start": [183, 1], "end": [188, 65], "traced_tactics": [{"tactic": "simp only [orderEmbOfFin, Set.range_comp ((\u2191) : _ \u2192 \u03b1) (s.orderIsoOfFin h),\nRelEmbedding.coe_trans, Set.image_univ, Finset.orderEmbOfFin, RelIso.range_eq,\n  OrderEmbedding.subtype_apply, OrderIso.coe_toOrderEmbedding, eq_self_iff_true,\n  Subtype.range_coe_subtype, Finset.setOf_mem, Finset.coe_inj]", "annotated_tactic": ["simp only [<a>orderEmbOfFin</a>, <a>Set.range_comp</a> ((\u2191) : _ \u2192 \u03b1) (s.orderIsoOfFin h),\n  <a>RelEmbedding.coe_trans</a>, <a>Set.image_univ</a>, <a>Finset.orderEmbOfFin</a>, <a>RelIso.range_eq</a>,\n    <a>OrderEmbedding.subtype_apply</a>, <a>OrderIso.coe_toOrderEmbedding</a>, <a>eq_self_iff_true</a>,\n    <a>Subtype.range_coe_subtype</a>, <a>Finset.setOf_mem</a>, <a>Finset.coe_inj</a>]", [{"full_name": "Finset.orderEmbOfFin", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [155, 5], "def_end_pos": [155, 18]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "RelEmbedding.coe_trans", "def_path": "Mathlib/Order/RelIso/Basic.lean", "def_pos": [311, 9], "def_end_pos": [311, 18]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}, {"full_name": "Finset.orderEmbOfFin", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [155, 5], "def_end_pos": [155, 18]}, {"full_name": "RelIso.range_eq", "def_path": "Mathlib/Order/RelIso/Set.lean", "def_pos": [47, 9], "def_end_pos": [47, 17]}, {"full_name": "OrderEmbedding.subtype_apply", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [729, 3], "def_end_pos": [729, 47]}, {"full_name": "OrderIso.coe_toOrderEmbedding", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 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": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 26]}, {"full_name": "Finset.setOf_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 18]}, {"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nk : \u2115\nh : card s = k\n\u22a2 Set.range \u2191(orderEmbOfFin s h) = \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.X_injective", "start": [293, 1], "end": [294, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.disjoint_compl_right_iff_subset", "start": [1774, 1], "end": [1775, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/AssocList.lean", "full_name": "Std.AssocList.modify_toList", "start": [195, 9], "end": [200, 49], "traced_tactics": [{"tactic": "simp [cond]", "annotated_tactic": ["simp [<a>cond</a>]", [{"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\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : BEq \u03b1\na : \u03b1\nl : AssocList \u03b1 \u03b2\n\u22a2 toList (modify a f l) =\n    List.replaceF\n      (fun x =>\n        match x with\n        | (k, v) => bif k == a then some (a, f k v) else none)\n      (toList l)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : BEq \u03b1\na : \u03b1\nl : AssocList \u03b1 \u03b2\n\u22a2 toList (modify a f l) =\n    List.replaceF\n      (fun x =>\n        match x.fst == a with\n        | true => some (a, f x.fst x.snd)\n        | false => none)\n      (toList l)"}, {"tactic": "induction l with simp [List.replaceF]\n| cons k v es ih => cases k == a <;> simp [ih]", "annotated_tactic": ["induction l with simp [<a>List.replaceF</a>]\n  | <a>cons</a> k v es ih => cases k == a <;> simp [ih]", [{"full_name": "List.replaceF", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [392, 13], "def_end_pos": [392, 21]}, {"full_name": "Std.AssocList.cons", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [20, 5], "def_end_pos": [20, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : BEq \u03b1\na : \u03b1\nl : AssocList \u03b1 \u03b2\n\u22a2 toList (modify a f l) =\n    List.replaceF\n      (fun x =>\n        match x.fst == a with\n        | true => some (a, f x.fst x.snd)\n        | false => none)\n      (toList l)", "state_after": "no goals"}, {"tactic": "cases k == a <;> simp [ih]", "annotated_tactic": ["cases k == a <;> simp [ih]", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : BEq \u03b1\na k : \u03b1\nv : \u03b2\nes : AssocList \u03b1 \u03b2\nih :\n  toList (modify a f es) =\n    List.replaceF\n      (fun x =>\n        match x.fst == a with\n        | true => some (a, f x.fst x.snd)\n        | false => none)\n      (toList es)\n\u22a2 toList\n      (match k == a with\n      | true => cons a (f k v) es\n      | false => cons k v (modify a f es)) =\n    match\n      match k == a with\n      | true => some (a, f k v)\n      | false => none with\n    | none =>\n      (k, v) ::\n        List.replaceF\n          (fun x =>\n            match x.fst == a with\n            | true => some (a, f x.fst x.snd)\n            | false => none)\n          (toList es)\n    | some a => a :: toList es", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.Fix.dest_mk", "start": [339, 1], "end": [345, 20], "traced_tactics": [{"tactic": "unfold Fix.dest", "annotated_tactic": ["unfold <a>Fix.dest</a>", [{"full_name": "QPF.Fix.dest", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [274, 5], "def_end_pos": [274, 13]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx : F (Fix F)\n\u22a2 dest (mk x) = x", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx : F (Fix F)\n\u22a2 rec (Functor.map 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": "QPF.Fix.rec_eq", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 19]}, {"full_name": "QPF.Fix.dest", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [274, 5], "def_end_pos": [274, 13]}, {"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\nx : F (Fix F)\n\u22a2 rec (Functor.map mk) (mk x) = x", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = x"}, {"tactic": "conv =>\n  rhs\n  rw [\u2190 id_map x]", "annotated_tactic": ["conv =>\n    rhs\n    rw [\u2190 <a>id_map</a> x]", [{"full_name": "QPF.id_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [71, 9], "def_end_pos": [71, 15]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = x", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = id <$> x"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = id <$> x", "state_after": "case e_a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx\u271d : F (Fix F)\nx : Fix F\n\u22a2 (mk \u2218 dest) x = id x"}, {"tactic": "apply Fix.mk_dest", "annotated_tactic": ["apply <a>Fix.mk_dest</a>", [{"full_name": "QPF.Fix.mk_dest", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [330, 9], "def_end_pos": [330, 20]}]], "state_before": "case e_a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nx\u271d : F (Fix F)\nx : Fix F\n\u22a2 (mk \u2218 dest) x = id x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snormEssSup_zero", "start": [195, 1], "end": [197, 25], "traced_tactics": [{"tactic": "simp_rw [snormEssSup, Pi.zero_apply, nnnorm_zero, ENNReal.coe_zero, \u2190 ENNReal.bot_eq_zero]", "annotated_tactic": ["simp_rw [<a>snormEssSup</a>, <a>Pi.zero_apply</a>, <a>nnnorm_zero</a>, <a>ENNReal.coe_zero</a>, \u2190 <a>ENNReal.bot_eq_zero</a>]", [{"full_name": "MeasureTheory.snormEssSup", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [78, 5], "def_end_pos": [78, 16]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "nnnorm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [932, 30], "def_end_pos": [932, 41]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "ENNReal.bot_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [682, 9], "def_end_pos": [682, 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\n\u22a2 snormEssSup 0 \u03bc = 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\n\u22a2 essSup (fun x => \u22a5) \u03bc = \u22a5"}, {"tactic": "exact essSup_const_bot", "annotated_tactic": ["exact <a>essSup_const_bot</a>", [{"full_name": "essSup_const_bot", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [173, 9], "def_end_pos": [173, 25]}]], "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 essSup (fun x => \u22a5) \u03bc = \u22a5", "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_mkMetric", "start": [530, 1], "end": [533, 59], "traced_tactics": [{"tactic": "rw [\u2190 toOuterMeasure_le, mkMetric_toOuterMeasure]", "annotated_tactic": ["rw [\u2190 <a>toOuterMeasure_le</a>, <a>mkMetric_toOuterMeasure</a>]", [{"full_name": "MeasureTheory.Measure.toOuterMeasure_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [976, 9], "def_end_pos": [976, 26]}, {"full_name": "MeasureTheory.Measure.mkMetric_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [464, 9], "def_end_pos": [464, 32]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u03bc : Measure X\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nh : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s)\n\u22a2 \u03bc \u2264 mkMetric m", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u03bc : Measure X\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nh : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s)\n\u22a2 \u2191\u03bc \u2264 OuterMeasure.mkMetric m"}, {"tactic": "exact OuterMeasure.le_mkMetric m \u03bc.toOuterMeasure \u03b5 h\u2080 h", "annotated_tactic": ["exact <a>OuterMeasure.le_mkMetric</a> m \u03bc.toOuterMeasure \u03b5 h\u2080 h", [{"full_name": "MeasureTheory.OuterMeasure.le_mkMetric", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [422, 9], "def_end_pos": [422, 20]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u03bc : Measure X\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nh : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s)\n\u22a2 \u2191\u03bc \u2264 OuterMeasure.mkMetric m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/NFA.lean", "full_name": "NFA.pumping_lemma", "start": [128, 1], "end": [134, 38], "traced_tactics": [{"tactic": "rw [\u2190 toDFA_correct] at hx \u22a2", "annotated_tactic": ["rw [\u2190 <a>toDFA_correct</a>] at hx \u22a2", [{"full_name": "NFA.toDFA_correct", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx : x \u2208 accepts M\nhlen : Fintype.card (Set \u03c3) \u2264 List.length x\n\u22a2 \u2203 a b c,\n    x = a ++ b ++ c \u2227 List.length a + List.length b \u2264 Fintype.card (Set \u03c3) \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 accepts M", "state_after": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx : x \u2208 DFA.accepts (toDFA M)\nhlen : Fintype.card (Set \u03c3) \u2264 List.length x\n\u22a2 \u2203 a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card (Set \u03c3) \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 DFA.accepts (toDFA M)"}, {"tactic": "exact M.toDFA.pumping_lemma hx hlen", "annotated_tactic": ["exact M.toDFA.pumping_lemma hx hlen", []], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx : x \u2208 DFA.accepts (toDFA M)\nhlen : Fintype.card (Set \u03c3) \u2264 List.length x\n\u22a2 \u2203 a b c,\n    x = a ++ b ++ c \u2227\n      List.length a + List.length b \u2264 Fintype.card (Set \u03c3) \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 DFA.accepts (toDFA M)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.union_mul_inter_subset", "start": [770, 1], "end": [771, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.disjoint_ordT5Nhd", "start": [197, 1], "end": [235, 23], "traced_tactics": [{"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": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\n\u22a2 Disjoint (ordT5Nhd s t) (ordT5Nhd t s)", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\n\u22a2 ordT5Nhd s t \u2293 ordT5Nhd t s \u2264 \u22a5"}, {"tactic": "rintro x \u27e8hx\u2081, hx\u2082\u27e9", "annotated_tactic": ["rintro x \u27e8hx\u2081, hx\u2082\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\n\u22a2 ordT5Nhd s t \u2293 ordT5Nhd t s \u2264 \u22a5", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2081 : x \u2208 ordT5Nhd s t\nhx\u2082 : x \u2208 ordT5Nhd t s\n\u22a2 x \u2208 \u22a5"}, {"tactic": "rcases mem_iUnion\u2082.1 hx\u2081 with \u27e8a, has, ha\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion\u2082</a>.1 hx\u2081 with \u27e8a, has, ha\u27e9", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2081 : x \u2208 ordT5Nhd s t\nhx\u2082 : x \u2208 ordT5Nhd t s\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2081 : x \u2208 ordT5Nhd s t\nhx\u2082 : x \u2208 ordT5Nhd t s\na : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\n\u22a2 x \u2208 \u22a5"}, {"tactic": "clear hx\u2081", "annotated_tactic": ["clear hx\u2081", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2081 : x \u2208 ordT5Nhd s t\nhx\u2082 : x \u2208 ordT5Nhd t s\na : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2082 : x \u2208 ordT5Nhd t s\na : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\n\u22a2 x \u2208 \u22a5"}, {"tactic": "rcases mem_iUnion\u2082.1 hx\u2082 with \u27e8b, hbt, hb\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion\u2082</a>.1 hx\u2082 with \u27e8b, hbt, hb\u27e9", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2082 : x \u2208 ordT5Nhd t s\na : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2082 : x \u2208 ordT5Nhd t s\na : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\nb : \u03b1\nhbt : b \u2208 t\nhb : x \u2208 ordConnectedComponent (s\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet t s))\u1d9c) b\n\u22a2 x \u2208 \u22a5"}, {"tactic": "clear hx\u2082", "annotated_tactic": ["clear hx\u2082", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x : \u03b1\nhx\u2082 : x \u2208 ordT5Nhd t s\na : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\nb : \u03b1\nhbt : b \u2208 t\nhb : x \u2208 ordConnectedComponent (s\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet t s))\u1d9c) b\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\nb : \u03b1\nhbt : b \u2208 t\nhb : x \u2208 ordConnectedComponent (s\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet t s))\u1d9c) b\n\u22a2 x \u2208 \u22a5"}, {"tactic": "rw [mem_ordConnectedComponent, subset_inter_iff] at ha hb", "annotated_tactic": ["rw [<a>mem_ordConnectedComponent</a>, <a>subset_inter_iff</a>] at ha hb", [{"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.subset_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [977, 9], "def_end_pos": [977, 25]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a\nb : \u03b1\nhbt : b \u2208 t\nhb : x \u2208 ordConnectedComponent (s\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet t s))\u1d9c) b\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\n\u22a2 x \u2208 \u22a5"}, {"tactic": "cases' le_total a b with hab hab", "annotated_tactic": ["cases' <a>le_total</a> a b with hab hab", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhab : a \u2264 b\n\u22a2 x \u2208 \u22a5\n\ncase intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhab : b \u2264 a\n\u22a2 x \u2208 \u22a5"}, {"tactic": "on_goal 2 => swap_var a \u2194 b, s \u2194 t, ha \u2194 hb, has \u2194 hbt", "annotated_tactic": ["on_goal 2 => swap_var a \u2194 b, s \u2194 t, ha \u2194 hb, has \u2194 hbt", []], "state_before": "case intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhab : a \u2264 b\n\u22a2 x \u2208 \u22a5\n\ncase intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhab : b \u2264 a\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhab : a \u2264 b\n\u22a2 x \u2208 \u22a5\n\ncase intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\na : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhab : a \u2264 b\n\u22a2 x \u2208 \u22a5"}, {"tactic": "swap_var a \u2194 b, s \u2194 t, ha \u2194 hb, has \u2194 hbt", "annotated_tactic": ["swap_var a \u2194 b, s \u2194 t, ha \u2194 hb, has \u2194 hbt", []], "state_before": "case intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx\u271d y z x a : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nb : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhab : b \u2264 a\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\na : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhab : a \u2264 b\n\u22a2 x \u2208 \u22a5"}, {"tactic": "cases' ha with ha ha'", "annotated_tactic": ["cases' ha with ha ha'", []], "state_before": "case intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\na : \u03b1\nhas : a \u2208 s\nha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhab : a \u2264 b\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 x \u2208 \u22a5"}, {"tactic": "cases' hb with hb hb'", "annotated_tactic": ["cases' hb with hb hb'", []], "state_before": "case intro.intro.intro.intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\nhb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\n\u22a2 x \u2208 \u22a5"}, {"tactic": "have hsub : [[a, b]] \u2286 (ordSeparatingSet s t).ordConnectedSection\u1d9c := by\n  rw [ordSeparatingSet_comm, uIcc_comm] at hb'\n  calc\n    [[a, b]] \u2286 [[a, x]] \u222a [[x, b]] := uIcc_subset_uIcc_union_uIcc\n    _ \u2286 (ordSeparatingSet s t).ordConnectedSection\u1d9c := union_subset ha' hb'", "annotated_tactic": ["have hsub : [[a, b]] \u2286 (<a>ordSeparatingSet</a> s t).<a>ordConnectedSection</a>\u1d9c := by\n      rw [<a>ordSeparatingSet_comm</a>, <a>uIcc_comm</a>] at hb'\n      calc\n        [[a, b]] \u2286 [[a, x]] \u222a [[x, b]] := <a>uIcc_subset_uIcc_union_uIcc</a>\n        _ \u2286 (<a>ordSeparatingSet</a> s t).<a>ordConnectedSection</a>\u1d9c := <a>union_subset</a> ha' hb'", [{"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"full_name": "Set.ordConnectedSection", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [137, 5], "def_end_pos": [137, 24]}, {"full_name": "Set.ordSeparatingSet_comm", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [172, 9], "def_end_pos": [172, 30]}, {"full_name": "Set.uIcc_comm", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Set.uIcc_subset_uIcc_union_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [249, 7], "def_end_pos": [249, 34]}, {"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"full_name": "Set.ordConnectedSection", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [137, 5], "def_end_pos": [137, 24]}, {"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.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 x \u2208 \u22a5"}, {"tactic": "clear ha' hb'", "annotated_tactic": ["clear ha' hb'", []], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 x \u2208 \u22a5"}, {"tactic": "cases' le_total x a with hxa hax", "annotated_tactic": ["cases' <a>le_total</a> x a with hxa hax", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhxa : x \u2264 a\n\u22a2 x \u2208 \u22a5\n\ncase intro.intro.intro.intro.intro.inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\n\u22a2 x \u2208 \u22a5"}, {"tactic": "cases' le_total b x with hbx hxb", "annotated_tactic": ["cases' <a>le_total</a> b x with hbx hxb", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhbx : b \u2264 x\n\u22a2 x \u2208 \u22a5\n\ncase intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\n\u22a2 x \u2208 \u22a5"}, {"tactic": "have h' : x \u2208 ordSeparatingSet s t := \u27e8mem_iUnion\u2082.2 \u27e8a, has, ha\u27e9, mem_iUnion\u2082.2 \u27e8b, hbt, hb\u27e9\u27e9", "annotated_tactic": ["have h' : x \u2208 <a>ordSeparatingSet</a> s t := \u27e8<a>mem_iUnion\u2082</a>.2 \u27e8a, has, ha\u27e9, <a>mem_iUnion\u2082</a>.2 \u27e8b, hbt, hb\u27e9\u27e9", [{"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}, {"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\n\u22a2 x \u2208 \u22a5"}, {"tactic": "suffices ordConnectedComponent (ordSeparatingSet s t) x \u2286 [[a, b]] from\n  hsub (this <| ordConnectedProj_mem_ordConnectedComponent _ \u27e8x, h'\u27e9) (mem_range_self _)", "annotated_tactic": ["suffices <a>ordConnectedComponent</a> (<a>ordSeparatingSet</a> s t) x \u2286 [[a, b]] from\n      hsub (this <| <a>ordConnectedProj_mem_ordConnectedComponent</a> _ \u27e8x, h'\u27e9) (<a>mem_range_self</a> _)", [{"full_name": "Set.ordConnectedComponent", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [28, 5], "def_end_pos": [28, 26]}, {"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"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_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.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\n\u22a2 x \u2208 \u22a5", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\n\u22a2 ordConnectedComponent (ordSeparatingSet s t) x \u2286 [[a, b]]"}, {"tactic": "rintro y (hy : [[x, y]] \u2286 ordSeparatingSet s t)", "annotated_tactic": ["rintro y (hy : [[x, y]] \u2286 <a>ordSeparatingSet</a> s t)", [{"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\n\u22a2 ordConnectedComponent (ordSeparatingSet s t) x \u2286 [[a, b]]", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\n\u22a2 y \u2208 [[a, b]]"}, {"tactic": "rw [uIcc_of_le hab, mem_Icc, \u2190 not_lt, \u2190 not_lt]", "annotated_tactic": ["rw [<a>uIcc_of_le</a> hab, <a>mem_Icc</a>, \u2190 <a>not_lt</a>, \u2190 <a>not_lt</a>]", [{"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": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\n\u22a2 y \u2208 [[a, b]]", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\n\u22a2 \u00acy < a \u2227 \u00acb < y"}, {"tactic": "have sol1 := fun (hya : y < a) =>\n    (disjoint_left (t := ordSeparatingSet s t)).1 disjoint_left_ordSeparatingSet has\n      (hy <| Icc_subset_uIcc' \u27e8hya.le, hax\u27e9)", "annotated_tactic": ["have sol1 := fun (hya : y < a) =>\n        (<a>disjoint_left</a> (t := <a>ordSeparatingSet</a> s t)).1 <a>disjoint_left_ordSeparatingSet</a> has\n          (hy <| <a>Icc_subset_uIcc'</a> \u27e8hya.le, hax\u27e9)", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}, {"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"full_name": "Set.disjoint_left_ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [176, 9], "def_end_pos": [176, 39]}, {"full_name": "Set.Icc_subset_uIcc'", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [94, 7], "def_end_pos": [94, 23]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\n\u22a2 \u00acy < a \u2227 \u00acb < y", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\nsol1 : y < a \u2192 False\n\u22a2 \u00acy < a \u2227 \u00acb < y"}, {"tactic": "have sol2 := fun (hby : b < y) =>\n    (disjoint_left (t := ordSeparatingSet s t)).1 disjoint_right_ordSeparatingSet hbt\n      (hy <| Icc_subset_uIcc \u27e8hxb, hby.le\u27e9)", "annotated_tactic": ["have sol2 := fun (hby : b < y) =>\n        (<a>disjoint_left</a> (t := <a>ordSeparatingSet</a> s t)).1 <a>disjoint_right_ordSeparatingSet</a> hbt\n          (hy <| <a>Icc_subset_uIcc</a> \u27e8hxb, hby.le\u27e9)", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}, {"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"full_name": "Set.disjoint_right_ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [182, 9], "def_end_pos": [182, 40]}, {"full_name": "Set.Icc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [92, 7], "def_end_pos": [92, 22]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\nsol1 : y < a \u2192 False\n\u22a2 \u00acy < a \u2227 \u00acb < y", "state_after": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\nsol1 : y < a \u2192 False\nsol2 : b < y \u2192 False\n\u22a2 \u00acy < a \u2227 \u00acb < y"}, {"tactic": "exact \u27e8sol1, sol2\u27e9", "annotated_tactic": ["exact \u27e8sol1, sol2\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y\u271d z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhxb : x \u2264 b\nh' : x \u2208 ordSeparatingSet s t\ny : \u03b1\nhy : [[x, y]] \u2286 ordSeparatingSet s t\nsol1 : y < a \u2192 False\nsol2 : b < y \u2192 False\n\u22a2 \u00acy < a \u2227 \u00acb < y", "state_after": "no goals"}, {"tactic": "rw [ordSeparatingSet_comm, uIcc_comm] at hb'", "annotated_tactic": ["rw [<a>ordSeparatingSet_comm</a>, <a>uIcc_comm</a>] at hb'", [{"full_name": "Set.ordSeparatingSet_comm", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [172, 9], "def_end_pos": [172, 30]}, {"full_name": "Set.uIcc_comm", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c\n\u22a2 [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[x, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c"}, {"tactic": "calc\n  [[a, b]] \u2286 [[a, x]] \u222a [[x, b]] := uIcc_subset_uIcc_union_uIcc\n  _ \u2286 (ordSeparatingSet s t).ordConnectedSection\u1d9c := union_subset ha' hb'", "annotated_tactic": ["calc\n        [[a, b]] \u2286 [[a, x]] \u222a [[x, b]] := <a>uIcc_subset_uIcc_union_uIcc</a>\n        _ \u2286 (<a>ordSeparatingSet</a> s t).<a>ordConnectedSection</a>\u1d9c := <a>union_subset</a> ha' hb'", [{"full_name": "Set.uIcc_subset_uIcc_union_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [249, 7], "def_end_pos": [249, 34]}, {"full_name": "Set.ordSeparatingSet", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [168, 5], "def_end_pos": [168, 21]}, {"full_name": "Set.ordConnectedSection", "def_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "def_pos": [137, 5], "def_end_pos": [137, 24]}, {"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 : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhb' : [[x, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\n\u22a2 [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c", "state_after": "no goals"}, {"tactic": "exact hb (Icc_subset_uIcc' \u27e8hxa, hab\u27e9) has", "annotated_tactic": ["exact hb (<a>Icc_subset_uIcc'</a> \u27e8hxa, hab\u27e9) has", [{"full_name": "Set.Icc_subset_uIcc'", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [94, 7], "def_end_pos": [94, 23]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhxa : x \u2264 a\n\u22a2 x \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "exact ha (Icc_subset_uIcc \u27e8hab, hbx\u27e9) hbt", "annotated_tactic": ["exact ha (<a>Icc_subset_uIcc</a> \u27e8hab, hbx\u27e9) hbt", [{"full_name": "Set.Icc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [92, 7], "def_end_pos": [92, 22]}]], "state_before": "case intro.intro.intro.intro.intro.inr.intro.intro.inr.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nt s : Set \u03b1\nx\u271d y z x b : \u03b1\nhbt : b \u2208 t\na : \u03b1\nhas : a \u2208 s\nhab : a \u2264 b\nha : [[a, x]] \u2286 t\u1d9c\nhb : [[b, x]] \u2286 s\u1d9c\nhsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c\nhax : a \u2264 x\nhbx : b \u2264 x\n\u22a2 x \u2208 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.Finite.countable", "start": [235, 1], "end": [236, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProd_eq_tsum_compProd", "start": [478, 1], "end": [481, 87], "traced_tactics": [{"tactic": "simp_rw [compProd_apply_eq_compProdFun _ _ _ hs]", "annotated_tactic": ["simp_rw [<a>compProd_apply_eq_compProdFun</a> _ _ _ hs]", [{"full_name": "ProbabilityTheory.kernel.compProd_apply_eq_compProdFun", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [216, 9], "def_end_pos": [216, 38]}]], "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 = \u2211' (n : \u2115) (m : \u2115), \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 seq \u03b7 m) 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 compProdFun \u03ba \u03b7 a s = \u2211' (n : \u2115) (m : \u2115), compProdFun (seq \u03ba n) (seq \u03b7 m) a s"}, {"tactic": "exact compProdFun_eq_tsum \u03ba \u03b7 a hs", "annotated_tactic": ["exact <a>compProdFun_eq_tsum</a> \u03ba \u03b7 a hs", [{"full_name": "ProbabilityTheory.kernel.compProdFun_eq_tsum", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [151, 9], "def_end_pos": [151, 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\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 = \u2211' (n : \u2115) (m : \u2115), compProdFun (seq \u03ba n) (seq \u03b7 m) a s", "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.reverse_sublist", "start": [435, 9], "end": [436, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.continuous_mass", "start": [471, 1], "end": [472, 71], "traced_tactics": [{"tactic": "simp_rw [\u2190 testAgainstNN_one]", "annotated_tactic": ["simp_rw [\u2190 <a>testAgainstNN_one</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_one", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [348, 9], "def_end_pos": [348, 26]}]], "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\u22a2 Continuous fun \u03bc => mass \u03bc", "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\u22a2 Continuous fun \u03bc => testAgainstNN \u03bc 1"}, {"tactic": "exact continuous_testAgainstNN_eval 1", "annotated_tactic": ["exact <a>continuous_testAgainstNN_eval</a> 1", [{"full_name": "MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [460, 9], "def_end_pos": [460, 38]}]], "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\u22a2 Continuous fun \u03bc => testAgainstNN \u03bc 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_eq_preimage_of_inverse", "start": [427, 1], "end": [430, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.Nonempty.inv", "start": [251, 1], "end": [252, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.mul", "start": [601, 1], "end": [605, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure", "start": [456, 1], "end": [469, 45], "traced_tactics": [{"tactic": "obtain \u27e8g, gpos, gmeas, hg\u27e9 :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x : \u03b1, 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b x : \u03b1, \u2191(g x) \u2202\u03bc < 1 :=\n  exists_pos_lintegral_lt_of_sigmaFinite \u03bc one_ne_zero", "annotated_tactic": ["obtain \u27e8g, gpos, gmeas, hg\u27e9 :\n    \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x : \u03b1, 0 < g x) \u2227 <a>Measurable</a> g \u2227 \u222b\u207b x : \u03b1, \u2191(g x) \u2202\u03bc < 1 :=\n    <a>exists_pos_lintegral_lt_of_sigmaFinite</a> \u03bc <a>one_ne_zero</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1576, 9], "def_end_pos": [1576, 47]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd"}, {"tactic": "refine' \u27e8\u03bc.withDensity fun x => g x, isFiniteMeasure_withDensity hg.ne_top, _\u27e9", "annotated_tactic": ["refine' \u27e8\u03bc.withDensity fun x => g x, <a>isFiniteMeasure_withDensity</a> hg.ne_top, _\u27e9", [{"full_name": "MeasureTheory.isFiniteMeasure_withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [111, 9], "def_end_pos": [111, 36]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u03bc \u226a withDensity \u03bc fun x => \u2191(g x)"}, {"tactic": "have : \u03bc = (\u03bc.withDensity fun x => g x).withDensity fun x => (g x)\u207b\u00b9 := by\n  have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 := by\n    ext1 x\n    exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top\n  rw [\u2190 withDensity_mul _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A,\n    withDensity_one]", "annotated_tactic": ["have : \u03bc = (\u03bc.withDensity fun x => g x).<a>withDensity</a> fun x => (g x)\u207b\u00b9 := by\n    have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 := by\n      ext1 x\n      exact <a>ENNReal.mul_inv_cancel</a> (<a>ENNReal.coe_ne_zero</a>.2 (gpos x).<a>ne'</a>) <a>ENNReal.coe_ne_top</a>\n    rw [\u2190 <a>withDensity_mul</a> _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A,\n      <a>withDensity_one</a>]", [{"full_name": "MeasureTheory.Measure.withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [33, 5], "def_end_pos": [33, 24]}, {"full_name": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 33]}, {"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": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.withDensity_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [411, 9], "def_end_pos": [411, 24]}, {"full_name": "MeasureTheory.withDensity_one", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [131, 9], "def_end_pos": [131, 24]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u03bc \u226a withDensity \u03bc fun x => \u2191(g x)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nthis : \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n\u22a2 \u03bc \u226a withDensity \u03bc fun x => \u2191(g x)"}, {"tactic": "nth_rw 1 [this]", "annotated_tactic": ["nth_rw 1 [this]", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nthis : \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n\u22a2 \u03bc \u226a withDensity \u03bc fun x => \u2191(g x)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nthis : \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n\u22a2 (withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9) \u226a withDensity \u03bc fun x => \u2191(g x)"}, {"tactic": "exact withDensity_absolutelyContinuous _ _", "annotated_tactic": ["exact <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": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nthis : \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n\u22a2 (withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9) \u226a withDensity \u03bc fun x => \u2191(g x)", "state_after": "no goals"}, {"tactic": "have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 := by\n  ext1 x\n  exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top", "annotated_tactic": ["have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 := by\n      ext1 x\n      exact <a>ENNReal.mul_inv_cancel</a> (<a>ENNReal.coe_ne_zero</a>.2 (gpos x).<a>ne'</a>) <a>ENNReal.coe_ne_top</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": "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": "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\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9", "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nA : ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1\n\u22a2 \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9"}, {"tactic": "rw [\u2190 withDensity_mul _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A,\n  withDensity_one]", "annotated_tactic": ["rw [\u2190 <a>withDensity_mul</a> _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A,\n      <a>withDensity_one</a>]", [{"full_name": "MeasureTheory.withDensity_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [411, 9], "def_end_pos": [411, 24]}, {"full_name": "MeasureTheory.withDensity_one", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [131, 9], "def_end_pos": [131, 24]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nA : ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1\n\u22a2 \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nx : \u03b1\n\u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) x = OfNat.ofNat 1 x"}, {"tactic": "exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top", "annotated_tactic": ["exact <a>ENNReal.mul_inv_cancel</a> (<a>ENNReal.coe_ne_zero</a>.2 (gpos x).<a>ne'</a>) <a>ENNReal.coe_ne_top</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": "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": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nx : \u03b1\n\u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) x = OfNat.ofNat 1 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "full_name": "Std.RBNode.all_iff", "start": [174, 1], "end": [175, 48], "traced_tactics": [{"tactic": "induction t <;> simp [*, all, All, and_assoc]", "annotated_tactic": ["induction t <;> simp [*, <a>all</a>, <a>All</a>, <a>and_assoc</a>]", [{"full_name": "Std.RBNode.all", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [156, 19], "def_end_pos": [156, 22]}, {"full_name": "Std.RBNode.All", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [166, 5], "def_end_pos": [166, 8]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nt : RBNode \u03b1\n\u22a2 all p t = true \u2194 All (fun x => p x = true) 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_uIcc_right", "start": [323, 1], "end": [325, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.withDensity_mul\u2080", "start": [404, 1], "end": [409, 69], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 withDensity \u03bc (f * g) = withDensity (withDensity \u03bc f) g", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (f * g)) s = \u2191\u2191(withDensity (withDensity \u03bc f) g) s"}, {"tactic": "rw [withDensity_apply _ hs, withDensity_apply _ hs, restrict_withDensity hs,\n  lintegral_withDensity_eq_lintegral_mul\u2080 hf.restrict hg.restrict]", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ hs, <a>withDensity_apply</a> _ hs, <a>restrict_withDensity</a> hs,\n    <a>lintegral_withDensity_eq_lintegral_mul\u2080</a> hf.restrict hg.restrict]", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"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\u2080", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [330, 9], "def_end_pos": [330, 48]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (f * g)) s = \u2191\u2191(withDensity (withDensity \u03bc f) g) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to1.tr_supports", "start": [1911, 1], "end": [1952, 61], "traced_tactics": [{"tactic": "suffices \u2200 q, SupportsStmt S q \u2192 (\u2200 q' \u2208 writes q, q' \u2208 trSupp M S) \u2192\n    SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n    \u2200 q' \u2208 writes q, SupportsStmt (trSupp M S) (tr enc dec M q') by\n  rcases Finset.mem_biUnion.1 h with \u27e8l, hl, h\u27e9\n  have :=\n    this _ (ss.2 _ hl) fun q' hq \u21a6 Finset.mem_biUnion.2 \u27e8_, hl, Finset.mem_insert_of_mem hq\u27e9\n  rcases Finset.mem_insert.1 h with (rfl | h)\n  exacts [this.1, this.2 _ h]", "annotated_tactic": ["suffices \u2200 q, <a>SupportsStmt</a> S q \u2192 (\u2200 q' \u2208 <a>writes</a> q, q' \u2208 <a>trSupp</a> M S) \u2192\n        <a>SupportsStmt</a> (<a>trSupp</a> M S) (<a>trNormal</a> dec q) \u2227\n        \u2200 q' \u2208 <a>writes</a> q, <a>SupportsStmt</a> (<a>trSupp</a> M S) (<a>tr</a> enc dec M q') by\n      rcases <a>Finset.mem_biUnion</a>.1 h with \u27e8l, hl, h\u27e9\n      have :=\n        this _ (ss.2 _ hl) fun q' hq \u21a6 <a>Finset.mem_biUnion</a>.2 \u27e8_, hl, <a>Finset.mem_insert_of_mem</a> hq\u27e9\n      rcases <a>Finset.mem_insert</a>.1 h with (rfl | h)\n      exacts [this.1, this.2 _ h]", [{"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"full_name": "Turing.TM1to1.trSupp", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1907, 19], "def_end_pos": [1907, 25]}, {"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM1to1.trSupp", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1907, 19], "def_end_pos": [1907, 25]}, {"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}, {"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM1to1.trSupp", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1907, 19], "def_end_pos": [1907, 25]}, {"full_name": "Turing.TM1to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1755, 5], "def_end_pos": [1755, 7]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"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", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\n\u22a2 \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "intro q hs hw", "annotated_tactic": ["intro q hs hw", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\n\u22a2 \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nq : Stmt\u2081\nhs : SupportsStmt S q\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "induction q", "annotated_tactic": ["induction q", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nq : Stmt\u2081\nhs : SupportsStmt S q\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "case move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  SupportsStmt S a\u271d\u00b9 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "case move d q IH =>\n  unfold writes at hw \u22a2\n  replace IH := IH hs hw; refine' \u27e8_, IH.2\u27e9\n  cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]", "annotated_tactic": ["case move d q IH =>\n      unfold <a>writes</a> at hw \u22a2\n      replace IH := IH hs hw; refine' \u27e8_, IH.2\u27e9\n      cases d <;> simp only [<a>trNormal</a>, <a>iterate</a>, <a>supportsStmt_move</a>, IH]", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"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.supportsStmt_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 26]}]], "state_before": "case move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  SupportsStmt S a\u271d\u00b9 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "case write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  SupportsStmt S a\u271d\u00b9 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "case load a q IH =>\n  unfold writes at hw \u22a2\n  replace IH := IH hs hw\n  refine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9", "annotated_tactic": ["case load a q IH =>\n      unfold <a>writes</a> at hw \u22a2\n      replace IH := IH hs hw\n      refine' \u27e8<a>supportsStmt_read</a> _ fun _ \u21a6 IH.1, IH.2\u27e9", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"full_name": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}]], "state_before": "case load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  SupportsStmt S a\u271d\u00b9 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "case branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  SupportsStmt S a\u271d\u00b9 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "case branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n  unfold writes at hw \u22a2\n  simp only [Finset.mem_union] at hw \u22a2\n  replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq)\n  replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq)\n  exact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9", "annotated_tactic": ["case branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n      unfold <a>writes</a> at hw \u22a2\n      simp only [<a>Finset.mem_union</a>] at hw \u22a2\n      replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (<a>Or.inl</a> hq)\n      replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (<a>Or.inr</a> hq)\n      exact \u27e8<a>supportsStmt_read</a> _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 <a>Or.rec</a> (IH\u2081.2 _) (IH\u2082.2 _)\u27e9", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"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": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}, {"full_name": "Or.rec", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [515, 11], "def_end_pos": [515, 13]}]], "state_before": "case branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  SupportsStmt S a\u271d\u00b9 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n  SupportsStmt S a\u271d \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "case goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "case goto l =>\n  simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n  refine' supportsStmt_read _ fun a _ s \u21a6 _\n  exact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9", "annotated_tactic": ["case goto l =>\n      simp only [<a>writes</a>, <a>Finset.not_mem_empty</a>]; refine' \u27e8_, fun _ \u21a6 <a>False.elim</a>\u27e9\n      refine' <a>supportsStmt_read</a> _ fun a _ s \u21a6 _\n      exact <a>Finset.mem_biUnion</a>.2 \u27e8_, hs _ _, <a>Finset.mem_insert_self</a> _ _\u27e9", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"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]}, {"full_name": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"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 goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "case halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "case halt =>\n  simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n  simp only [SupportsStmt, supportsStmt_move, trNormal]", "annotated_tactic": ["case halt =>\n      simp only [<a>writes</a>, <a>Finset.not_mem_empty</a>]; refine' \u27e8_, fun _ \u21a6 <a>False.elim</a>\u27e9\n      simp only [<a>SupportsStmt</a>, <a>supportsStmt_move</a>, <a>trNormal</a>]", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"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]}, {"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM1to1.supportsStmt_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 26]}, {"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}]], "state_before": "case halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "no goals"}, {"tactic": "rcases Finset.mem_biUnion.1 h with \u27e8l, hl, h\u27e9", "annotated_tactic": ["rcases <a>Finset.mem_biUnion</a>.1 h with \u27e8l, hl, h\u27e9", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nthis :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)", "state_after": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d : q \u2208 trSupp M S\nthis :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh : q \u2208 insert (\u039b'.normal l) (writes (M l))\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)"}, {"tactic": "have :=\n  this _ (ss.2 _ hl) fun q' hq \u21a6 Finset.mem_biUnion.2 \u27e8_, hl, Finset.mem_insert_of_mem hq\u27e9", "annotated_tactic": ["have :=\n        this _ (ss.2 _ hl) fun q' hq \u21a6 <a>Finset.mem_biUnion</a>.2 \u27e8_, hl, <a>Finset.mem_insert_of_mem</a> hq\u27e9", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"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 intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d : q \u2208 trSupp M S\nthis :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh : q \u2208 insert (\u039b'.normal l) (writes (M l))\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)", "state_after": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d : q \u2208 trSupp M S\nthis\u271d :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh : q \u2208 insert (\u039b'.normal l) (writes (M l))\nthis :\n  SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)"}, {"tactic": "rcases Finset.mem_insert.1 h with (rfl | h)", "annotated_tactic": ["rcases <a>Finset.mem_insert</a>.1 h with (rfl | h)", [{"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}]], "state_before": "case intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d : q \u2208 trSupp M S\nthis\u271d :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh : q \u2208 insert (\u039b'.normal l) (writes (M l))\nthis :\n  SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)", "state_after": "case intro.intro.inl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nthis\u271d :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nthis :\n  SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nh\u271d : \u039b'.normal l \u2208 trSupp M S\nh : \u039b'.normal l \u2208 insert (\u039b'.normal l) (writes (M l))\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.normal l))\n\ncase intro.intro.inr\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d\u00b9 : q \u2208 trSupp M S\nthis\u271d :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh\u271d : q \u2208 insert (\u039b'.normal l) (writes (M l))\nthis :\n  SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nh : q \u2208 writes (M l)\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)"}, {"tactic": "exacts [this.1, this.2 _ h]", "annotated_tactic": ["exacts [this.1, this.2 _ h]", []], "state_before": "case intro.intro.inl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nthis\u271d :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nthis :\n  SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nh\u271d : \u039b'.normal l \u2208 trSupp M S\nh : \u039b'.normal l \u2208 insert (\u039b'.normal l) (writes (M l))\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.normal l))\n\ncase intro.intro.inr\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d\u00b9 : q \u2208 trSupp M S\nthis\u271d :\n  \u2200 (q : Stmt\u2081),\n    SupportsStmt S q \u2192\n      (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n        SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n          \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh\u271d : q \u2208 insert (\u039b'.normal l) (writes (M l))\nthis :\n  SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nh : q \u2208 writes (M l)\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)", "state_after": "no goals"}, {"tactic": "unfold writes at hw \u22a2", "annotated_tactic": ["unfold <a>writes</a> at hw \u22a2", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "replace IH := IH hs hw", "annotated_tactic": ["replace IH := IH hs hw", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "refine' \u27e8_, IH.2\u27e9", "annotated_tactic": ["refine' \u27e8_, IH.2\u27e9", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q))"}, {"tactic": "cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]", "annotated_tactic": ["cases d <;> simp only [<a>trNormal</a>, <a>iterate</a>, <a>supportsStmt_move</a>, 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.supportsStmt_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 26]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q))", "state_after": "no goals"}, {"tactic": "unfold writes at hw \u22a2", "annotated_tactic": ["unfold <a>writes</a> at hw \u22a2", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'),\n      q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and_iff]\n  at hw \u22a2", "annotated_tactic": ["simp only [<a>Finset.mem_image</a>, <a>Finset.mem_union</a>, <a>Finset.mem_univ</a>, <a>exists_prop</a>, <a>true_and_iff</a>]\n        at hw \u22a2", [{"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_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'),\n      q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "replace IH := IH hs fun q hq \u21a6 hw q (Or.inr hq)", "annotated_tactic": ["replace IH := IH hs fun q hq \u21a6 hw q (<a>Or.inr</a> hq)", [{"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": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "refine' \u27e8supportsStmt_read _ fun a _ s \u21a6 hw _ (Or.inl \u27e8_, rfl\u27e9), fun q' hq \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8<a>supportsStmt_read</a> _ fun a _ s \u21a6 hw _ (<a>Or.inl</a> \u27e8_, <a>rfl</a>\u27e9), fun q' hq \u21a6 _\u27e9", [{"full_name": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}, {"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": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n    \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nq' : \u039b'\nhq : (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "rcases hq with (\u27e8a, q\u2082, rfl\u27e9 | hq)", "annotated_tactic": ["rcases hq with (\u27e8a, q\u2082, rfl\u27e9 | hq)", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nq' : \u039b'\nhq : (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "case inl.intro.refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na : \u0393\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.write a q))\n\ncase inr\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nq' : \u039b'\nhq : q' \u2208 writes q\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "simp only [tr, supportsStmt_write, supportsStmt_move, IH.1]", "annotated_tactic": ["simp only [<a>tr</a>, <a>supportsStmt_write</a>, <a>supportsStmt_move</a>, IH.1]", [{"full_name": "Turing.TM1to1.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1755, 5], "def_end_pos": [1755, 7]}, {"full_name": "Turing.TM1to1.supportsStmt_write", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1712, 9], "def_end_pos": [1712, 27]}, {"full_name": "Turing.TM1to1.supportsStmt_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 26]}]], "state_before": "case inl.intro.refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na : \u0393\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.write a q))", "state_after": "no goals"}, {"tactic": "exact IH.2 _ hq", "annotated_tactic": ["exact IH.2 _ hq", []], "state_before": "case inr\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nq' : \u039b'\nhq : q' \u2208 writes q\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "no goals"}, {"tactic": "unfold writes at hw \u22a2", "annotated_tactic": ["unfold <a>writes</a> at hw \u22a2", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "replace IH := IH hs hw", "annotated_tactic": ["replace IH := IH hs hw", []], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  SupportsStmt S q \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "refine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9", "annotated_tactic": ["refine' \u27e8<a>supportsStmt_read</a> _ fun _ \u21a6 IH.1, IH.2\u27e9", [{"full_name": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n  SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "no goals"}, {"tactic": "unfold writes at hw \u22a2", "annotated_tactic": ["unfold <a>writes</a> at hw \u22a2", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  SupportsStmt S q\u2081 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  SupportsStmt S q\u2081 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "simp only [Finset.mem_union] at hw \u22a2", "annotated_tactic": ["simp only [<a>Finset.mem_union</a>] at hw \u22a2", [{"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  SupportsStmt S q\u2081 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  SupportsStmt S q\u2081 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq)", "annotated_tactic": ["replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (<a>Or.inl</a> hq)", [{"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": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  SupportsStmt S q\u2081 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\nIH\u2081 :\n  SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq)", "annotated_tactic": ["replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (<a>Or.inr</a> hq)", [{"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": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2082 :\n  SupportsStmt S q\u2082 \u2192\n    (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n      SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n        \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\nIH\u2081 :\n  SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\nIH\u2081 :\n  SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "exact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9", "annotated_tactic": ["exact \u27e8<a>supportsStmt_read</a> _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 <a>Or.rec</a> (IH\u2081.2 _) (IH\u2082.2 _)\u27e9", [{"full_name": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}, {"full_name": "Or.rec", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [515, 11], "def_end_pos": [515, 13]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\nIH\u2081 :\n  SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n  SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "no goals"}, {"tactic": "simp only [writes, Finset.not_mem_empty]", "annotated_tactic": ["simp only [<a>writes</a>, <a>Finset.not_mem_empty</a>]", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227\n    \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "refine' \u27e8_, fun _ \u21a6 False.elim\u27e9", "annotated_tactic": ["refine' \u27e8_, fun _ \u21a6 <a>False.elim</a>\u27e9", [{"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": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227\n    \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l))"}, {"tactic": "refine' supportsStmt_read _ fun a _ s \u21a6 _", "annotated_tactic": ["refine' <a>supportsStmt_read</a> _ fun a _ s \u21a6 _", [{"full_name": "Turing.TM1to1.supportsStmt_read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 26]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l))", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\na : \u0393\nx\u271d : Bool\ns : \u03c3\n\u22a2 (fun x s => \u039b'.normal (l a s)) x\u271d s \u2208 trSupp M S"}, {"tactic": "exact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9", "annotated_tactic": ["exact <a>Finset.mem_biUnion</a>.2 \u27e8_, hs _ _, <a>Finset.mem_insert_self</a> _ _\u27e9", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\na : \u0393\nx\u271d : Bool\ns : \u03c3\n\u22a2 (fun x s => \u039b'.normal (l a s)) x\u271d s \u2208 trSupp M S", "state_after": "no goals"}, {"tactic": "simp only [writes, Finset.not_mem_empty]", "annotated_tactic": ["simp only [<a>writes</a>, <a>Finset.not_mem_empty</a>]", [{"full_name": "Turing.TM1to1.writes", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1896, 19], "def_end_pos": [1896, 25]}, {"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [548, 9], "def_end_pos": [548, 22]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n    \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227 \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')"}, {"tactic": "refine' \u27e8_, fun _ \u21a6 False.elim\u27e9", "annotated_tactic": ["refine' \u27e8_, fun _ \u21a6 <a>False.elim</a>\u27e9", [{"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": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227 \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt)"}, {"tactic": "simp only [SupportsStmt, supportsStmt_move, trNormal]", "annotated_tactic": ["simp only [<a>SupportsStmt</a>, <a>supportsStmt_move</a>, <a>trNormal</a>]", [{"full_name": "Turing.TM1.SupportsStmt", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 17]}, {"full_name": "Turing.TM1to1.supportsStmt_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 26]}, {"full_name": "Turing.TM1to1.trNormal", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1687, 5], "def_end_pos": [1687, 13]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : 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\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt)", "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_smul_left'", "start": [937, 1], "end": [941, 38], "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": [114, 1], "end": [118, 45], "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": [75, 15], "def_end_pos": [75, 24]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 NoSibling (tail le s)", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 NoSibling (Option.getD (tail? le s) nil)"}, {"tactic": "match eq : s.tail? le with\n| none => cases s with cases eq | nil => constructor\n| some tl => exact Heap.noSibling_tail? eq", "annotated_tactic": ["match eq : s.tail? le with\n  | <a>none</a> => cases s with cases eq | <a>nil</a> => constructor\n  | <a>some</a> tl => exact <a>Heap.noSibling_tail?</a> 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.PairingHeapImp.Heap.nil", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [23, 5], "def_end_pos": [23, 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.PairingHeapImp.Heap.noSibling_tail?", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [108, 9], "def_end_pos": [108, 29]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 NoSibling (Option.getD (tail? le s) nil)", "state_after": "no goals"}, {"tactic": "cases s with cases eq | nil => constructor", "annotated_tactic": ["cases s with cases eq | <a>nil</a> => constructor", [{"full_name": "Std.PairingHeapImp.Heap.nil", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [23, 5], "def_end_pos": [23, 8]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\neq : tail? le s = none\n\u22a2 NoSibling (Option.getD none nil)", "state_after": "no goals"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case nil.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\n\u22a2 NoSibling (Option.getD none nil)", "state_after": "no goals"}, {"tactic": "exact Heap.noSibling_tail? eq", "annotated_tactic": ["exact <a>Heap.noSibling_tail?</a> eq", [{"full_name": "Std.PairingHeapImp.Heap.noSibling_tail?", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [108, 9], "def_end_pos": [108, 29]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns tl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 NoSibling (Option.getD (some tl) nil)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.norm_setToL1_le_mul_norm'", "start": [1227, 1], "end": [1233, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_mul", "start": [171, 1], "end": [173, 79], "traced_tactics": [{"tactic": "cases a <;> cases b <;>\n  simp only [\u2190 Int.mul_def, Int.mul, natAbs_negOfNat] <;> simp only [natAbs]", "annotated_tactic": ["cases a <;> cases b <;>\n    simp only [\u2190 <a>Int.mul_def</a>, <a>Int.mul</a>, <a>natAbs_negOfNat</a>] <;> simp only [<a>natAbs</a>]", [{"full_name": "Int.mul_def", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [56, 17], "def_end_pos": [56, 24]}, {"full_name": "Int.mul", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [122, 15], "def_end_pos": [122, 18]}, {"full_name": "Int.natAbs_negOfNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [168, 9], "def_end_pos": [168, 24]}, {"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": "a b : Int\n\u22a2 natAbs (a * b) = natAbs a * natAbs b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pairwise.lean", "full_name": "List.pairwiseDisjoint_iff_coe_toFinset_pairwise_disjoint", "start": [98, 1], "end": [101, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.condexpIndSMul_empty", "start": [461, 1], "end": [464, 63], "traced_tactics": [{"tactic": "rw [condexpIndSMul, indicatorConstLp_empty]", "annotated_tactic": ["rw [<a>condexpIndSMul</a>, <a>indicatorConstLp_empty</a>]", [{"full_name": "MeasureTheory.condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [384, 19], "def_end_pos": [384, 33]}, {"full_name": "MeasureTheory.indicatorConstLp_empty", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [803, 9], "def_end_pos": [803, 31]}]], "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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nx : G\n\u22a2 condexpIndSMul hm (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) x = 0", "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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nx : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) 0) = 0"}, {"tactic": "simp only [Submodule.coe_zero, ContinuousLinearMap.map_zero]", "annotated_tactic": ["simp only [<a>Submodule.coe_zero</a>, <a>ContinuousLinearMap.map_zero</a>]", [{"full_name": "Submodule.coe_zero", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [319, 9], "def_end_pos": [319, 17]}, {"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\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\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : 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\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nx : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) 0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.le_sub_of_le_upcrossingsBefore", "start": [579, 1], "end": [585, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.measure_ge_le_exp_mul_mgf", "start": [329, 1], "end": [348, 66], "traced_tactics": [{"tactic": "cases' ht.eq_or_lt with ht_zero_eq ht_pos", "annotated_tactic": ["cases' ht.eq_or_lt with ht_zero_eq ht_pos", []], "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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t\n\ncase inr\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t"}, {"tactic": "calc\n  (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal = (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal := by\n    congr with \u03c9\n    simp only [Set.mem_setOf_eq, exp_le_exp, gt_iff_lt]\n    exact \u27e8fun h => mul_le_mul_of_nonneg_left h ht_pos.le,\n      fun h => le_of_mul_le_mul_left h ht_pos\u27e9\n  _ \u2264 (exp (t * \u03b5))\u207b\u00b9 * \u03bc[fun \u03c9 => exp (t * X \u03c9)] := by\n    have : exp (t * \u03b5) * (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal \u2264\n        \u03bc[fun \u03c9 => exp (t * X \u03c9)] :=\n      mul_meas_ge_le_integral_of_nonneg (ae_of_all _ fun x => (exp_pos _).le) h_int _\n    rwa [mul_comm (exp (t * \u03b5))\u207b\u00b9, \u2190 div_eq_mul_inv, le_div_iff' (exp_pos _)]\n  _ = exp (-t * \u03b5) * mgf X \u03bc t := by rw [neg_mul, exp_neg]; rfl", "annotated_tactic": ["calc\n    (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).<a>toReal</a> = (\u03bc {\u03c9 | <a>exp</a> (t * \u03b5) \u2264 <a>exp</a> (t * X \u03c9)}).<a>toReal</a> := by\n      congr with \u03c9\n      simp only [<a>Set.mem_setOf_eq</a>, <a>exp_le_exp</a>, <a>gt_iff_lt</a>]\n      exact \u27e8fun h => <a>mul_le_mul_of_nonneg_left</a> h ht_pos.le,\n        fun h => <a>le_of_mul_le_mul_left</a> h ht_pos\u27e9\n    _ \u2264 (<a>exp</a> (t * \u03b5))\u207b\u00b9 * \u03bc[fun \u03c9 => <a>exp</a> (t * X \u03c9)] := by\n      have : <a>exp</a> (t * \u03b5) * (\u03bc {\u03c9 | <a>exp</a> (t * \u03b5) \u2264 <a>exp</a> (t * X \u03c9)}).<a>toReal</a> \u2264\n          \u03bc[fun \u03c9 => <a>exp</a> (t * X \u03c9)] :=\n        <a>mul_meas_ge_le_integral_of_nonneg</a> (<a>ae_of_all</a> _ fun x => (<a>exp_pos</a> _).<a>le</a>) h_int _\n      rwa [<a>mul_comm</a> (<a>exp</a> (t * \u03b5))\u207b\u00b9, \u2190 <a>div_eq_mul_inv</a>, <a>le_div_iff'</a> (<a>exp_pos</a> _)]\n    _ = <a>exp</a> (-t * \u03b5) * <a>mgf</a> X \u03bc t := by rw [<a>neg_mul</a>, <a>exp_neg</a>]; rfl", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 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": "Real.exp_le_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1534, 9], "def_end_pos": [1534, 19]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"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": "le_of_mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [176, 9], "def_end_pos": [176, 30]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "MeasureTheory.mul_meas_ge_le_integral_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1695, 9], "def_end_pos": [1695, 42]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 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": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 20]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 16]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "ProbabilityTheory.mgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [103, 5], "def_end_pos": [103, 8]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "Real.exp_neg", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1169, 16], "def_end_pos": [1169, 23]}]], "state_before": "case inr\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "no goals"}, {"tactic": "rw [ht_zero_eq.symm]", "annotated_tactic": ["rw [ht_zero_eq.symm]", []], "state_before": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-0 * \u03b5) * mgf X \u03bc 0"}, {"tactic": "simp only [neg_zero, zero_mul, exp_zero, mgf_zero', one_mul]", "annotated_tactic": ["simp only [<a>neg_zero</a>, <a>zero_mul</a>, <a>exp_zero</a>, <a>mgf_zero'</a>, <a>one_mul</a>]", [{"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1014, 3], "def_end_pos": [1014, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Real.exp_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 17]}, {"full_name": "ProbabilityTheory.mgf_zero'", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-0 * \u03b5) * mgf X \u03bc 0", "state_after": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 ENNReal.toReal (\u2191\u2191\u03bc Set.univ)"}, {"tactic": "rw [ENNReal.toReal_le_toReal (measure_ne_top \u03bc _) (measure_ne_top \u03bc _)]", "annotated_tactic": ["rw [<a>ENNReal.toReal_le_toReal</a> (<a>measure_ne_top</a> \u03bc _) (<a>measure_ne_top</a> \u03bc _)]", [{"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_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 inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 ENNReal.toReal (\u2191\u2191\u03bc Set.univ)", "state_after": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 \u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9} \u2264 \u2191\u2191\u03bc Set.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": "case inl\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_zero_eq : 0 = t\n\u22a2 \u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9} \u2264 \u2191\u2191\u03bc Set.univ", "state_after": "no goals"}, {"tactic": "congr with \u03c9", "annotated_tactic": ["congr with \u03c9", []], "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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) = ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)})", "state_after": "case e_a.e_a.h\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03b5 \u2264 X \u03c9} \u2194 \u03c9 \u2208 {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}"}, {"tactic": "simp only [Set.mem_setOf_eq, exp_le_exp, gt_iff_lt]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>, <a>exp_le_exp</a>, <a>gt_iff_lt</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": "Real.exp_le_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1534, 9], "def_end_pos": [1534, 19]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}]], "state_before": "case e_a.e_a.h\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03b5 \u2264 X \u03c9} \u2194 \u03c9 \u2208 {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}", "state_after": "case e_a.e_a.h\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03b5 \u2264 X \u03c9 \u2194 t * \u03b5 \u2264 t * X \u03c9"}, {"tactic": "exact \u27e8fun h => mul_le_mul_of_nonneg_left h ht_pos.le,\n  fun h => le_of_mul_le_mul_left h ht_pos\u27e9", "annotated_tactic": ["exact \u27e8fun h => <a>mul_le_mul_of_nonneg_left</a> h ht_pos.le,\n        fun h => <a>le_of_mul_le_mul_left</a> h ht_pos\u27e9", [{"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": "le_of_mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [176, 9], "def_end_pos": [176, 30]}]], "state_before": "case e_a.e_a.h\n\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03b5 \u2264 X \u03c9 \u2194 t * \u03b5 \u2264 t * X \u03c9", "state_after": "no goals"}, {"tactic": "have : exp (t * \u03b5) * (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal \u2264\n    \u03bc[fun \u03c9 => exp (t * X \u03c9)] :=\n  mul_meas_ge_le_integral_of_nonneg (ae_of_all _ fun x => (exp_pos _).le) h_int _", "annotated_tactic": ["have : <a>exp</a> (t * \u03b5) * (\u03bc {\u03c9 | <a>exp</a> (t * \u03b5) \u2264 <a>exp</a> (t * X \u03c9)}).<a>toReal</a> \u2264\n          \u03bc[fun \u03c9 => <a>exp</a> (t * X \u03c9)] :=\n        <a>mul_meas_ge_le_integral_of_nonneg</a> (<a>ae_of_all</a> _ fun x => (<a>exp_pos</a> _).<a>le</a>) h_int _", [{"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "MeasureTheory.mul_meas_ge_le_integral_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1695, 9], "def_end_pos": [1695, 42]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264\n    (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc", "state_after": "\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\nthis :\n  rexp (t * \u03b5) * ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264 \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264\n    (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc"}, {"tactic": "rwa [mul_comm (exp (t * \u03b5))\u207b\u00b9, \u2190 div_eq_mul_inv, le_div_iff' (exp_pos _)]", "annotated_tactic": ["rwa [<a>mul_comm</a> (<a>exp</a> (t * \u03b5))\u207b\u00b9, \u2190 <a>div_eq_mul_inv</a>, <a>le_div_iff'</a> (<a>exp_pos</a> _)]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 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": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 20]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\nthis :\n  rexp (t * \u03b5) * ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264 \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264\n    (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [neg_mul, exp_neg]", "annotated_tactic": ["rw [<a>neg_mul</a>, <a>exp_neg</a>]", [{"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "Real.exp_neg", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1169, 16], "def_end_pos": [1169, 23]}]], "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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "\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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = (rexp (t * \u03b5))\u207b\u00b9 * mgf X \u03bc t"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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 : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable fun \u03c9 => rexp (t * X \u03c9)\nht_pos : 0 < t\n\u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = (rexp (t * \u03b5))\u207b\u00b9 * mgf X \u03bc t", "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_smul", "start": [1089, 8], "end": [1105, 89], "traced_tactics": [{"tactic": "by_cases hr : 0 \u2264 r", "annotated_tactic": ["by_cases hr : 0 \u2264 r", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\n\u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc"}, {"tactic": "lift r to \u211d\u22650 using hr", "annotated_tactic": ["lift r to \u211d\u22650 using hr", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\n\u22a2 singularPart (\u2191r \u2022 s) \u03bc = \u2191r \u2022 singularPart s \u03bc"}, {"tactic": "exact singularPart_smul_nnreal s \u03bc r", "annotated_tactic": ["exact <a>singularPart_smul_nnreal</a> s \u03bc r", [{"full_name": "MeasureTheory.SignedMeasure.singularPart_smul_nnreal", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [1078, 9], "def_end_pos": [1078, 33]}]], "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\n\u22a2 singularPart (\u2191r \u2022 s) \u03bc = \u2191r \u2022 singularPart s \u03bc", "state_after": "no goals"}, {"tactic": "rw [singularPart, singularPart]", "annotated_tactic": ["rw [<a>singularPart</a>, <a>singularPart</a>]", [{"full_name": "MeasureTheory.SignedMeasure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [839, 5], "def_end_pos": [839, 17]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [839, 5], "def_end_pos": [839, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 toSignedMeasure (Measure.singularPart (toJordanDecomposition (r \u2022 s)).posPart \u03bc) -\n      toSignedMeasure (Measure.singularPart (toJordanDecomposition (r \u2022 s)).negPart \u03bc) =\n    r \u2022\n      (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n        toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))"}, {"tactic": "rw [toSignedMeasure_smul, toSignedMeasure_smul, \u2190 neg_sub, \u2190 smul_sub]", "annotated_tactic": ["rw [<a>toSignedMeasure_smul</a>, <a>toSignedMeasure_smul</a>, \u2190 <a>neg_sub</a>, \u2190 <a>smul_sub</a>]", [{"full_name": "MeasureTheory.Measure.toSignedMeasure_smul", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [475, 9], "def_end_pos": [475, 29]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_smul", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [475, 9], "def_end_pos": [475, 29]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 toSignedMeasure (Real.toNNReal (-r) \u2022 Measure.singularPart (toJordanDecomposition s).negPart \u03bc) -\n      toSignedMeasure (Real.toNNReal (-r) \u2022 Measure.singularPart (toJordanDecomposition s).posPart \u03bc) =\n    r \u2022\n      (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n        toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 -(Real.toNNReal (-r) \u2022\n        (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n          toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))) =\n    r \u2022\n      (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n        toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))"}, {"tactic": "change -(((-r).toNNReal : \u211d) \u2022 (_ : SignedMeasure \u03b1)) = _", "annotated_tactic": ["change -(((-r).<a>toNNReal</a> : \u211d) \u2022 (_ : <a>SignedMeasure</a> \u03b1)) = _", [{"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}, {"full_name": "MeasureTheory.SignedMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [70, 8], "def_end_pos": [70, 21]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 -(Real.toNNReal (-r) \u2022\n        (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n          toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))) =\n    r \u2022\n      (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n        toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 -(\u2191(Real.toNNReal (-r)) \u2022\n        (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n          toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))) =\n    r \u2022\n      (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n        toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))"}, {"tactic": "rw [\u2190 neg_smul, Real.coe_toNNReal _ (le_of_lt (neg_pos.mpr (not_le.1 hr))), neg_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_smul</a>, <a>Real.coe_toNNReal</a> _ (<a>le_of_lt</a> (neg_pos.mpr (<a>not_le</a>.1 hr))), <a>neg_neg</a>]", [{"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "Real.coe_toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [122, 9], "def_end_pos": [122, 33]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns\u271d t s : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 -(\u2191(Real.toNNReal (-r)) \u2022\n        (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n          toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))) =\n    r \u2022\n      (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -\n        toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_smul_measure_of_ne_zero", "start": [651, 1], "end": [657, 61], "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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : p = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc"}, {"tactic": "exact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_top c", "annotated_tactic": ["exact <a>snorm_smul_measure_of_ne_zero_of_ne_top</a> hp0 hp_top c", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSeminorm.0.MeasureTheory.snorm_smul_measure_of_ne_zero_of_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [642, 17], "def_end_pos": [642, 56]}]], "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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f 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\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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : p = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc", "state_after": "no goals"}, {"tactic": "simp [hp_top, snormEssSup_smul_measure hc]", "annotated_tactic": ["simp [hp_top, <a>snormEssSup_smul_measure</a> hc]", [{"full_name": "MeasureTheory.snormEssSup_smul_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [635, 9], "def_end_pos": [635, 33]}]], "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 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \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_iff_forall_indepSet", "start": [518, 1], "end": [521, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.algHom_ext", "start": [497, 1], "end": [499, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.lt_next'", "start": [272, 1], "end": [272, 69], "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_Ioi_rat", "start": [1869, 1], "end": [1879, 64], "traced_tactics": [{"tactic": "rw [borel_eq_generateFrom_Ioi]", "annotated_tactic": ["rw [<a>borel_eq_generateFrom_Ioi</a>]", [{"full_name": "borel_eq_generateFrom_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [157, 9], "def_end_pos": [157, 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, {Ioi \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 Ioi) = MeasurableSpace.generateFrom (\u22c3 a, {Ioi \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 Ioi) = MeasurableSpace.generateFrom (\u22c3 a, {Ioi \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 Ioi \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 Ioi \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 (Ioi a)"}, {"tactic": "have : IsGLB (range ((\u2191) : \u211a \u2192 \u211d) \u2229 Ioi a) a := by\n  simp [isGLB_iff_le_iff, mem_lowerBounds, \u2190 le_iff_forall_lt_rat_imp_le]", "annotated_tactic": ["have : <a>IsGLB</a> (<a>range</a> ((\u2191) : \u211a \u2192 \u211d) \u2229 <a>Ioi</a> a) a := by\n    simp [<a>isGLB_iff_le_iff</a>, <a>mem_lowerBounds</a>, \u2190 <a>le_iff_forall_lt_rat_imp_le</a>]", [{"full_name": "IsGLB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [81, 5], "def_end_pos": [81, 10]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "isGLB_iff_le_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [323, 9], "def_end_pos": [323, 25]}, {"full_name": "mem_lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}, {"full_name": "le_iff_forall_lt_rat_imp_le", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [299, 9], "def_end_pos": [299, 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 (Ioi 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 : IsGLB (range Rat.cast \u2229 Ioi a) a\n\u22a2 MeasurableSet (Ioi a)"}, {"tactic": "rw [\u2190 this.biUnion_Ioi_eq, \u2190 image_univ, \u2190 image_inter_preimage, univ_inter, biUnion_image]", "annotated_tactic": ["rw [\u2190 this.biUnion_Ioi_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 : IsGLB (range Rat.cast \u2229 Ioi a) a\n\u22a2 MeasurableSet (Ioi 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 : IsGLB (range Rat.cast \u2229 Ioi a) a\n\u22a2 MeasurableSet (\u22c3 y \u2208 Rat.cast \u207b\u00b9' Ioi a, Ioi \u2191y)"}, {"tactic": "exact MeasurableSet.biUnion (to_countable _)\n  fun b _ => GenerateMeasurable.basic (Ioi (b : \u211d)) (by simp)", "annotated_tactic": ["exact <a>MeasurableSet.biUnion</a> (<a>to_countable</a> _)\n    fun b _ => <a>GenerateMeasurable.basic</a> (<a>Ioi</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.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 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 : IsGLB (range Rat.cast \u2229 Ioi a) a\n\u22a2 MeasurableSet (\u22c3 y \u2208 Rat.cast \u207b\u00b9' Ioi a, Ioi \u2191y)", "state_after": "no goals"}, {"tactic": "simp [isGLB_iff_le_iff, mem_lowerBounds, \u2190 le_iff_forall_lt_rat_imp_le]", "annotated_tactic": ["simp [<a>isGLB_iff_le_iff</a>, <a>mem_lowerBounds</a>, \u2190 <a>le_iff_forall_lt_rat_imp_le</a>]", [{"full_name": "isGLB_iff_le_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [323, 9], "def_end_pos": [323, 25]}, {"full_name": "mem_lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}, {"full_name": "le_iff_forall_lt_rat_imp_le", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [299, 9], "def_end_pos": [299, 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 IsGLB (range Rat.cast \u2229 Ioi 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 : IsGLB (range Rat.cast \u2229 Ioi a) a\nb : \u211a\nx\u271d : b \u2208 Rat.cast \u207b\u00b9' Ioi a\n\u22a2 Ioi \u2191b \u2208 \u22c3 a, {Ioi \u2191a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.mod'_to_nat", "start": [1622, 1], "end": [1623, 29], "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": [1313, 1], "end": [1327, 97], "traced_tactics": [{"tactic": "rw [snorm'_lim_eq_lintegral_liminf hp_pos.le h_lim]", "annotated_tactic": ["rw [<a>snorm'_lim_eq_lintegral_liminf</a> hp_pos.le h_lim]", [{"full_name": "MeasureTheory.Lp.snorm'_lim_eq_lintegral_liminf", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1298, 9], "def_end_pos": [1298, 39]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (1 / p) \u2264 liminf (fun n => snorm' (f n) p \u03bc) atTop"}, {"tactic": "rw [\u2190 ENNReal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.le_rpow_one_div_iff</a> (by simp [hp_pos] : 0 < 1 / p), <a>one_div_one_div</a>]", [{"full_name": "ENNReal.le_rpow_one_div_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [658, 9], "def_end_pos": [658, 28]}, {"full_name": "one_div_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [411, 9], "def_end_pos": [411, 24]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (1 / p) \u2264 liminf (fun n => snorm' (f n) p \u03bc) atTop", "state_after": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc \u2264 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p"}, {"tactic": "refine (lintegral_liminf_le' fun m => (hf m).ennnorm.pow_const _).trans_eq ?_", "annotated_tactic": ["refine (<a>lintegral_liminf_le'</a> fun m => (hf m).ennnorm.pow_const _).<a>trans_eq</a> ?_", [{"full_name": "MeasureTheory.lintegral_liminf_le'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 29]}, {"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\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc \u2264 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p", "state_after": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p"}, {"tactic": "have h_pow_liminf :\n  (atTop.liminf fun n => snorm' (f n) p \u03bc) ^ p = atTop.liminf fun n => snorm' (f n) p \u03bc ^ p := by\n  have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hp_pos\n  have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2\n  refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply _ _ _ _\n  all_goals isBoundedDefault", "annotated_tactic": ["have h_pow_liminf :\n    (atTop.liminf fun n => <a>snorm'</a> (f n) p \u03bc) ^ p = atTop.liminf fun n => <a>snorm'</a> (f n) p \u03bc ^ p := by\n    have h_rpow_mono := <a>ENNReal.strictMono_rpow_of_pos</a> hp_pos\n    have h_rpow_surj := (<a>ENNReal.rpow_left_bijective</a> hp_pos.ne.symm).2\n    refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).<a>liminf_apply</a> _ _ _ _\n    all_goals isBoundedDefault", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "ENNReal.strictMono_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 31]}, {"full_name": "ENNReal.rpow_left_bijective", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [845, 9], "def_end_pos": [845, 28]}, {"full_name": "OrderIso.liminf_apply", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1468, 9], "def_end_pos": [1468, 30]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p", "state_after": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_pow_liminf : liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop\n\u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p"}, {"tactic": "rw [h_pow_liminf]", "annotated_tactic": ["rw [h_pow_liminf]", []], "state_before": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_pow_liminf : liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop\n\u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p", "state_after": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_pow_liminf : liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop\n\u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop"}, {"tactic": "simp_rw [snorm', \u2190 ENNReal.rpow_mul, one_div, inv_mul_cancel hp_pos.ne.symm, ENNReal.rpow_one]", "annotated_tactic": ["simp_rw [<a>snorm'</a>, \u2190 <a>ENNReal.rpow_mul</a>, <a>one_div</a>, <a>inv_mul_cancel</a> hp_pos.ne.symm, <a>ENNReal.rpow_one</a>]", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"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": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"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\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_pow_liminf : liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop\n\u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop", "state_after": "no goals"}, {"tactic": "simp [hp_pos]", "annotated_tactic": ["simp [hp_pos]", []], "state_before": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 0 < 1 / p", "state_after": "no goals"}, {"tactic": "have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hp_pos", "annotated_tactic": ["have h_rpow_mono := <a>ENNReal.strictMono_rpow_of_pos</a> hp_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": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop", "state_after": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\n\u22a2 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop"}, {"tactic": "have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2", "annotated_tactic": ["have h_rpow_surj := (<a>ENNReal.rpow_left_bijective</a> hp_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": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\n\u22a2 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop", "state_after": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop"}, {"tactic": "refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply _ _ _ _", "annotated_tactic": ["refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).<a>liminf_apply</a> _ _ _ _", [{"full_name": "OrderIso.liminf_apply", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1468, 9], "def_end_pos": [1468, 30]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => snorm' (f n) p \u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => snorm' (f n) p \u03bc\n\ncase refine'_3\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>\n    \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc)\n\ncase refine'_4\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>\n    \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc)"}, {"tactic": "all_goals isBoundedDefault", "annotated_tactic": ["all_goals isBoundedDefault", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => snorm' (f n) p \u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => snorm' (f n) p \u03bc\n\ncase refine'_3\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>\n    \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc)\n\ncase refine'_4\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>\n    \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc)", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "case refine'_4\n\u03b1 : Type u_1\nE\u271d : 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\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\np : \u211d\nhp_pos : 0 < p\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))\nh_rpow_mono : StrictMono fun x => x ^ p\nh_rpow_surj : Function.Surjective fun y => y ^ p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>\n    \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.exists_le_lintegral", "start": [738, 1], "end": [739, 98], "traced_tactics": [{"tactic": "simpa only [laverage_eq_lintegral] using exists_le_laverage (IsProbabilityMeasure.ne_zero \u03bc) hf", "annotated_tactic": ["simpa only [<a>laverage_eq_lintegral</a>] using <a>exists_le_laverage</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hf", [{"full_name": "MeasureTheory.laverage_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [95, 9], "def_end_pos": [95, 30]}, {"full_name": "MeasureTheory.exists_le_laverage", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [702, 9], "def_end_pos": [702, 27]}, {"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\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : AEMeasurable f\n\u22a2 \u2203 x, f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "AEMeasurable.isLUB", "start": [1194, 1], "end": [1218, 59], "traced_tactics": [{"tactic": "nontriviality \u03b1", "annotated_tactic": ["nontriviality \u03b1", []], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\n\u22a2 AEMeasurable g"}, {"tactic": "haveI h\u03b1 : Nonempty \u03b1 := inferInstance", "annotated_tactic": ["haveI h\u03b1 : <a>Nonempty</a> \u03b1 := <a>inferInstance</a>", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}, {"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\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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\n\u22a2 AEMeasurable g"}, {"tactic": "cases' isEmpty_or_nonempty \u03b9 with h\u03b9 h\u03b9", "annotated_tactic": ["cases' <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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : IsEmpty \u03b9\n\u22a2 AEMeasurable g\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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\n\u22a2 AEMeasurable g"}, {"tactic": "let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB { a | \u2203 i, f' i = a } (g x)", "annotated_tactic": ["let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => <a>IsLUB</a> { a | \u2203 i, f' i = a } (g x)", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}]], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 AEMeasurable g"}, {"tactic": "let g_seq := (aeSeqSet hf p).piecewise g fun _ => h\u03b1.some", "annotated_tactic": ["let g_seq := (<a>aeSeqSet</a> hf p).<a>piecewise</a> g fun _ => h\u03b1.some", [{"full_name": "aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [36, 5], "def_end_pos": [36, 13]}, {"full_name": "Set.piecewise", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [993, 5], "def_end_pos": [993, 18]}]], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\n\u22a2 AEMeasurable g"}, {"tactic": "refine' \u27e8g_seq, Measurable.isLUB (aeSeq.measurable hf p) hg_seq, _\u27e9", "annotated_tactic": ["refine' \u27e8g_seq, <a>Measurable.isLUB</a> (<a>aeSeq.measurable</a> hf p) hg_seq, _\u27e9", [{"full_name": "Measurable.isLUB", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1145, 9], "def_end_pos": [1145, 25]}, {"full_name": "aeSeq.measurable", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [95, 9], "def_end_pos": [95, 19]}]], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nhg_seq : \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b)\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nhg_seq : \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b)\n\u22a2 g =\u1d50[\u03bc] g_seq"}, {"tactic": "exact\n  (ite_ae_eq_of_measure_compl_zero g (fun _ => h\u03b1.some) (aeSeqSet hf p)\n      (aeSeq.measure_compl_aeSeqSet_eq_zero hf hg)).symm", "annotated_tactic": ["exact\n    (<a>ite_ae_eq_of_measure_compl_zero</a> g (fun _ => h\u03b1.some) (<a>aeSeqSet</a> hf p)\n        (<a>aeSeq.measure_compl_aeSeqSet_eq_zero</a> hf hg)).<a>symm</a>", [{"full_name": "MeasureTheory.ite_ae_eq_of_measure_compl_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3235, 9], "def_end_pos": [3235, 40]}, {"full_name": "aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [36, 5], "def_end_pos": [36, 13]}, {"full_name": "aeSeq.measure_compl_aeSeqSet_eq_zero", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [100, 9], "def_end_pos": [100, 39]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nhg_seq : \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b)\n\u22a2 g =\u1d50[\u03bc] g_seq", "state_after": "no goals"}, {"tactic": "simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg", "annotated_tactic": ["simp only [<a>IsEmpty.exists_iff</a>, <a>setOf_false</a>, <a>isLUB_empty_iff</a>] at hg", [{"full_name": "IsEmpty.exists_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [126, 9], "def_end_pos": [126, 19]}, {"full_name": "Set.setOf_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 20]}, {"full_name": "isLUB_empty_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [887, 17], "def_end_pos": [887, 32]}]], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : IsEmpty \u03b9\n\u22a2 AEMeasurable g", "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : IsEmpty \u03b9\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsBot (g b)\n\u22a2 AEMeasurable g"}, {"tactic": "exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (ha _).antisymm (hb _))", "annotated_tactic": ["exact <a>aemeasurable_const'</a> (hg.mono fun a ha => hg.mono fun b hb => (ha _).<a>antisymm</a> (hb _))", [{"full_name": "aemeasurable_const'", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [244, 9], "def_end_pos": [244, 28]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [128, 7], "def_end_pos": [128, 21]}]], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : IsEmpty \u03b9\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsBot (g b)\n\u22a2 AEMeasurable g", "state_after": "no goals"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\n\u22a2 \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\n\u22a2 IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b)"}, {"tactic": "simp only [aeSeq, Set.piecewise]", "annotated_tactic": ["simp only [<a>aeSeq</a>, <a>Set.piecewise</a>]", [{"full_name": "aeSeq", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [42, 19], "def_end_pos": [42, 24]}, {"full_name": "Set.piecewise", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [993, 5], "def_end_pos": [993, 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\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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\n\u22a2 IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\n\u22a2 IsLUB\n    {a |\n      \u2203 i,\n        (if b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) then mk (f i) (_ : AEMeasurable (f i)) b\n          else Nonempty.some (_ : Nonempty \u03b1)) =\n          a}\n    (if b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) then g b else Nonempty.some h\u03b1)"}, {"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\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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\n\u22a2 IsLUB\n    {a |\n      \u2203 i,\n        (if b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) then mk (f i) (_ : AEMeasurable (f i)) b\n          else Nonempty.some (_ : Nonempty \u03b1)) =\n          a}\n    (if b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) then g b else Nonempty.some h\u03b1)", "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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 IsLUB {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g 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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : \u00acb \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 IsLUB {a | \u2203 i, Nonempty.some (_ : Nonempty \u03b1) = a} (Nonempty.some h\u03b1)"}, {"tactic": "have h_set_eq : { a : \u03b1 | \u2203 i : \u03b9, (hf i).mk (f i) b = a } =\n  { a : \u03b1 | \u2203 i : \u03b9, f i b = a } := by\n  ext x\n  simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]", "annotated_tactic": ["have h_set_eq : { a : \u03b1 | \u2203 i : \u03b9, (hf i).<a>mk</a> (f i) b = a } =\n        { a : \u03b1 | \u2203 i : \u03b9, f i b = a } := by\n        ext x\n        simp_rw [<a>Set.mem_setOf_eq</a>, <a>aeSeq.mk_eq_fun_of_mem_aeSeqSet</a> hf h]", [{"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "aeSeq.mk_eq_fun_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [50, 9], "def_end_pos": [50, 34]}]], "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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 IsLUB {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g 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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\nh_set_eq : {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a}\n\u22a2 IsLUB {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b)"}, {"tactic": "rw [h_set_eq]", "annotated_tactic": ["rw [h_set_eq]", []], "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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\nh_set_eq : {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a}\n\u22a2 IsLUB {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g 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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\nh_set_eq : {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a}\n\u22a2 IsLUB {a | \u2203 i, f i b = a} (g b)"}, {"tactic": "exact aeSeq.fun_prop_of_mem_aeSeqSet hf h", "annotated_tactic": ["exact <a>aeSeq.fun_prop_of_mem_aeSeqSet</a> hf h", [{"full_name": "aeSeq.fun_prop_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [81, 9], "def_end_pos": [81, 33]}]], "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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\nh_set_eq : {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a}\n\u22a2 IsLUB {a | \u2203 i, f i b = a} (g b)", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a}", "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\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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\nx : \u03b1\n\u22a2 x \u2208 {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} \u2194 x \u2208 {a | \u2203 i, f i b = a}"}, {"tactic": "simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]", "annotated_tactic": ["simp_rw [<a>Set.mem_setOf_eq</a>, <a>aeSeq.mk_eq_fun_of_mem_aeSeqSet</a> hf h]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "aeSeq.mk_eq_fun_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [50, 9], "def_end_pos": [50, 34]}]], "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\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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\nx : \u03b1\n\u22a2 x \u2208 {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} \u2194 x \u2208 {a | \u2203 i, f i b = a}", "state_after": "no goals"}, {"tactic": "exact IsGreatest.isLUB \u27e8(@exists_const (h\u03b1.some = h\u03b1.some) \u03b9 _).2 rfl, fun x \u27e8i, hi\u27e9 => hi.ge\u27e9", "annotated_tactic": ["exact <a>IsGreatest.isLUB</a> \u27e8(@<a>exists_const</a> (h\u03b1.some = h\u03b1.some) \u03b9 _).2 <a>rfl</a>, fun x \u27e8i, hi\u27e9 => hi.ge\u27e9", [{"full_name": "IsGreatest.isLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [281, 9], "def_end_pos": [281, 25]}, {"full_name": "exists_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [369, 17], "def_end_pos": [369, 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 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 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 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\n\u03b9 : Sort u_6\n\u03bc : Measure \u03b4\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\ng : \u03b4 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b)\n\u271d : Nontrivial \u03b1\nh\u03b1 : Nonempty \u03b1\nh\u03b9 : Nonempty \u03b9\np : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\ng_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1\nb : \u03b4\nh : \u00acb \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x)\n\u22a2 IsLUB {a | \u2203 i, Nonempty.some (_ : Nonempty \u03b1) = a} (Nonempty.some h\u03b1)", "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.forM_cons'", "start": [278, 9], "end": [280, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurableSet_Ico", "start": [594, 1], "end": [595, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_congr_measure_of_integrable", "start": [1605, 1], "end": [1631, 71], "traced_tactics": [{"tactic": "have h_int : \u2200 g : \u03b1 \u2192 E, Integrable g \u03bc \u2192 Integrable g \u03bc' := fun g hg =>\n  Integrable.of_measure_le_smul c' hc' h\u03bc'_le hg", "annotated_tactic": ["have h_int : \u2200 g : \u03b1 \u2192 E, <a>Integrable</a> g \u03bc \u2192 <a>Integrable</a> g \u03bc' := fun g hg =>\n    <a>Integrable.of_measure_le_smul</a> c' hc' h\u03bc'_le hg", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Integrable.of_measure_le_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [535, 9], "def_end_pos": [535, 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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\n\u22a2 setToFun \u03bc T hT f = 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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc' T hT' f"}, {"tactic": "apply hf\u03bc.induction (P := fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f)", "annotated_tactic": ["apply hf\u03bc.induction (P := fun f => <a>setToFun</a> \u03bc T hT f = <a>setToFun</a> \u03bc' T hT' f)", [{"full_name": "MeasureTheory.setToFun", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1270, 5], "def_end_pos": [1270, 13]}, {"full_name": "MeasureTheory.setToFun", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1270, 5], "def_end_pos": [1270, 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 : 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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc' T hT' f", "state_after": "case h_ind\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) (indicator s fun x => c)\n\ncase h_add\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Integrable f \u2192\n        Integrable g \u2192\n          (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) f \u2192\n            (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) g \u2192\n              (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) (f + g)\n\ncase h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 IsClosed {f | (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) \u2191\u2191f}\n\ncase h_ae\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984,\n    f =\u1d50[\u03bc] g \u2192\n      Integrable f \u2192\n        (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) f \u2192 (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) g"}, {"tactic": "intro c s hs h\u03bcs", "annotated_tactic": ["intro c s hs h\u03bcs", []], "state_before": "case h_ind\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) (indicator s fun x => c)", "state_after": "case h_ind\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 setToFun \u03bc T hT (indicator s fun x => c) = setToFun \u03bc' T hT' (indicator s fun x => c)"}, {"tactic": "have h\u03bc's : \u03bc' s \u2260 \u221e := by\n  refine' ((h\u03bc'_le s hs).trans_lt _).ne\n  rw [Measure.smul_apply, smul_eq_mul]\n  exact ENNReal.mul_lt_top hc' h\u03bcs.ne", "annotated_tactic": ["have h\u03bc's : \u03bc' s \u2260 \u221e := by\n      refine' ((h\u03bc'_le s hs).<a>trans_lt</a> _).<a>ne</a>\n      rw [<a>Measure.smul_apply</a>, <a>smul_eq_mul</a>]\n      exact <a>ENNReal.mul_lt_top</a> hc' h\u03bcs.ne", [{"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.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": "ENNReal.mul_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [612, 9], "def_end_pos": [612, 19]}]], "state_before": "case h_ind\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 setToFun \u03bc T hT (indicator s fun x => c) = setToFun \u03bc' T hT' (indicator s fun x => c)", "state_after": "case h_ind\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nh\u03bc's : \u2191\u2191\u03bc' s \u2260 \u22a4\n\u22a2 setToFun \u03bc T hT (indicator s fun x => c) = setToFun \u03bc' T hT' (indicator s fun x => c)"}, {"tactic": "rw [setToFun_indicator_const hT hs h\u03bcs.ne, setToFun_indicator_const hT' hs h\u03bc's]", "annotated_tactic": ["rw [<a>setToFun_indicator_const</a> hT hs h\u03bcs.ne, <a>setToFun_indicator_const</a> hT' hs h\u03bc's]", [{"full_name": "MeasureTheory.setToFun_indicator_const", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 33]}, {"full_name": "MeasureTheory.setToFun_indicator_const", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 33]}]], "state_before": "case h_ind\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\nh\u03bc's : \u2191\u2191\u03bc' s \u2260 \u22a4\n\u22a2 setToFun \u03bc T hT (indicator s fun x => c) = setToFun \u03bc' T hT' (indicator s fun x => c)", "state_after": "no goals"}, {"tactic": "refine' ((h\u03bc'_le s hs).trans_lt _).ne", "annotated_tactic": ["refine' ((h\u03bc'_le s hs).<a>trans_lt</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": "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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc' s \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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191(c' \u2022 \u03bc) s < \u22a4"}, {"tactic": "rw [Measure.smul_apply, smul_eq_mul]", "annotated_tactic": ["rw [<a>Measure.smul_apply</a>, <a>smul_eq_mul</a>]", [{"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]}]], "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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191(c' \u2022 \u03bc) s < \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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 c' * \u2191\u2191\u03bc s < \u22a4"}, {"tactic": "exact ENNReal.mul_lt_top hc' h\u03bcs.ne", "annotated_tactic": ["exact <a>ENNReal.mul_lt_top</a> hc' h\u03bcs.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": "\u03b1 : Type u_1\nE : Type u_2\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 c' * \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "intro f\u2082 g\u2082 _ hf\u2082 hg\u2082 h_eq_f h_eq_g", "annotated_tactic": ["intro f\u2082 g\u2082 _ hf\u2082 hg\u2082 h_eq_f h_eq_g", []], "state_before": "case h_add\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984,\n    Disjoint (Function.support f) (Function.support g) \u2192\n      Integrable f \u2192\n        Integrable g \u2192\n          (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) f \u2192\n            (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) g \u2192\n              (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) (f + g)", "state_after": "case h_add\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf\u2082 g\u2082 : \u03b1 \u2192 E\na\u271d : Disjoint (Function.support f\u2082) (Function.support g\u2082)\nhf\u2082 : Integrable f\u2082\nhg\u2082 : Integrable g\u2082\nh_eq_f : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082\nh_eq_g : setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082\n\u22a2 setToFun \u03bc T hT (f\u2082 + g\u2082) = setToFun \u03bc' T hT' (f\u2082 + g\u2082)"}, {"tactic": "rw [setToFun_add hT hf\u2082 hg\u2082, setToFun_add hT' (h_int f\u2082 hf\u2082) (h_int g\u2082 hg\u2082), h_eq_f, h_eq_g]", "annotated_tactic": ["rw [<a>setToFun_add</a> hT hf\u2082 hg\u2082, <a>setToFun_add</a> hT' (h_int f\u2082 hf\u2082) (h_int g\u2082 hg\u2082), h_eq_f, h_eq_g]", [{"full_name": "MeasureTheory.setToFun_add", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1365, 9], "def_end_pos": [1365, 21]}, {"full_name": "MeasureTheory.setToFun_add", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1365, 9], "def_end_pos": [1365, 21]}]], "state_before": "case h_add\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf\u2082 g\u2082 : \u03b1 \u2192 E\na\u271d : Disjoint (Function.support f\u2082) (Function.support g\u2082)\nhf\u2082 : Integrable f\u2082\nhg\u2082 : Integrable g\u2082\nh_eq_f : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082\nh_eq_g : setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082\n\u22a2 setToFun \u03bc T hT (f\u2082 + g\u2082) = setToFun \u03bc' T hT' (f\u2082 + g\u2082)", "state_after": "no goals"}, {"tactic": "refine' isClosed_eq (continuous_setToFun hT) _", "annotated_tactic": ["refine' <a>isClosed_eq</a> (<a>continuous_setToFun</a> hT) _", [{"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}, {"full_name": "MeasureTheory.continuous_setToFun", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1506, 9], "def_end_pos": [1506, 28]}]], "state_before": "case h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 IsClosed {f | (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) \u2191\u2191f}", "state_after": "case h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191f"}, {"tactic": "have :\n  (fun f : \u03b1 \u2192\u2081[\u03bc] E => setToFun \u03bc' T hT' f) = fun f : \u03b1 \u2192\u2081[\u03bc] E =>\n    setToFun \u03bc' T hT' ((h_int f (L1.integrable_coeFn f)).toL1 f) := by\n  ext1 f; exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm", "annotated_tactic": ["have :\n      (fun f : \u03b1 \u2192\u2081[\u03bc] E => <a>setToFun</a> \u03bc' T hT' f) = fun f : \u03b1 \u2192\u2081[\u03bc] E =>\n        <a>setToFun</a> \u03bc' T hT' ((h_int f (<a>L1.integrable_coeFn</a> f)).<a>toL1</a> f) := by\n      ext1 f; exact <a>setToFun_congr_ae</a> hT' (<a>Integrable.coeFn_toL1</a> _).<a>symm</a>", [{"full_name": "MeasureTheory.setToFun", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1270, 5], "def_end_pos": [1270, 13]}, {"full_name": "MeasureTheory.setToFun", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1270, 5], "def_end_pos": [1270, 13]}, {"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", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1402, 5], "def_end_pos": [1402, 9]}, {"full_name": "MeasureTheory.setToFun_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1418, 9], "def_end_pos": [1418, 26]}, {"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 h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191f", "state_after": "case h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nthis : (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))\n\u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191f"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nthis : (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))\n\u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191f", "state_after": "case h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nthis : (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))\n\u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))"}, {"tactic": "exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' h\u03bc'_le)", "annotated_tactic": ["exact (<a>continuous_setToFun</a> hT').<a>comp</a> (<a>continuous_L1_toL1</a> c' hc' h\u03bc'_le)", [{"full_name": "MeasureTheory.continuous_setToFun", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1506, 9], "def_end_pos": [1506, 28]}, {"full_name": "Continuous.comp", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1673, 9], "def_end_pos": [1673, 24]}, {"full_name": "MeasureTheory.continuous_L1_toL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1560, 9], "def_end_pos": [1560, 27]}]], "state_before": "case h_closed\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nthis : (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))\n\u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))", "state_after": "no goals"}, {"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\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))", "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\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\u00b9 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf\u271d : \u03b1 \u2192 E\nhf\u03bc : Integrable f\u271d\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToFun \u03bc' T hT' \u2191\u2191f = setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))"}, {"tactic": "exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm", "annotated_tactic": ["exact <a>setToFun_congr_ae</a> hT' (<a>Integrable.coeFn_toL1</a> _).<a>symm</a>", [{"full_name": "MeasureTheory.setToFun_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1418, 9], "def_end_pos": [1418, 26]}, {"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 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\u00b9 g : \u03b1 \u2192 E\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf\u271d : \u03b1 \u2192 E\nhf\u03bc : Integrable f\u271d\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToFun \u03bc' T hT' \u2191\u2191f = setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))", "state_after": "no goals"}, {"tactic": "intro f\u2082 g\u2082 hfg _ hf_eq", "annotated_tactic": ["intro f\u2082 g\u2082 hfg _ hf_eq", []], "state_before": "case h_ae\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984,\n    f =\u1d50[\u03bc] g \u2192\n      Integrable f \u2192\n        (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) f \u2192 (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) g", "state_after": "case h_ae\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf\u2082 g\u2082 : \u03b1 \u2192 E\nhfg : f\u2082 =\u1d50[\u03bc] g\u2082\na\u271d : Integrable f\u2082\nhf_eq : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082\n\u22a2 setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082"}, {"tactic": "have hfg' : f\u2082 =\u1d50[\u03bc'] g\u2082 := (Measure.absolutelyContinuous_of_le_smul h\u03bc'_le).ae_eq hfg", "annotated_tactic": ["have hfg' : f\u2082 =\u1d50[\u03bc'] g\u2082 := (<a>Measure.absolutelyContinuous_of_le_smul</a> h\u03bc'_le).<a>ae_eq</a> hfg", [{"full_name": "MeasureTheory.Measure.absolutelyContinuous_of_le_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2169, 9], "def_end_pos": [2169, 40]}, {"full_name": "MeasureTheory.Measure.AbsolutelyContinuous.ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2188, 9], "def_end_pos": [2188, 35]}]], "state_before": "case h_ae\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf\u2082 g\u2082 : \u03b1 \u2192 E\nhfg : f\u2082 =\u1d50[\u03bc] g\u2082\na\u271d : Integrable f\u2082\nhf_eq : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082\n\u22a2 setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082", "state_after": "case h_ae\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf\u2082 g\u2082 : \u03b1 \u2192 E\nhfg : f\u2082 =\u1d50[\u03bc] g\u2082\na\u271d : Integrable f\u2082\nhf_eq : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082\nhfg' : f\u2082 =\u1d50[\u03bc'] g\u2082\n\u22a2 setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082"}, {"tactic": "rw [\u2190 setToFun_congr_ae hT hfg, hf_eq, setToFun_congr_ae hT' hfg']", "annotated_tactic": ["rw [\u2190 <a>setToFun_congr_ae</a> hT hfg, hf_eq, <a>setToFun_congr_ae</a> hT' hfg']", [{"full_name": "MeasureTheory.setToFun_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1418, 9], "def_end_pos": [1418, 26]}, {"full_name": "MeasureTheory.setToFun_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1418, 9], "def_end_pos": [1418, 26]}]], "state_before": "case h_ae\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\n\u03bc' : Measure \u03b1\nc' : \u211d\u22650\u221e\nhc' : c' \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc' T C'\nf : \u03b1 \u2192 E\nhf\u03bc : Integrable f\nh_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g\nf\u2082 g\u2082 : \u03b1 \u2192 E\nhfg : f\u2082 =\u1d50[\u03bc] g\u2082\na\u271d : Integrable f\u2082\nhf_eq : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082\nhfg' : f\u2082 =\u1d50[\u03bc'] g\u2082\n\u22a2 setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.strictAnti_iff'", "start": [212, 1], "end": [214, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_smul_left", "start": [741, 1], "end": [743, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.cast_sub'", "start": [1175, 1], "end": [1200, 64], "traced_tactics": [{"tactic": "rw [sub'_one, Num.cast_toZNum, \u2190 Num.cast_to_nat, pred'_to_nat, \u2190 Nat.sub_one]", "annotated_tactic": ["rw [<a>sub'_one</a>, <a>Num.cast_toZNum</a>, \u2190 <a>Num.cast_to_nat</a>, <a>pred'_to_nat</a>, \u2190 <a>Nat.sub_one</a>]", [{"full_name": "PosNum.sub'_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [817, 9], "def_end_pos": [817, 17]}, {"full_name": "Num.cast_toZNum", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [785, 9], "def_end_pos": [785, 20]}, {"full_name": "Num.cast_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [301, 9], "def_end_pos": [301, 20]}, {"full_name": "PosNum.pred'_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [533, 9], "def_end_pos": [533, 21]}, {"full_name": "Nat.sub_one", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [374, 19], "def_end_pos": [374, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\n\u22a2 \u2191(sub' a 1) = \u2191a - \u21911", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\n\u22a2 \u2191(\u2191a - 1) = \u2191a - \u21911"}, {"tactic": "simp [PosNum.cast_pos]", "annotated_tactic": ["simp [<a>PosNum.cast_pos</a>]", [{"full_name": "PosNum.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [681, 9], "def_end_pos": [681, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\n\u22a2 \u2191(\u2191a - 1) = \u2191a - \u21911", "state_after": "no goals"}, {"tactic": "rw [one_sub', Num.cast_toZNumNeg, \u2190 neg_sub, neg_inj, \u2190 Num.cast_to_nat, pred'_to_nat,\n    \u2190 Nat.sub_one]", "annotated_tactic": ["rw [<a>one_sub'</a>, <a>Num.cast_toZNumNeg</a>, \u2190 <a>neg_sub</a>, <a>neg_inj</a>, \u2190 <a>Num.cast_to_nat</a>, <a>pred'_to_nat</a>,\n        \u2190 <a>Nat.sub_one</a>]", [{"full_name": "PosNum.one_sub'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [820, 9], "def_end_pos": [820, 17]}, {"full_name": "Num.cast_toZNumNeg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [791, 9], "def_end_pos": [791, 23]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [256, 3], "def_end_pos": [256, 14]}, {"full_name": "Num.cast_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [301, 9], "def_end_pos": [301, 20]}, {"full_name": "PosNum.pred'_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [533, 9], "def_end_pos": [533, 21]}, {"full_name": "Nat.sub_one", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [374, 19], "def_end_pos": [374, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nb : PosNum\n\u22a2 \u2191(sub' 1 b) = \u21911 - \u2191b", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nb : PosNum\n\u22a2 \u2191(\u2191b - 1) = \u2191b - \u21911"}, {"tactic": "simp [PosNum.cast_pos]", "annotated_tactic": ["simp [<a>PosNum.cast_pos</a>]", [{"full_name": "PosNum.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [681, 9], "def_end_pos": [681, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nb : PosNum\n\u22a2 \u2191(\u2191b - 1) = \u2191b - \u21911", "state_after": "no goals"}, {"tactic": "rw [sub', ZNum.cast_bit0, cast_sub' a b]", "annotated_tactic": ["rw [<a>sub'</a>, <a>ZNum.cast_bit0</a>, cast_sub' a b]", [{"full_name": "PosNum.sub'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 9]}, {"full_name": "ZNum.cast_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit0 a) (bit0 b)) = \u2191(bit0 a) - \u2191(bit0 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b)"}, {"tactic": "have : ((a + -b + (a + -b) : \u2124) : \u03b1) = a + a + (-b + -b) := by simp [add_left_comm]", "annotated_tactic": ["have : ((a + -b + (a + -b) : \u2124) : \u03b1) = a + a + (-b + -b) := by simp [<a>add_left_comm</a>]", [{"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b)\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b)"}, {"tactic": "simpa [_root_.bit0, sub_eq_add_neg] using this", "annotated_tactic": ["simpa [<a>_root_.bit0</a>, <a>sub_eq_add_neg</a>] using this", [{"full_name": "bit0", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [36, 34], "def_end_pos": [36, 38]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b)\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b)", "state_after": "no goals"}, {"tactic": "simp [add_left_comm]", "annotated_tactic": ["simp [<a>add_left_comm</a>]", [{"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b)", "state_after": "no goals"}, {"tactic": "rw [sub', ZNum.cast_bitm1, cast_sub' a b]", "annotated_tactic": ["rw [<a>sub'</a>, <a>ZNum.cast_bitm1</a>, cast_sub' a b]", [{"full_name": "PosNum.sub'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 9]}, {"full_name": "ZNum.cast_bitm1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1142, 9], "def_end_pos": [1142, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit0 a) (bit1 b)) = \u2191(bit0 a) - \u2191(bit1 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b)"}, {"tactic": "have : ((-b + (a + (-b + -1)) : \u2124) : \u03b1) = (a + -1 + (-b + -b) : \u2124) := by\n  simp [add_comm, add_left_comm]", "annotated_tactic": ["have : ((-b + (a + (-b + -1)) : \u2124) : \u03b1) = (a + -1 + (-b + -b) : \u2124) := by\n      simp [<a>add_comm</a>, <a>add_left_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b))\n\u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b)"}, {"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this", "annotated_tactic": ["simpa [<a>_root_.bit1</a>, <a>_root_.bit0</a>, <a>sub_eq_add_neg</a>] using this", [{"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": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b))\n\u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b)", "state_after": "no goals"}, {"tactic": "simp [add_comm, add_left_comm]", "annotated_tactic": ["simp [<a>add_comm</a>, <a>add_left_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b))", "state_after": "no goals"}, {"tactic": "rw [sub', ZNum.cast_bit1, cast_sub' a b]", "annotated_tactic": ["rw [<a>sub'</a>, <a>ZNum.cast_bit1</a>, cast_sub' a b]", [{"full_name": "PosNum.sub'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 9]}, {"full_name": "ZNum.cast_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit1 a) (bit0 b)) = \u2191(bit1 a) - \u2191(bit0 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b)"}, {"tactic": "have : ((-b + (a + (-b + 1)) : \u2124) : \u03b1) = (a + 1 + (-b + -b) : \u2124) := by\n  simp [add_comm, add_left_comm]", "annotated_tactic": ["have : ((-b + (a + (-b + 1)) : \u2124) : \u03b1) = (a + 1 + (-b + -b) : \u2124) := by\n      simp [<a>add_comm</a>, <a>add_left_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b))\n\u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b)"}, {"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this", "annotated_tactic": ["simpa [<a>_root_.bit1</a>, <a>_root_.bit0</a>, <a>sub_eq_add_neg</a>] using this", [{"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": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b))\n\u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b)", "state_after": "no goals"}, {"tactic": "simp [add_comm, add_left_comm]", "annotated_tactic": ["simp [<a>add_comm</a>, <a>add_left_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b))", "state_after": "no goals"}, {"tactic": "rw [sub', ZNum.cast_bit0, cast_sub' a b]", "annotated_tactic": ["rw [<a>sub'</a>, <a>ZNum.cast_bit0</a>, cast_sub' a b]", [{"full_name": "PosNum.sub'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 9]}, {"full_name": "ZNum.cast_bit0", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit1 a) (bit1 b)) = \u2191(bit1 a) - \u2191(bit1 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b)"}, {"tactic": "have : ((-b + (a + -b) : \u2124) : \u03b1) = a + (-b + -b) := by simp [add_left_comm]", "annotated_tactic": ["have : ((-b + (a + -b) : \u2124) : \u03b1) = a + (-b + -b) := by simp [<a>add_left_comm</a>]", [{"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b)", "state_after": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b)\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b)"}, {"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this", "annotated_tactic": ["simpa [<a>_root_.bit1</a>, <a>_root_.bit0</a>, <a>sub_eq_add_neg</a>] using this", [{"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": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b)\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b)", "state_after": "no goals"}, {"tactic": "simp [add_left_comm]", "annotated_tactic": ["simp [<a>add_left_comm</a>]", [{"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b)", "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.map_eq", "start": [756, 1], "end": [757, 39], "traced_tactics": [{"tactic": "simpa using mapAux_of_valid f [] s.1", "annotated_tactic": ["simpa using <a>mapAux_of_valid</a> f [] s.1", [{"full_name": "String.mapAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [748, 9], "def_end_pos": [748, 24]}]], "state_before": "f : Char \u2192 Char\ns : String\n\u22a2 map f s = { data := List.map f s.data }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Quotient.lean", "full_name": "MulAction.orbitZpowersEquiv_symm_apply'", "start": [160, 1], "end": [164, 56], "traced_tactics": [{"tactic": "rw [orbitZpowersEquiv_symm_apply, ZMod.coe_int_cast]", "annotated_tactic": ["rw [<a>orbitZpowersEquiv_symm_apply</a>, <a>ZMod.coe_int_cast</a>]", [{"full_name": "MulAction.orbitZpowersEquiv_symm_apply", "def_path": "Mathlib/Data/ZMod/Quotient.lean", "def_pos": [153, 9], "def_end_pos": [153, 37]}, {"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\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\na : \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 \u2191(orbitZpowersEquiv a b).symm \u2191k =\n    { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022\n      { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) }", "state_after": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\na : \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 { val := a, property := (_ : a \u2208 zpowers a) } ^ (k % \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b)) \u2022\n      { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) } =\n    { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022\n      { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) }"}, {"tactic": "exact Subtype.ext (zpow_smul_mod_minimalPeriod _ _ k)", "annotated_tactic": ["exact <a>Subtype.ext</a> (<a>zpow_smul_mod_minimalPeriod</a> _ _ k)", [{"full_name": "Subtype.ext", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [65, 19], "def_end_pos": [65, 22]}, {"full_name": "MulAction.zpow_smul_mod_minimalPeriod", "def_path": "Mathlib/Dynamics/PeriodicPts.lean", "def_pos": [663, 9], "def_end_pos": [663, 36]}]], "state_before": "n : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\na : \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 { val := a, property := (_ : a \u2208 zpowers a) } ^ (k % \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b)) \u2022\n      { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) } =\n    { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022\n      { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers 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": "WithTop.image_coe_Ioo", "start": [124, 1], "end": [126, 94], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ioo, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ioo_subset_Iio_self <| Iio_subset_Iio le_top)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ioo</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Subset.trans</a> <a>Ioo_subset_Iio_self</a> <| <a>Iio_subset_Iio</a> <a>le_top</a>)]", [{"full_name": "WithTop.preimage_coe_Ioo", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [71, 9], "def_end_pos": [71, 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.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.Ioo_subset_Iio_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [523, 9], "def_end_pos": [523, 28]}, {"full_name": "Set.Iio_subset_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 23]}, {"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\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Ioo a b = Ioo \u2191a \u2191b", "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.diff_erase", "start": [1674, 1], "end": [1675, 36], "traced_tactics": [{"tactic": "rw [\u2190 diff_cons_right, diff_cons]", "annotated_tactic": ["rw [\u2190 <a>diff_cons_right</a>, <a>diff_cons</a>]", [{"full_name": "List.diff_cons_right", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1671, 9], "def_end_pos": [1671, 24]}, {"full_name": "List.diff_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1668, 17], "def_end_pos": [1668, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\n\u22a2 List.erase (List.diff l\u2081 l\u2082) a = List.diff (List.erase l\u2081 a) l\u2082", "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_mk", "start": [227, 1], "end": [230, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.lintegral_prod_of_measurable", "start": [821, 1], "end": [846, 94], "traced_tactics": [{"tactic": "have m := @measurable_prod_mk_left", "annotated_tactic": ["have m := @<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": "\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\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "refine' Measurable.ennreal_induction\n  (P := fun f => \u222b\u207b z, f z \u2202\u03bc.prod \u03bd = \u222b\u207b x, \u222b\u207b y, f (x, y) \u2202\u03bd \u2202\u03bc) _ _ _", "annotated_tactic": ["refine' <a>Measurable.ennreal_induction</a>\n    (P := fun f => \u222b\u207b z, f z \u2202\u03bc.prod \u03bd = \u222b\u207b x, \u222b\u207b y, f (x, y) \u2202\u03bd \u2202\u03bc) _ _ _", [{"full_name": "Measurable.ennreal_induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 37]}]], "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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_1\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n    MeasurableSet s \u2192\n      (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (indicator s fun x => c)\n\ncase refine'_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\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n    Disjoint (support f) (support g) \u2192\n      Measurable f \u2192\n        Measurable g \u2192\n          (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) f \u2192\n            (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) g \u2192\n              (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f + g)\n\ncase refine'_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\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n    (\u2200 (n : \u2115), Measurable (f n)) \u2192\n      Monotone f \u2192\n        (\u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)) \u2192\n          (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) fun x => \u2a06 n, f n x"}, {"tactic": "intro c s hs", "annotated_tactic": ["intro c s hs", []], "state_before": "case refine'_1\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n    MeasurableSet s \u2192\n      (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (indicator s fun x => c)", "state_after": "case refine'_1\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "conv_rhs =>\n  enter [2, x, 2, y]\n  rw [\u2190 indicator_comp_right, const_def, const_comp, \u2190 const_def]", "annotated_tactic": ["conv_rhs =>\n      enter [2, x, 2, y]\n      rw [\u2190 <a>indicator_comp_right</a>, <a>const_def</a>, <a>const_comp</a>, \u2190 <a>const_def</a>]", [{"full_name": "Set.indicator_comp_right", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [245, 3], "def_end_pos": [245, 14]}, {"full_name": "Function.const_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}, {"full_name": "Function.const_comp", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 19]}, {"full_name": "Function.const_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}]], "state_before": "case refine'_1\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_1\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc"}, {"tactic": "conv_rhs =>\n  enter [2, x]\n  rw [lintegral_indicator _ (m (x := x) hs), lintegral_const,\n    Measure.restrict_apply MeasurableSet.univ, univ_inter]", "annotated_tactic": ["conv_rhs =>\n      enter [2, x]\n      rw [<a>lintegral_indicator</a> _ (m (x := x) hs), <a>lintegral_const</a>,\n        <a>Measure.restrict_apply</a> <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"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'_1\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\nm :\n  \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n    Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_1\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), c * \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc"}, {"tactic": "simp [hs, lintegral_const_mul, measurable_measure_prod_mk_left (\u03bd := \u03bd) hs, prod_apply]", "annotated_tactic": ["simp [hs, <a>lintegral_const_mul</a>, <a>measurable_measure_prod_mk_left</a> (\u03bd := \u03bd) hs, <a>prod_apply</a>]", [{"full_name": "MeasureTheory.lintegral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [671, 9], "def_end_pos": [671, 28]}, {"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": "MeasureTheory.Measure.prod_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 19]}]], "state_before": "case refine'_1\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), c * \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rintro f g - hf _ h2f h2g", "annotated_tactic": ["rintro f g - hf _ h2f h2g", []], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n    Disjoint (support f) (support g) \u2192\n      Measurable f \u2192\n        Measurable g \u2192\n          (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) f \u2192\n            (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) g \u2192\n              (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f + g)", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp only [Pi.add_apply]", "annotated_tactic": ["simp only [<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 refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "conv_lhs => rw [lintegral_add_left hf]", "annotated_tactic": ["conv_lhs => rw [<a>lintegral_add_left</a> hf]", [{"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 refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "conv_rhs => enter [2, x]; erw [lintegral_add_left (hf.comp (m (x := x)))]", "annotated_tactic": ["conv_rhs => enter [2, x]; erw [<a>lintegral_add_left</a> (hf.comp (m (x := x)))]", [{"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 refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (a : \u03b2), (f \u2218 Prod.mk x) a \u2202\u03bd + \u222b\u207b (a : \u03b2), g (x, a) \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp [lintegral_add_left, Measurable.lintegral_prod_right', hf, h2f, h2g]", "annotated_tactic": ["simp [<a>lintegral_add_left</a>, <a>Measurable.lintegral_prod_right'</a>, hf, h2f, h2g]", [{"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.lintegral_prod_right'", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [240, 9], "def_end_pos": [240, 41]}]], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =\n    \u222b\u207b (x : \u03b1), \u222b\u207b (a : \u03b2), (f \u2218 Prod.mk x) a \u2202\u03bd + \u222b\u207b (a : \u03b2), g (x, a) \u2202\u03bd \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro f hf h2f h3f", "annotated_tactic": ["intro f hf h2f h3f", []], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n    (\u2200 (n : \u2115), Measurable (f n)) \u2192\n      Monotone f \u2192\n        (\u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)) \u2192\n          (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) fun x => \u2a06 n, f n x", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "have kf : \u2200 x n, Measurable fun y => f n (x, y) := fun x n => (hf n).comp m", "annotated_tactic": ["have kf : \u2200 x n, <a>Measurable</a> fun y => f n (x, y) := fun x n => (hf n).<a>comp</a> m", [{"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]}]], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "have k2f : \u2200 x, Monotone fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y)", "annotated_tactic": ["have k2f : \u2200 x, <a>Monotone</a> fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y)", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}]], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "have lf : \u2200 n, Measurable fun x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun n => (hf n).lintegral_prod_right'", "annotated_tactic": ["have lf : \u2200 n, <a>Measurable</a> fun x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun n => (hf n).<a>lintegral_prod_right'</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Measurable.lintegral_prod_right'", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [240, 9], "def_end_pos": [240, 41]}]], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nlf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "have l2f : Monotone fun n x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun i j hij x =>\n  lintegral_mono (k2f x hij)", "annotated_tactic": ["have l2f : <a>Monotone</a> fun n x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun i j hij x =>\n      <a>lintegral_mono</a> (k2f x hij)", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"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'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nlf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nlf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\nl2f : Monotone fun n x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp only [lintegral_iSup hf h2f, lintegral_iSup (kf _), k2f, lintegral_iSup lf l2f, h3f]", "annotated_tactic": ["simp only [<a>lintegral_iSup</a> hf h2f, <a>lintegral_iSup</a> (kf _), k2f, <a>lintegral_iSup</a> lf l2f, h3f]", [{"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "MeasureTheory.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}]], "state_before": "case refine'_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\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\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nlf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\nl2f : Monotone fun n x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\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.boundedBy_eq", "start": [856, 1], "end": [858, 72], "traced_tactics": [{"tactic": "rw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd]", "annotated_tactic": ["rw [<a>boundedBy_eq_ofFunction</a> m_empty, <a>ofFunction_eq</a> s m_mono m_subadd]", [{"full_name": "MeasureTheory.OuterMeasure.boundedBy_eq_ofFunction", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [842, 9], "def_end_pos": [842, 32]}, {"full_name": "MeasureTheory.OuterMeasure.ofFunction_eq", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [705, 9], "def_end_pos": [705, 22]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nm_empty : m \u2205 = 0\nm_mono : \u2200 \u2983t : Set \u03b1\u2984, s \u2286 t \u2192 m s \u2264 m t\nm_subadd : \u2200 (s : \u2115 \u2192 Set \u03b1), m (\u22c3 i, s i) \u2264 \u2211' (i : \u2115), m (s i)\n\u22a2 \u2191(boundedBy m) s = m s", "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.find?_eq", "start": [125, 9], "end": [126, 87], "traced_tactics": [{"tactic": "simp [find?_eq_findEntry?]", "annotated_tactic": ["simp [<a>find?_eq_findEntry?</a>]", [{"full_name": "Std.AssocList.find?_eq_findEntry?", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [121, 9], "def_end_pos": [121, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : BEq \u03b1\na : \u03b1\nl : AssocList \u03b1 \u03b2\n\u22a2 find? a l = Option.map (fun x => x.snd) (List.find? (fun x => x.fst == a) (toList l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.FinMeasAdditive.map_empty_eq_zero", "start": [144, 1], "end": [150, 34], "traced_tactics": [{"tactic": "have h_empty : \u03bc \u2205 \u2260 \u221e := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne", "annotated_tactic": ["have h_empty : \u03bc \u2205 \u2260 \u221e := (measure_empty.le.trans_lt <a>ENNReal.coe_lt_top</a>).<a>ne</a>", [{"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": "\u03b1 : Type u_1\nE : Type u_2\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nhT : FinMeasAdditive \u03bc T\n\u22a2 T \u2205 = 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\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nhT : FinMeasAdditive \u03bc T\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\n\u22a2 T \u2205 = 0"}, {"tactic": "specialize hT \u2205 \u2205 MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty \u2205)", "annotated_tactic": ["specialize hT \u2205 \u2205 <a>MeasurableSet.empty</a> <a>MeasurableSet.empty</a> h_empty h_empty (<a>Set.inter_empty</a> \u2205)", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "Set.inter_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [931, 9], "def_end_pos": [931, 20]}]], "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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nhT : FinMeasAdditive \u03bc T\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\n\u22a2 T \u2205 = 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\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nhT : T (\u2205 \u222a \u2205) = T \u2205 + T \u2205\n\u22a2 T \u2205 = 0"}, {"tactic": "rw [Set.union_empty] at hT", "annotated_tactic": ["rw [<a>Set.union_empty</a>] at hT", [{"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\nE : Type u_2\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nhT : T (\u2205 \u222a \u2205) = T \u2205 + T \u2205\n\u22a2 T \u2205 = 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\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nhT : T \u2205 = T \u2205 + T \u2205\n\u22a2 T \u2205 = 0"}, {"tactic": "nth_rw 1 [\u2190 add_zero (T \u2205)] at hT", "annotated_tactic": ["nth_rw 1 [\u2190 <a>add_zero</a> (T \u2205)] at hT", [{"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\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nhT : T \u2205 = T \u2205 + T \u2205\n\u22a2 T \u2205 = 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\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nhT : T \u2205 + 0 = T \u2205 + T \u2205\n\u22a2 T \u2205 = 0"}, {"tactic": "exact (add_left_cancel hT).symm", "annotated_tactic": ["exact (<a>add_left_cancel</a> hT).<a>symm</a>", [{"full_name": "add_left_cancel", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [171, 3], "def_end_pos": [171, 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": "\u03b1 : Type u_1\nE : Type u_2\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\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : AddCommMonoid \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\n\u03b2 : Type u_8\ninst\u271d : AddCancelMonoid \u03b2\nT : Set \u03b1 \u2192 \u03b2\nh_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nhT : T \u2205 + 0 = T \u2205 + T \u2205\n\u22a2 T \u2205 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "AntilipschitzWith.hausdorffMeasure_preimage_le", "start": [828, 1], "end": [857, 59], "traced_tactics": [{"tactic": "rcases eq_or_ne K 0 with (rfl | h0)", "annotated_tactic": ["rcases <a>eq_or_ne</a> K 0 with (rfl | h0)", [{"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\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inl\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n\ncase inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "have hKd0 : (K : \u211d\u22650\u221e) ^ d \u2260 0 := by simp [h0]", "annotated_tactic": ["have hKd0 : (K : \u211d\u22650\u221e) ^ d \u2260 0 := by simp [h0]", []], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "have hKd : (K : \u211d\u22650\u221e) ^ d \u2260 \u221e := by simp [hd]", "annotated_tactic": ["have hKd : (K : \u211d\u22650\u221e) ^ d \u2260 \u221e := by simp [hd]", []], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hKd0 hKd,\n  \u2190 ENNReal.tsum_mul_left]", "annotated_tactic": ["simp only [<a>hausdorffMeasure_apply</a>, <a>ENNReal.mul_iSup</a>, <a>ENNReal.mul_iInf_of_ne</a> hKd0 hKd,\n    \u2190 <a>ENNReal.tsum_mul_left</a>]", [{"full_name": "MeasureTheory.Measure.hausdorffMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [591, 9], "def_end_pos": [591, 31]}, {"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}, {"full_name": "ENNReal.mul_iInf_of_ne", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2527, 9], "def_end_pos": [2527, 23]}, {"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u22a2 \u2a06 r,\n      \u2a06 (_ : 0 < r),\n        \u2a05 t,\n          \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 i,\n      \u2a06 (_ : 0 < i),\n        \u2a05 i_1,\n          \u2a05 (_ : s \u2286 \u22c3 n, i_1 n),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i), \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d"}, {"tactic": "refine' iSup\u2082_le fun \u03b5 \u03b50 => _", "annotated_tactic": ["refine' <a>iSup\u2082_le</a> fun \u03b5 \u03b50 => _", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u22a2 \u2a06 r,\n      \u2a06 (_ : 0 < r),\n        \u2a05 t,\n          \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 i,\n      \u2a06 (_ : 0 < i),\n        \u2a05 i_1,\n          \u2a05 (_ : s \u2286 \u22c3 n, i_1 n),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i), \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 i,\n      \u2a06 (_ : 0 < i),\n        \u2a05 i_1,\n          \u2a05 (_ : s \u2286 \u22c3 n, i_1 n),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i), \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d"}, {"tactic": "refine' le_iSup\u2082_of_le (\u03b5 / K) (by simp [\u03b50.ne']) _", "annotated_tactic": ["refine' <a>le_iSup\u2082_of_le</a> (\u03b5 / K) (by simp [\u03b50.ne']) _", [{"full_name": "le_iSup\u2082_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [865, 9], "def_end_pos": [865, 23]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 i,\n      \u2a06 (_ : 0 < i),\n        \u2a05 i_1,\n          \u2a05 (_ : s \u2286 \u22c3 n, i_1 n),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i), \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a05 i,\n      \u2a05 (_ : s \u2286 \u22c3 n, i n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (i n) \u2264 \u03b5 / \u2191K), \u2211' (i_1 : \u2115), \u2a06 (_ : Set.Nonempty (i i_1)), \u2191K ^ d * diam (i i_1) ^ d"}, {"tactic": "refine' le_iInf\u2082 fun t hst => le_iInf fun ht\u03b5 => _", "annotated_tactic": ["refine' <a>le_iInf\u2082</a> fun t hst => <a>le_iInf</a> fun ht\u03b5 => _", [{"full_name": "le_iInf\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [887, 9], "def_end_pos": [887, 17]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a05 i,\n      \u2a05 (_ : s \u2286 \u22c3 n, i n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (i n) \u2264 \u03b5 / \u2191K), \u2211' (i_1 : \u2115), \u2a06 (_ : Set.Nonempty (i i_1)), \u2191K ^ d * diam (i i_1) ^ d", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nhst : s \u2286 \u22c3 n, t n\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d"}, {"tactic": "replace hst : f \u207b\u00b9' s \u2286 _ := preimage_mono hst", "annotated_tactic": ["replace hst : f \u207b\u00b9' s \u2286 _ := <a>preimage_mono</a> hst", [{"full_name": "Set.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nhst : s \u2286 \u22c3 n, t n\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 f \u207b\u00b9' \u22c3 n, t n\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d"}, {"tactic": "rw [preimage_iUnion] at hst", "annotated_tactic": ["rw [<a>preimage_iUnion</a>] at hst", [{"full_name": "Set.preimage_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1854, 9], "def_end_pos": [1854, 24]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 f \u207b\u00b9' \u22c3 n, t n\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d", "state_after": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d"}, {"tactic": "refine' iInf\u2082_le_of_le _ hst (iInf_le_of_le (fun n => _) _)", "annotated_tactic": ["refine' <a>iInf\u2082_le_of_le</a> _ hst (<a>iInf_le_of_le</a> (fun n => _) _)", [{"full_name": "iInf\u2082_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [870, 9], "def_end_pos": [870, 23]}, {"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}]], "state_before": "case inr\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\n\u22a2 \u2a05 t,\n      \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),\n        \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d", "state_after": "case inr.refine'_1\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\n\u22a2 diam (f \u207b\u00b9' t n) \u2264 \u03b5\n\ncase inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (f \u207b\u00b9' t n)), diam (f \u207b\u00b9' t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d"}, {"tactic": "rcases eq_empty_or_nonempty (f \u207b\u00b9' s) with (hs | \u27e8x, hx\u27e9)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> (f \u207b\u00b9' s) with (hs | \u27e8x, hx\u27e9)", [{"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 inl\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inl.inl\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nhs : f \u207b\u00b9' s = \u2205\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n\ncase inl.inr.intro\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "have : f \u207b\u00b9' s = {x} := by\n  haveI : Subsingleton X := hf.subsingleton\n  have : (f \u207b\u00b9' s).Subsingleton := subsingleton_univ.anti (subset_univ _)\n  exact (subsingleton_iff_singleton hx).1 this", "annotated_tactic": ["have : f \u207b\u00b9' s = {x} := by\n      haveI : <a>Subsingleton</a> X := hf.subsingleton\n      have : (f \u207b\u00b9' s).<a>Subsingleton</a> := subsingleton_univ.anti (<a>subset_univ</a> _)\n      exact (<a>subsingleton_iff_singleton</a> hx).1 this", [{"full_name": "Subsingleton", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [867, 7], "def_end_pos": [867, 19]}, {"full_name": "Set.Subsingleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2356, 15], "def_end_pos": [2356, 27]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}, {"full_name": "Set.subsingleton_iff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2385, 9], "def_end_pos": [2385, 35]}]], "state_before": "case inl.inr.intro\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inl.inr.intro\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case inl.inr.intro\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inl.inr.intro\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "rcases eq_or_lt_of_le hd with (rfl | h'd)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hd with (rfl | h'd)", [{"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 inl.inr.intro\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inl.inr.intro.inl\n\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\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u21910 ^ 0 * \u2191\u2191\u03bcH[0] s\n\ncase inl.inr.intro.inr\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nh'd : 0 < d\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "simp only [hs, measure_empty, zero_le]", "annotated_tactic": ["simp only [hs, <a>measure_empty</a>, <a>zero_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": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case inl.inl\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nhs : f \u207b\u00b9' s = \u2205\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "no goals"}, {"tactic": "haveI : Subsingleton X := hf.subsingleton", "annotated_tactic": ["haveI : <a>Subsingleton</a> X := hf.subsingleton", [{"full_name": "Subsingleton", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [867, 7], "def_end_pos": [867, 19]}]], "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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 f \u207b\u00b9' s = {x}", "state_after": "\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : Subsingleton X\n\u22a2 f \u207b\u00b9' s = {x}"}, {"tactic": "have : (f \u207b\u00b9' s).Subsingleton := subsingleton_univ.anti (subset_univ _)", "annotated_tactic": ["have : (f \u207b\u00b9' s).<a>Subsingleton</a> := subsingleton_univ.anti (<a>subset_univ</a> _)", [{"full_name": "Set.Subsingleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2356, 15], "def_end_pos": [2356, 27]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : Subsingleton X\n\u22a2 f \u207b\u00b9' s = {x}", "state_after": "\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis\u271d : Subsingleton X\nthis : Set.Subsingleton (f \u207b\u00b9' s)\n\u22a2 f \u207b\u00b9' s = {x}"}, {"tactic": "exact (subsingleton_iff_singleton hx).1 this", "annotated_tactic": ["exact (<a>subsingleton_iff_singleton</a> hx).1 this", [{"full_name": "Set.subsingleton_iff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2385, 9], "def_end_pos": [2385, 35]}]], "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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis\u271d : Subsingleton X\nthis : Set.Subsingleton (f \u207b\u00b9' s)\n\u22a2 f \u207b\u00b9' s = {x}", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.rpow_zero, one_mul, mul_zero]", "annotated_tactic": ["simp only [<a>ENNReal.rpow_zero</a>, <a>one_mul</a>, <a>mul_zero</a>]", [{"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": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case inl.inr.intro.inl\n\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\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u21910 ^ 0 * \u2191\u2191\u03bcH[0] s", "state_after": "case inl.inr.intro.inl\n\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\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u2191\u2191\u03bcH[0] s"}, {"tactic": "rw [hausdorffMeasure_zero_singleton]", "annotated_tactic": ["rw [<a>hausdorffMeasure_zero_singleton</a>]", [{"full_name": "MeasureTheory.Measure.hausdorffMeasure_zero_singleton", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [668, 9], "def_end_pos": [668, 40]}]], "state_before": "case inl.inr.intro.inl\n\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\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u2191\u2191\u03bcH[0] s", "state_after": "case inl.inr.intro.inl\n\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\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s"}, {"tactic": "exact one_le_hausdorffMeasure_zero_of_nonempty \u27e8f x, hx\u27e9", "annotated_tactic": ["exact <a>one_le_hausdorffMeasure_zero_of_nonempty</a> \u27e8f x, hx\u27e9", [{"full_name": "MeasureTheory.Measure.one_le_hausdorffMeasure_zero_of_nonempty", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [693, 9], "def_end_pos": [693, 49]}]], "state_before": "case inl.inr.intro.inl\n\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\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s", "state_after": "no goals"}, {"tactic": "haveI := noAtoms_hausdorff X h'd", "annotated_tactic": ["haveI := <a>noAtoms_hausdorff</a> X h'd", [{"full_name": "MeasureTheory.Measure.noAtoms_hausdorff", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [656, 9], "def_end_pos": [656, 26]}]], "state_before": "case inl.inr.intro.inr\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nh'd : 0 < d\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "case inl.inr.intro.inr\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis\u271d : f \u207b\u00b9' s = {x}\nh'd : 0 < d\nthis : NoAtoms \u03bcH[d]\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s"}, {"tactic": "simp only [zero_le, measure_singleton]", "annotated_tactic": ["simp only [<a>zero_le</a>, <a>measure_singleton</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": "MeasureTheory.NoAtoms.measure_singleton", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3103, 3], "def_end_pos": [3103, 20]}]], "state_before": "case inl.inr.intro.inr\n\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\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis\u271d : f \u207b\u00b9' s = {x}\nh'd : 0 < d\nthis : NoAtoms \u03bcH[d]\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s", "state_after": "no goals"}, {"tactic": "simp [h0]", "annotated_tactic": ["simp [h0]", []], "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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\n\u22a2 \u2191K ^ d \u2260 0", "state_after": "no goals"}, {"tactic": "simp [hd]", "annotated_tactic": ["simp [hd]", []], "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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\n\u22a2 \u2191K ^ d \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp [\u03b50.ne']", "annotated_tactic": ["simp [\u03b50.ne']", []], "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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 0 < \u03b5 / \u2191K", "state_after": "no goals"}, {"tactic": "exact (hf.ediam_preimage_le _).trans (ENNReal.mul_le_of_le_div' <| ht\u03b5 n)", "annotated_tactic": ["exact (hf.ediam_preimage_le _).<a>trans</a> (<a>ENNReal.mul_le_of_le_div'</a> <| ht\u03b5 n)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "ENNReal.mul_le_of_le_div'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1654, 9], "def_end_pos": [1654, 26]}]], "state_before": "case inr.refine'_1\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\n\u22a2 diam (f \u207b\u00b9' t n) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "refine' ENNReal.tsum_le_tsum fun n => iSup_le_iff.2 fun hft => _", "annotated_tactic": ["refine' <a>ENNReal.tsum_le_tsum</a> fun n => <a>iSup_le_iff</a>.2 fun hft => _", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "iSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [964, 9], "def_end_pos": [964, 20]}]], "state_before": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (f \u207b\u00b9' t n)), diam (f \u207b\u00b9' t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d", "state_after": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2a06 (_ : Set.Nonempty (t n)), \u2191K ^ d * diam (t n) ^ d"}, {"tactic": "simp only [nonempty_of_nonempty_preimage hft, ciSup_pos]", "annotated_tactic": ["simp only [<a>nonempty_of_nonempty_preimage</a> hft, <a>ciSup_pos</a>]", [{"full_name": "Set.nonempty_of_nonempty_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [188, 9], "def_end_pos": [188, 38]}, {"full_name": "ciSup_pos", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [875, 9], "def_end_pos": [875, 18]}]], "state_before": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2a06 (_ : Set.Nonempty (t n)), \u2191K ^ d * diam (t n) ^ d", "state_after": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2191K ^ d * diam (t n) ^ d"}, {"tactic": "rw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ hd]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.mul_rpow_of_nonneg</a> _ _ hd]", [{"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": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2191K ^ d * diam (t n) ^ d", "state_after": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 (\u2191K * diam (t n)) ^ d"}, {"tactic": "exact ENNReal.rpow_le_rpow (hf.ediam_preimage_le _) hd", "annotated_tactic": ["exact <a>ENNReal.rpow_le_rpow</a> (hf.ediam_preimage_le _) hd", [{"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": "case inr.refine'_2\n\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\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 (\u2191K * diam (t n)) ^ d", "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\u2082", "start": [724, 1], "end": [726, 55], "traced_tactics": [{"tactic": "simp only [Tape.mk\u2082, Tape.map_mk', ListBlank.map_mk]", "annotated_tactic": ["simp only [<a>Tape.mk\u2082</a>, <a>Tape.map_mk'</a>, <a>ListBlank.map_mk</a>]", [{"full_name": "Turing.Tape.mk\u2082", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [595, 5], "def_end_pos": [595, 13]}, {"full_name": "Turing.Tape.map_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [718, 9], "def_end_pos": [718, 21]}, {"full_name": "Turing.ListBlank.map_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [389, 9], "def_end_pos": [389, 25]}]], "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 R : List \u0393\n\u22a2 map f (mk\u2082 L R) = mk\u2082 (List.map f.f L) (List.map f.f R)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.cardinal_generateMeasurableRec_le", "start": [91, 1], "end": [113, 49], "traced_tactics": [{"tactic": "apply (aleph 1).ord.out.wo.wf.induction i", "annotated_tactic": ["apply (<a>aleph</a> 1).ord.out.wo.wf.induction i", [{"full_name": "Cardinal.aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [245, 5], "def_end_pos": [245, 10]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2200 (x : (Quotient.out (ord (aleph 1))).\u03b1),\n    (\u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n        WellOrder.r (Quotient.out (ord (aleph 1))) y x \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080) \u2192\n      #\u2191(generateMeasurableRec s x) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "intro i IH", "annotated_tactic": ["intro i IH", []], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni : (Quotient.out (ord (aleph 1))).\u03b1\n\u22a2 \u2200 (x : (Quotient.out (ord (aleph 1))).\u03b1),\n    (\u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n        WellOrder.r (Quotient.out (ord (aleph 1))) y x \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080) \u2192\n      #\u2191(generateMeasurableRec s x) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "have A := aleph0_le_aleph 1", "annotated_tactic": ["have A := <a>aleph0_le_aleph</a> 1", [{"full_name": "Cardinal.aleph0_le_aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [291, 9], "def_end_pos": [291, 24]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "have B : aleph 1 \u2264 max #s 2 ^ aleph0.{u} :=\n  aleph_one_le_continuum.trans (power_le_power_right (le_max_right _ _))", "annotated_tactic": ["have B : <a>aleph</a> 1 \u2264 <a>max</a> #s 2 ^ <a>aleph0</a>.{u} :=\n    aleph_one_le_continuum.trans (<a>power_le_power_right</a> (<a>le_max_right</a> _ _))", [{"full_name": "Cardinal.aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [245, 5], "def_end_pos": [245, 10]}, {"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": "Cardinal.aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1235, 5], "def_end_pos": [1235, 11]}, {"full_name": "Cardinal.power_le_power_right", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 29]}, {"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\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "have C : \u2135\u2080 \u2264 max #s 2 ^ aleph0.{u} := A.trans B", "annotated_tactic": ["have C : \u2135\u2080 \u2264 <a>max</a> #s 2 ^ <a>aleph0</a>.{u} := A.trans B", [{"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": "Cardinal.aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1235, 5], "def_end_pos": [1235, 11]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "have J : #(\u22c3 j : Iio i, generateMeasurableRec s j.1) \u2264 max #s 2 ^ aleph0.{u} := by\n  refine (mk_iUnion_le _).trans ?_\n  have D : \u2a06 j : Iio i, #(generateMeasurableRec s j) \u2264 _ := ciSup_le' fun \u27e8j, hj\u27e9 => IH j hj\n  apply (mul_le_mul' ((mk_subtype_le _).trans (aleph 1).mk_ord_out.le) D).trans\n  rw [mul_eq_max A C]\n  exact max_le B le_rfl", "annotated_tactic": ["have J : #(\u22c3 j : <a>Iio</a> i, <a>generateMeasurableRec</a> s j.1) \u2264 <a>max</a> #s 2 ^ <a>aleph0</a>.{u} := by\n    refine (<a>mk_iUnion_le</a> _).<a>trans</a> ?_\n    have D : \u2a06 j : <a>Iio</a> i, #(<a>generateMeasurableRec</a> s j) \u2264 _ := <a>ciSup_le'</a> fun \u27e8j, hj\u27e9 => IH j hj\n    apply (<a>mul_le_mul'</a> ((<a>mk_subtype_le</a> _).<a>trans</a> (<a>aleph</a> 1).mk_ord_out.le) D).<a>trans</a>\n    rw [<a>mul_eq_max</a> A C]\n    exact <a>max_le</a> B <a>le_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": "MeasurableSpace.generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [47, 5], "def_end_pos": [47, 26]}, {"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": "Cardinal.aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1235, 5], "def_end_pos": [1235, 11]}, {"full_name": "Cardinal.mk_iUnion_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2171, 9], "def_end_pos": [2171, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "MeasurableSpace.generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [47, 5], "def_end_pos": [47, 26]}, {"full_name": "ciSup_le'", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1252, 9], "def_end_pos": [1252, 18]}, {"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "Cardinal.mk_subtype_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [287, 9], "def_end_pos": [287, 22]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Cardinal.aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [245, 5], "def_end_pos": [245, 10]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Cardinal.mul_eq_max", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [549, 9], "def_end_pos": [549, 19]}, {"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "rw [generateMeasurableRec]", "annotated_tactic": ["rw [<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\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191((s \u222a {\u2205} \u222a compl '' \u22c3 j, generateMeasurableRec s \u2191j) \u222a range fun f => \u22c3 n, \u2191(f n)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "apply_rules [(mk_union_le _ _).trans, add_le_of_le C, mk_image_le.trans]", "annotated_tactic": ["apply_rules [(<a>mk_union_le</a> _ _).<a>trans</a>, <a>add_le_of_le</a> C, mk_image_le.trans]", [{"full_name": "Cardinal.mk_union_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2221, 9], "def_end_pos": [2221, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Cardinal.add_le_of_le", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [757, 9], "def_end_pos": [757, 21]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191((s \u222a {\u2205} \u222a compl '' \u22c3 j, generateMeasurableRec s \u2191j) \u222a range fun f => \u22c3 n, \u2191(f n)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "case h1.h1.h1\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191s \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\ncase h1.h1.h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191{\u2205} \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\ncase h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(range fun f => \u22c3 n, \u2191(f n)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "refine (mk_iUnion_le _).trans ?_", "annotated_tactic": ["refine (<a>mk_iUnion_le</a> _).<a>trans</a> ?_", [{"full_name": "Cardinal.mk_iUnion_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2171, 9], "def_end_pos": [2171, 21]}, {"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\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(Iio i) * \u2a06 i_1, #\u2191(generateMeasurableRec s \u2191i_1) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "have D : \u2a06 j : Iio i, #(generateMeasurableRec s j) \u2264 _ := ciSup_le' fun \u27e8j, hj\u27e9 => IH j hj", "annotated_tactic": ["have D : \u2a06 j : <a>Iio</a> i, #(<a>generateMeasurableRec</a> s j) \u2264 _ := <a>ciSup_le'</a> fun \u27e8j, hj\u27e9 => IH j hj", [{"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "MeasurableSpace.generateMeasurableRec", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "def_pos": [47, 5], "def_end_pos": [47, 26]}, {"full_name": "ciSup_le'", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [1252, 9], "def_end_pos": [1252, 18]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(Iio i) * \u2a06 i_1, #\u2191(generateMeasurableRec s \u2191i_1) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nD : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(Iio i) * \u2a06 i_1, #\u2191(generateMeasurableRec s \u2191i_1) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "apply (mul_le_mul' ((mk_subtype_le _).trans (aleph 1).mk_ord_out.le) D).trans", "annotated_tactic": ["apply (<a>mul_le_mul'</a> ((<a>mk_subtype_le</a> _).<a>trans</a> (<a>aleph</a> 1).mk_ord_out.le) D).<a>trans</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": "Cardinal.mk_subtype_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [287, 9], "def_end_pos": [287, 22]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Cardinal.aleph", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [245, 5], "def_end_pos": [245, 10]}, {"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\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nD : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(Iio i) * \u2a06 i_1, #\u2191(generateMeasurableRec s \u2191i_1) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nD : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 aleph 1 * max (#\u2191s) 2 ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "rw [mul_eq_max A C]", "annotated_tactic": ["rw [<a>mul_eq_max</a> A C]", [{"full_name": "Cardinal.mul_eq_max", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [549, 9], "def_end_pos": [549, 19]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nD : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 aleph 1 * max (#\u2191s) 2 ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nD : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 max (aleph 1) (max (#\u2191s) 2 ^ \u2135\u2080) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "exact max_le B le_rfl", "annotated_tactic": ["exact <a>max_le</a> B <a>le_rfl</a>", [{"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nD : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 max (aleph 1) (max (#\u2191s) 2 ^ \u2135\u2080) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "no goals"}, {"tactic": "exact (le_max_left _ _).trans (self_le_power _ one_lt_aleph0.le)", "annotated_tactic": ["exact (<a>le_max_left</a> _ _).<a>trans</a> (<a>self_le_power</a> _ one_lt_aleph0.le)", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Cardinal.self_le_power", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [730, 9], "def_end_pos": [730, 22]}]], "state_before": "case h1.h1.h1\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191s \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "no goals"}, {"tactic": "rw [mk_singleton]", "annotated_tactic": ["rw [<a>mk_singleton</a>]", [{"full_name": "Cardinal.mk_singleton", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2057, 9], "def_end_pos": [2057, 21]}]], "state_before": "case h1.h1.h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191{\u2205} \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "case h1.h1.h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "exact one_lt_aleph0.le.trans C", "annotated_tactic": ["exact one_lt_aleph0.le.trans C", []], "state_before": "case h1.h1.h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "no goals"}, {"tactic": "apply mk_range_le.trans", "annotated_tactic": ["apply mk_range_le.trans", []], "state_before": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(range fun f => \u22c3 n, \u2191(f n)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #(\u2115 \u2192 \u2191(\u22c3 j, generateMeasurableRec s \u2191j)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "simp only [mk_pi, prod_const, lift_uzero, mk_denumerable, lift_aleph0]", "annotated_tactic": ["simp only [<a>mk_pi</a>, <a>prod_const</a>, <a>lift_uzero</a>, <a>mk_denumerable</a>, <a>lift_aleph0</a>]", [{"full_name": "Cardinal.mk_pi", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 14]}, {"full_name": "Cardinal.prod_const", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1053, 9], "def_end_pos": [1053, 19]}, {"full_name": "Cardinal.lift_uzero", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 19]}, {"full_name": "Cardinal.mk_denumerable", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1635, 9], "def_end_pos": [1635, 23]}, {"full_name": "Cardinal.lift_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1256, 9], "def_end_pos": [1256, 20]}]], "state_before": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #(\u2115 \u2192 \u2191(\u22c3 j, generateMeasurableRec s \u2191j)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "have := @power_le_power_right _ _ \u2135\u2080 J", "annotated_tactic": ["have := @<a>power_le_power_right</a> _ _ \u2135\u2080 J", [{"full_name": "Cardinal.power_le_power_right", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 29]}]], "state_before": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\n\u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nthis : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 (max (#\u2191s) 2 ^ \u2135\u2080) ^ \u2135\u2080\n\u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080"}, {"tactic": "rwa [\u2190 power_mul, aleph0_mul_aleph0] at this", "annotated_tactic": ["rwa [\u2190 <a>power_mul</a>, <a>aleph0_mul_aleph0</a>] at this", [{"full_name": "Cardinal.power_mul", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [599, 9], "def_end_pos": [599, 18]}, {"full_name": "Cardinal.aleph0_mul_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1644, 9], "def_end_pos": [1644, 26]}]], "state_before": "case h2\n\u03b1 : Type u\ns : Set (Set \u03b1)\ni\u271d i : (Quotient.out (ord (aleph 1))).\u03b1\nIH :\n  \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),\n    WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nA : \u2135\u2080 \u2264 aleph 1\nB : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nC : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nJ : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080\nthis : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 (max (#\u2191s) 2 ^ \u2135\u2080) ^ \u2135\u2080\n\u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.innerContent_iSup_nat", "start": [170, 1], "end": [193, 6], "traced_tactics": [{"tactic": "refine' iSup\u2082_le fun K hK => _", "annotated_tactic": ["refine' <a>iSup\u2082_le</a> fun K hK => _", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\n\u22a2 innerContent \u03bc (\u2a06 i, U i) \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)", "state_after": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)"}, {"tactic": "obtain \u27e8t, ht\u27e9 :=\n  K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [\u2190 Opens.coe_iSup])", "annotated_tactic": ["obtain \u27e8t, ht\u27e9 :=\n    K.isCompact.elim_finite_subcover _ (fun i => (U i).<a>isOpen</a>) (by rwa [\u2190 <a>Opens.coe_iSup</a>])", [{"full_name": "TopologicalSpace.Opens.isOpen", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [117, 19], "def_end_pos": [117, 25]}, {"full_name": "TopologicalSpace.Opens.coe_iSup", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [232, 9], "def_end_pos": [232, 17]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)", "state_after": "case intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)"}, {"tactic": "rcases K.isCompact.finite_compact_cover t (SetLike.coe \u2218 U) (fun i _ => (U i).isOpen) ht with\n  \u27e8K', h1K', h2K', h3K'\u27e9", "annotated_tactic": ["rcases K.isCompact.finite_compact_cover t (<a>SetLike.coe</a> \u2218 U) (fun i _ => (U i).<a>isOpen</a>) ht with\n    \u27e8K', h1K', h2K', h3K'\u27e9", [{"full_name": "SetLike.coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [99, 13], "def_end_pos": [99, 16]}, {"full_name": "TopologicalSpace.Opens.isOpen", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [117, 19], "def_end_pos": [117, 25]}]], "state_before": "case intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)", "state_after": "case intro.intro.intro.intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)"}, {"tactic": "let L : \u2115 \u2192 Compacts G := fun n => \u27e8K' n, h1K' n\u27e9", "annotated_tactic": ["let L : \u2115 \u2192 <a>Compacts</a> G := fun n => \u27e8K' n, h1K' n\u27e9", [{"full_name": "TopologicalSpace.Compacts", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [36, 11], "def_end_pos": [36, 19]}]], "state_before": "case intro.intro.intro.intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)", "state_after": "case intro.intro.intro.intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)"}, {"tactic": "convert le_trans (h3 t L) _", "annotated_tactic": ["convert <a>le_trans</a> (h3 t L) _", [{"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.intro.intro\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)", "state_after": "case h.e'_3.h.e'_1\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 K = Finset.sup t L\n\ncase intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 (Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (L i)) \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)"}, {"tactic": "refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>Finset.sum_le_sum</a> _) (<a>ENNReal.sum_le_tsum</a> t)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "ENNReal.sum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [831, 19], "def_end_pos": [831, 30]}]], "state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 (Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (L i)) \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i)", "state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 \u2200 (i : \u2115), i \u2208 t \u2192 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 innerContent \u03bc (U i)"}, {"tactic": "intro i _", "annotated_tactic": ["intro i _", []], "state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 \u2200 (i : \u2115), i \u2208 t \u2192 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 innerContent \u03bc (U i)", "state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : \u2115\na\u271d : i \u2208 t\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 innerContent \u03bc (U i)"}, {"tactic": "refine' le_trans _ (le_iSup _ (L i))", "annotated_tactic": ["refine' <a>le_trans</a> _ (<a>le_iSup</a> _ (L i))", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}]], "state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : \u2115\na\u271d : i \u2208 t\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 innerContent \u03bc (U i)", "state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : \u2115\na\u271d : i \u2208 t\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 \u2a06 (_ : \u2191(L i) \u2286 \u2191(U i)), (fun s => \u2191(toFun \u03bc s)) (L i)"}, {"tactic": "refine' le_trans _ (le_iSup _ (h2K' i))", "annotated_tactic": ["refine' <a>le_trans</a> _ (<a>le_iSup</a> _ (h2K' i))", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}]], "state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : \u2115\na\u271d : i \u2208 t\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 \u2a06 (_ : \u2191(L i) \u2286 \u2191(U i)), (fun s => \u2191(toFun \u03bc s)) (L i)", "state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : \u2115\na\u271d : i \u2208 t\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 (fun s => \u2191(toFun \u03bc s)) (L i)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : \u2115\na\u271d : i \u2208 t\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 (fun s => \u2191(toFun \u03bc s)) (L i)", "state_after": "no goals"}, {"tactic": "intro t K", "annotated_tactic": ["intro t K", []], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\n\u22a2 \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)", "state_after": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)"}, {"tactic": "refine' Finset.induction_on t _ _", "annotated_tactic": ["refine' <a>Finset.induction_on</a> t _ _", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)", "state_after": "case refine'_1\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup \u2205 K) \u2264 Finset.sum \u2205 fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\n\ncase refine'_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\n\u22a2 \u2200 \u2983a : \u2115\u2984 {s : Finset \u2115},\n    \u00aca \u2208 s \u2192\n      ((fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)) \u2192\n        (fun s => \u2191(toFun \u03bc s)) (Finset.sup (insert a s) K) \u2264\n          Finset.sum (insert a s) fun i => (fun s => \u2191(toFun \u03bc s)) (K i)"}, {"tactic": "simp only [\u03bc.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]", "annotated_tactic": ["simp only [\u03bc.empty, <a>nonpos_iff_eq_zero</a>, <a>Finset.sum_empty</a>, <a>Finset.sup_empty</a>]", [{"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": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "Finset.sup_empty", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}]], "state_before": "case refine'_1\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup \u2205 K) \u2264 Finset.sum \u2205 fun i => (fun s => \u2191(toFun \u03bc s)) (K i)", "state_after": "no goals"}, {"tactic": "intro n s hn ih", "annotated_tactic": ["intro n s hn ih", []], "state_before": "case refine'_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\n\u22a2 \u2200 \u2983a : \u2115\u2984 {s : Finset \u2115},\n    \u00aca \u2208 s \u2192\n      ((fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)) \u2192\n        (fun s => \u2191(toFun \u03bc s)) (Finset.sup (insert a s) K) \u2264\n          Finset.sum (insert a s) fun i => (fun s => \u2191(toFun \u03bc s)) (K i)", "state_after": "case refine'_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\nn : \u2115\ns : Finset \u2115\nhn : \u00acn \u2208 s\nih : (fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup (insert n s) K) \u2264 Finset.sum (insert n s) fun i => (fun s => \u2191(toFun \u03bc s)) (K i)"}, {"tactic": "rw [Finset.sup_insert, Finset.sum_insert hn]", "annotated_tactic": ["rw [<a>Finset.sup_insert</a>, <a>Finset.sum_insert</a> hn]", [{"full_name": "Finset.sup_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [59, 9], "def_end_pos": [59, 19]}, {"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\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\nn : \u2115\ns : Finset \u2115\nhn : \u00acn \u2208 s\nih : (fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup (insert n s) K) \u2264 Finset.sum (insert n s) fun i => (fun s => \u2191(toFun \u03bc s)) (K i)", "state_after": "case refine'_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\nn : \u2115\ns : Finset \u2115\nhn : \u00acn \u2208 s\nih : (fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (K n \u2294 Finset.sup s K) \u2264\n    (fun s => \u2191(toFun \u03bc s)) (K n) + Finset.sum s fun x => (fun s => \u2191(toFun \u03bc s)) (K x)"}, {"tactic": "exact le_trans (\u03bc.sup_le _ _) (add_le_add_left ih _)", "annotated_tactic": ["exact <a>le_trans</a> (\u03bc.sup_le _ _) (<a>add_le_add_left</a> ih _)", [{"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]}]], "state_before": "case refine'_2\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nt : Finset \u2115\nK : \u2115 \u2192 Compacts G\nn : \u2115\ns : Finset \u2115\nhn : \u00acn \u2208 s\nih : (fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\n\u22a2 (fun s => \u2191(toFun \u03bc s)) (K n \u2294 Finset.sup s K) \u2264\n    (fun s => \u2191(toFun \u03bc s)) (K n) + Finset.sum s fun x => (fun s => \u2191(toFun \u03bc s)) (K x)", "state_after": "no goals"}, {"tactic": "rwa [\u2190 Opens.coe_iSup]", "annotated_tactic": ["rwa [\u2190 <a>Opens.coe_iSup</a>]", [{"full_name": "TopologicalSpace.Opens.coe_iSup", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [232, 9], "def_end_pos": [232, 17]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\n\u22a2 \u2191K \u2286 \u22c3 i, \u2191(U i)", "state_after": "no goals"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case h.e'_3.h.e'_1\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 K = Finset.sup t L", "state_after": "case h.e'_3.h.e'_1.h\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 \u2191K = \u2191(Finset.sup t L)"}, {"tactic": "rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]", "annotated_tactic": ["rw [<a>Compacts.coe_finset_sup</a>, <a>Finset.sup_eq_iSup</a>]", [{"full_name": "TopologicalSpace.Compacts.coe_finset_sup", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [124, 9], "def_end_pos": [124, 23]}, {"full_name": "Finset.sup_eq_iSup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [274, 9], "def_end_pos": [274, 20]}]], "state_before": "case h.e'_3.h.e'_1.h\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 \u2191K = \u2191(Finset.sup t L)", "state_after": "case h.e'_3.h.e'_1.h\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 \u2191K = \u2a06 a \u2208 t, \u2191(L a)"}, {"tactic": "exact h3K'", "annotated_tactic": ["exact h3K'", []], "state_before": "case h.e'_3.h.e'_1.h\nG : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nU : \u2115 \u2192 Opens G\nh3 :\n  \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),\n    (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i)\nK : Compacts G\nhK : \u2191K \u2286 \u2191(\u2a06 i, U i)\nt : Finset \u2115\nht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i)\nK' : \u2115 \u2192 Set G\nh1K' : \u2200 (i : \u2115), IsCompact (K' i)\nh2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i\nh3K' : \u2191K = \u22c3 i \u2208 t, K' i\nL : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n\u22a2 \u2191K = \u2a06 a \u2208 t, \u2191(L a)", "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_smul", "start": [461, 1], "end": [464, 30], "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\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toSignedMeasure (toJordanDecomposition (r \u2022 s)) = toSignedMeasure (r \u2022 toJordanDecomposition s)"}, {"tactic": "simp [toSignedMeasure_smul]", "annotated_tactic": ["simp [<a>toSignedMeasure_smul</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_smul", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [187, 9], "def_end_pos": [187, 29]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toSignedMeasure (toJordanDecomposition (r \u2022 s)) = toSignedMeasure (r \u2022 toJordanDecomposition s)", "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", "start": [638, 1], "end": [640, 26], "traced_tactics": [{"tactic": "simp only [integral_const', Real.volume_Ioc, ENNReal.toReal_ofReal', \u2190 neg_sub b,\n  max_zero_sub_eq_self]", "annotated_tactic": ["simp only [<a>integral_const'</a>, <a>Real.volume_Ioc</a>, <a>ENNReal.toReal_ofReal'</a>, \u2190 <a>neg_sub</a> b,\n    <a>max_zero_sub_eq_self</a>]", [{"full_name": "intervalIntegral.integral_const'", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [632, 9], "def_end_pos": [632, 24]}, {"full_name": "Real.volume_Ioc", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 19]}, {"full_name": "ENNReal.toReal_ofReal'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [195, 9], "def_end_pos": [195, 23]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "max_zero_sub_eq_self", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [26, 7], "def_end_pos": [26, 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\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\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c = (b - a) \u2022 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_mem_insert_top_image_coe", "start": [1629, 1], "end": [1631, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_coe_eq_coe_finsetCard", "start": [84, 1], "end": [85, 72], "traced_tactics": [{"tactic": "rw [Finite.encard_eq_coe_toFinset_card (Finset.finite_toSet s)]", "annotated_tactic": ["rw [<a>Finite.encard_eq_coe_toFinset_card</a> (<a>Finset.finite_toSet</a> s)]", [{"full_name": "Set.Finite.encard_eq_coe_toFinset_card", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [75, 9], "def_end_pos": [75, 43]}, {"full_name": "Finset.finite_toSet", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [545, 9], "def_end_pos": [545, 21]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 encard \u2191s = \u2191(Finset.card s)", "state_after": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2191(Finset.card (Finite.toFinset (_ : Set.Finite \u2191s))) = \u2191(Finset.card s)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2191(Finset.card (Finite.toFinset (_ : Set.Finite \u2191s))) = \u2191(Finset.card s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.map\u2082_mk", "start": [266, 1], "end": [268, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snormEssSup_add_le", "start": [795, 1], "end": [799, 26], "traced_tactics": [{"tactic": "refine' le_trans (essSup_mono_ae (eventually_of_forall fun x => _)) (ENNReal.essSup_add_le _ _)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>essSup_mono_ae</a> (<a>eventually_of_forall</a> fun x => _)) (<a>ENNReal.essSup_add_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": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "ENNReal.essSup_add_le", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [321, 9], "def_end_pos": [321, 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 F\n\u22a2 snormEssSup (f + g) \u03bc \u2264 snormEssSup f \u03bc + snormEssSup g \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 F\nx : \u03b1\n\u22a2 (fun x => \u2191\u2016(f + g) x\u2016\u208a) x \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) + fun x => \u2191\u2016g x\u2016\u208a) x"}, {"tactic": "simp_rw [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe]", "annotated_tactic": ["simp_rw [<a>Pi.add_apply</a>, \u2190 <a>ENNReal.coe_add</a>, <a>ENNReal.coe_le_coe</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]}]], "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 F\nx : \u03b1\n\u22a2 (fun x => \u2191\u2016(f + g) x\u2016\u208a) x \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) + fun x => \u2191\u2016g x\u2016\u208a) x", "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 F\nx : \u03b1\n\u22a2 \u2016f x + g x\u2016\u208a \u2264 \u2016f x\u2016\u208a + \u2016g x\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": "\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 F\nx : \u03b1\n\u22a2 \u2016f x + g x\u2016\u208a \u2264 \u2016f x\u2016\u208a + \u2016g x\u2016\u208a", "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_get?", "start": [46, 1], "end": [50, 83], "traced_tactics": [{"tactic": "rw [\u2190 range_list_nthLe, \u2190 range_comp]", "annotated_tactic": ["rw [\u2190 <a>range_list_nthLe</a>, \u2190 <a>range_comp</a>]", [{"full_name": "Set.range_list_nthLe", "def_path": "Mathlib/Data/Set/List.lean", "def_pos": [40, 9], "def_end_pos": [40, 25]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 range (get? l) = insert none (some '' {x | x \u2208 l})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 range (get? l) = insert none (range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)))"}, {"tactic": "refine' (range_subset_iff.2 fun n => _).antisymm (insert_subset_iff.2 \u27e8_, _\u27e9)", "annotated_tactic": ["refine' (<a>range_subset_iff</a>.2 fun n => _).<a>antisymm</a> (<a>insert_subset_iff</a>.2 \u27e8_, _\u27e9)", [{"full_name": "Set.range_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [749, 9], "def_end_pos": [749, 25]}, {"full_name": "HasSubset.Subset.antisymm", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [667, 7], "def_end_pos": [667, 32]}, {"full_name": "Set.insert_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1169, 9], "def_end_pos": [1169, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 range (get? l) = insert none (range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)))", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nn : \u2115\n\u22a2 get? l n \u2208 insert none (range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)))\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 none \u2208 range (get? l)\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)) \u2286 range (get? l)"}, {"tactic": "exacts [(le_or_lt l.length n).imp get?_eq_none.2 (fun hlt => \u27e8\u27e8_, hlt\u27e9, (get?_eq_get hlt).symm\u27e9),\n  \u27e8_, get?_eq_none.2 le_rfl\u27e9, range_subset_iff.2 <| fun k => \u27e8_, get?_eq_get _\u27e9]", "annotated_tactic": ["exacts [(<a>le_or_lt</a> l.length n).<a>imp</a> <a>get?_eq_none</a>.2 (fun hlt => \u27e8\u27e8_, hlt\u27e9, (<a>get?_eq_get</a> hlt).<a>symm</a>\u27e9),\n    \u27e8_, <a>get?_eq_none</a>.2 <a>le_rfl</a>\u27e9, <a>range_subset_iff</a>.2 <| fun k => \u27e8_, <a>get?_eq_get</a> _\u27e9]", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}, {"full_name": "Or.imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [249, 9], "def_end_pos": [249, 15]}, {"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_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 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": "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": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.range_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [749, 9], "def_end_pos": [749, 25]}, {"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": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nn : \u2115\n\u22a2 get? l n \u2208 insert none (range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)))\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 none \u2208 range (get? l)\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)) \u2286 range (get? l)", "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_closedBall", "start": [502, 1], "end": [504, 81], "traced_tactics": [{"tactic": "rw [addHaar_closedBall' \u03bc x hr, addHaar_closed_unit_ball_eq_addHaar_unit_ball]", "annotated_tactic": ["rw [<a>addHaar_closedBall'</a> \u03bc x hr, <a>addHaar_closed_unit_ball_eq_addHaar_unit_ball</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_closedBall'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [481, 9], "def_end_pos": [481, 28]}, {"full_name": "MeasureTheory.Measure.addHaar_closed_unit_ball_eq_addHaar_unit_ball", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [486, 9], "def_end_pos": [486, 54]}]], "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\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (closedBall 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/Image.lean", "full_name": "Function.Injective.subsingleton_image_iff", "start": [1310, 1], "end": [1312, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.castNum_shiftLeft", "start": [948, 1], "end": [957, 67], "traced_tactics": [{"tactic": "cases m <;> dsimp only [\u2190shiftl_eq_shiftLeft, shiftl]", "annotated_tactic": ["cases m <;> dsimp only [\u2190<a>shiftl_eq_shiftLeft</a>, <a>shiftl</a>]", [{"full_name": "Num.shiftl_eq_shiftLeft", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [199, 15], "def_end_pos": [199, 34]}, {"full_name": "Num.shiftl", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [192, 5], "def_end_pos": [192, 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\na\u271d : PosNum\n\u22a2 \u2191(pos (a\u271d <<< n)) = \u2191(pos a\u271d) <<< n"}, {"tactic": "simp only [cast_pos]", "annotated_tactic": ["simp only [<a>cast_pos</a>]", [{"full_name": "Num.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [287, 9], "def_end_pos": [287, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 \u2191(pos (a\u271d <<< n)) = \u2191(pos a\u271d) <<< n", "state_after": "case pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 \u2191(a\u271d <<< n) = \u2191a\u271d <<< n"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "case pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 \u2191(a\u271d <<< n) = \u2191a\u271d <<< n", "state_after": "case pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(a\u271d <<< Nat.zero) = \u2191a\u271d <<< Nat.zero\n\ncase pos.succ\n\u03b1 : Type u_1\na\u271d : PosNum\nn : \u2115\nIH : \u2191(a\u271d <<< n) = \u2191a\u271d <<< n\n\u22a2 \u2191(a\u271d <<< Nat.succ n) = \u2191a\u271d <<< Nat.succ n"}, {"tactic": "simp [PosNum.shiftl_succ_eq_bit0_shiftl, Nat.shiftLeft_succ, IH,\n      Nat.bit0_val, pow_succ, \u2190 mul_assoc, mul_comm,\n      -shiftl_eq_shiftLeft, -PosNum.shiftl_eq_shiftLeft, shiftl]", "annotated_tactic": ["simp [<a>PosNum.shiftl_succ_eq_bit0_shiftl</a>, <a>Nat.shiftLeft_succ</a>, IH,\n        <a>Nat.bit0_val</a>, <a>pow_succ</a>, \u2190 <a>mul_assoc</a>, <a>mul_comm</a>,\n        -<a>shiftl_eq_shiftLeft</a>, -<a>PosNum.shiftl_eq_shiftLeft</a>, <a>shiftl</a>]", [{"full_name": "PosNum.shiftl_succ_eq_bit0_shiftl", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [124, 9], "def_end_pos": [124, 35]}, {"full_name": "Nat.shiftLeft_succ", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [208, 9], "def_end_pos": [208, 23]}, {"full_name": "Nat.bit0_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [152, 9], "def_end_pos": [152, 17]}, {"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_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Num.shiftl_eq_shiftLeft", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [199, 15], "def_end_pos": [199, 34]}, {"full_name": "PosNum.shiftl_eq_shiftLeft", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [119, 15], "def_end_pos": [119, 34]}, {"full_name": "Num.shiftl", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [192, 5], "def_end_pos": [192, 11]}]], "state_before": "case pos.succ\n\u03b1 : Type u_1\na\u271d : PosNum\nn : \u2115\nIH : \u2191(a\u271d <<< n) = \u2191a\u271d <<< n\n\u22a2 \u2191(a\u271d <<< Nat.succ n) = \u2191a\u271d <<< Nat.succ n", "state_after": "no goals"}, {"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_shiftLeft", "annotated_tactic": ["apply <a>Nat.zero_shiftLeft</a>", [{"full_name": "Nat.zero_shiftLeft", "def_path": "Mathlib/Data/Nat/Size.lean", "def_pos": [36, 9], "def_end_pos": [36, 23]}]], "state_before": "case zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u2191zero <<< n = \u21910", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(a\u271d <<< Nat.zero) = \u2191a\u271d <<< Nat.zero", "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_Icc", "start": [1157, 1], "end": [1158, 80], "traced_tactics": [{"tactic": "rw [\u2190 map_add_right_Icc, map_eq_image, addRightEmbedding, Embedding.coeFn_mk]", "annotated_tactic": ["rw [\u2190 <a>map_add_right_Icc</a>, <a>map_eq_image</a>, <a>addRightEmbedding</a>, <a>Embedding.coeFn_mk</a>]", [{"full_name": "Finset.map_add_right_Icc", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1086, 9], "def_end_pos": [1086, 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": 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<a>filter_length_eq_length</a>]", [{"full_name": "List.countP_eq_length_filter", "def_path": "lake-packages/std/Std/Data/List/Count.lean", "def_pos": [60, 9], "def_end_pos": [60, 32]}, {"full_name": "List.filter_length_eq_length", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1372, 9], "def_end_pos": [1372, 32]}]], "state_before": "\u03b1 : Type u_1\np q : \u03b1 \u2192 Bool\nl : List \u03b1\n\u22a2 countP p l = length l \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "Monotone.intervalIntegrable", "start": [389, 1], "end": [391, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", 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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'\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc (T + T') (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) 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 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"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_comp_smul_deriv'''", "start": [1354, 1], "end": [1392, 88], "traced_tactics": [{"tactic": "rw [hf.image_uIcc, \u2190 intervalIntegrable_iff'] at hg1", "annotated_tactic": ["rw [hf.image_uIcc, \u2190 <a>intervalIntegrable_iff'</a>] at hg1", [{"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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]])\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u"}, {"tactic": "have h_der :\n  \u2200 x \u2208 Ioo (min a b) (max a b),\n    HasDerivWithinAt (fun u => \u222b t in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x := by\n  intro x hx\n  obtain \u27e8c, hc\u27e9 := nonempty_Ioo.mpr hx.1\n  obtain \u27e8d, hd\u27e9 := nonempty_Ioo.mpr hx.2\n  have cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) := by\n    rw [uIcc_of_le (hc.2.trans hd.1).le]\n    exact Icc_subset_Ioo hc.1 hd.2\n  replace hg_cont := hg_cont.mono (image_subset f cdsub)\n  let J := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\n  have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc\n  rw [hJ] at hg_cont\n  have h2x : f x \u2208 J := by rw [\u2190 hJ]; exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le)\n  have h2g : IntervalIntegrable g volume (f a) (f x) := by\n    refine' hg1.mono_set _\n    rw [\u2190 hf.image_uIcc]\n    exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx)\n  have h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) :=\n    hg_cont.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc (f x)\n  haveI : Fact (f x \u2208 J) := \u27e8h2x\u27e9\n  have : HasDerivWithinAt (fun u => \u222b x in f a..u, g x) (g (f x)) J (f x) :=\n    intervalIntegral.integral_hasDerivWithinAt_right h2g h3g (hg_cont (f x) h2x)\n  refine' (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) _).Ioi_of_Ioo hd.1\n  rw [\u2190 hJ]\n  refine' (mapsTo_image _ _).mono _ Subset.rfl\n  exact Ioo_subset_Icc_self.trans ((Icc_subset_Icc_left hc.2.le).trans Icc_subset_uIcc)", "annotated_tactic": ["have h_der :\n    \u2200 x \u2208 <a>Ioo</a> (<a>min</a> a b) (<a>max</a> a b),\n      <a>HasDerivWithinAt</a> (fun u => \u222b t in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (<a>Ioi</a> x) x := by\n    intro x hx\n    obtain \u27e8c, hc\u27e9 := nonempty_Ioo.mpr hx.1\n    obtain \u27e8d, hd\u27e9 := nonempty_Ioo.mpr hx.2\n    have cdsub : [[c, d]] \u2286 <a>Ioo</a> (<a>min</a> a b) (<a>max</a> a b) := by\n      rw [<a>uIcc_of_le</a> (hc.2.<a>trans</a> hd.1).<a>le</a>]\n      exact <a>Icc_subset_Ioo</a> hc.1 hd.2\n    replace hg_cont := hg_cont.mono (<a>image_subset</a> f cdsub)\n    let J := [[<a>sInf</a> (f '' [[c, d]]), <a>sSup</a> (f '' [[c, d]])]]\n    have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans <a>Ioo_subset_Icc_self</a>)).<a>image_uIcc</a>\n    rw [hJ] at hg_cont\n    have h2x : f x \u2208 J := by rw [\u2190 hJ]; exact <a>mem_image_of_mem</a> _ (<a>mem_uIcc_of_le</a> hc.2.<a>le</a> hd.1.<a>le</a>)\n    have h2g : <a>IntervalIntegrable</a> g <a>volume</a> (f a) (f x) := by\n      refine' hg1.mono_set _\n      rw [\u2190 hf.image_uIcc]\n      exact hf.surjOn_uIcc <a>left_mem_uIcc</a> (<a>Ioo_subset_Icc_self</a> hx)\n    have h3g : <a>StronglyMeasurableAtFilter</a> g (\ud835\udcdd[J] f x) :=\n      hg_cont.stronglyMeasurableAtFilter_nhdsWithin <a>measurableSet_Icc</a> (f x)\n    haveI : <a>Fact</a> (f x \u2208 J) := \u27e8h2x\u27e9\n    have : <a>HasDerivWithinAt</a> (fun u => \u222b x in f a..u, g x) (g (f x)) J (f x) :=\n      <a>intervalIntegral.integral_hasDerivWithinAt_right</a> h2g h3g (hg_cont (f x) h2x)\n    refine' (this.scomp x ((hff' x hx).<a>Ioo_of_Ioi</a> hd.1) _).<a>Ioi_of_Ioo</a> hd.1\n    rw [\u2190 hJ]\n    refine' (<a>mapsTo_image</a> _ _).<a>mono</a> _ <a>Subset.rfl</a>\n    exact Ioo_subset_Icc_self.trans ((<a>Icc_subset_Icc_left</a> hc.2.<a>le</a>).<a>trans</a> <a>Icc_subset_uIcc</a>)", [{"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": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "HasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 21]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"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": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"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": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Set.Icc_subset_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [467, 9], "def_end_pos": [467, 23]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "InfSet.sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [62, 3], "def_end_pos": [62, 7]}, {"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}, {"full_name": "Set.Ioo_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 28]}, {"full_name": "ContinuousOn.image_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [564, 9], "def_end_pos": [564, 19]}, {"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": "Set.mem_uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [102, 7], "def_end_pos": [102, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"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": "Set.left_mem_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [97, 15], "def_end_pos": [97, 28]}, {"full_name": "Set.Ioo_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 28]}, {"full_name": "StronglyMeasurableAtFilter", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [38, 5], "def_end_pos": [38, 31]}, {"full_name": "measurableSet_Icc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 26]}, {"full_name": "Fact", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 11]}, {"full_name": "HasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 21]}, {"full_name": "intervalIntegral.integral_hasDerivWithinAt_right", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [911, 9], "def_end_pos": [911, 40]}, {"full_name": "HasDerivWithinAt.Ioo_of_Ioi", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [358, 37], "def_end_pos": [358, 64]}, {"full_name": "HasDerivWithinAt.Ioi_of_Ioo", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [358, 8], "def_end_pos": [358, 35]}, {"full_name": "Set.mapsTo_image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [512, 9], "def_end_pos": [512, 21]}, {"full_name": "Set.MapsTo.mono", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [464, 9], "def_end_pos": [464, 20]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "Set.Icc_subset_Icc_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [459, 9], "def_end_pos": [459, 28]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.Icc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [92, 7], "def_end_pos": [92, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nh_der :\n  \u2200 (x : \u211d),\n    x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u"}, {"tactic": "rw [\u2190 intervalIntegrable_iff'] at hg2", "annotated_tactic": ["rw [\u2190 <a>intervalIntegrable_iff'</a>] at hg2", [{"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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nh_der :\n  \u2200 (x : \u211d),\n    x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntervalIntegrable (fun x => f' x \u2022 (g \u2218 f) x) volume a b\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nh_der :\n  \u2200 (x : \u211d),\n    x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u"}, {"tactic": "simp_rw [integral_eq_sub_of_hasDeriv_right h_cont h_der hg2, integral_same, sub_zero]", "annotated_tactic": ["simp_rw [<a>integral_eq_sub_of_hasDeriv_right</a> h_cont h_der hg2, <a>integral_same</a>, <a>sub_zero</a>]", [{"full_name": "intervalIntegral.integral_eq_sub_of_hasDeriv_right", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 42]}, {"full_name": "intervalIntegral.integral_same", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [470, 9], "def_end_pos": [470, 22]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 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\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntervalIntegrable (fun x => f' x \u2022 (g \u2218 f) x) volume a b\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nh_der :\n  \u2200 (x : \u211d),\n    x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u", "state_after": "no goals"}, {"tactic": "refine' (continuousOn_primitive_interval' hg1 _).comp hf _", "annotated_tactic": ["refine' (<a>continuousOn_primitive_interval'</a> hg1 _).<a>comp</a> hf _", [{"full_name": "intervalIntegral.continuousOn_primitive_interval'", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1218, 9], "def_end_pos": [1218, 41]}, {"full_name": "ContinuousOn.comp", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [929, 9], "def_end_pos": [929, 26]}]], "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' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]", "state_after": "case refine'_1\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 f a \u2208 [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]\n\ncase refine'_2\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 MapsTo (fun u => f u) [[a, b]] [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]"}, {"tactic": "rw [\u2190 hf.image_uIcc]", "annotated_tactic": ["rw [\u2190 hf.image_uIcc]", []], "state_before": "case refine'_1\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 f a \u2208 [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]", "state_after": "case refine'_1\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 f a \u2208 f '' [[a, b]]"}, {"tactic": "exact mem_image_of_mem f left_mem_uIcc", "annotated_tactic": ["exact <a>mem_image_of_mem</a> f <a>left_mem_uIcc</a>", [{"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": "Set.left_mem_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [97, 15], "def_end_pos": [97, 28]}]], "state_before": "case refine'_1\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 f a \u2208 f '' [[a, b]]", "state_after": "no goals"}, {"tactic": "rw [\u2190 hf.image_uIcc]", "annotated_tactic": ["rw [\u2190 hf.image_uIcc]", []], "state_before": "case refine'_2\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 MapsTo (fun u => f u) [[a, b]] [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]", "state_after": "case refine'_2\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 MapsTo (fun u => f u) [[a, b]] (f '' [[a, b]])"}, {"tactic": "exact mapsTo_image _ _", "annotated_tactic": ["exact <a>mapsTo_image</a> _ _", [{"full_name": "Set.mapsTo_image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [512, 9], "def_end_pos": [512, 21]}]], "state_before": "case refine'_2\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\n\u22a2 MapsTo (fun u => f u) [[a, b]] (f '' [[a, b]])", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\n\u22a2 \u2200 (x : \u211d),\n    x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "obtain \u27e8c, hc\u27e9 := nonempty_Ioo.mpr hx.1", "annotated_tactic": ["obtain \u27e8c, hc\u27e9 := nonempty_Ioo.mpr hx.1", []], "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' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "state_after": "case 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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "obtain \u27e8d, hd\u27e9 := nonempty_Ioo.mpr hx.2", "annotated_tactic": ["obtain \u27e8d, hd\u27e9 := nonempty_Ioo.mpr hx.2", []], "state_before": "case 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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "have cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) := by\n  rw [uIcc_of_le (hc.2.trans hd.1).le]\n  exact Icc_subset_Ioo hc.1 hd.2", "annotated_tactic": ["have cdsub : [[c, d]] \u2286 <a>Ioo</a> (<a>min</a> a b) (<a>max</a> a b) := by\n      rw [<a>uIcc_of_le</a> (hc.2.<a>trans</a> hd.1).<a>le</a>]\n      exact <a>Icc_subset_Ioo</a> hc.1 hd.2", [{"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": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"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": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Set.Icc_subset_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [467, 9], "def_end_pos": [467, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "replace hg_cont := hg_cont.mono (image_subset f cdsub)", "annotated_tactic": ["replace hg_cont := hg_cont.mono (<a>image_subset</a> f cdsub)", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nhg_cont : ContinuousOn g (f '' [[c, d]])\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "let J := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]", "annotated_tactic": ["let J := [[<a>sInf</a> (f '' [[c, d]]), <a>sSup</a> (f '' [[c, d]])]]", [{"full_name": "InfSet.sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [62, 3], "def_end_pos": [62, 7]}, {"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\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nhg_cont : ContinuousOn g (f '' [[c, d]])\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nhg_cont : ContinuousOn g (f '' [[c, d]])\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc", "annotated_tactic": ["have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans <a>Ioo_subset_Icc_self</a>)).<a>image_uIcc</a>", [{"full_name": "Set.Ioo_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 28]}, {"full_name": "ContinuousOn.image_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [564, 9], "def_end_pos": [564, 19]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nhg_cont : ContinuousOn g (f '' [[c, d]])\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nhg_cont : ContinuousOn g (f '' [[c, d]])\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhJ : f '' [[c, d]] = J\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "rw [hJ] at hg_cont", "annotated_tactic": ["rw [hJ] at hg_cont", []], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nhg_cont : ContinuousOn g (f '' [[c, d]])\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhJ : f '' [[c, d]] = J\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "have h2x : f x \u2208 J := by rw [\u2190 hJ]; exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le)", "annotated_tactic": ["have h2x : f x \u2208 J := by rw [\u2190 hJ]; exact <a>mem_image_of_mem</a> _ (<a>mem_uIcc_of_le</a> hc.2.<a>le</a> hd.1.<a>le</a>)", [{"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": "Set.mem_uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [102, 7], "def_end_pos": [102, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"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\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "have h2g : IntervalIntegrable g volume (f a) (f x) := by\n  refine' hg1.mono_set _\n  rw [\u2190 hf.image_uIcc]\n  exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx)", "annotated_tactic": ["have h2g : <a>IntervalIntegrable</a> g <a>volume</a> (f a) (f x) := by\n      refine' hg1.mono_set _\n      rw [\u2190 hf.image_uIcc]\n      exact hf.surjOn_uIcc <a>left_mem_uIcc</a> (<a>Ioo_subset_Icc_self</a> hx)", [{"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": "Set.left_mem_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [97, 15], "def_end_pos": [97, 28]}, {"full_name": "Set.Ioo_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 28]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "have h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) :=\n  hg_cont.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc (f x)", "annotated_tactic": ["have h3g : <a>StronglyMeasurableAtFilter</a> g (\ud835\udcdd[J] f x) :=\n      hg_cont.stronglyMeasurableAtFilter_nhdsWithin <a>measurableSet_Icc</a> (f x)", [{"full_name": "StronglyMeasurableAtFilter", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [38, 5], "def_end_pos": [38, 31]}, {"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\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\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "haveI : Fact (f x \u2208 J) := \u27e8h2x\u27e9", "annotated_tactic": ["haveI : <a>Fact</a> (f x \u2208 J) := \u27e8h2x\u27e9", [{"full_name": "Fact", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis : Fact (f x \u2208 J)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "have : HasDerivWithinAt (fun u => \u222b x in f a..u, g x) (g (f x)) J (f x) :=\n  intervalIntegral.integral_hasDerivWithinAt_right h2g h3g (hg_cont (f x) h2x)", "annotated_tactic": ["have : <a>HasDerivWithinAt</a> (fun u => \u222b x in f a..u, g x) (g (f x)) J (f x) :=\n      <a>intervalIntegral.integral_hasDerivWithinAt_right</a> h2g h3g (hg_cont (f x) h2x)", [{"full_name": "HasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 21]}, {"full_name": "intervalIntegral.integral_hasDerivWithinAt_right", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [911, 9], "def_end_pos": [911, 40]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis : Fact (f x \u2208 J)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x"}, {"tactic": "refine' (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) _).Ioi_of_Ioo hd.1", "annotated_tactic": ["refine' (this.scomp x ((hff' x hx).<a>Ioo_of_Ioi</a> hd.1) _).<a>Ioi_of_Ioo</a> hd.1", [{"full_name": "HasDerivWithinAt.Ioo_of_Ioi", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [358, 37], "def_end_pos": [358, 64]}, {"full_name": "HasDerivWithinAt.Ioi_of_Ioo", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [358, 8], "def_end_pos": [358, 35]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 MapsTo f (Ioo x d) J"}, {"tactic": "rw [\u2190 hJ]", "annotated_tactic": ["rw [\u2190 hJ]", []], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 MapsTo f (Ioo x d) J", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 MapsTo f (Ioo x d) (f '' [[c, d]])"}, {"tactic": "refine' (mapsTo_image _ _).mono _ Subset.rfl", "annotated_tactic": ["refine' (<a>mapsTo_image</a> _ _).<a>mono</a> _ <a>Subset.rfl</a>", [{"full_name": "Set.mapsTo_image", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [512, 9], "def_end_pos": [512, 21]}, {"full_name": "Set.MapsTo.mono", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [464, 9], "def_end_pos": [464, 20]}, {"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 MapsTo f (Ioo x d) (f '' [[c, d]])", "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 Ioo x d \u2286 [[c, d]]"}, {"tactic": "exact Ioo_subset_Icc_self.trans ((Icc_subset_Icc_left hc.2.le).trans Icc_subset_uIcc)", "annotated_tactic": ["exact Ioo_subset_Icc_self.trans ((<a>Icc_subset_Icc_left</a> hc.2.<a>le</a>).<a>trans</a> <a>Icc_subset_uIcc</a>)", [{"full_name": "Set.Icc_subset_Icc_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [459, 9], "def_end_pos": [459, 28]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.Icc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [92, 7], "def_end_pos": [92, 22]}]], "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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\nh2g : IntervalIntegrable g volume (f a) (f x)\nh3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x)\nthis\u271d : Fact (f x \u2208 J)\nthis : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x)\n\u22a2 Ioo x d \u2286 [[c, d]]", "state_after": "no goals"}, {"tactic": "rw [uIcc_of_le (hc.2.trans hd.1).le]", "annotated_tactic": ["rw [<a>uIcc_of_le</a> (hc.2.<a>trans</a> hd.1).<a>le</a>]", [{"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": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\n\u22a2 [[c, d]] \u2286 Ioo (min a b) (max 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\n\u22a2 Icc c d \u2286 Ioo (min a b) (max a b)"}, {"tactic": "exact Icc_subset_Ioo hc.1 hd.2", "annotated_tactic": ["exact <a>Icc_subset_Ioo</a> hc.1 hd.2", [{"full_name": "Set.Icc_subset_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [467, 9], "def_end_pos": [467, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\n\u22a2 Icc c d \u2286 Ioo (min a b) (max a b)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hJ]", "annotated_tactic": ["rw [\u2190 hJ]", []], "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' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\n\u22a2 f x \u2208 J", "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\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\n\u22a2 f x \u2208 f '' [[c, d]]"}, {"tactic": "exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le)", "annotated_tactic": ["exact <a>mem_image_of_mem</a> _ (<a>mem_uIcc_of_le</a> hc.2.<a>le</a> hd.1.<a>le</a>)", [{"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": "Set.mem_uIcc_of_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [102, 7], "def_end_pos": [102, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\n\u22a2 f x \u2208 f '' [[c, d]]", "state_after": "no goals"}, {"tactic": "refine' hg1.mono_set _", "annotated_tactic": ["refine' hg1.mono_set _", []], "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' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 IntervalIntegrable g volume (f a) (f x)", "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\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 [[f a, f x]] \u2286 [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]"}, {"tactic": "rw [\u2190 hf.image_uIcc]", "annotated_tactic": ["rw [\u2190 hf.image_uIcc]", []], "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' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 [[f a, f x]] \u2286 [[sInf (f '' [[a, b]]), sSup (f '' [[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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 [[f a, f x]] \u2286 f '' [[a, b]]"}, {"tactic": "exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx)", "annotated_tactic": ["exact hf.surjOn_uIcc <a>left_mem_uIcc</a> (<a>Ioo_subset_Icc_self</a> hx)", [{"full_name": "Set.left_mem_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [97, 15], "def_end_pos": [97, 28]}, {"full_name": "Set.Ioo_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 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\u271d\u00b9 : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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\nhg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]]\nh_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]]\nx : \u211d\nhx : x \u2208 Ioo (min a b) (max a b)\nc : \u211d\nhc : c \u2208 Ioo (min a b) x\nd : \u211d\nhd : d \u2208 Ioo x (max a b)\ncdsub : [[c, d]] \u2286 Ioo (min a b) (max a b)\nJ : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\nhg_cont : ContinuousOn g J\nhJ : f '' [[c, d]] = J\nh2x : f x \u2208 J\n\u22a2 [[f a, f x]] \u2286 f '' [[a, b]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "full_name": "SatisfiesM.seqLeft", "start": [157, 11], "end": [160, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_coe_unit_coprime", "start": [744, 1], "end": [752, 44], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\n\u22a2 Nat.Coprime (val \u2191u) n", "state_after": "case zero\nu : (ZMod Nat.zero)\u02e3\n\u22a2 Nat.Coprime (val \u2191u) Nat.zero\n\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 Nat.Coprime (val \u2191u) (Nat.succ n)"}, {"tactic": "apply Nat.coprime_of_mul_modEq_one ((u\u207b\u00b9 : Units (ZMod (n + 1))) : ZMod (n + 1)).val", "annotated_tactic": ["apply <a>Nat.coprime_of_mul_modEq_one</a> ((u\u207b\u00b9 : <a>Units</a> (<a>ZMod</a> (n + 1))) : <a>ZMod</a> (n + 1)).<a>val</a>", [{"full_name": "Nat.coprime_of_mul_modEq_one", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [395, 9], "def_end_pos": [395, 33]}, {"full_name": "Units", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [45, 11], "def_end_pos": [45, 16]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod.val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 Nat.Coprime (val \u2191u) (Nat.succ n)", "state_after": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]"}, {"tactic": "have := Units.ext_iff.1 (mul_right_inv u)", "annotated_tactic": ["have := <a>Units.ext_iff</a>.1 (<a>mul_right_inv</a> u)", [{"full_name": "Units.ext_iff", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [153, 9], "def_end_pos": [153, 16]}, {"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}]], "state_before": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]", "state_after": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = \u21911\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]"}, {"tactic": "rw [Units.val_one] at this", "annotated_tactic": ["rw [<a>Units.val_one</a>] at this", [{"full_name": "Units.val_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [235, 9], "def_end_pos": [235, 16]}]], "state_before": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = \u21911\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]", "state_after": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = 1\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]"}, {"tactic": "rw [\u2190 eq_iff_modEq_nat, Nat.cast_one, \u2190 this]", "annotated_tactic": ["rw [\u2190 <a>eq_iff_modEq_nat</a>, <a>Nat.cast_one</a>, \u2190 this]", [{"full_name": "ZMod.eq_iff_modEq_nat", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [719, 9], "def_end_pos": [719, 25]}, {"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 succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = 1\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]", "state_after": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = 1\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9)"}, {"tactic": "clear this", "annotated_tactic": ["clear this", []], "state_before": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = 1\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9)", "state_after": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9)"}, {"tactic": "rw [\u2190 nat_cast_zmod_val ((u * u\u207b\u00b9 : Units (ZMod (n + 1))) : ZMod (n + 1))]", "annotated_tactic": ["rw [\u2190 <a>nat_cast_zmod_val</a> ((u * u\u207b\u00b9 : <a>Units</a> (<a>ZMod</a> (n + 1))) : <a>ZMod</a> (n + 1))]", [{"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": "Units", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [45, 11], "def_end_pos": [45, 16]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}]], "state_before": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9)", "state_after": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(val \u2191(u * u\u207b\u00b9))"}, {"tactic": "rw [Units.val_mul, val_mul, nat_cast_mod]", "annotated_tactic": ["rw [<a>Units.val_mul</a>, <a>val_mul</a>, <a>nat_cast_mod</a>]", [{"full_name": "Units.val_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [229, 9], "def_end_pos": [229, 16]}, {"full_name": "ZMod.val_mul", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [643, 9], "def_end_pos": [643, 16]}, {"full_name": "ZMod.nat_cast_mod", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [712, 9], "def_end_pos": [712, 21]}]], "state_before": "case succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(val \u2191(u * u\u207b\u00b9))", "state_after": "no goals"}, {"tactic": "rcases Int.units_eq_one_or u with (rfl | rfl) <;> simp", "annotated_tactic": ["rcases <a>Int.units_eq_one_or</a> u with (rfl | rfl) <;> simp", [{"full_name": "Int.units_eq_one_or", "def_path": "Mathlib/Data/Int/Units.lean", "def_pos": [30, 9], "def_end_pos": [30, 24]}]], "state_before": "case zero\nu : (ZMod Nat.zero)\u02e3\n\u22a2 Nat.Coprime (val \u2191u) Nat.zero", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Filtration.lean", "full_name": "MeasureTheory.Filtration.filtrationOfSet_eq_natural", "start": [281, 1], "end": [305, 69], "traced_tactics": [{"tactic": "simp only [filtrationOfSet, natural, measurableSpace_iSup_eq, exists_prop, mk.injEq]", "annotated_tactic": ["simp only [<a>filtrationOfSet</a>, <a>natural</a>, <a>measurableSpace_iSup_eq</a>, <a>exists_prop</a>, mk.injEq]", [{"full_name": "MeasureTheory.filtrationOfSet", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [242, 5], "def_end_pos": [242, 20]}, {"full_name": "MeasureTheory.Filtration.natural", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [268, 5], "def_end_pos": [268, 12]}, {"full_name": "MeasurableSpace.measurableSpace_iSup_eq", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [517, 9], "def_end_pos": [517, 32]}, {"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\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 filtrationOfSet hsm =\n    natural (fun i => Set.indicator (s i) fun x => 1) (_ : \u2200 (i : \u03b9), StronglyMeasurable (Set.indicator (s i) 1))", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 (fun i => MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t}) = fun i =>\n    MeasurableSpace.generateFrom {s_1 | \u2203 n, MeasurableSet s_1}"}, {"tactic": "ext1 i", "annotated_tactic": ["ext1 i", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 (fun i => MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t}) = fun i =>\n    MeasurableSpace.generateFrom {s_1 | \u2203 n, MeasurableSet s_1}", "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 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\n\u22a2 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t} = MeasurableSpace.generateFrom {s_1 | \u2203 n, MeasurableSet s_1}"}, {"tactic": "refine' le_antisymm (generateFrom_le _) (generateFrom_le _)", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>generateFrom_le</a> _) (<a>generateFrom_le</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_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}]], "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 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\n\u22a2 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t} = MeasurableSpace.generateFrom {s_1 | \u2203 n, MeasurableSet s_1}", "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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 {t | \u2203 j, j \u2264 i \u2227 s j = t} \u2192 MeasurableSet t\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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 {s_1 | \u2203 n, MeasurableSet s_1} \u2192 MeasurableSet t"}, {"tactic": "rintro _ \u27e8j, hij, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8j, hij, rfl\u27e9", []], "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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 {t | \u2203 j, j \u2264 i \u2227 s j = t} \u2192 MeasurableSet t", "state_after": "case h.refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni j : \u03b9\nhij : j \u2264 i\n\u22a2 MeasurableSet (s j)"}, {"tactic": "refine' measurableSet_generateFrom \u27e8{1}, measurableSet_singleton 1, _\u27e9", "annotated_tactic": ["refine' <a>measurableSet_generateFrom</a> \u27e8{1}, <a>measurableSet_singleton</a> 1, _\u27e9", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"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.refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni j : \u03b9\nhij : j \u2264 i\n\u22a2 MeasurableSet (s j)", "state_after": "case h.refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni j : \u03b9\nhij : j \u2264 i\n\u22a2 (Set.indicator (s j) fun x => 1) \u207b\u00b9' {1} = s j"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni j : \u03b9\nhij : j \u2264 i\n\u22a2 (Set.indicator (s j) fun x => 1) \u207b\u00b9' {1} = s j", "state_after": "case h.refine'_1.intro.intro.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni j : \u03b9\nhij : j \u2264 i\nx : \u03a9\n\u22a2 x \u2208 (Set.indicator (s j) fun x => 1) \u207b\u00b9' {1} \u2194 x \u2208 s j"}, {"tactic": "simp [Set.indicator_const_preimage_eq_union]", "annotated_tactic": ["simp [<a>Set.indicator_const_preimage_eq_union</a>]", [{"full_name": "Set.indicator_const_preimage_eq_union", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [293, 3], "def_end_pos": [293, 14]}]], "state_before": "case h.refine'_1.intro.intro.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni j : \u03b9\nhij : j \u2264 i\nx : \u03a9\n\u22a2 x \u2208 (Set.indicator (s j) fun x => 1) \u207b\u00b9' {1} \u2194 x \u2208 s j", "state_after": "no goals"}, {"tactic": "rintro t \u27e8n, ht\u27e9", "annotated_tactic": ["rintro t \u27e8n, ht\u27e9", []], "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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 {s_1 | \u2203 n, MeasurableSet s_1} \u2192 MeasurableSet t", "state_after": "case h.refine'_2.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\n\u22a2 MeasurableSet t"}, {"tactic": "suffices MeasurableSpace.generateFrom {t | n \u2264 i \u2227\n  MeasurableSet[MeasurableSpace.comap ((s n).indicator (fun _ => 1 : \u03a9 \u2192 \u03b2)) m\u03b2] t} \u2264\n    MeasurableSpace.generateFrom {t | \u2203 (j : \u03b9), j \u2264 i \u2227 s j = t} by\n  exact this _ ht", "annotated_tactic": ["suffices <a>MeasurableSpace.generateFrom</a> {t | n \u2264 i \u2227\n      MeasurableSet[<a>MeasurableSpace.comap</a> ((s n).<a>indicator</a> (fun _ => 1 : \u03a9 \u2192 \u03b2)) m\u03b2] t} \u2264\n        <a>MeasurableSpace.generateFrom</a> {t | \u2203 (j : \u03b9), j \u2264 i \u2227 s j = t} by\n      exact this _ ht", [{"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}, {"full_name": "MeasurableSpace.comap", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [102, 15], "def_end_pos": [102, 20]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case h.refine'_2.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\n\u22a2 MeasurableSpace.generateFrom {t | n \u2264 i \u2227 MeasurableSet t} \u2264 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t}"}, {"tactic": "refine' generateFrom_le _", "annotated_tactic": ["refine' <a>generateFrom_le</a> _", [{"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}]], "state_before": "case h.refine'_2.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\n\u22a2 MeasurableSpace.generateFrom {t | n \u2264 i \u2227 MeasurableSet t} \u2264 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t}", "state_after": "case h.refine'_2.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 {t | n \u2264 i \u2227 MeasurableSet t} \u2192 MeasurableSet t"}, {"tactic": "rintro t \u27e8hn, u, _, hu'\u27e9", "annotated_tactic": ["rintro t \u27e8hn, u, _, hu'\u27e9", []], "state_before": "case h.refine'_2.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u03a9), t \u2208 {t | n \u2264 i \u2227 MeasurableSet t} \u2192 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\n\u22a2 MeasurableSet t"}, {"tactic": "obtain heq | heq | heq | heq := Set.indicator_const_preimage (s n) u (1 : \u03b2)", "annotated_tactic": ["obtain heq | heq | heq | heq := <a>Set.indicator_const_preimage</a> (s n) u (1 : \u03b2)", [{"full_name": "Set.indicator_const_preimage", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [302, 3], "def_end_pos": [302, 14]}]], "state_before": "case h.refine'_2.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u \u2208 {\u2205}\n\u22a2 MeasurableSet t"}, {"tactic": "pick_goal 4", "annotated_tactic": ["pick_goal 4", []], "state_before": "case h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u \u2208 {\u2205}\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u \u2208 {\u2205}\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t"}, {"tactic": "rw [Set.mem_singleton_iff] at heq", "annotated_tactic": ["rw [<a>Set.mem_singleton_iff</a>] at heq", [{"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.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u \u2208 {\u2205}\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = \u2205\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t"}, {"tactic": "all_goals rw [heq] at hu'; rw [\u2190 hu']", "annotated_tactic": ["all_goals rw [heq] at hu'; rw [\u2190 hu']", []], "state_before": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = \u2205\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet t\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : \u2205 = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = \u2205\n\u22a2 MeasurableSet \u2205\n\ncase h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : Set.univ = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet Set.univ\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : s n = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet (s n)\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (s n)\u1d9c = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet (s n)\u1d9c"}, {"tactic": "exacts [measurableSet_empty _, MeasurableSet.univ, measurableSet_generateFrom \u27e8n, hn, rfl\u27e9,\n  MeasurableSet.compl (measurableSet_generateFrom \u27e8n, hn, rfl\u27e9)]", "annotated_tactic": ["exacts [<a>measurableSet_empty</a> _, <a>MeasurableSet.univ</a>, <a>measurableSet_generateFrom</a> \u27e8n, hn, <a>rfl</a>\u27e9,\n      <a>MeasurableSet.compl</a> (<a>measurableSet_generateFrom</a> \u27e8n, hn, <a>rfl</a>\u27e9)]", [{"full_name": "MeasurableSpace.measurableSet_empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [53, 3], "def_end_pos": [53, 22]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"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.refine'_2.intro.intro.intro.intro.inr.inr.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : \u2205 = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = \u2205\n\u22a2 MeasurableSet \u2205\n\ncase h.refine'_2.intro.intro.intro.intro.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : Set.univ = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ\n\u22a2 MeasurableSet Set.univ\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : s n = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n\n\u22a2 MeasurableSet (s n)\n\ncase h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (s n)\u1d9c = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet (s n)\u1d9c", "state_after": "no goals"}, {"tactic": "exact this _ ht", "annotated_tactic": ["exact this _ ht", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\nthis :\n  MeasurableSpace.generateFrom {t | n \u2264 i \u2227 MeasurableSet t} \u2264 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t}\n\u22a2 MeasurableSet t", "state_after": "no goals"}, {"tactic": "rw [heq] at hu'", "annotated_tactic": ["rw [heq] at hu'", []], "state_before": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (s n)\u1d9c = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t"}, {"tactic": "rw [\u2190 hu']", "annotated_tactic": ["rw [\u2190 hu']", []], "state_before": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (s n)\u1d9c = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet t", "state_after": "case h.refine'_2.intro.intro.intro.intro.inr.inr.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : MulZeroOneClass \u03b2\ninst\u271d : Nontrivial \u03b2\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\ni : \u03b9\nt\u271d : Set \u03a9\nn : \u03b9\nht : MeasurableSet t\u271d\nt : Set \u03a9\nhn : n \u2264 i\nu : Set \u03b2\nleft\u271d : MeasurableSet u\nhu' : (s n)\u1d9c = t\nheq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c\n\u22a2 MeasurableSet (s n)\u1d9c"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.step_run", "start": [2506, 1], "end": [2510, 23], "traced_tactics": [{"tactic": "unfold stWrite", "annotated_tactic": ["unfold <a>stWrite</a>", [{"full_name": "Turing.TM2to1.stWrite", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2435, 5], "def_end_pos": [2435, 12]}]], "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\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =\n    TM2.stepAux q (stVar v (S k) (StAct.peek f)) (update S k (stWrite v (S k) (StAct.peek f)))", "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\nk : K\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =\n    TM2.stepAux q (stVar v (S k) (StAct.peek f))\n      (update S k\n        (match StAct.peek f with\n        | StAct.push f => f v :: S k\n        | StAct.peek a => S k\n        | StAct.pop a => List.tail (S k)))"}, {"tactic": "rw [Function.update_eq_self]", "annotated_tactic": ["rw [<a>Function.update_eq_self</a>]", [{"full_name": "Function.update_eq_self", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 23]}]], "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\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =\n    TM2.stepAux q (stVar v (S k) (StAct.peek f))\n      (update S k\n        (match StAct.peek f with\n        | StAct.push f => f v :: S k\n        | StAct.peek a => S k\n        | StAct.pop a => List.tail (S 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\nk : K\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S = TM2.stepAux q (stVar v (S k) (StAct.peek f)) S"}, {"tactic": "rfl", "annotated_tactic": ["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\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S = TM2.stepAux q (stVar v (S k) (StAct.peek f)) S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.cgf_zero", "start": [170, 1], "end": [171, 67], "traced_tactics": [{"tactic": "simp only [cgf_zero', measure_univ, ENNReal.one_toReal, log_one]", "annotated_tactic": ["simp only [<a>cgf_zero'</a>, <a>measure_univ</a>, <a>ENNReal.one_toReal</a>, <a>log_one</a>]", [{"full_name": "ProbabilityTheory.cgf_zero'", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [166, 9], "def_end_pos": [166, 18]}, {"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": "Real.log_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 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\n\u22a2 cgf X \u03bc 0 = 0", "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.take_length", "start": [169, 9], "end": [169, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.size_eq", "start": [680, 1], "end": [680, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrableOn.smul_continuousOn", "start": [601, 1], "end": [606, 88], "traced_tactics": [{"tactic": "rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2", "annotated_tactic": ["rw [<a>MeasureTheory.locallyIntegrableOn_iff</a> (<a>Or.inr</a> hs)] at hf \u22a2", [{"full_name": "MeasureTheory.locallyIntegrableOn_iff", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"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": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 \ud835\udd5c\ng : X \u2192 E\ns : Set X\nhs : IsOpen s\nhf : LocallyIntegrableOn f s\nhg : ContinuousOn g s\n\u22a2 LocallyIntegrableOn (fun x => f x \u2022 g x) s", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 \ud835\udd5c\ng : X \u2192 E\ns : Set X\nhs : IsOpen s\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nhg : ContinuousOn g s\n\u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => f x \u2022 g x) k"}, {"tactic": "exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c", "annotated_tactic": ["exact fun k hk_sub hk_c => (hf k hk_sub hk_c).<a>smul_continuousOn</a> (hg.mono hk_sub) hk_c", [{"full_name": "MeasureTheory.IntegrableOn.smul_continuousOn", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [565, 9], "def_end_pos": [565, 39]}]], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d g\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 \ud835\udd5c\ng : X \u2192 E\ns : Set X\nhs : IsOpen s\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nhg : ContinuousOn g s\n\u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => f x \u2022 g x) k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "exists_partition_approximatesLinearOn_of_hasFDerivWithinAt", "start": [254, 1], "end": [269, 94], "traced_tactics": [{"tactic": "rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with\n  \u27e8t, A, t_closed, st, t_approx, ht\u27e9", "annotated_tactic": ["rcases <a>exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt</a> f s f' hf' r rpos with\n    \u27e8t, A, t_closed, st, t_approx, ht\u27e9", [{"full_name": "exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [111, 9], "def_end_pos": [111, 70]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\n\u22a2 \u2203 t A,\n    Pairwise (Disjoint on t) \u2227\n      (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n        s \u2286 \u22c3 n, t n \u2227\n          (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n            (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)", "state_after": "case intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 t A,\n    Pairwise (Disjoint on t) \u2227\n      (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n        s \u2286 \u22c3 n, t n \u2227\n          (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n            (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)"}, {"tactic": "refine'\n  \u27e8disjointed t, A, disjoint_disjointed _,\n    MeasurableSet.disjointed fun n => (t_closed n).measurableSet, _, _, ht\u27e9", "annotated_tactic": ["refine'\n    \u27e8<a>disjointed</a> t, A, <a>disjoint_disjointed</a> _,\n      <a>MeasurableSet.disjointed</a> fun n => (t_closed n).<a>measurableSet</a>, _, _, ht\u27e9", [{"full_name": "disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [49, 5], "def_end_pos": [49, 15]}, {"full_name": "disjoint_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [74, 9], "def_end_pos": [74, 28]}, {"full_name": "MeasurableSet.disjointed", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [236, 19], "def_end_pos": [236, 43]}, {"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.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 t A,\n    Pairwise (Disjoint on t) \u2227\n      (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n        s \u2286 \u22c3 n, t n \u2227\n          (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n            (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)", "state_after": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, disjointed t n\n\ncase intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))"}, {"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": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, disjointed t n", "state_after": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, t n"}, {"tactic": "exact st", "annotated_tactic": ["exact st", []], "state_before": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, t n", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))", "state_after": "case intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : 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 disjointed t n) (r (A n))"}, {"tactic": "exact (t_approx n).mono_set (inter_subset_inter_right _ (disjointed_subset _ _))", "annotated_tactic": ["exact (t_approx n).<a>mono_set</a> (<a>inter_subset_inter_right</a> _ (<a>disjointed_subset</a> _ _))", [{"full_name": "ApproximatesLinearOn.mono_set", "def_path": "Mathlib/Analysis/Calculus/Inverse.lean", "def_pos": [142, 9], "def_end_pos": [142, 17]}, {"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": "disjointed_subset", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : 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 disjointed t n) (r (A n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.range_extend", "start": [139, 1], "end": [143, 69], "traced_tactics": [{"tactic": "refine' (range_extend_subset _ _ _).antisymm _", "annotated_tactic": ["refine' (<a>range_extend_subset</a> _ _ _).<a>antisymm</a> _", [{"full_name": "Set.range_extend_subset", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [130, 9], "def_end_pos": [130, 28]}, {"full_name": "HasSubset.Subset.antisymm", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [667, 7], "def_end_pos": [667, 32]}]], "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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\n\u22a2 range (extend f g g') = range g \u222a g' '' (range f)\u1d9c", "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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\n\u22a2 range g \u222a g' '' (range f)\u1d9c \u2286 range (extend f g g')"}, {"tactic": "rintro z (\u27e8x, rfl\u27e9 | \u27e8y, hy, rfl\u27e9)", "annotated_tactic": ["rintro z (\u27e8x, rfl\u27e9 | \u27e8y, hy, rfl\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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\n\u22a2 range g \u222a g' '' (range f)\u1d9c \u2286 range (extend f g g')", "state_after": "case inl.intro\n\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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nx : \u03b1\n\u22a2 g x \u2208 range (extend f g g')\n\ncase inr.intro.intro\n\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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\ny : \u03b2\nhy : y \u2208 (range f)\u1d9c\n\u22a2 g' y \u2208 range (extend f g g')"}, {"tactic": "exacts [\u27e8f x, hf.extend_apply _ _ _\u27e9, \u27e8y, extend_apply' _ _ _ hy\u27e9]", "annotated_tactic": ["exacts [\u27e8f x, hf.extend_apply _ _ _\u27e9, \u27e8y, <a>extend_apply'</a> _ _ _ hy\u27e9]", [{"full_name": "Function.extend_apply'", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [743, 9], "def_end_pos": [743, 22]}]], "state_before": "case inl.intro\n\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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nx : \u03b1\n\u22a2 g x \u2208 range (extend f g g')\n\ncase inr.intro.intro\n\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\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\ny : \u03b2\nhy : y \u2208 (range f)\u1d9c\n\u22a2 g' y \u2208 range (extend f g g')", "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_finite", "start": [69, 1], "end": [70, 84], "traced_tactics": [{"tactic": "rw [\u2190 count_apply_finset, Finite.coe_toFinset]", "annotated_tactic": ["rw [\u2190 <a>count_apply_finset</a>, <a>Finite.coe_toFinset</a>]", [{"full_name": "MeasureTheory.Measure.count_apply_finset", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [58, 9], "def_end_pos": [58, 27]}, {"full_name": "Set.Finite.coe_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [178, 19], "def_end_pos": [178, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.8455\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Set \u03b1\nhs : Set.Finite s\n\u22a2 \u2191\u2191count s = \u2191(Finset.card (Finite.toFinset hs))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean", "full_name": "Set.MulAntidiagonal.finite_of_isWf", "start": [136, 1], "end": [138, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet", "start": [202, 1], "end": [206, 55], "traced_tactics": [{"tactic": "obtain \u27e8t', ht', rfl\u27e9 := ht", "annotated_tactic": ["obtain \u27e8t', ht', rfl\u27e9 := ht", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : AEMeasurable Y\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s \u2202\u03bc = \u2191\u2191\u03bc (t \u2229 Y \u207b\u00b9' s)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : AEMeasurable Y\nhs : MeasurableSet s\nt' : Set \u03b2\nht' : MeasurableSet t'\n\u22a2 \u222b\u207b (a : \u03b1) in X \u207b\u00b9' t', \u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s \u2202\u03bc = \u2191\u2191\u03bc (X \u207b\u00b9' t' \u2229 Y \u207b\u00b9' s)"}, {"tactic": "rw [set_lintegral_preimage_condDistrib hX hY hs ht']", "annotated_tactic": ["rw [<a>set_lintegral_preimage_condDistrib</a> hX hY hs ht']", [{"full_name": "ProbabilityTheory.set_lintegral_preimage_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [191, 9], "def_end_pos": [191, 43]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\ninst\u271d\u00b9 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : 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\nhX : Measurable X\nhY : AEMeasurable Y\nhs : MeasurableSet s\nt' : Set \u03b2\nht' : MeasurableSet t'\n\u22a2 \u222b\u207b (a : \u03b1) in X \u207b\u00b9' t', \u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s \u2202\u03bc = \u2191\u2191\u03bc (X \u207b\u00b9' t' \u2229 Y \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive", "start": [326, 1], "end": [340, 80], "traced_tactics": [{"tactic": "have hwuv : s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 := by\n  refine'\n    subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv))\n      (hu.symmDiff hv) (restrict_le_restrict_union _ _ hu hsu hv hsv) hs\n      Set.symmDiff_subset_union _\n  rw [\u2190 Set.inter_symmDiff_distrib_left]\n  exact Set.inter_subset_right _ _", "annotated_tactic": ["have hwuv : s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 := by\n    refine'\n      <a>subset_positive_null_set</a> (hu.union hv) ((hw.inter hu).<a>symmDiff</a> (hw.inter hv))\n        (hu.symmDiff hv) (<a>restrict_le_restrict_union</a> _ _ hu hsu hv hsv) hs\n        <a>Set.symmDiff_subset_union</a> _\n    rw [\u2190 <a>Set.inter_symmDiff_distrib_left</a>]\n    exact <a>Set.inter_subset_right</a> _ _", [{"full_name": "MeasureTheory.SignedMeasure.subset_positive_null_set", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [278, 9], "def_end_pos": [278, 33]}, {"full_name": "MeasurableSet.symmDiff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [211, 19], "def_end_pos": [211, 41]}, {"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_union", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [970, 9], "def_end_pos": [970, 35]}, {"full_name": "Set.symmDiff_subset_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2124, 9], "def_end_pos": [2124, 30]}, {"full_name": "Set.inter_symmDiff_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2138, 9], "def_end_pos": [2138, 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\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\nhwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)"}, {"tactic": "obtain \u27e8huv, hvu\u27e9 :=\n  of_diff_eq_zero_of_symmDiff_eq_zero_positive (hw.inter hu) (hw.inter hv)\n    (restrict_le_restrict_subset _ _ hu hsu (w.inter_subset_right u))\n    (restrict_le_restrict_subset _ _ hv hsv (w.inter_subset_right v)) hwuv", "annotated_tactic": ["obtain \u27e8huv, hvu\u27e9 :=\n    <a>of_diff_eq_zero_of_symmDiff_eq_zero_positive</a> (hw.inter hu) (hw.inter hv)\n      (<a>restrict_le_restrict_subset</a> _ _ hu hsu (w.inter_subset_right u))\n      (<a>restrict_le_restrict_subset</a> _ _ hv hsv (w.inter_subset_right v)) hwuv", [{"full_name": "MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [303, 9], "def_end_pos": [303, 53]}, {"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [883, 9], "def_end_pos": [883, 36]}, {"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_subset", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [883, 9], "def_end_pos": [883, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\nhwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\nhwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\nhuv : \u2191s ((w \u2229 u) \\ (w \u2229 v)) = 0\nhvu : \u2191s ((w \u2229 v) \\ (w \u2229 u)) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)"}, {"tactic": "rw [\u2190 of_diff_of_diff_eq_zero (hw.inter hu) (hw.inter hv) hvu, huv, zero_add]", "annotated_tactic": ["rw [\u2190 <a>of_diff_of_diff_eq_zero</a> (hw.inter hu) (hw.inter hv) hvu, huv, <a>zero_add</a>]", [{"full_name": "MeasureTheory.VectorMeasure.of_diff_of_diff_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [202, 9], "def_end_pos": [202, 32]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\nhwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\nhuv : \u2191s ((w \u2229 u) \\ (w \u2229 v)) = 0\nhvu : \u2191s ((w \u2229 v) \\ (w \u2229 u)) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)", "state_after": "no goals"}, {"tactic": "refine'\n  subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv))\n    (hu.symmDiff hv) (restrict_le_restrict_union _ _ hu hsu hv hsv) hs\n    Set.symmDiff_subset_union _", "annotated_tactic": ["refine'\n      <a>subset_positive_null_set</a> (hu.union hv) ((hw.inter hu).<a>symmDiff</a> (hw.inter hv))\n        (hu.symmDiff hv) (<a>restrict_le_restrict_union</a> _ _ hu hsu hv hsv) hs\n        <a>Set.symmDiff_subset_union</a> _", [{"full_name": "MeasureTheory.SignedMeasure.subset_positive_null_set", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [278, 9], "def_end_pos": [278, 33]}, {"full_name": "MeasurableSet.symmDiff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [211, 19], "def_end_pos": [211, 41]}, {"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_union", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [970, 9], "def_end_pos": [970, 35]}, {"full_name": "Set.symmDiff_subset_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2124, 9], "def_end_pos": [2124, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 (w \u2229 u) \u2206 (w \u2229 v) \u2286 u \u2206 v"}, {"tactic": "rw [\u2190 Set.inter_symmDiff_distrib_left]", "annotated_tactic": ["rw [\u2190 <a>Set.inter_symmDiff_distrib_left</a>]", [{"full_name": "Set.inter_symmDiff_distrib_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2138, 9], "def_end_pos": [2138, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 (w \u2229 u) \u2206 (w \u2229 v) \u2286 u \u2206 v", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 w \u2229 u \u2206 v \u2286 u \u2206 v"}, {"tactic": "exact Set.inter_subset_right _ _", "annotated_tactic": ["exact <a>Set.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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 w \u2229 u \u2206 v \u2286 u \u2206 v", "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.eq_one_of_mul_eq_self_left", "start": [1322, 1], "end": [1323, 60], "traced_tactics": [{"tactic": "rw [Int.one_mul, H]", "annotated_tactic": ["rw [<a>Int.one_mul</a>, H]", [{"full_name": "Int.one_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [521, 27], "def_end_pos": [521, 34]}]], "state_before": "a b : Int\nHpos : a \u2260 0\nH : b * a = a\n\u22a2 b * a = 1 * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quot.lift\u2082_mk", "start": [124, 1], "end": [127, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "IsUnit.measurable_const_smul_iff", "start": [738, 8], "end": [741, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/EpsilonNFA.lean", "full_name": "NFA.to\u03b5NFA_evalFrom_match", "start": [193, 1], "end": [203, 6], "traced_tactics": [{"tactic": "rw [evalFrom, \u03b5NFA.evalFrom, to\u03b5NFA_\u03b5Closure]", "annotated_tactic": ["rw [<a>evalFrom</a>, <a>\u03b5NFA.evalFrom</a>, <a>to\u03b5NFA_\u03b5Closure</a>]", [{"full_name": "NFA.evalFrom", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [66, 5], "def_end_pos": [66, 13]}, {"full_name": "\u03b5NFA.evalFrom", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [90, 5], "def_end_pos": [90, 13]}, {"full_name": "NFA.to\u03b5NFA_\u03b5Closure", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [184, 9], "def_end_pos": [184, 24]}]], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart : Set \u03c3\n\u22a2 \u03b5NFA.evalFrom (to\u03b5NFA M) start = evalFrom M start", "state_after": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart : Set \u03c3\n\u22a2 List.foldl (\u03b5NFA.stepSet (to\u03b5NFA M)) start = List.foldl (stepSet M) start"}, {"tactic": "suffices \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M by rw [this]", "annotated_tactic": ["suffices <a>\u03b5NFA.stepSet</a> (<a>to\u03b5NFA</a> M) = <a>stepSet</a> M by rw [this]", [{"full_name": "\u03b5NFA.stepSet", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [70, 5], "def_end_pos": [70, 12]}, {"full_name": "NFA.to\u03b5NFA", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [177, 5], "def_end_pos": [177, 11]}, {"full_name": "NFA.stepSet", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [52, 5], "def_end_pos": [52, 12]}]], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart : Set \u03c3\n\u22a2 List.foldl (\u03b5NFA.stepSet (to\u03b5NFA M)) start = List.foldl (stepSet M) start", "state_after": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart : Set \u03c3\n\u22a2 \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M"}, {"tactic": "ext S s", "annotated_tactic": ["ext S s", []], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart : Set \u03c3\n\u22a2 \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M", "state_after": "case h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 x\u271d \u2208 \u03b5NFA.stepSet (to\u03b5NFA M) S s \u2194 x\u271d \u2208 stepSet M S s"}, {"tactic": "simp only [stepSet, \u03b5NFA.stepSet, exists_prop, Set.mem_iUnion]", "annotated_tactic": ["simp only [<a>stepSet</a>, <a>\u03b5NFA.stepSet</a>, <a>exists_prop</a>, <a>Set.mem_iUnion</a>]", [{"full_name": "NFA.stepSet", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [52, 5], "def_end_pos": [52, 12]}, {"full_name": "\u03b5NFA.stepSet", "def_path": "Mathlib/Computability/EpsilonNFA.lean", "def_pos": [70, 5], "def_end_pos": [70, 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]}]], "state_before": "case h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 x\u271d \u2208 \u03b5NFA.stepSet (to\u03b5NFA M) S s \u2194 x\u271d \u2208 stepSet M S s", "state_after": "case h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 (\u2203 i, i \u2208 S \u2227 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) i (some s))) \u2194 \u2203 i, i \u2208 S \u2227 x\u271d \u2208 step M i s"}, {"tactic": "apply exists_congr", "annotated_tactic": ["apply <a>exists_congr</a>", [{"full_name": "exists_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [379, 9], "def_end_pos": [379, 21]}]], "state_before": "case h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 (\u2203 i, i \u2208 S \u2227 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) i (some s))) \u2194 \u2203 i, i \u2208 S \u2227 x\u271d \u2208 step M i s", "state_after": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 \u2200 (a : \u03c3), a \u2208 S \u2227 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a (some s)) \u2194 a \u2208 S \u2227 x\u271d \u2208 step M a s"}, {"tactic": "simp only [and_congr_right_iff]", "annotated_tactic": ["simp only [<a>and_congr_right_iff</a>]", [{"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 h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 \u2200 (a : \u03c3), a \u2208 S \u2227 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a (some s)) \u2194 a \u2208 S \u2227 x\u271d \u2208 step M a s", "state_after": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 \u2200 (a : \u03c3), a \u2208 S \u2192 (x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a (some s)) \u2194 x\u271d \u2208 step M a s)"}, {"tactic": "intro _ _", "annotated_tactic": ["intro _ _", []], "state_before": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d : \u03c3\n\u22a2 \u2200 (a : \u03c3), a \u2208 S \u2192 (x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a (some s)) \u2194 x\u271d \u2208 step M a s)", "state_after": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d a\u271d\u00b9 : \u03c3\na\u271d : a\u271d\u00b9 \u2208 S\n\u22a2 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a\u271d\u00b9 (some s)) \u2194 x\u271d \u2208 step M a\u271d\u00b9 s"}, {"tactic": "rw [M.to\u03b5NFA_\u03b5Closure]", "annotated_tactic": ["rw [M.to\u03b5NFA_\u03b5Closure]", []], "state_before": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d a\u271d\u00b9 : \u03c3\na\u271d : a\u271d\u00b9 \u2208 S\n\u22a2 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a\u271d\u00b9 (some s)) \u2194 x\u271d \u2208 step M a\u271d\u00b9 s", "state_after": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d a\u271d\u00b9 : \u03c3\na\u271d : a\u271d\u00b9 \u2208 S\n\u22a2 x\u271d \u2208 \u03b5NFA.step (to\u03b5NFA M) a\u271d\u00b9 (some s) \u2194 x\u271d \u2208 step M a\u271d\u00b9 s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h.h.h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS\u271d : Set \u03c3\nx : List \u03b1\ns\u271d : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart S : Set \u03c3\ns : \u03b1\nx\u271d a\u271d\u00b9 : \u03c3\na\u271d : a\u271d\u00b9 \u2208 S\n\u22a2 x\u271d \u2208 \u03b5NFA.step (to\u03b5NFA M) a\u271d\u00b9 (some s) \u2194 x\u271d \u2208 step M a\u271d\u00b9 s", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM\u271d : \u03b5NFA \u03b1 \u03c3\nS : Set \u03c3\nx : List \u03b1\ns : \u03c3\na : \u03b1\nM : NFA \u03b1 \u03c3\nstart : Set \u03c3\nthis : \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M\n\u22a2 List.foldl (\u03b5NFA.stepSet (to\u03b5NFA M)) start = List.foldl (stepSet M) start", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.card_insert", "start": [1218, 1], "end": [1220, 76], "traced_tactics": [{"tactic": "rw [\u2190 card_fintypeInsertOfNotMem s h]", "annotated_tactic": ["rw [\u2190 <a>card_fintypeInsertOfNotMem</a> s h]", [{"full_name": "Set.card_fintypeInsertOfNotMem", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1209, 9], "def_end_pos": [1209, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\na : \u03b1\ns : Set \u03b1\ninst\u271d : Fintype \u2191s\nh : \u00aca \u2208 s\nd : Fintype \u2191(insert a s)\n\u22a2 Fintype.card \u2191(insert a s) = Fintype.card \u2191s + 1", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\na : \u03b1\ns : Set \u03b1\ninst\u271d : Fintype \u2191s\nh : \u00aca \u2208 s\nd : Fintype \u2191(insert a s)\n\u22a2 Fintype.card \u2191(insert a s) = Fintype.card \u2191(insert a s)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\na : \u03b1\ns : Set \u03b1\ninst\u271d : Fintype \u2191s\nh : \u00aca \u2208 s\nd : Fintype \u2191(insert a s)\n\u22a2 Fintype.card \u2191(insert a s) = Fintype.card \u2191(insert a s)", "state_after": "case h.e_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\na : \u03b1\ns : Set \u03b1\ninst\u271d : Fintype \u2191s\nh : \u00aca \u2208 s\nd : Fintype \u2191(insert a s)\n\u22a2 d = fintypeInsertOfNotMem s h"}, {"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\na : \u03b1\ns : Set \u03b1\ninst\u271d : Fintype \u2191s\nh : \u00aca \u2208 s\nd : Fintype \u2191(insert a s)\n\u22a2 d = fintypeInsertOfNotMem s h", "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.mod_add_div", "start": [550, 1], "end": [553, 93], "traced_tactics": [{"tactic": "induction m, k using mod.inductionOn with rw [div_eq, mod_eq]\n| base x y h => simp [h]\n| ind x y h IH => simp [h]; rw [Nat.mul_succ, \u2190 Nat.add_assoc, IH, Nat.sub_add_cancel h.2]", "annotated_tactic": ["induction m, k using <a>mod.inductionOn</a> with rw [<a>div_eq</a>, <a>mod_eq</a>]\n  | base x y h => simp [h]\n  | ind x y h IH => simp [h]; rw [<a>Nat.mul_succ</a>, \u2190 <a>Nat.add_assoc</a>, IH, <a>Nat.sub_add_cancel</a> h.2]", [{"full_name": "Nat.mod.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [97, 9], "def_end_pos": [97, 24]}, {"full_name": "Nat.div_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [25, 9], "def_end_pos": [25, 15]}, {"full_name": "Nat.mod_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [94, 9], "def_end_pos": [94, 15]}, {"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.sub_add_cancel", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [583, 19], "def_end_pos": [583, 33]}]], "state_before": "m k : Nat\n\u22a2 m % k + k * (m / k) = m", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case base\nx y : Nat\nh : \u00ac(0 < y \u2227 y \u2264 x)\n\u22a2 ((if 0 < y \u2227 y \u2264 x then (x - y) % y else x) + y * if 0 < y \u2227 y \u2264 x then (x - y) / y + 1 else 0) = x", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case ind\nx y : Nat\nh : 0 < y \u2227 y \u2264 x\nIH : (x - y) % y + y * ((x - y) / y) = x - y\n\u22a2 ((if 0 < y \u2227 y \u2264 x then (x - y) % y else x) + y * if 0 < y \u2227 y \u2264 x then (x - y) / y + 1 else 0) = x", "state_after": "case ind\nx y : Nat\nh : 0 < y \u2227 y \u2264 x\nIH : (x - y) % y + y * ((x - y) / y) = x - y\n\u22a2 (x - y) % y + y * ((x - y) / y + 1) = x"}, {"tactic": "rw [Nat.mul_succ, \u2190 Nat.add_assoc, IH, Nat.sub_add_cancel h.2]", "annotated_tactic": ["rw [<a>Nat.mul_succ</a>, \u2190 <a>Nat.add_assoc</a>, IH, <a>Nat.sub_add_cancel</a> h.2]", [{"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.sub_add_cancel", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [583, 19], "def_end_pos": [583, 33]}]], "state_before": "case ind\nx y : Nat\nh : 0 < y \u2227 y \u2264 x\nIH : (x - y) % y + y * ((x - y) / y) = x - y\n\u22a2 (x - y) % y + y * ((x - y) / y + 1) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Set.IciExtend_self", "start": [252, 1], "end": [253, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "isPiSystem_generatePiSystem", "start": [243, 1], "end": [244, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Floor.lean", "full_name": "MeasurableSet.image_fract", "start": [59, 1], "end": [62, 90], "traced_tactics": [{"tactic": "simp only [Int.image_fract, sub_eq_add_neg, image_add_right']", "annotated_tactic": ["simp only [<a>Int.image_fract</a>, <a>sub_eq_add_neg</a>, <a>image_add_right'</a>]", [{"full_name": "Int.image_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1068, 9], "def_end_pos": [1068, 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": [1215, 3], "def_end_pos": [1215, 14]}]], "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 '' 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 m => (measurable_add_const _ hs).inter measurableSet_Ico", "annotated_tactic": ["exact <a>MeasurableSet.iUnion</a> fun m => (<a>measurable_add_const</a> _ hs).<a>inter</a> <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": "MeasurableAdd.measurable_add_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [63, 3], "def_end_pos": [63, 23]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"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/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mem_iff_find?", "start": [672, 1], "end": [673, 64], "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.expand_WF.foldl", "start": [113, 1], "end": [141, 25], "traced_tactics": [{"tactic": "induction l generalizing target with\n| nil => exact \u27e8ht\u2081, fun _ h\u2081 _ h\u2082 => (ht\u2082 _ h\u2081 _ h\u2082).1\u27e9\n| cons _ _ ih =>\n  simp at hl\u2081 hl\u2082 ht\u2082\n  refine ih hl\u2081.2 hl\u2082.2\n    (reinsertAux_WF ht\u2081 fun _ h => (ht\u2082 _ (Array.getElem_mem_data ..) _ h).2.1)\n    (fun _ h => ?_)\n  simp [reinsertAux, Buckets.update] at h\n  match List.mem_or_eq_of_mem_set h with\n  | .inl h =>\n    intro _ hf\n    have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf\n    exact \u27e8h\u2081, h\u2082.2\u27e9\n  | .inr h => subst h; intro\n    | _, .head .. =>\n      exact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9\n    | _, .tail _ h =>\n      have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h\n      exact \u27e8h\u2081, h\u2082.2\u27e9", "annotated_tactic": ["induction l generalizing target with\n  | <a>nil</a> => exact \u27e8ht\u2081, fun _ h\u2081 _ h\u2082 => (ht\u2082 _ h\u2081 _ h\u2082).1\u27e9\n  | <a>cons</a> _ _ ih =>\n    simp at hl\u2081 hl\u2082 ht\u2082\n    refine ih hl\u2081.2 hl\u2082.2\n      (<a>reinsertAux_WF</a> ht\u2081 fun _ h => (ht\u2082 _ (<a>Array.getElem_mem_data</a> ..) _ h).2.1)\n      (fun _ h => ?_)\n    simp [<a>reinsertAux</a>, <a>Buckets.update</a>] at h\n    match <a>List.mem_or_eq_of_mem_set</a> h with\n    | .inl h =>\n      intro _ hf\n      have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf\n      exact \u27e8h\u2081, h\u2082.2\u27e9\n    | .inr h => subst h; intro\n      | _, .head .. =>\n        exact \u27e8hl\u2082.1 \u25b8 <a>Nat.le_refl</a> _, fun _ h h' => hl\u2081.1 _ h (<a>PartialEquivBEq.symm</a> h')\u27e9\n      | _, .tail _ h =>\n        have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (<a>Array.getElem_mem_data</a> ..) _ h\n        exact \u27e8h\u2081, h\u2082.2\u27e9", [{"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": "Std.HashMap.Imp.reinsertAux_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [70, 9], "def_end_pos": [70, 23]}, {"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}, {"full_name": "Std.HashMap.Imp.reinsertAux", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [102, 15], "def_end_pos": [102, 26]}, {"full_name": "Std.HashMap.Imp.Buckets.update", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [32, 5], "def_end_pos": [32, 11]}, {"full_name": "List.mem_or_eq_of_mem_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [965, 9], "def_end_pos": [965, 29]}, {"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": "PartialEquivBEq.symm", "def_path": "lake-packages/std/Std/Classes/BEq.lean", "def_pos": [16, 3], "def_end_pos": [16, 7]}, {"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\nl : List (\u03b1 \u00d7 \u03b2)\ni : Nat\nhl\u2081 : \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l\nhl\u2082 : \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 l \u2192 rank x.fst = i\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 l \u2192 \u00ac(x.fst == k) = true)\n        bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target l) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target l).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket", "state_after": "no goals"}, {"tactic": "exact \u27e8ht\u2081, fun _ h\u2081 _ h\u2082 => (ht\u2082 _ h\u2081 _ h\u2082).1\u27e9", "annotated_tactic": ["exact \u27e8ht\u2081, fun _ h\u2081 _ h\u2082 => (ht\u2082 _ h\u2081 _ h\u2082).1\u27e9", []], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhl\u2081 : \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) []\nhl\u2082 : \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 [] \u2192 rank x.fst = i\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 [] \u2192 \u00ac(x.fst == k) = true)\n        bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target []) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target []).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket", "state_after": "no goals"}, {"tactic": "simp at hl\u2081 hl\u2082 ht\u2082", "annotated_tactic": ["simp at hl\u2081 hl\u2082 ht\u2082", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (head\u271d :: tail\u271d)\nhl\u2082 : \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 head\u271d :: tail\u271d \u2192 rank x.fst = i\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n              x \u2208 head\u271d :: tail\u271d \u2192 \u00ac(x.fst == k) = true)\n        bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket"}, {"tactic": "refine ih hl\u2081.2 hl\u2082.2\n  (reinsertAux_WF ht\u2081 fun _ h => (ht\u2082 _ (Array.getElem_mem_data ..) _ h).2.1)\n  (fun _ h => ?_)", "annotated_tactic": ["refine ih hl\u2081.2 hl\u2082.2\n      (<a>reinsertAux_WF</a> ht\u2081 fun _ h => (ht\u2082 _ (<a>Array.getElem_mem_data</a> ..) _ h).2.1)\n      (fun _ h => ?_)", [{"full_name": "Std.HashMap.Imp.reinsertAux_WF", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [70, 9], "def_end_pos": [70, 23]}, {"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\n\u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)) \u2227\n    \u2200 (bucket : AssocList \u03b1 \u03b2),\n      bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)).val.data \u2192\n        AssocList.All (fun k x => rank k \u2264 i) bucket", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh : x\u271d \u2208 ((fun d x => reinsertAux d x.fst x.snd) target head\u271d).val.data\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    x\u271d"}, {"tactic": "simp [reinsertAux, Buckets.update] at h", "annotated_tactic": ["simp [<a>reinsertAux</a>, <a>Buckets.update</a>] at h", [{"full_name": "Std.HashMap.Imp.reinsertAux", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [102, 15], "def_end_pos": [102, 26]}, {"full_name": "Std.HashMap.Imp.Buckets.update", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [32, 5], "def_end_pos": [32, 11]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh : x\u271d \u2208 ((fun d x => reinsertAux d x.fst x.snd) target head\u271d).val.data\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    x\u271d", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    x\u271d"}, {"tactic": "match List.mem_or_eq_of_mem_set h with\n| .inl h =>\n  intro _ hf\n  have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf\n  exact \u27e8h\u2081, h\u2082.2\u27e9\n| .inr h => subst h; intro\n  | _, .head .. =>\n    exact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9\n  | _, .tail _ h =>\n    have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h\n    exact \u27e8h\u2081, h\u2082.2\u27e9", "annotated_tactic": ["match <a>List.mem_or_eq_of_mem_set</a> h with\n    | .inl h =>\n      intro _ hf\n      have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf\n      exact \u27e8h\u2081, h\u2082.2\u27e9\n    | .inr h => subst h; intro\n      | _, .head .. =>\n        exact \u27e8hl\u2082.1 \u25b8 <a>Nat.le_refl</a> _, fun _ h h' => hl\u2081.1 _ h (<a>PartialEquivBEq.symm</a> h')\u27e9\n      | _, .tail _ h =>\n        have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (<a>Array.getElem_mem_data</a> ..) _ h\n        exact \u27e8h\u2081, h\u2082.2\u27e9", [{"full_name": "List.mem_or_eq_of_mem_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [965, 9], "def_end_pos": [965, 29]}, {"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": "PartialEquivBEq.symm", "def_path": "lake-packages/std/Std/Classes/BEq.lean", "def_pos": [16, 3], "def_end_pos": [16, 7]}, {"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    x\u271d", "state_after": "no goals"}, {"tactic": "intro _ hf", "annotated_tactic": ["intro _ hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh\u271d :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh : x\u271d \u2208 target.val.data\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    x\u271d", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh\u271d :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh : x\u271d \u2208 target.val.data\na\u271d : \u03b1 \u00d7 \u03b2\nhf : a\u271d \u2208 AssocList.toList x\u271d\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true"}, {"tactic": "have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf", "annotated_tactic": ["have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh\u271d :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh : x\u271d \u2208 target.val.data\na\u271d : \u03b1 \u00d7 \u03b2\nhf : a\u271d \u2208 AssocList.toList x\u271d\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh\u271d :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh : x\u271d \u2208 target.val.data\na\u271d : \u03b1 \u00d7 \u03b2\nhf : a\u271d \u2208 AssocList.toList x\u271d\nh\u2081 : rank a\u271d.fst \u2264 i\nh\u2082 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    \u00ac(head\u271d.fst == a\u271d.fst) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == a\u271d.fst) = true\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true"}, {"tactic": "exact \u27e8h\u2081, h\u2082.2\u27e9", "annotated_tactic": ["exact \u27e8h\u2081, h\u2082.2\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh\u271d :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh : x\u271d \u2208 target.val.data\na\u271d : \u03b1 \u00d7 \u03b2\nhf : a\u271d \u2208 AssocList.toList x\u271d\nh\u2081 : rank a\u271d.fst \u2264 i\nh\u2082 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    \u00ac(head\u271d.fst == a\u271d.fst) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == a\u271d.fst) = true\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true", "state_after": "no goals"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nx\u271d : AssocList \u03b1 \u03b2\nh\u271d :\n  x\u271d \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh :\n  x\u271d =\n    AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    x\u271d", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nh :\n  AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    (AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])"}, {"tactic": "intro\n| _, .head .. =>\nexact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9\n| _, .tail _ h =>\nhave \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h\nexact \u27e8h\u2081, h\u2082.2\u27e9", "annotated_tactic": ["intro\n      | _, .head .. =>\n        exact \u27e8hl\u2082.1 \u25b8 <a>Nat.le_refl</a> _, fun _ h h' => hl\u2081.1 _ h (<a>PartialEquivBEq.symm</a> h')\u27e9\n      | _, .tail _ h =>\n        have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (<a>Array.getElem_mem_data</a> ..) _ h\n        exact \u27e8h\u2081, h\u2082.2\u27e9", [{"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": "PartialEquivBEq.symm", "def_path": "lake-packages/std/Std/Classes/BEq.lean", "def_pos": [16, 3], "def_end_pos": [16, 7]}, {"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nh :\n  AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\n\u22a2 AssocList.All\n    (fun k x =>\n      rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n    (AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])", "state_after": "no goals"}, {"tactic": "exact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9", "annotated_tactic": ["exact \u27e8hl\u2082.1 \u25b8 <a>Nat.le_refl</a> _, fun _ h h' => hl\u2081.1 _ h (<a>PartialEquivBEq.symm</a> h')\u27e9", [{"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": "PartialEquivBEq.symm", "def_path": "lake-packages/std/Std/Classes/BEq.lean", "def_pos": [16, 3], "def_end_pos": [16, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nh :\n  AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx\u271d :\n  x\u271d\u00b9 \u2208\n    AssocList.toList\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\n\u22a2 rank (head\u271d.fst, head\u271d.snd).fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n      x \u2208 tail\u271d \u2192 \u00ac(x.fst == (head\u271d.fst, head\u271d.snd).fst) = true", "state_after": "no goals"}, {"tactic": "have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h", "annotated_tactic": ["have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (<a>Array.getElem_mem_data</a> ..) _ h", [{"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nh\u271d :\n  AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx\u271d :\n  x\u271d\u00b9 \u2208\n    AssocList.toList\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\na\u271d : \u03b1 \u00d7 \u03b2\nh :\n  List.Mem a\u271d\n    (AssocList.toList\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nh\u271d :\n  AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx\u271d :\n  x\u271d\u00b9 \u2208\n    AssocList.toList\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\na\u271d : \u03b1 \u00d7 \u03b2\nh :\n  List.Mem a\u271d\n    (AssocList.toList\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh\u2081 : rank a\u271d.fst \u2264 i\nh\u2082 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    \u00ac(head\u271d.fst == a\u271d.fst) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == a\u271d.fst) = true\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true"}, {"tactic": "exact \u27e8h\u2081, h\u2082.2\u27e9", "annotated_tactic": ["exact \u27e8h\u2081, h\u2082.2\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nrank : \u03b1 \u2192 Nat\ni : Nat\nhead\u271d : \u03b1 \u00d7 \u03b2\ntail\u271d : List (\u03b1 \u00d7 \u03b2)\nih :\n  (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192\n    (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192\n      \u2200 {target : Buckets \u03b1 \u03b2},\n        Buckets.WF target \u2192\n          (\u2200 (bucket : AssocList \u03b1 \u03b2),\n              bucket \u2208 target.val.data \u2192\n                AssocList.All\n                  (fun k x =>\n                    rank k \u2264 i \u2227\n                      \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),\n                        x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)\n                  bucket) \u2192\n            Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227\n              \u2200 (bucket : AssocList \u03b1 \u03b2),\n                bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192\n                  AssocList.All (fun k x => rank k \u2264 i) bucket\ntarget : Buckets \u03b1 \u03b2\nht\u2081 : Buckets.WF target\nhl\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d\nhl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i\nht\u2082 :\n  \u2200 (bucket : AssocList \u03b1 \u03b2),\n    bucket \u2208 target.val.data \u2192\n      AssocList.All\n        (fun k x =>\n          rank k \u2264 i \u2227\n            \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n              \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)\n        bucket\nh\u271d :\n  AssocList.cons head\u271d.fst head\u271d.snd\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208\n    List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx\u271d :\n  x\u271d\u00b9 \u2208\n    AssocList.toList\n      (AssocList.cons head\u271d.fst head\u271d.snd\n        target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\na\u271d : \u03b1 \u00d7 \u03b2\nh :\n  List.Mem a\u271d\n    (AssocList.toList\n      target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val])\nh\u2081 : rank a\u271d.fst \u2264 i\nh\u2082 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    \u00ac(head\u271d.fst == a\u271d.fst) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == a\u271d.fst) = true\n\u22a2 rank a\u271d.fst \u2264 i \u2227\n    \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true", "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.weaklyRegular_of_finite", "start": [374, 1], "end": [434, 86], "traced_tactics": [{"tactic": "have hfin : \u2200 {s}, \u03bc s \u2260 \u22a4 := @(measure_ne_top \u03bc)", "annotated_tactic": ["have hfin : \u2200 {s}, \u03bc s \u2260 \u22a4 := @(<a>measure_ne_top</a> \u03bc)", [{"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\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\n\u22a2 WeaklyRegular \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 WeaklyRegular \u03bc"}, {"tactic": "suffices \u2200 s, MeasurableSet s \u2192 \u2200 \u03b5, \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227\n    IsClosed F \u2227 IsOpen U \u2227 \u03bc s \u2264 \u03bc F + \u03b5 \u2227 \u03bc U \u2264 \u03bc s + \u03b5 by\n  refine'\n    { outerRegular := fun s hs r hr => _\n      innerRegular := H }\n  rcases exists_between hr with \u27e8r', hsr', hr'r\u27e9\n  rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9\n  refine' \u27e8U, hsU, hUo, _\u27e9\n  rw [add_tsub_cancel_of_le hsr'.le] at H\n  exact H.trans_lt hr'r", "annotated_tactic": ["suffices \u2200 s, <a>MeasurableSet</a> s \u2192 \u2200 \u03b5, \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227\n      <a>IsClosed</a> F \u2227 <a>IsOpen</a> U \u2227 \u03bc s \u2264 \u03bc F + \u03b5 \u2227 \u03bc U \u2264 \u03bc s + \u03b5 by\n    refine'\n      { outerRegular := fun s hs r hr => _\n        innerRegular := H }\n    rcases <a>exists_between</a> hr with \u27e8r', hsr', hr'r\u27e9\n    rcases this s hs _ (<a>tsub_pos_iff_lt</a>.2 hsr').<a>ne'</a> with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9\n    refine' \u27e8U, hsU, hUo, _\u27e9\n    rw [<a>add_tsub_cancel_of_le</a> hsr'.le] at H\n    exact H.trans_lt hr'r", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}, {"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}, {"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": "add_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [24, 9], "def_end_pos": [24, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 WeaklyRegular \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "refine' MeasurableSet.induction_on_open _ _ _", "annotated_tactic": ["refine' <a>MeasurableSet.induction_on_open</a> _ _ _", [{"full_name": "MeasurableSet.induction_on_open", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (U : Set \u03b1),\n    IsOpen U \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e),\n        \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (t : Set \u03b1),\n    MeasurableSet t \u2192\n      (\u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t \u2227 \u2203 U, U \u2287 t \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t + \u03b5) \u2192\n        \u2200 (\u03b5 : \u211d\u22650\u221e),\n          \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t\u1d9c \u2227 \u2203 U, U \u2287 t\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t\u1d9c + \u03b5\n\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\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) (\u03b5 : \u211d\u22650\u221e),\n            \u03b5 \u2260 0 \u2192\n              \u2203 F, F \u2286 f i \u2227 \u2203 U, U \u2287 f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (f i) + \u03b5) \u2192\n          \u2200 (\u03b5 : \u211d\u22650\u221e),\n            \u03b5 \u2260 0 \u2192\n              \u2203 F,\n                F \u2286 \u22c3 i, f i \u2227\n                  \u2203 U, U \u2287 \u22c3 i, f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, f i) + \u03b5"}, {"tactic": "refine'\n  { outerRegular := fun s hs r hr => _\n    innerRegular := H }", "annotated_tactic": ["refine'\n      { outerRegular := fun s hs r hr => _\n        innerRegular := H }", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 WeaklyRegular \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r"}, {"tactic": "rcases exists_between hr with \u27e8r', hsr', hr'r\u27e9", "annotated_tactic": ["rcases <a>exists_between</a> hr with \u27e8r', hsr', hr'r\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\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r"}, {"tactic": "rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9", "annotated_tactic": ["rcases this s hs _ (<a>tsub_pos_iff_lt</a>.2 hsr').<a>ne'</a> with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9", [{"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]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r", "state_after": "case intro.intro.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s)\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r"}, {"tactic": "refine' \u27e8U, hsU, hUo, _\u27e9", "annotated_tactic": ["refine' \u27e8U, hsU, hUo, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s)\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r", "state_after": "case intro.intro.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc U < r"}, {"tactic": "rw [add_tsub_cancel_of_le hsr'.le] at H", "annotated_tactic": ["rw [<a>add_tsub_cancel_of_le</a> hsr'.le] at H", [{"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 intro.intro.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc U < r", "state_after": "case intro.intro.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 r'\n\u22a2 \u2191\u2191\u03bc U < r"}, {"tactic": "exact H.trans_lt hr'r", "annotated_tactic": ["exact H.trans_lt hr'r", []], "state_before": "case intro.intro.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 r'\n\u22a2 \u2191\u2191\u03bc U < r", "state_after": "no goals"}, {"tactic": "intro U hU \u03b5 h\u03b5", "annotated_tactic": ["intro U hU \u03b5 h\u03b5", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (U : Set \u03b1),\n    IsOpen U \u2192\n      \u2200 (\u03b5 : \u211d\u22650\u221e),\n        \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nU : Set \u03b1\nhU : IsOpen U\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5"}, {"tactic": "rcases H.exists_subset_lt_add isClosed_empty hU hfin h\u03b5 with \u27e8F, hsF, hFc, hF\u27e9", "annotated_tactic": ["rcases H.exists_subset_lt_add <a>isClosed_empty</a> hU hfin h\u03b5 with \u27e8F, hsF, hFc, hF\u27e9", [{"full_name": "isClosed_empty", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [212, 17], "def_end_pos": [212, 31]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nU : Set \u03b1\nhU : IsOpen U\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5", "state_after": "case refine'_1.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nU : Set \u03b1\nhU : IsOpen U\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhsF : F \u2286 U\nhFc : IsClosed F\nhF : \u2191\u2191\u03bc U < \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5"}, {"tactic": "exact \u27e8F, hsF, U, Subset.rfl, hFc, hU, hF.le, le_self_add\u27e9", "annotated_tactic": ["exact \u27e8F, hsF, U, <a>Subset.rfl</a>, hFc, hU, hF.le, <a>le_self_add</a>\u27e9", [{"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "le_self_add", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [154, 3], "def_end_pos": [154, 14]}]], "state_before": "case refine'_1.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nU : Set \u03b1\nhU : IsOpen U\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhsF : F \u2286 U\nhFc : IsClosed F\nhF : \u2191\u2191\u03bc U < \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5", "state_after": "no goals"}, {"tactic": "rintro s hs H \u03b5 h\u03b5", "annotated_tactic": ["rintro s hs H \u03b5 h\u03b5", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (t : Set \u03b1),\n    MeasurableSet t \u2192\n      (\u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t \u2227 \u2203 U, U \u2287 t \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t + \u03b5) \u2192\n        \u2200 (\u03b5 : \u211d\u22650\u221e),\n          \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t\u1d9c \u2227 \u2203 U, U \u2287 t\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t\u1d9c + \u03b5", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5"}, {"tactic": "rcases H \u03b5 h\u03b5 with \u27e8F, hFs, U, hsU, hFc, hUo, hF, hU\u27e9", "annotated_tactic": ["rcases H \u03b5 h\u03b5 with \u27e8F, hFs, U, hsU, hFc, hUo, hF, hU\u27e9", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5", "state_after": "case refine'_2.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFs : F \u2286 s\nU : Set \u03b1\nhsU : U \u2287 s\nhFc : IsClosed F\nhUo : IsOpen U\nhF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5\nhU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5"}, {"tactic": "refine'\n  \u27e8U\u1d9c, compl_subset_compl.2 hsU, F\u1d9c, compl_subset_compl.2 hFs, hUo.isClosed_compl,\n    hFc.isOpen_compl, _\u27e9", "annotated_tactic": ["refine'\n      \u27e8U\u1d9c, <a>compl_subset_compl</a>.2 hsU, F\u1d9c, <a>compl_subset_compl</a>.2 hFs, hUo.isClosed_compl,\n        hFc.isOpen_compl, _\u27e9", [{"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1756, 9], "def_end_pos": [1756, 27]}, {"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1756, 9], "def_end_pos": [1756, 27]}]], "state_before": "case refine'_2.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFs : F \u2286 s\nU : Set \u03b1\nhsU : U \u2287 s\nhFc : IsClosed F\nhUo : IsOpen U\nhF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5\nhU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5", "state_after": "case refine'_2.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFs : F \u2286 s\nU : Set \u03b1\nhsU : U \u2287 s\nhFc : IsClosed F\nhUo : IsOpen U\nhF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5\nhU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc U\u1d9c + \u03b5 \u2227 \u2191\u2191\u03bc F\u1d9c \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5"}, {"tactic": "simp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and_iff]", "annotated_tactic": ["simp only [<a>measure_compl_le_add_iff</a>, *, hUo.measurableSet, hFc.measurableSet, <a>true_and_iff</a>]", [{"full_name": "MeasureTheory.measure_compl_le_add_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2890, 9], "def_end_pos": [2890, 33]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "case refine'_2.intro.intro.intro.intro.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFs : F \u2286 s\nU : Set \u03b1\nhsU : U \u2287 s\nhFc : IsClosed F\nhUo : IsOpen U\nhF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5\nhU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc U\u1d9c + \u03b5 \u2227 \u2191\u2191\u03bc F\u1d9c \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5", "state_after": "no goals"}, {"tactic": "intro s hsd hsm H \u03b5 \u03b50", "annotated_tactic": ["intro s hsd hsm H \u03b5 \u03b50", []], "state_before": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\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) (\u03b5 : \u211d\u22650\u221e),\n            \u03b5 \u2260 0 \u2192\n              \u2203 F, F \u2286 f i \u2227 \u2203 U, U \u2287 f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (f i) + \u03b5) \u2192\n          \u2200 (\u03b5 : \u211d\u22650\u221e),\n            \u03b5 \u2260 0 \u2192\n              \u2203 F,\n                F \u2286 \u22c3 i, f i \u2227\n                  \u2203 U, U \u2287 \u22c3 i, f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, f i) + \u03b5", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "have \u03b50' : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).ne'", "annotated_tactic": ["have \u03b50' : \u03b5 / 2 \u2260 0 := (<a>ENNReal.half_pos</a> \u03b50).<a>ne'</a>", [{"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 refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "rcases ENNReal.exists_pos_sum_of_countable' \u03b50' \u2115 with \u27e8\u03b4, \u03b40, h\u03b4\u03b5\u27e9", "annotated_tactic": ["rcases <a>ENNReal.exists_pos_sum_of_countable'</a> \u03b50' \u2115 with \u27e8\u03b4, \u03b40, h\u03b4\u03b5\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": "case refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "case refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "choose F hFs U hsU hFc hUo hF hU using fun n => H n (\u03b4 n) (\u03b40 n).ne'", "annotated_tactic": ["choose F hFs U hsU hFc hUo hF hU using fun n => H n (\u03b4 n) (\u03b40 n).<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 refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "case refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "have : Tendsto (fun t => (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2) atTop (\ud835\udcdd <| \u03bc (\u22c3 n, s n) + \u03b5 / 2) := by\n  rw [measure_iUnion hsd hsm]\n  exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds", "annotated_tactic": ["have : <a>Tendsto</a> (fun t => (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2) <a>atTop</a> (\ud835\udcdd <| \u03bc (\u22c3 n, s n) + \u03b5 / 2) := by\n      rw [<a>measure_iUnion</a> hsd hsm]\n      exact <a>Tendsto.add</a> ENNReal.summable.hasSum <a>tendsto_const_nhds</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": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "Filter.Tendsto.add", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [118, 3], "def_end_pos": [118, 14]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "case refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "case refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "rcases (this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin \u03b50').exists with \u27e8t, ht\u27e9", "annotated_tactic": ["rcases (this.eventually <| <a>lt_mem_nhds</a> <| <a>ENNReal.lt_add_right</a> hfin \u03b50').<a>exists</a> with \u27e8t, ht\u27e9", [{"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": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "case refine'_3.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "refine'\n  \u27e8\u22c3 k \u2208 t, F k, iUnion_mono fun k => iUnion_subset fun _ => hFs _, \u22c3 n, U n, iUnion_mono hsU,\n    isClosed_biUnion_finset fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _\u27e9", "annotated_tactic": ["refine'\n      \u27e8\u22c3 k \u2208 t, F k, <a>iUnion_mono</a> fun k => <a>iUnion_subset</a> fun _ => hFs _, \u22c3 n, U n, <a>iUnion_mono</a> hsU,\n        <a>isClosed_biUnion_finset</a> fun k _ => hFc k, <a>isOpen_iUnion</a> hUo, ht.le.trans _, _\u27e9", [{"full_name": "Set.iUnion_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [478, 9], "def_end_pos": [478, 20]}, {"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_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [478, 9], "def_end_pos": [478, 20]}, {"full_name": "isClosed_biUnion_finset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [264, 7], "def_end_pos": [264, 30]}, {"full_name": "isOpen_iUnion", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [147, 9], "def_end_pos": [147, 22]}]], "state_before": "case refine'_3.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\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2203 F,\n    F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "case refine'_3.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2191\u2191\u03bc (\u22c3 k \u2208 t, F k) + \u03b5\n\ncase refine'_3.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, U n) \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5"}, {"tactic": "rw [measure_iUnion hsd hsm]", "annotated_tactic": ["rw [<a>measure_iUnion</a> hsd hsm]", [{"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\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2211' (i : \u2115), \u2191\u2191\u03bc (s i) + \u03b5 / 2))"}, {"tactic": "exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds", "annotated_tactic": ["exact <a>Tendsto.add</a> ENNReal.summable.hasSum <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": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2211' (i : \u2115), \u2191\u2191\u03bc (s i) + \u03b5 / 2))", "state_after": "no goals"}, {"tactic": "calc\n  (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2 \u2264 ((\u2211 k in t, \u03bc (F k)) + \u2211 k in t, \u03b4 k) + \u03b5 / 2 := by\n    rw [\u2190 sum_add_distrib]\n    exact add_le_add_right (sum_le_sum fun k _ => hF k) _\n  _ \u2264 (\u2211 k in t, \u03bc (F k)) + \u03b5 / 2 + \u03b5 / 2 :=\n    (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans h\u03b4\u03b5.le) _) _)\n  _ = \u03bc (\u22c3 k \u2208 t, F k) + \u03b5 := by\n    rw [measure_biUnion_finset, add_assoc, ENNReal.add_halves]\n    exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n),\n      fun k _ => (hFc k).measurableSet]", "annotated_tactic": ["calc\n        (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2 \u2264 ((\u2211 k in t, \u03bc (F k)) + \u2211 k in t, \u03b4 k) + \u03b5 / 2 := by\n          rw [\u2190 <a>sum_add_distrib</a>]\n          exact <a>add_le_add_right</a> (<a>sum_le_sum</a> fun k _ => hF k) _\n        _ \u2264 (\u2211 k in t, \u03bc (F k)) + \u03b5 / 2 + \u03b5 / 2 :=\n          (<a>add_le_add_right</a> (<a>add_le_add_left</a> ((<a>ENNReal.sum_le_tsum</a> _).<a>trans</a> h\u03b4\u03b5.le) _) _)\n        _ = \u03bc (\u22c3 k \u2208 t, F k) + \u03b5 := by\n          rw [<a>measure_biUnion_finset</a>, <a>add_assoc</a>, <a>ENNReal.add_halves</a>]\n          exacts [fun k _ n _ hkn => (hsd hkn).<a>mono</a> (hFs k) (hFs n),\n            fun k _ => (hFc k).<a>measurableSet</a>]", [{"full_name": "Finset.sum_add_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [657, 3], "def_end_pos": [657, 14]}, {"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": "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_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": "ENNReal.sum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [831, 19], "def_end_pos": [831, 30]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.measure_biUnion_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [194, 9], "def_end_pos": [194, 31]}, {"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": "Disjoint.mono", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}]], "state_before": "case refine'_3.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2191\u2191\u03bc (\u22c3 k \u2208 t, F k) + \u03b5", "state_after": "no goals"}, {"tactic": "rw [\u2190 sum_add_distrib]", "annotated_tactic": ["rw [\u2190 <a>sum_add_distrib</a>]", [{"full_name": "Finset.sum_add_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [657, 3], "def_end_pos": [657, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 k in t, \u2191\u2191\u03bc (F k) + \u2211 k in t, \u03b4 k + \u03b5 / 2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 x in t, (\u2191\u2191\u03bc (F x) + \u03b4 x) + \u03b5 / 2"}, {"tactic": "exact add_le_add_right (sum_le_sum fun k _ => hF k) _", "annotated_tactic": ["exact <a>add_le_add_right</a> (<a>sum_le_sum</a> fun k _ => hF k) _", [{"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": "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\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 x in t, (\u2191\u2191\u03bc (F x) + \u03b4 x) + \u03b5 / 2", "state_after": "no goals"}, {"tactic": "rw [measure_biUnion_finset, add_assoc, ENNReal.add_halves]", "annotated_tactic": ["rw [<a>measure_biUnion_finset</a>, <a>add_assoc</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "MeasureTheory.measure_biUnion_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [194, 9], "def_end_pos": [194, 31]}, {"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\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (F k) + \u03b5 / 2 + \u03b5 / 2 = \u2191\u2191\u03bc (\u22c3 k \u2208 t, F k) + \u03b5", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 PairwiseDisjoint \u2191t fun k => F k\n\ncase hm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2200 (b : \u2115), b \u2208 t \u2192 MeasurableSet (F b)"}, {"tactic": "exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n),\n  fun k _ => (hFc k).measurableSet]", "annotated_tactic": ["exacts [fun k _ n _ hkn => (hsd hkn).<a>mono</a> (hFs k) (hFs n),\n            fun k _ => (hFc k).<a>measurableSet</a>]", [{"full_name": "Disjoint.mono", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 PairwiseDisjoint \u2191t fun k => F k\n\ncase hm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2200 (b : \u2115), b \u2208 t \u2192 MeasurableSet (F b)", "state_after": "no goals"}, {"tactic": "calc\n  \u03bc (\u22c3 n, U n) \u2264 \u2211' n, \u03bc (U n) := measure_iUnion_le _\n  _ \u2264 \u2211' n, (\u03bc (s n) + \u03b4 n) := (ENNReal.tsum_le_tsum hU)\n  _ = \u03bc (\u22c3 n, s n) + \u2211' n, \u03b4 n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]\n  _ \u2264 \u03bc (\u22c3 n, s n) + \u03b5 := add_le_add_left (h\u03b4\u03b5.le.trans ENNReal.half_le_self) _", "annotated_tactic": ["calc\n        \u03bc (\u22c3 n, U n) \u2264 \u2211' n, \u03bc (U n) := <a>measure_iUnion_le</a> _\n        _ \u2264 \u2211' n, (\u03bc (s n) + \u03b4 n) := (<a>ENNReal.tsum_le_tsum</a> hU)\n        _ = \u03bc (\u22c3 n, s n) + \u2211' n, \u03b4 n := by rw [<a>measure_iUnion</a> hsd hsm, <a>ENNReal.tsum_add</a>]\n        _ \u2264 \u03bc (\u22c3 n, s n) + \u03b5 := <a>add_le_add_left</a> (h\u03b4\u03b5.le.trans <a>ENNReal.half_le_self</a>) _", [{"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"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_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"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_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "ENNReal.half_le_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1810, 19], "def_end_pos": [1810, 31]}]], "state_before": "case refine'_3.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2191\u2191\u03bc (\u22c3 n, U n) \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5", "state_after": "no goals"}, {"tactic": "rw [measure_iUnion hsd hsm, ENNReal.tsum_add]", "annotated_tactic": ["rw [<a>measure_iUnion</a> hsd hsm, <a>ENNReal.tsum_add</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.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\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n  \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n    \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (s n) + \u03b4 n) = \u2191\u2191\u03bc (\u22c3 n, s n) + \u2211' (n : \u2115), \u03b4 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.powersetCard_card_add", "start": [322, 1], "end": [324, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one", "start": [858, 1], "end": [872, 54], "traced_tactics": [{"tactic": "obtain \u27e8t', t'_meas, t't, t'pos, t'top\u27e9 : \u2203 t', MeasurableSet t' \u2227 t' \u2286 t \u2227 0 < \u03bc t' \u2227 \u03bc t' < \u22a4 :=\n  exists_subset_measure_lt_top ht h't.bot_lt", "annotated_tactic": ["obtain \u27e8t', t'_meas, t't, t'pos, t'top\u27e9 : \u2203 t', <a>MeasurableSet</a> t' \u2227 t' \u2286 t \u2227 0 < \u03bc t' \u2227 \u03bc t' < \u22a4 :=\n    <a>exists_subset_measure_lt_top</a> ht h't.bot_lt", [{"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": "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t))", "state_after": "case intro.intro.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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t))"}, {"tactic": "filter_upwards [(tendsto_order.1\n        (tendsto_addHaar_inter_smul_one_of_density_one \u03bc s x h t' t'_meas t'pos.ne' t'top.ne)).1\n    0 zero_lt_one]", "annotated_tactic": ["filter_upwards [(<a>tendsto_order</a>.1\n          (<a>tendsto_addHaar_inter_smul_one_of_density_one</a> \u03bc s x h t' t'_meas t'pos.ne' t'top.ne)).1\n      0 <a>zero_lt_one</a>]", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [831, 9], "def_end_pos": [831, 54]}, {"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\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t))", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\n\u22a2 \u2200 (a : \u211d), 0 < \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t')) / \u2191\u2191\u03bc ({x} + a \u2022 t') \u2192 Set.Nonempty (s \u2229 ({x} + a \u2022 t))"}, {"tactic": "intro r hr", "annotated_tactic": ["intro r hr", []], "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\n\u22a2 \u2200 (a : \u211d), 0 < \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t')) / \u2191\u2191\u03bc ({x} + a \u2022 t') \u2192 Set.Nonempty (s \u2229 ({x} + a \u2022 t))", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))"}, {"tactic": "have : \u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0 := fun h' => by\n  simp only [ENNReal.not_lt_zero, ENNReal.zero_div, h'] at hr", "annotated_tactic": ["have : \u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0 := fun h' => by\n    simp only [<a>ENNReal.not_lt_zero</a>, <a>ENNReal.zero_div</a>, h'] at hr", [{"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}, {"full_name": "ENNReal.zero_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1604, 27], "def_end_pos": [1604, 35]}]], "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))"}, {"tactic": "have : (s \u2229 ({x} + r \u2022 t')).Nonempty := nonempty_of_measure_ne_zero this", "annotated_tactic": ["have : (s \u2229 ({x} + r \u2022 t')).<a>Nonempty</a> := <a>nonempty_of_measure_ne_zero</a> this", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "MeasureTheory.nonempty_of_measure_ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [189, 9], "def_end_pos": [189, 36]}]], "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))", "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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\nthis : Set.Nonempty (s \u2229 ({x} + r \u2022 t'))\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))"}, {"tactic": "apply this.mono (inter_subset_inter Subset.rfl _)", "annotated_tactic": ["apply this.mono (<a>inter_subset_inter</a> <a>Subset.rfl</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.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\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\nthis : Set.Nonempty (s \u2229 ({x} + r \u2022 t'))\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\nthis : Set.Nonempty (s \u2229 ({x} + r \u2022 t'))\n\u22a2 {x} + r \u2022 t' \u2286 {x} + r \u2022 t"}, {"tactic": "exact add_subset_add Subset.rfl (smul_set_mono t't)", "annotated_tactic": ["exact <a>add_subset_add</a> <a>Subset.rfl</a> (<a>smul_set_mono</a> t't)", [{"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.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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\nthis : Set.Nonempty (s \u2229 ({x} + r \u2022 t'))\n\u22a2 {x} + r \u2022 t' \u2286 {x} + r \u2022 t", "state_after": "no goals"}, {"tactic": "simp only [ENNReal.not_lt_zero, ENNReal.zero_div, h'] at hr", "annotated_tactic": ["simp only [<a>ENNReal.not_lt_zero</a>, <a>ENNReal.zero_div</a>, h'] at hr", [{"full_name": "ENNReal.not_lt_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}, {"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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nh' : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) = 0\n\u22a2 False", "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_tendsto_of_bdd", "start": [196, 1], "end": [200, 43], "traced_tactics": [{"tactic": "filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_atTop_of_snorm_bdd one_ne_zero\n  (fun n => (hf.stronglyMeasurable n).measurable.mono (\u2131.le n) le_rfl) hbdd] with \u03c9 h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [hf.upcrossings_ae_lt_top hbdd, <a>ae_bdd_liminf_atTop_of_snorm_bdd</a> <a>one_ne_zero</a>\n    (fun n => (hf.stronglyMeasurable n).measurable.mono (\u2131.le n) <a>le_rfl</a>) hbdd] with \u03c9 h\u2081 h\u2082", [{"full_name": "MeasureTheory.ae_bdd_liminf_atTop_of_snorm_bdd", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 41]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"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\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) \u2202\u03bc, \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h\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\n\u03c9 : \u03a9\nh\u2081 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh\u2082 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "exact tendsto_of_uncrossing_lt_top h\u2082 h\u2081", "annotated_tactic": ["exact <a>tendsto_of_uncrossing_lt_top</a> h\u2082 h\u2081", [{"full_name": "MeasureTheory.tendsto_of_uncrossing_lt_top", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [142, 9], "def_end_pos": [142, 37]}]], "state_before": "case h\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\n\u03c9 : \u03a9\nh\u2081 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4\nh\u2082 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4\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/Data/Bundle.lean", "full_name": "Bundle.TotalSpace.mk_injective", "start": [77, 1], "end": [78, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Set.IccExtend_range", "start": [229, 1], "end": [230, 65], "traced_tactics": [{"tactic": "simp only [IccExtend, range_comp f, range_projIcc, image_univ]", "annotated_tactic": ["simp only [<a>IccExtend</a>, <a>range_comp</a> f, <a>range_projIcc</a>, <a>image_univ</a>]", [{"full_name": "Set.IccExtend", "def_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "def_pos": [202, 5], "def_end_pos": [202, 14]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "Set.range_projIcc", "def_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "def_pos": [165, 9], "def_end_pos": [165, 22]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u2191(Icc a b) \u2192 \u03b2\n\u22a2 range (IccExtend h f) = range f", "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.All.append", "start": [303, 11], "end": [311, 36], "traced_tactics": [{"tactic": "unfold append", "annotated_tactic": ["unfold <a>append</a>", [{"full_name": "Std.RBNode.append", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [351, 5], "def_end_pos": [351, 11]}]], "state_before": "\u03b1\u271d : Type u_1\nl r : RBNode \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\nhl : All p l\nhr : All p r\n\u22a2 All p (append l r)", "state_after": "\u03b1\u271d : Type u_1\nl r : RBNode \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\nhl : All p l\nhr : All p r\n\u22a2 All p\n    (match l, r with\n    | nil, x => x\n    | x, nil => x\n    | node red a x b, node red c y d =>\n      match append b c with\n      | node red b' z c' => node red (node red a x b') z (node red c' y d)\n      | bc => node red a x (node red bc y d)\n    | node black a x b, node black c y d =>\n      match append b c with\n      | node red b' z c' => node red (node black a x b') z (node black c' y d)\n      | bc => balLeft a x (node black bc y d)\n    | a@h:(node black l v r), node red b x c => node red (append a b) x c\n    | node red a x b, c@h:(node black l v r) => node red a x (append b c))"}, {"tactic": "split <;> try simp [*]", "annotated_tactic": ["split <;> try simp [*]", []], "state_before": "\u03b1\u271d : Type u_1\nl r : RBNode \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\nhl : All p l\nhr : All p r\n\u22a2 All p\n    (match l, r with\n    | nil, x => x\n    | x, nil => x\n    | node red a x b, node red c y d =>\n      match append b c with\n      | node red b' z c' => node red (node red a x b') z (node red c' y d)\n      | bc => node red a x (node red bc y d)\n    | node black a x b, node black c y d =>\n      match append b c with\n      | node red b' z c' => node red (node black a x b') z (node black c' y d)\n      | bc => balLeft a x (node black bc y d)\n    | a@h:(node black l v r), node red b x c => node red (append a b) x c\n    | node red a x b, c@h:(node black l v r) => node red a x (append b c))", "state_after": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))\n\ncase h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))\n\ncase h_5\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 l\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d b\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nc\u271d : RBNode \u03b1\u271d\nhl : All p (node black l\u271d v\u271d r\u271d)\nhr : All p (node red b\u271d x\u271d c\u271d)\n\u22a2 p x\u271d \u2227 All p (append (node black l\u271d v\u271d r\u271d) b\u271d) \u2227 All p c\u271d\n\ncase h_6\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d l\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node black l\u271d v\u271d r\u271d)\n\u22a2 p x\u271d \u2227 All p a\u271d \u2227 All p (append b\u271d (node black l\u271d v\u271d r\u271d))"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case h_6\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d l\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node black l\u271d v\u271d r\u271d)\n\u22a2 All p (node red a\u271d x\u271d (append b\u271d (node black l\u271d v\u271d r\u271d)))", "state_after": "case h_6\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d l\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node black l\u271d v\u271d r\u271d)\n\u22a2 p x\u271d \u2227 All p a\u271d \u2227 All p (append b\u271d (node black l\u271d v\u271d r\u271d))"}, {"tactic": "have \u27e8hx, ha, hb\u27e9 := hl", "annotated_tactic": ["have \u27e8hx, ha, hb\u27e9 := hl", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))", "state_after": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))"}, {"tactic": "have \u27e8hy, hc, hd\u27e9 := hr", "annotated_tactic": ["have \u27e8hy, hc, hd\u27e9 := hr", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))", "state_after": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))"}, {"tactic": "have := hb.append hc", "annotated_tactic": ["have := hb.append hc", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))", "state_after": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\nthis : All p (append b\u271d c\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))"}, {"tactic": "split <;> simp_all", "annotated_tactic": ["split <;> simp_all", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node red c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\nthis : All p (append b\u271d c\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)\n    | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d))", "state_after": "no goals"}, {"tactic": "have \u27e8hx, ha, hb\u27e9 := hl", "annotated_tactic": ["have \u27e8hx, ha, hb\u27e9 := hl", []], "state_before": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))", "state_after": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))"}, {"tactic": "have \u27e8hy, hc, hd\u27e9 := hr", "annotated_tactic": ["have \u27e8hy, hc, hd\u27e9 := hr", []], "state_before": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))", "state_after": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))"}, {"tactic": "have := hb.append hc", "annotated_tactic": ["have := hb.append hc", []], "state_before": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))", "state_after": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\nthis : All p (append b\u271d c\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))"}, {"tactic": "split <;> simp_all [All.balLeft]", "annotated_tactic": ["split <;> simp_all [<a>All.balLeft</a>]", [{"full_name": "Std.RBNode.All.balLeft", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [235, 19], "def_end_pos": [235, 30]}]], "state_before": "case h_4\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d c\u271d : RBNode \u03b1\u271d\ny\u271d : \u03b1\u271d\nd\u271d : RBNode \u03b1\u271d\nhl : All p (node black a\u271d x\u271d b\u271d)\nhr : All p (node black c\u271d y\u271d d\u271d)\nhx : p x\u271d\nha : All p a\u271d\nhb : All p b\u271d\nhy : p y\u271d\nhc : All p c\u271d\nhd : All p d\u271d\nthis : All p (append b\u271d c\u271d)\n\u22a2 All p\n    (match append b\u271d c\u271d with\n    | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)\n    | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d))", "state_after": "no goals"}, {"tactic": "simp_all [hl.append hr.2.1]", "annotated_tactic": ["simp_all [hl.append hr.2.1]", []], "state_before": "case h_5\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 l\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d b\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nc\u271d : RBNode \u03b1\u271d\nhl : All p (node black l\u271d v\u271d r\u271d)\nhr : All p (node red b\u271d x\u271d c\u271d)\n\u22a2 p x\u271d \u2227 All p (append (node black l\u271d v\u271d r\u271d) b\u271d) \u2227 All p c\u271d", "state_after": "no goals"}, {"tactic": "simp_all [hl.2.2.append hr]", "annotated_tactic": ["simp_all [hl.2.2.append hr]", []], "state_before": "case h_6\n\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d\nx\u271d : \u03b1\u271d\nb\u271d l\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nhl : All p (node red a\u271d x\u271d b\u271d)\nhr : All p (node black l\u271d v\u271d r\u271d)\n\u22a2 p x\u271d \u2227 All p a\u271d \u2227 All p (append b\u271d (node black l\u271d v\u271d r\u271d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated", "start": [1897, 1], "end": [1905, 89], "traced_tactics": [{"tactic": "rcases exists_seq_separating \u03b1 MeasurableSet.empty univ with \u27e8e, hem, he\u27e9", "annotated_tactic": ["rcases <a>exists_seq_separating</a> \u03b1 <a>MeasurableSet.empty</a> <a>univ</a> with \u27e8e, hem, he\u27e9", [{"full_name": "exists_seq_separating", "def_path": "Mathlib/Order/Filter/CountableSeparatingOn.lean", "def_pos": [103, 9], "def_end_pos": [103, 30]}, {"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"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\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\n\u22a2 \u2203 f, Measurable f \u2227 Injective f", "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 t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\n\u22a2 \u2203 f, Measurable f \u2227 Injective f"}, {"tactic": "rw [measurable_pi_iff]", "annotated_tactic": ["rw [<a>measurable_pi_iff</a>]", [{"full_name": "measurable_pi_iff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [890, 9], "def_end_pos": [890, 26]}]], "state_before": "case intro.intro.refine_1\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\n\u22a2 Measurable fun x x_1 => decide (x \u2208 e x_1)", "state_after": "case intro.intro.refine_1\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\n\u22a2 \u2200 (a : \u2115), Measurable fun x => decide (x \u2208 e a)"}, {"tactic": "refine fun n \u21a6 measurable_to_bool ?_", "annotated_tactic": ["refine fun n \u21a6 <a>measurable_to_bool</a> ?_", [{"full_name": "measurable_to_bool", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 27]}]], "state_before": "case intro.intro.refine_1\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\n\u22a2 \u2200 (a : \u2115), Measurable fun x => decide (x \u2208 e a)", "state_after": "case intro.intro.refine_1\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\nn : \u2115\n\u22a2 MeasurableSet ((fun x => decide (x \u2208 e n)) \u207b\u00b9' {true})"}, {"tactic": "simpa only [preimage, mem_singleton_iff, Bool.decide_iff, setOf_mem_eq] using hem n", "annotated_tactic": ["simpa only [<a>preimage</a>, <a>mem_singleton_iff</a>, <a>Bool.decide_iff</a>, <a>setOf_mem_eq</a>] using hem n", [{"full_name": "Set.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "Bool.decide_iff", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [126, 9], "def_end_pos": [126, 19]}, {"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.refine_1\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\nn : \u2115\n\u22a2 MeasurableSet ((fun x => decide (x \u2208 e n)) \u207b\u00b9' {true})", "state_after": "no goals"}, {"tactic": "exact fun x y h \u21a6 he x trivial y trivial fun n \u21a6 decide_eq_decide.1 <| congr_fun h _", "annotated_tactic": ["exact fun x y h \u21a6 he x <a>trivial</a> y <a>trivial</a> fun n \u21a6 <a>decide_eq_decide</a>.1 <| <a>congr_fun</a> h _", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"full_name": "decide_eq_decide", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [542, 17], "def_end_pos": [542, 33]}, {"full_name": "congr_fun", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [42, 7], "def_end_pos": [42, 16]}]], "state_before": "case intro.intro.refine_2\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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ\ne : \u2115 \u2192 Set \u03b1\nhem : \u2200 (n : \u2115), MeasurableSet (e n)\nhe : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y\n\u22a2 Injective fun x x_1 => decide (x \u2208 e x_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Floor.lean", "full_name": "Nat.measurable_floor", "start": [72, 1], "end": [74, 100], "traced_tactics": [{"tactic": "cases' eq_or_ne \u230an\u230b\u208a 0 with h h <;> simp_all [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]", "annotated_tactic": ["cases' <a>eq_or_ne</a> \u230an\u230b\u208a 0 with h h <;> simp_all [h, <a>Nat.preimage_floor_of_ne_zero</a>, -<a>floor_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_floor_of_ne_zero", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [262, 9], "def_end_pos": [262, 34]}, {"full_name": "Nat.floor_eq_zero", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [232, 9], "def_end_pos": [232, 22]}]], "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 (floor \u207b\u00b9' {\u230an\u230b\u208a})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snormEssSup_trim", "start": [1015, 1], "end": [1017, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "torusIntegral_succ", "start": [260, 1], "end": [263, 43], "traced_tactics": [{"tactic": "simpa using torusIntegral_succAbove hf 0", "annotated_tactic": ["simpa using <a>torusIntegral_succAbove</a> hf 0", [{"full_name": "torusIntegral_succAbove", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [235, 9], "def_end_pos": [235, 32]}]], "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\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c 0, R 0), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succ, R \u2218 Fin.succ), f (Fin.cons x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.preimage_of_equiv", "start": [147, 1], "end": [158, 17], "traced_tactics": [{"tactic": "rwa [mem_preimage, hef g x]", "annotated_tactic": ["rwa [<a>mem_preimage</a>, hef g x]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\ne : G \u2192 H\nhe : Bijective e\nhef : \u2200 (g : G), Semiconj f (fun x => e g \u2022 x) fun x => g \u2022 x\nx : \u03b2\nx\u271d : \u2203 g, g \u2022 f x \u2208 s\ng : G\nhg : g \u2022 f x \u2208 s\n\u22a2 e g \u2022 x \u2208 f \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "lift e to G \u2243 H using he", "annotated_tactic": ["lift e to G \u2243 H using he", []], "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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\ne : G \u2192 H\nhe : Bijective e\nhef : \u2200 (g : G), Semiconj f (fun x => e g \u2022 x) fun x => g \u2022 x\na b : H\nhab : a \u2260 b\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b", "state_after": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b"}, {"tactic": "have : (e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (e.symm b\u207b\u00b9)\u207b\u00b9 := by simp [hab]", "annotated_tactic": ["have : (e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (e.symm b\u207b\u00b9)\u207b\u00b9 := by simp [hab]", []], "state_before": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b", "state_after": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\nthis : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b"}, {"tactic": "have := (h.aedisjoint this).preimage hf", "annotated_tactic": ["have := (h.aedisjoint this).<a>preimage</a> hf", [{"full_name": "MeasureTheory.AEDisjoint.preimage", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2433, 9], "def_end_pos": [2433, 28]}]], "state_before": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\nthis : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b", "state_after": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\nthis\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\nthis : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9)\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b"}, {"tactic": "simp only [Semiconj] at hef", "annotated_tactic": ["simp only [<a>Semiconj</a>] at hef", [{"full_name": "Function.Semiconj", "def_path": "Mathlib/Logic/Function/Conjugate.lean", "def_pos": [31, 5], "def_end_pos": [31, 13]}]], "state_before": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\nthis\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\nthis : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9)\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b", "state_after": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G) (x : \u03b2), f (\u2191e g \u2022 x) = g \u2022 f x\nthis\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\nthis : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9)\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b"}, {"tactic": "simpa only [onFun, \u2190 preimage_smul_inv, preimage_preimage, \u2190 hef, e.apply_symm_apply, inv_inv]\n  using this", "annotated_tactic": ["simpa only [<a>onFun</a>, \u2190 <a>preimage_smul_inv</a>, <a>preimage_preimage</a>, \u2190 hef, e.apply_symm_apply, <a>inv_inv</a>]\n      using this", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.preimage_smul_inv", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [905, 9], "def_end_pos": [905, 26]}, {"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [176, 9], "def_end_pos": [176, 26]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "case 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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G) (x : \u03b2), f (\u2191e g \u2022 x) = g \u2022 f x\nthis\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\nthis : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9)\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b", "state_after": "no goals"}, {"tactic": "simp [hab]", "annotated_tactic": ["simp [hab]", []], "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\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9", "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_or_top_of_indepSet_self", "start": [32, 1], "end": [42, 37], "traced_tactics": [{"tactic": "rw [IndepSet_iff] at h_indep", "annotated_tactic": ["rw [<a>IndepSet_iff</a>] at h_indep", [{"full_name": "ProbabilityTheory.IndepSet_iff", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [190, 7], "def_end_pos": [190, 19]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : IndepSet t t\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4"}, {"tactic": "specialize h_indep t t (measurableSet_generateFrom (Set.mem_singleton t))\n  (measurableSet_generateFrom (Set.mem_singleton t))", "annotated_tactic": ["specialize h_indep t t (<a>measurableSet_generateFrom</a> (<a>Set.mem_singleton</a> t))\n    (<a>measurableSet_generateFrom</a> (<a>Set.mem_singleton</a> t))", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}, {"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4"}, {"tactic": "by_cases h0 : \u03bc t = 0", "annotated_tactic": ["by_cases h0 : \u03bc t = 0", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4"}, {"tactic": "by_cases h_top : \u03bc t = \u221e", "annotated_tactic": ["by_cases h_top : \u03bc t = \u221e", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\nh_top : \u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\nh_top : \u00ac\u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4"}, {"tactic": "rw [\u2190 one_mul (\u03bc (t \u2229 t)), Set.inter_self, ENNReal.mul_eq_mul_right h0 h_top] at h_indep", "annotated_tactic": ["rw [\u2190 <a>one_mul</a> (\u03bc (t \u2229 t)), <a>Set.inter_self</a>, <a>ENNReal.mul_eq_mul_right</a> h0 h_top] at h_indep", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [926, 9], "def_end_pos": [926, 19]}, {"full_name": "ENNReal.mul_eq_mul_right", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1060, 9], "def_end_pos": [1060, 25]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\nh_top : \u00ac\u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : 1 = \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\nh_top : \u00ac\u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4"}, {"tactic": "exact Or.inr (Or.inl h_indep.symm)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>Or.inl</a> h_indep.symm)", [{"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]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : 1 = \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\nh_top : \u00ac\u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "no goals"}, {"tactic": "exact Or.inl h0", "annotated_tactic": ["exact <a>Or.inl</a> h0", [{"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\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "no goals"}, {"tactic": "exact Or.inr (Or.inr h_top)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>Or.inr</a> h_top)", [{"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.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\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nt : Set \u03a9\nh_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t\nh0 : \u00ac\u2191\u2191\u03bc t = 0\nh_top : \u2191\u2191\u03bc t = \u22a4\n\u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "isAddFundamentalDomain_Ioc'", "start": [49, 1], "end": [55, 52], "traced_tactics": [{"tactic": "refine' IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => _", "annotated_tactic": ["refine' <a>IsAddFundamentalDomain.mk'</a> measurableSet_Ioc.nullMeasurableSet fun x => _", [{"full_name": "MeasureTheory.IsAddFundamentalDomain.mk'", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [77, 3], "def_end_pos": [77, 14]}]], "state_before": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\n\u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.op (zmultiples T) } (Ioc t (t + T))", "state_after": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\nx : \u211d\n\u22a2 \u2203! g, g +\u1d65 x \u2208 Ioc t (t + T)"}, {"tactic": "have : Bijective (codRestrict (fun n : \u2124 => n \u2022 T) (AddSubgroup.zmultiples T) _) :=\n  (Equiv.ofInjective (fun n : \u2124 => n \u2022 T) (zsmul_strictMono_left hT).injective).bijective", "annotated_tactic": ["have : <a>Bijective</a> (<a>codRestrict</a> (fun n : \u2124 => n \u2022 T) (<a>AddSubgroup.zmultiples</a> T) _) :=\n    (<a>Equiv.ofInjective</a> (fun n : \u2124 => n \u2022 T) (<a>zsmul_strictMono_left</a> hT).<a>injective</a>).<a>bijective</a>", [{"full_name": "Function.Bijective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [131, 5], "def_end_pos": [131, 14]}, {"full_name": "Set.codRestrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [148, 5], "def_end_pos": [148, 16]}, {"full_name": "AddSubgroup.zmultiples", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}, {"full_name": "Equiv.ofInjective", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [606, 19], "def_end_pos": [606, 30]}, {"full_name": "zsmul_strictMono_left", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [328, 15], "def_end_pos": [328, 36]}, {"full_name": "StrictMono.injective", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [880, 9], "def_end_pos": [880, 29]}, {"full_name": "Equiv.bijective", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [203, 19], "def_end_pos": [203, 28]}]], "state_before": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\nx : \u211d\n\u22a2 \u2203! g, g +\u1d65 x \u2208 Ioc t (t + T)", "state_after": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\nx : \u211d\nthis : Bijective (codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))\n\u22a2 \u2203! g, g +\u1d65 x \u2208 Ioc t (t + T)"}, {"tactic": "refine' (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2 _", "annotated_tactic": ["refine' (<a>AddSubgroup.equivOp</a> _).bijective.comp this |>.<a>existsUnique_iff</a>.2 _", [{"full_name": "AddSubgroup.equivOp", "def_path": "Mathlib/GroupTheory/Subgroup/MulOpposite.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "Function.Bijective.existsUnique_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 35]}]], "state_before": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\nx : \u211d\nthis : Bijective (codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))\n\u22a2 \u2203! g, g +\u1d65 x \u2208 Ioc t (t + T)", "state_after": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\nx : \u211d\nthis : Bijective (codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))\n\u22a2 \u2203! x_1,\n    (\u2191(equivOp (zmultiples T)) \u2218 codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))\n          x_1 +\u1d65\n        x \u2208\n      Ioc t (t + T)"}, {"tactic": "simpa using existsUnique_add_zsmul_mem_Ioc hT x t", "annotated_tactic": ["simpa using <a>existsUnique_add_zsmul_mem_Ioc</a> hT x t", [{"full_name": "existsUnique_add_zsmul_mem_Ioc", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [101, 9], "def_end_pos": [101, 39]}]], "state_before": "T : \u211d\nhT : 0 < T\nt : \u211d\n\u03bc : autoParam (Measure \u211d) _auto\u271d\nx : \u211d\nthis : Bijective (codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))\n\u22a2 \u2203! x_1,\n    (\u2191(equivOp (zmultiples T)) \u2218 codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))\n          x_1 +\u1d65\n        x \u2208\n      Ioc t (t + T)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.measurable_condCdfRat", "start": [598, 1], "end": [603, 23], "traced_tactics": [{"tactic": "simp_rw [condCdfRat, ite_apply]", "annotated_tactic": ["simp_rw [<a>condCdfRat</a>, <a>ite_apply</a>]", [{"full_name": "ProbabilityTheory.condCdfRat", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [570, 19], "def_end_pos": [570, 29]}, {"full_name": "ite_apply", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 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)\nq : \u211a\n\u22a2 Measurable fun a => condCdfRat \u03c1 a q", "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)\nq : \u211a\n\u22a2 Measurable fun a => if a \u2208 condCdfSet \u03c1 then ENNReal.toReal (preCdf \u03c1 q a) else if q < 0 then 0 else 1"}, {"tactic": "exact\n  Measurable.ite (measurableSet_condCdfSet \u03c1) measurable_preCdf.ennreal_toReal\n    measurable_const", "annotated_tactic": ["exact\n    <a>Measurable.ite</a> (<a>measurableSet_condCdfSet</a> \u03c1) measurable_preCdf.ennreal_toReal\n      <a>measurable_const</a>", [{"full_name": "Measurable.ite", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 23]}, {"full_name": "ProbabilityTheory.measurableSet_condCdfSet", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [545, 9], "def_end_pos": [545, 33]}, {"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\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nq : \u211a\n\u22a2 Measurable fun a => if a \u2208 condCdfSet \u03c1 then ENNReal.toReal (preCdf \u03c1 q a) else if q < 0 then 0 else 1", "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.bisim'", "start": [498, 1], "end": [508, 26], "traced_tactics": [{"tactic": "rcases h x' Qx' with \u27e8a, f, f', ux'eq, vx'eq, h'\u27e9", "annotated_tactic": ["rcases h x' Qx' with \u27e8a, f, f', ux'eq, vx'eq, h'\u27e9", []], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n  \u2200 (x : \u03b1),\n    Q x \u2192\n      \u2203 a f f',\n        dest (u x) = abs { fst := a, snd := f } \u2227\n          dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\n\u22a2 Liftr R (dest x) (dest y)", "state_after": "case intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n  \u2200 (x : \u03b1),\n    Q x \u2192\n      \u2203 a f f',\n        dest (u x) = abs { fst := a, snd := f } \u2227\n          dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\n\u22a2 Liftr R (dest x) (dest y)"}, {"tactic": "rw [liftr_iff]", "annotated_tactic": ["rw [<a>liftr_iff</a>]", [{"full_name": "QPF.liftr_iff", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [134, 9], "def_end_pos": [134, 18]}]], "state_before": "case intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n  \u2200 (x : \u03b1),\n    Q x \u2192\n      \u2203 a f f',\n        dest (u x) = abs { fst := a, snd := f } \u2227\n          dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\n\u22a2 Liftr R (dest x) (dest y)", "state_after": "case intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n  \u2200 (x : \u03b1),\n    Q x \u2192\n      \u2203 a f f',\n        dest (u x) = abs { fst := a, snd := f } \u2227\n          dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\n\u22a2 \u2203 a f\u2080 f\u2081,\n    dest x = abs { fst := a, snd := f\u2080 } \u2227\n      dest y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), R (f\u2080 i) (f\u2081 i)"}, {"tactic": "refine' \u27e8a, f, f', xeq.symm \u25b8 ux'eq, yeq.symm \u25b8 vx'eq, h'\u27e9", "annotated_tactic": ["refine' \u27e8a, f, f', xeq.symm \u25b8 ux'eq, yeq.symm \u25b8 vx'eq, h'\u27e9", []], "state_before": "case intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n  \u2200 (x : \u03b1),\n    Q x \u2192\n      \u2203 a f f',\n        dest (u x) = abs { fst := a, snd := f } \u2227\n          dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\n\u22a2 \u2203 a f\u2080 f\u2081,\n    dest x = abs { fst := a, snd := f\u2080 } \u2227\n      dest y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), R (f\u2080 i) (f\u2081 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.norm_le_mul_norm_of_ae_le_mul", "start": [377, 1], "end": [384, 16], "traced_tactics": [{"tactic": "cases' le_or_lt 0 c with hc hc", "annotated_tactic": ["cases' <a>le_or_lt</a> 0 c with hc hc", [{"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\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\n\u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016", "state_after": "case inl\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : 0 \u2264 c\n\u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016\n\ncase inr\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : c < 0\n\u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016"}, {"tactic": "lift c to \u211d\u22650 using hc", "annotated_tactic": ["lift c to \u211d\u22650 using hc", []], "state_before": "case inl\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : 0 \u2264 c\n\u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016", "state_after": "case inl.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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 \u2191c * \u2016\u2191\u2191g x\u2016\n\u22a2 \u2016f\u2016 \u2264 \u2191c * \u2016g\u2016"}, {"tactic": "exact NNReal.coe_le_coe.mpr (nnnorm_le_mul_nnnorm_of_ae_le_mul h)", "annotated_tactic": ["exact NNReal.coe_le_coe.mpr (<a>nnnorm_le_mul_nnnorm_of_ae_le_mul</a> h)", [{"full_name": "MeasureTheory.Lp.nnnorm_le_mul_nnnorm_of_ae_le_mul", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [367, 9], "def_end_pos": [367, 42]}]], "state_before": "case inl.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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 \u2191c * \u2016\u2191\u2191g x\u2016\n\u22a2 \u2016f\u2016 \u2264 \u2191c * \u2016g\u2016", "state_after": "no goals"}, {"tactic": "simp only [norm_def]", "annotated_tactic": ["simp only [<a>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 inr\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : c < 0\n\u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016", "state_after": "case inr\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : c < 0\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191f) p \u03bc) \u2264 c * ENNReal.toReal (snorm (\u2191\u2191g) p \u03bc)"}, {"tactic": "have := snorm_eq_zero_and_zero_of_ae_le_mul_neg h hc p", "annotated_tactic": ["have := <a>snorm_eq_zero_and_zero_of_ae_le_mul_neg</a> h hc p", [{"full_name": "MeasureTheory.snorm_eq_zero_and_zero_of_ae_le_mul_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1311, 9], "def_end_pos": [1311, 48]}]], "state_before": "case inr\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : c < 0\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191f) p \u03bc) \u2264 c * ENNReal.toReal (snorm (\u2191\u2191g) p \u03bc)", "state_after": "case inr\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : c < 0\nthis : snorm (fun x => \u2191\u2191f x) p \u03bc = 0 \u2227 snorm (fun x => \u2191\u2191g x) p \u03bc = 0\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191f) p \u03bc) \u2264 c * ENNReal.toReal (snorm (\u2191\u2191g) p \u03bc)"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case inr\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 : \u211d\nf : { x // x \u2208 Lp E p }\ng : { x // x \u2208 Lp F p }\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016\nhc : c < 0\nthis : snorm (fun x => \u2191\u2191f x) p \u03bc = 0 \u2227 snorm (fun x => \u2191\u2191g x) p \u03bc = 0\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191f) p \u03bc) \u2264 c * ENNReal.toReal (snorm (\u2191\u2191g) p \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_push", "start": [1107, 1], "end": [1108, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector3.lean", "full_name": "exists_vector_succ", "start": [238, 1], "end": [239, 91], "traced_tactics": [{"tactic": "rw [cons_head_tail v]", "annotated_tactic": ["rw [<a>cons_head_tail</a> v]", [{"full_name": "Vector3.cons_head_tail", "def_path": "Mathlib/Data/Vector3.lean", "def_pos": [108, 9], "def_end_pos": [108, 23]}]], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 (succ n) \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 (succ n)\nfv : f v\n\u22a2 f (?m.41426 f x\u271d v fv :: ?m.41427 f x\u271d v fv)", "state_after": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 (succ n) \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 (succ n)\nfv : f v\n\u22a2 f v"}, {"tactic": "exact fv", "annotated_tactic": ["exact fv", []], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 (succ n) \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 (succ n)\nfv : f v\n\u22a2 f 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.drop", "start": [898, 1], "end": [906, 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.drop 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.drop 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.drop 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.drop s n)"}, {"tactic": "simp [Substring.drop, this]", "annotated_tactic": ["simp [<a>Substring.drop</a>, this]", [{"full_name": "Substring.drop", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [575, 15], "def_end_pos": [575, 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.drop 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 + { byteIdx := utf8Len (List.take n m) }, stopPos := s.stopPos }"}, {"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 + { byteIdx := utf8Len (List.take n m) }, stopPos := s.stopPos }", "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 } + { byteIdx := utf8Len (List.take n m) },\n      stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },\n      stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },\n      stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },\n      stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },\n          stopPos := { byteIdx := utf8Len l + utf8Len m } }.stopPos.byteIdx =\n    utf8Len (l ++ List.take n m) + utf8Len (List.drop n m)"}, {"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 } + { byteIdx := utf8Len (List.take n m) },\n          stopPos := { byteIdx := utf8Len l + utf8Len m } }.stopPos.byteIdx =\n    utf8Len (l ++ List.take n m) + utf8Len (List.drop 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 ++ (List.take n m ++ List.drop n m ++ r) },\n          startPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) },\n          stopPos := { byteIdx := utf8Len l + utf8Len (List.take n m ++ List.drop n m) } }.stopPos.byteIdx =\n    utf8Len (l ++ List.take n m) + utf8Len (List.drop n m)"}, {"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) },\n          startPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) },\n          stopPos := { byteIdx := utf8Len l + utf8Len (List.take n m ++ List.drop n m) } }.stopPos.byteIdx =\n    utf8Len (l ++ List.take n m) + utf8Len (List.drop n m)", "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 } + { byteIdx := utf8Len (List.take n m) },\n          stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },\n          stopPos := { byteIdx := utf8Len l + utf8Len m } }.startPos.byteIdx =\n    utf8Len (l ++ List.take n m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.sum_unitVec_mul_slice", "start": [290, 1], "end": [305, 46], "traced_tactics": [{"tactic": "apply slice_eq _ _ _", "annotated_tactic": ["apply <a>slice_eq</a> _ _ _", [{"full_name": "Holor.slice_eq", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [250, 9], "def_end_pos": [250, 17]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\n\u22a2 \u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d) = x", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\n\u22a2 slice (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) = slice x"}, {"tactic": "ext i hid", "annotated_tactic": ["ext i hid", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\n\u22a2 slice (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) = slice x", "state_after": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 slice (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) i hid = slice x i hid"}, {"tactic": "rw [\u2190 slice_sum]", "annotated_tactic": ["rw [\u2190 <a>slice_sum</a>]", [{"full_name": "Holor.slice_sum", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [278, 9], "def_end_pos": [278, 18]}]], "state_before": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 slice (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) i hid = slice x i hid", "state_after": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 \u2211 x_1 in Finset.attach (Finset.range d), slice (unitVec d \u2191x_1 \u2297 slice x \u2191x_1 (_ : Nat.succ \u2191x_1 \u2264 d)) i hid =\n    slice x i hid"}, {"tactic": "simp only [slice_unitVec_mul hid]", "annotated_tactic": ["simp only [<a>slice_unitVec_mul</a> hid]", [{"full_name": "Holor.slice_unitVec_mul", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [262, 9], "def_end_pos": [262, 26]}]], "state_before": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 \u2211 x_1 in Finset.attach (Finset.range d), slice (unitVec d \u2191x_1 \u2297 slice x \u2191x_1 (_ : Nat.succ \u2191x_1 \u2264 d)) i hid =\n    slice x i hid", "state_after": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 (\u2211 x_1 in Finset.attach (Finset.range d), if i = \u2191x_1 then slice x \u2191x_1 (_ : Nat.succ \u2191x_1 \u2264 d) else 0) =\n    slice x i hid"}, {"tactic": "rw [Finset.sum_eq_single (Subtype.mk i <| Finset.mem_range.2 hid)]", "annotated_tactic": ["rw [<a>Finset.sum_eq_single</a> (<a>Subtype.mk</a> i <| <a>Finset.mem_range</a>.2 hid)]", [{"full_name": "Finset.sum_eq_single", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [798, 3], "def_end_pos": [798, 14]}, {"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": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}]], "state_before": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 (\u2211 x_1 in Finset.attach (Finset.range d), if i = \u2191x_1 then slice x \u2191x_1 (_ : Nat.succ \u2191x_1 \u2264 d) else 0) =\n    slice x i hid", "state_after": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then\n      slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }\n        (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)\n    else 0) =\n    slice x i hid\n\ncase h.h.h\u2080\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 \u2200 (b : { x // x \u2208 Finset.range d }),\n    b \u2208 Finset.attach (Finset.range d) \u2192\n      b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) } \u2192\n        (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0\n\ncase h.h.h\u2081\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d) \u2192\n    (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then\n        slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }\n          (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)\n      else 0) =\n      0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then\n      slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }\n        (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)\n    else 0) =\n    slice x i hid", "state_after": "no goals"}, {"tactic": "intro (b : { x // x \u2208 Finset.range d }) (_ : b \u2208 (Finset.range d).attach) (hbi : b \u2260 \u27e8i, _\u27e9)", "annotated_tactic": ["intro (b : { x // x \u2208 <a>Finset.range</a> d }) (_ : b \u2208 (Finset.range d).attach) (hbi : b \u2260 \u27e8i, _\u27e9)", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "case h.h.h\u2080\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 \u2200 (b : { x // x \u2208 Finset.range d }),\n    b \u2208 Finset.attach (Finset.range d) \u2192\n      b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) } \u2192\n        (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0", "state_after": "case h.h.h\u2080\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nb : { x // x \u2208 Finset.range d }\nx\u271d : b \u2208 Finset.attach (Finset.range d)\nhbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) }\n\u22a2 (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0"}, {"tactic": "have hbi' : i \u2260 b := by simpa only [Ne.def, Subtype.ext_iff, Subtype.coe_mk] using hbi.symm", "annotated_tactic": ["have hbi' : i \u2260 b := by simpa only [<a>Ne.def</a>, <a>Subtype.ext_iff</a>, <a>Subtype.coe_mk</a>] using hbi.symm", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case h.h.h\u2080\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nb : { x // x \u2208 Finset.range d }\nx\u271d : b \u2208 Finset.attach (Finset.range d)\nhbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) }\n\u22a2 (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0", "state_after": "case h.h.h\u2080\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nb : { x // x \u2208 Finset.range d }\nx\u271d : b \u2208 Finset.attach (Finset.range d)\nhbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) }\nhbi' : i \u2260 \u2191b\n\u22a2 (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0"}, {"tactic": "simp [hbi']", "annotated_tactic": ["simp [hbi']", []], "state_before": "case h.h.h\u2080\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nb : { x // x \u2208 Finset.range d }\nx\u271d : b \u2208 Finset.attach (Finset.range d)\nhbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) }\nhbi' : i \u2260 \u2191b\n\u22a2 (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0", "state_after": "no goals"}, {"tactic": "simpa only [Ne.def, Subtype.ext_iff, Subtype.coe_mk] using hbi.symm", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>Subtype.ext_iff</a>, <a>Subtype.coe_mk</a>] using hbi.symm", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nb : { x // x \u2208 Finset.range d }\nx\u271d : b \u2208 Finset.attach (Finset.range d)\nhbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) }\n\u22a2 i \u2260 \u2191b", "state_after": "no goals"}, {"tactic": "intro (hid' : Subtype.mk i _ \u2209 Finset.attach (Finset.range d))", "annotated_tactic": ["intro (hid' : <a>Subtype.mk</a> i _ \u2209 <a>Finset.attach</a> (<a>Finset.range</a> d))", [{"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": "Finset.attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2440, 5], "def_end_pos": [2440, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "case h.h.h\u2081\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\n\u22a2 \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d) \u2192\n    (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then\n        slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }\n          (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)\n      else 0) =\n      0", "state_after": "case h.h.h\u2081\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nhid' : \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d)\n\u22a2 (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then\n      slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }\n        (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)\n    else 0) =\n    0"}, {"tactic": "exfalso", "annotated_tactic": ["exfalso", []], "state_before": "case h.h.h\u2081\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nhid' : \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d)\n\u22a2 (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then\n      slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }\n        (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)\n    else 0) =\n    0", "state_after": "case h.h.h\u2081.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nhid' : \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d)\n\u22a2 False"}, {"tactic": "exact absurd (Finset.mem_attach _ _) hid'", "annotated_tactic": ["exact <a>absurd</a> (<a>Finset.mem_attach</a> _ _) hid'", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}, {"full_name": "Finset.mem_attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2459, 9], "def_end_pos": [2459, 19]}]], "state_before": "case h.h.h\u2081.h\n\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d : Ring \u03b1\nx : Holor \u03b1 (d :: ds)\ni : \u2115\nhid : i < d\nhid' : \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d)\n\u22a2 False", "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_const'", "start": [157, 1], "end": [162, 38], "traced_tactics": [{"tactic": "cases' isEmpty_or_nonempty \u03b1 with _ h", "annotated_tactic": ["cases' <a>isEmpty_or_nonempty</a> \u03b1 with _ h", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\n\u22a2 StronglyMeasurable f", "state_after": "case inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh\u271d : IsEmpty \u03b1\n\u22a2 StronglyMeasurable f\n\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh : Nonempty \u03b1\n\u22a2 StronglyMeasurable f"}, {"tactic": "exact stronglyMeasurable_of_isEmpty f", "annotated_tactic": ["exact <a>stronglyMeasurable_of_isEmpty</a> f", [{"full_name": "MeasureTheory.stronglyMeasurable_of_isEmpty", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 38]}]], "state_before": "case inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh\u271d : IsEmpty \u03b1\n\u22a2 StronglyMeasurable f", "state_after": "no goals"}, {"tactic": "convert stronglyMeasurable_const (\u03b2 := \u03b2) using 1", "annotated_tactic": ["convert <a>stronglyMeasurable_const</a> (\u03b2 := \u03b2) using 1", [{"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}]], "state_before": "case inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh : Nonempty \u03b1\n\u22a2 StronglyMeasurable f", "state_after": "case h.e'_5\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh : Nonempty \u03b1\n\u22a2 f = fun x => ?inr.convert_3\n\ncase inr.convert_3\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh : Nonempty \u03b1\n\u22a2 \u03b2"}, {"tactic": "exact funext fun x => hf x h.some", "annotated_tactic": ["exact <a>funext</a> fun x => hf x h.some", [{"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.e'_5\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh : Nonempty \u03b1\n\u22a2 f = fun x => ?inr.convert_3\n\ncase inr.convert_3\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nm : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x y : \u03b1), f x = f y\nh : Nonempty \u03b1\n\u22a2 \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff'", "start": [3002, 1], "end": [3014, 7], "traced_tactics": [{"tactic": "have hst : \u03bc (s \\ t) \u2260 \u221e := (measure_lt_top_of_subset (diff_subset s t) hs').ne", "annotated_tactic": ["have hst : \u03bc (s \\ t) \u2260 \u221e := (<a>measure_lt_top_of_subset</a> (<a>diff_subset</a> s t) hs').<a>ne</a>", [{"full_name": "MeasureTheory.measure_lt_top_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [342, 9], "def_end_pos": [342, 33]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))"}, {"tactic": "have hts : \u03bc (t \\ s) \u2260 \u221e := (measure_lt_top_of_subset (diff_subset t s) ht').ne", "annotated_tactic": ["have hts : \u03bc (t \\ s) \u2260 \u221e := (<a>measure_lt_top_of_subset</a> (<a>diff_subset</a> t s) ht').<a>ne</a>", [{"full_name": "MeasureTheory.measure_lt_top_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [342, 9], "def_end_pos": [342, 33]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))"}, {"tactic": "suffices : (\u03bc s).toReal - (\u03bc t).toReal = (\u03bc (s \\ t)).toReal - (\u03bc (t \\ s)).toReal", "annotated_tactic": ["suffices : (\u03bc s).<a>toReal</a> - (\u03bc t).<a>toReal</a> = (\u03bc (s \\ t)).<a>toReal</a> - (\u03bc (t \\ s)).<a>toReal</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": "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]}]], "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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))\n\ncase this\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 : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))"}, {"tactic": "rw [measure_diff' s ht ht', measure_diff' t hs hs',\n  ENNReal.toReal_sub_of_le measure_le_measure_union_right (measure_union_ne_top hs' ht'),\n  ENNReal.toReal_sub_of_le measure_le_measure_union_right (measure_union_ne_top ht' hs'),\n  union_comm t s]", "annotated_tactic": ["rw [<a>measure_diff'</a> s ht ht', <a>measure_diff'</a> t hs hs',\n    <a>ENNReal.toReal_sub_of_le</a> <a>measure_le_measure_union_right</a> (<a>measure_union_ne_top</a> hs' ht'),\n    <a>ENNReal.toReal_sub_of_le</a> <a>measure_le_measure_union_right</a> (<a>measure_union_ne_top</a> ht' hs'),\n    <a>union_comm</a> t s]", [{"full_name": "MeasureTheory.measure_diff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [247, 9], "def_end_pos": [247, 22]}, {"full_name": "MeasureTheory.measure_diff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [247, 9], "def_end_pos": [247, 22]}, {"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": "MeasureTheory.measure_le_measure_union_right", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [209, 9], "def_end_pos": [209, 39]}, {"full_name": "MeasureTheory.measure_union_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [324, 9], "def_end_pos": [324, 29]}, {"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": "MeasureTheory.measure_le_measure_union_right", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [209, 9], "def_end_pos": [209, 39]}, {"full_name": "MeasureTheory.measure_union_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [324, 9], "def_end_pos": [324, 29]}, {"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}]], "state_before": "case this\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 : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))", "state_after": "case this\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 : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) =\n    ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc t) - (ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc s))"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case this\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 : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) =\n    ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc t) - (ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc s))", "state_after": "no goals"}, {"tactic": "rw [this, measure_symmDiff_eq hs ht, ENNReal.toReal_add hst hts]", "annotated_tactic": ["rw [this, <a>measure_symmDiff_eq</a> hs ht, <a>ENNReal.toReal_add</a> hst hts]", [{"full_name": "MeasureTheory.measure_symmDiff_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [152, 7], "def_end_pos": [152, 26]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 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\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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))| \u2264\n    ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) + ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))"}, {"tactic": "convert abs_sub (\u03bc (s \\ t)).toReal (\u03bc (t \\ s)).toReal <;> simp", "annotated_tactic": ["convert <a>abs_sub</a> (\u03bc (s \\ t)).<a>toReal</a> (\u03bc (t \\ s)).<a>toReal</a> <;> simp", [{"full_name": "abs_sub", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [276, 9], "def_end_pos": [276, 16]}, {"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]}]], "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\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))| \u2264\n    ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) + ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Finsupp.lean", "full_name": "Finset.mem_finsupp_iff", "start": [47, 1], "end": [56, 88], "traced_tactics": [{"tactic": "refine' mem_map.trans \u27e8_, _\u27e9", "annotated_tactic": ["refine' mem_map.trans \u27e8_, _\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 f \u2208 Finset.finsupp s t \u2194 f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i", "state_after": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f) \u2192\n    f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\n\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i) \u2192\n    \u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f"}, {"tactic": "rintro \u27e8f, hf, rfl\u27e9", "annotated_tactic": ["rintro \u27e8f, hf, rfl\u27e9", []], "state_before": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f) \u2192\n    f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i", "state_after": "case refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\n\u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f).support \u2286 s \u2227\n    \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i"}, {"tactic": "refine' \u27e8support_indicator_subset _ _, fun i hi => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>support_indicator_subset</a> _ _, fun i hi => _\u27e9", [{"full_name": "Finsupp.support_indicator_subset", "def_path": "Mathlib/Data/Finsupp/Indicator.lean", "def_pos": [67, 9], "def_end_pos": [67, 33]}]], "state_before": "case refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\n\u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f).support \u2286 s \u2227\n    \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i", "state_after": "case refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i"}, {"tactic": "convert mem_pi.1 hf i hi", "annotated_tactic": ["convert <a>mem_pi</a>.1 hf i hi", [{"full_name": "Finset.mem_pi", "def_path": "Mathlib/Data/Finset/Pi.lean", "def_pos": [49, 9], "def_end_pos": [49, 15]}]], "state_before": "case refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i", "state_after": "case h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i = f i hi"}, {"tactic": "exact indicator_of_mem hi _", "annotated_tactic": ["exact <a>indicator_of_mem</a> hi _", [{"full_name": "Finsupp.indicator_of_mem", "def_path": "Mathlib/Data/Finsupp/Indicator.lean", "def_pos": [44, 9], "def_end_pos": [44, 25]}]], "state_before": "case h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i = f i hi", "state_after": "no goals"}, {"tactic": "refine' fun h => \u27e8fun i _ => f i, mem_pi.2 h.2, _\u27e9", "annotated_tactic": ["refine' fun h => \u27e8fun i _ => f i, <a>mem_pi</a>.2 h.2, _\u27e9", [{"full_name": "Finset.mem_pi", "def_path": "Mathlib/Data/Finset/Pi.lean", "def_pos": [49, 9], "def_end_pos": [49, 15]}]], "state_before": "case refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i) \u2192\n    \u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f", "state_after": "case refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\n\u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) = f"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "case refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\n\u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) = f", "state_after": "case refine'_2.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\ni : \u03b9\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) i = \u2191f i"}, {"tactic": "exact ite_eq_left_iff.2 fun hi => (not_mem_support_iff.1 fun H => hi <| h.1 H).symm", "annotated_tactic": ["exact <a>ite_eq_left_iff</a>.2 fun hi => (<a>not_mem_support_iff</a>.1 fun H => hi <| h.1 H).<a>symm</a>", [{"full_name": "ite_eq_left_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1159, 17], "def_end_pos": [1159, 32]}, {"full_name": "Finsupp.not_mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [195, 9], "def_end_pos": [195, 28]}, {"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.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\ni : \u03b9\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) i = \u2191f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableSet.exists_measurable_proj", "start": [1256, 1], "end": [1261, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "full_name": "MeasureTheory.Measure.withDensity\u1d65_absolutelyContinuous", "start": [132, 1], "end": [139, 52], "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\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc"}, {"tactic": "refine' VectorMeasure.AbsolutelyContinuous.mk fun i hi\u2081 hi\u2082 => _", "annotated_tactic": ["refine' <a>VectorMeasure.AbsolutelyContinuous.mk</a> fun i hi\u2081 hi\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 pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191(toENNRealVectorMeasure \u03bc) i = 0\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0"}, {"tactic": "rw [toENNRealVectorMeasure_apply_measurable hi\u2081] at hi\u2082", "annotated_tactic": ["rw [<a>toENNRealVectorMeasure_apply_measurable</a> hi\u2081] at hi\u2082", [{"full_name": "MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [495, 9], "def_end_pos": [495, 48]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191(toENNRealVectorMeasure \u03bc) i = 0\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191\u2191\u03bc i = 0\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0"}, {"tactic": "rw [withDensity\u1d65_apply hf hi\u2081, Measure.restrict_zero_set hi\u2082, integral_zero_measure]", "annotated_tactic": ["rw [<a>withDensity\u1d65_apply</a> hf hi\u2081, <a>Measure.restrict_zero_set</a> hi\u2082, <a>integral_zero_measure</a>]", [{"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.Measure.restrict_zero_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 26]}, {"full_name": "MeasureTheory.integral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 30]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191\u2191\u03bc i = 0\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0", "state_after": "no goals"}, {"tactic": "rw [withDensity\u1d65, dif_neg hf]", "annotated_tactic": ["rw [<a>withDensity\u1d65</a>, <a>dif_neg</a> hf]", [{"full_name": "MeasureTheory.Measure.withDensity\u1d65", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [43, 5], "def_end_pos": [43, 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\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f\n\u22a2 0 \u226a\u1d65 toENNRealVectorMeasure \u03bc"}, {"tactic": "exact VectorMeasure.AbsolutelyContinuous.zero _", "annotated_tactic": ["exact <a>VectorMeasure.AbsolutelyContinuous.zero</a> _", [{"full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.zero", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1099, 9], "def_end_pos": [1099, 13]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f\n\u22a2 0 \u226a\u1d65 toENNRealVectorMeasure \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.setLaverage_one", "start": [214, 1], "end": [215, 29], "traced_tactics": []}, {"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_of_equiv", "start": [319, 1], "end": [358, 34], "traced_tactics": [{"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\n\u22a2 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b", "state_after": "case h\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nx : F\n\u22a2 x \u2208 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) \u2194 x \u2208 Set.Icc a b"}, {"tactic": "simp only [Set.mem_preimage, Set.mem_Icc, he_ord]", "annotated_tactic": ["simp only [<a>Set.mem_preimage</a>, <a>Set.mem_Icc</a>, he_ord]", [{"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]}]], "state_before": "case h\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nx : F\n\u22a2 x \u2208 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) \u2194 x \u2208 Set.Icc a b", "state_after": "no goals"}, {"tactic": "rw [\u2190 hIcc, eL.symm_preimage_preimage]", "annotated_tactic": ["rw [\u2190 hIcc, eL.symm_preimage_preimage]", []], "state_before": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\n\u22a2 Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b", "state_after": "no goals"}, {"tactic": "simp only [hDF]", "annotated_tactic": ["simp only [hDF]", []], "state_before": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, DF x =\n    \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))", "state_after": "no goals"}, {"tactic": "rw [\u2190 he_vol.set_integral_preimage_emb he_emb]", "annotated_tactic": ["rw [\u2190 he_vol.set_integral_preimage_emb he_emb]", []], "state_before": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n    \u222b (x : (fun x => Fin (n + 1) \u2192 \u211d) a) in Set.Icc (\u2191eL a) (\u2191eL b),\n      \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) (\u2191(ContinuousLinearEquiv.symm eL) (e i))", "state_after": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n    \u222b (x : F) in \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b),\n      \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) (\u2191eL x))) (\u2191(ContinuousLinearEquiv.symm eL) (e i))"}, {"tactic": "simp only [hIcc, eL.symm_apply_apply]", "annotated_tactic": ["simp only [hIcc, eL.symm_apply_apply]", []], "state_before": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n    \u222b (x : F) in \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b),\n      \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) (\u2191eL x))) (\u2191(ContinuousLinearEquiv.symm eL) (e i))", "state_after": "no goals"}, {"tactic": "refine integral_divergence_of_hasFDerivWithinAt_off_countable' (eL a) (eL b)\n  ((he_ord _ _).2 hle) (fun i x => f i (eL.symm x))\n  (fun i x => f' i (eL.symm x) \u2218L (eL.symm : \u211d\u207f\u207a\u00b9 \u2192L[\u211d] F)) (eL.symm \u207b\u00b9' s)\n  (hs.preimage eL.symm.injective) ?_ ?_ ?_", "annotated_tactic": ["refine <a>integral_divergence_of_hasFDerivWithinAt_off_countable'</a> (eL a) (eL b)\n        ((he_ord _ _).2 hle) (fun i x => f i (eL.symm x))\n        (fun i x => f' i (eL.symm x) \u2218L (eL.symm : \u211d\u207f\u207a\u00b9 \u2192L[\u211d] F)) (eL.symm \u207b\u00b9' s)\n        (hs.preimage eL.symm.injective) ?_ ?_ ?_", [{"full_name": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable'", "def_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "def_pos": [301, 9], "def_end_pos": [301, 64]}]], "state_before": "E : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : (fun x => Fin (n + 1) \u2192 \u211d) a) in Set.Icc (\u2191eL a) (\u2191eL b),\n      \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n    \u2211 i : Fin (n + 1),\n      ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (\u2191eL a \u2218 Fin.succAbove i) (\u2191eL b \u2218 Fin.succAbove i),\n          f i (\u2191(ContinuousLinearEquiv.symm eL) (Fin.insertNth i (\u2191eL b i) x))) -\n        \u222b (x : Fin n \u2192 \u211d) in Set.Icc (\u2191eL a \u2218 Fin.succAbove i) (\u2191eL b \u2218 Fin.succAbove i),\n          f i (\u2191(ContinuousLinearEquiv.symm eL) (Fin.insertNth i (\u2191eL a i) x)))", "state_after": "case refine_1\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u2200 (i : Fin (n + 1)), ContinuousOn ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i) (Set.Icc (\u2191eL a) (\u2191eL b))\n\ncase refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s \u2192\n      \u2200 (i : Fin (n + 1)),\n        HasFDerivAt ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i)\n          ((fun i x =>\n              ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n            i x)\n          x\n\ncase refine_3\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 IntegrableOn\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        \u2191((fun i x =>\n                ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n              i x)\n          (e i))\n    (Set.Icc (\u2191eL a) (\u2191eL b))"}, {"tactic": "exact fun i => (Hc i).comp eL.symm.continuousOn hIcc'.subset", "annotated_tactic": ["exact fun i => (Hc i).<a>comp</a> eL.symm.continuousOn hIcc'.subset", [{"full_name": "ContinuousOn.comp", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [929, 9], "def_end_pos": [929, 26]}]], "state_before": "case refine_1\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u2200 (i : Fin (n + 1)), ContinuousOn ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i) (Set.Icc (\u2191eL a) (\u2191eL b))", "state_after": "no goals"}, {"tactic": "refine' fun x hx i => (Hd (eL.symm x) \u27e8_, hx.2\u27e9 i).comp x eL.symm.hasFDerivAt", "annotated_tactic": ["refine' fun x hx i => (Hd (eL.symm x) \u27e8_, hx.2\u27e9 i).<a>comp</a> x eL.symm.hasFDerivAt", [{"full_name": "HasFDerivAt.comp", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Comp.lean", "def_pos": [106, 9], "def_end_pos": [106, 25]}]], "state_before": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s \u2192\n      \u2200 (i : Fin (n + 1)),\n        HasFDerivAt ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i)\n          ((fun i x =>\n              ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n            i x)\n          x", "state_after": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (Set.Icc a b)"}, {"tactic": "rw [\u2190 hIcc]", "annotated_tactic": ["rw [\u2190 hIcc]", []], "state_before": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (Set.Icc a b)", "state_after": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (\u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b))"}, {"tactic": "refine' preimage_interior_subset_interior_preimage eL.continuous _", "annotated_tactic": ["refine' <a>preimage_interior_subset_interior_preimage</a> eL.continuous _", [{"full_name": "preimage_interior_subset_interior_preimage", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1659, 9], "def_end_pos": [1659, 51]}]], "state_before": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (\u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b))", "state_after": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 \u2191eL \u207b\u00b9' interior (Set.Icc (\u2191eL a) (\u2191eL b))"}, {"tactic": "simpa only [Set.mem_preimage, eL.apply_symm_apply, \u2190 pi_univ_Icc,\n  interior_pi_set (@finite_univ (Fin _) _), interior_Icc] using hx.1", "annotated_tactic": ["simpa only [<a>Set.mem_preimage</a>, eL.apply_symm_apply, \u2190 <a>pi_univ_Icc</a>,\n          <a>interior_pi_set</a> (@<a>finite_univ</a> (<a>Fin</a> _) _), <a>interior_Icc</a>] using hx.1", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"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": "interior_pi_set", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1400, 9], "def_end_pos": [1400, 24]}, {"full_name": "Set.finite_univ", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [722, 9], "def_end_pos": [722, 20]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "interior_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2371, 9], "def_end_pos": [2371, 21]}]], "state_before": "case refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 \u2191eL \u207b\u00b9' interior (Set.Icc (\u2191eL a) (\u2191eL b))", "state_after": "no goals"}, {"tactic": "rw [\u2190 he_vol.integrableOn_comp_preimage he_emb, hIcc]", "annotated_tactic": ["rw [\u2190 he_vol.integrableOn_comp_preimage he_emb, hIcc]", []], "state_before": "case refine_3\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 IntegrableOn\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        \u2191((fun i x =>\n                ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n              i x)\n          (e i))\n    (Set.Icc (\u2191eL a) (\u2191eL b))", "state_after": "case refine_3\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 IntegrableOn\n    ((fun x =>\n        \u2211 i : Fin (n + 1),\n          \u2191((fun i x =>\n                  ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n                i x)\n            (e i)) \u2218\n      \u2191eL)\n    (Set.Icc a b)"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_union_inter", "start": [2183, 1], "end": [2184, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithTop.image_coe_Ico", "start": [113, 1], "end": [115, 94], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ico_subset_Iio_self <| Iio_subset_Iio le_top)]", "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> (<a>Subset.trans</a> <a>Ico_subset_Iio_self</a> <| <a>Iio_subset_Iio</a> <a>le_top</a>)]", [{"full_name": "WithTop.preimage_coe_Ico", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [63, 9], "def_end_pos": [63, 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.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.Ico_subset_Iio_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 28]}, {"full_name": "Set.Iio_subset_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 23]}, {"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\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/MeasureTheory/Measure/OpenPos.lean", "full_name": "Continuous.isOpenPosMeasure_map", "start": [154, 1], "end": [160, 81], "traced_tactics": [{"tactic": "refine' \u27e8fun U hUo hUne => _\u27e9", "annotated_tactic": ["refine' \u27e8fun U hUo hUne => _\u27e9", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u2077 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d\u2074 : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\ninst\u271d\u00b3 : OpensMeasurableSpace X\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : MeasurableSpace Z\ninst\u271d : BorelSpace Z\nf : X \u2192 Z\nhf : Continuous f\nhf_surj : Surjective f\n\u22a2 IsOpenPosMeasure (map f \u03bc)", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u2077 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d\u2074 : IsOpenPosMeasure \u03bc\ns U\u271d F : Set X\nx : X\ninst\u271d\u00b3 : OpensMeasurableSpace X\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : MeasurableSpace Z\ninst\u271d : BorelSpace Z\nf : X \u2192 Z\nhf : Continuous f\nhf_surj : Surjective f\nU : Set Z\nhUo : IsOpen U\nhUne : Set.Nonempty U\n\u22a2 \u2191\u2191(map f \u03bc) U \u2260 0"}, {"tactic": "rw [Measure.map_apply hf.measurable hUo.measurableSet]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> hf.measurable hUo.measurableSet]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u2077 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d\u2074 : IsOpenPosMeasure \u03bc\ns U\u271d F : Set X\nx : X\ninst\u271d\u00b3 : OpensMeasurableSpace X\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : MeasurableSpace Z\ninst\u271d : BorelSpace Z\nf : X \u2192 Z\nhf : Continuous f\nhf_surj : Surjective f\nU : Set Z\nhUo : IsOpen U\nhUne : Set.Nonempty 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"MeasureTheory.martingale_const", "start": [70, 1], "end": [72, 67], "traced_tactics": [{"tactic": "rw [condexp_const (\u2131.le _)]", "annotated_tactic": ["rw [<a>condexp_const</a> (\u2131.le _)]", [{"full_name": "MeasureTheory.condexp_const", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 22]}]], "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\u271d : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nx : E\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 \u03bc[(fun x_1 x_2 => x) j|\u2191\u2131 i] =\u1d50[\u03bc] (fun x_1 x_2 => x) i", "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.exists_insert_toList_zoom_nil", "start": [546, 1], "end": [549, 77], "traced_tactics": [{"tactic": "simp [\u2190 zoom_toList e, insert_toList_zoom_nil ht e]", "annotated_tactic": ["simp [\u2190 <a>zoom_toList</a> e, <a>insert_toList_zoom_nil</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_nil", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [542, 9], "def_end_pos": [542, 31]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\np : Path \u03b1\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\ne : zoom (cmp v) t Path.root = (nil, p)\n\u22a2 toList t = Path.listL p ++ Path.listR p \u2227 toList (insert cmp t v) = Path.listL p ++ v :: Path.listR p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "full_name": "Std.Range.forIn_eq_forIn_range'", "start": [88, 1], "end": [100, 68], "traced_tactics": [{"tactic": "refine Eq.trans ?_ <| (forIn'_eq_forIn_range' r init (fun x _ => f x)).trans ?_", "annotated_tactic": ["refine <a>Eq.trans</a> ?_ <| (<a>forIn'_eq_forIn_range'</a> r init (fun x _ => f x)).<a>trans</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": "Std.Range.forIn'_eq_forIn_range'", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [43, 9], "def_end_pos": [43, 31]}, {"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": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn r init f = forIn (List.range' r.start (numElems r) r.step) init f", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn r init f = forIn' r init fun x x_1 => f x\n\ncase refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 (forIn\n      (List.pmap Subtype.mk (List.range' r.start (numElems r) r.step)\n        (_ : \u2200 (x : Nat), x \u2208 List.range' r.start (numElems r) r.step \u2192 x \u2208 r))\n      init fun x =>\n      match (motive := { x // x \u2208 r } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a) =\n    forIn (List.range' r.start (numElems r) r.step) init f"}, {"tactic": "simp [forIn, forIn', Range.forIn, Range.forIn']", "annotated_tactic": ["simp [<a>forIn</a>, <a>forIn'</a>, <a>Range.forIn</a>, <a>Range.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": "ForIn'.forIn'", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [264, 3], "def_end_pos": [264, 9]}, {"full_name": "Std.Range.forIn", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [23, 25], "def_end_pos": [23, 30]}, {"full_name": "Std.Range.forIn'", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [38, 25], "def_end_pos": [38, 31]}]], "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn r init f = forIn' r init fun x x_1 => f x", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn.loop f r.stop r.start r.stop r.step init =\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) r.stop r.start (_ : r.start \u2264 r.start) init"}, {"tactic": "suffices \u2200 fuel i hl b, forIn'.loop r.start r.stop r.step (fun x _ => f x) fuel i hl b =\n    forIn.loop f fuel i r.stop r.step b from (this _ ..).symm", "annotated_tactic": ["suffices \u2200 fuel i hl b, <a>forIn'.loop</a> r.start r.stop r.step (fun x _ => f x) fuel i hl b =\n        <a>forIn.loop</a> f fuel i r.stop r.step b from (this _ ..).<a>symm</a>", [{"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}, {"full_name": "Std.Range.forIn.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [25, 25], "def_end_pos": [25, 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": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn.loop f r.stop r.start r.stop r.step init =\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) r.stop r.start (_ : r.start \u2264 r.start) init", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (fuel i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b"}, {"tactic": "intro fuel", "annotated_tactic": ["intro fuel", []], "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (fuel i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nfuel : Nat\n\u22a2 \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b"}, {"tactic": "induction fuel <;> intro i hl b <;>\nunfold forIn.loop forIn'.loop <;> simp [*] <;> split <;> try simp", "annotated_tactic": ["induction fuel <;> intro i hl b <;>\n      unfold <a>forIn.loop</a> <a>forIn'.loop</a> <;> simp [*] <;> split <;> try simp", [{"full_name": "Std.Range.forIn.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [25, 25], "def_end_pos": [25, 29]}, {"full_name": "Std.Range.forIn'.loop", "def_path": "lake-packages/lean4/src/lean/Init/Data/Range.lean", "def_pos": [39, 25], "def_end_pos": [39, 29]}]], "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nfuel : Nat\n\u22a2 \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b", "state_after": "case refine_1.succ.inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : i < r.stop\n\u22a2 (do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b) =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b\n\ncase refine_1.succ.inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : \u00aci < r.stop\n\u22a2 pure b =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_1.zero.inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : \u00aci < r.stop\n\u22a2 pure b = pure b", "state_after": "no goals"}, {"tactic": "simp [if_neg (Nat.not_le.2 \u2039_\u203a)]", "annotated_tactic": ["simp [<a>if_neg</a> (<a>Nat.not_le</a>.2 \u2039_\u203a)]", [{"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": "Nat.not_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [147, 27], "def_end_pos": [147, 33]}]], "state_before": "case refine_1.succ.inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : i < r.stop\n\u22a2 (do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b) =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b", "state_after": "no goals"}, {"tactic": "simp [if_pos (Nat.not_lt.1 \u2039_\u203a)]", "annotated_tactic": ["simp [<a>if_pos</a> (<a>Nat.not_lt</a>.1 \u2039_\u203a)]", [{"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": "Nat.not_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [150, 27], "def_end_pos": [150, 33]}]], "state_before": "case refine_1.succ.inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : \u00aci < r.stop\n\u22a2 pure b =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b", "state_after": "no goals"}, {"tactic": "intro L", "annotated_tactic": ["intro L", []], "state_before": "case refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (L : List Nat) (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r),\n    (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f", "state_after": "case refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r), (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f"}, {"tactic": "induction L generalizing init <;> intro H <;> simp [*]", "annotated_tactic": ["induction L generalizing init <;> intro H <;> simp [*]", []], "state_before": "case refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r), (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indepSet_empty_left", "start": [247, 1], "end": [249, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Antitone.measurable", "start": [1255, 11], "end": [1257, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.inter_mul_singleton", "start": [1914, 1], "end": [1915, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.toNat_lt", "start": [1386, 9], "end": [1387, 50], "traced_tactics": [{"tactic": "rw [\u2190 Int.not_le, \u2190 Nat.not_le, Int.le_toNat h]", "annotated_tactic": ["rw [\u2190 <a>Int.not_le</a>, \u2190 <a>Nat.not_le</a>, <a>Int.le_toNat</a> h]", [{"full_name": "Int.not_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [663, 19], "def_end_pos": [663, 25]}, {"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": "Int.le_toNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1383, 17], "def_end_pos": [1383, 25]}]], "state_before": "n : Nat\nz : Int\nh : 0 \u2264 z\n\u22a2 toNat z < n \u2194 z < \u2191n", "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_ae", "start": [1418, 1], "end": [1424, 54], "traced_tactics": [{"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\nE : Type u_2\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 : f =\u1d50[\u03bc] g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "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 : f =\u1d50[\u03bc] g\nhfi : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g\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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g"}, {"tactic": "have hgi : Integrable g \u03bc := hfi.congr h", "annotated_tactic": ["have hgi : <a>Integrable</a> g \u03bc := hfi.congr h", [{"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\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 : f =\u1d50[\u03bc] g\nhfi : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "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 : f =\u1d50[\u03bc] g\nhfi : Integrable f\nhgi : Integrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g"}, {"tactic": "rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h]", "annotated_tactic": ["rw [<a>setToFun_eq</a> hT hfi, <a>setToFun_eq</a> hT hgi, (<a>Integrable.toL1_eq_toL1_iff</a> f g hfi hgi).2 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.setToFun_eq", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 20]}, {"full_name": "MeasureTheory.Integrable.toL1_eq_toL1_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1427, 9], "def_end_pos": [1427, 25]}]], "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 : f =\u1d50[\u03bc] g\nhfi : Integrable f\nhgi : Integrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "no goals"}, {"tactic": "have hgi : \u00acIntegrable g \u03bc := by rw [integrable_congr h] at hfi; exact hfi", "annotated_tactic": ["have hgi : \u00ac<a>Integrable</a> g \u03bc := by rw [<a>integrable_congr</a> h] at hfi; exact hfi", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}]], "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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "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\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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\nhgi : \u00acIntegrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g"}, {"tactic": "rw [setToFun_undef hT hfi, setToFun_undef hT hgi]", "annotated_tactic": ["rw [<a>setToFun_undef</a> hT hfi, <a>setToFun_undef</a> hT hgi]", [{"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]}]], "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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\nhgi : \u00acIntegrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "no goals"}, {"tactic": "rw [integrable_congr h] at hfi", "annotated_tactic": ["rw [<a>integrable_congr</a> h] at hfi", [{"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\nE : Type u_2\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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\n\u22a2 \u00acIntegrable 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\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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable g\n\u22a2 \u00acIntegrable g"}, {"tactic": "exact hfi", "annotated_tactic": ["exact hfi", []], "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 : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable g\n\u22a2 \u00acIntegrable g", "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_prehaar", "start": [352, 1], "end": [355, 82], "traced_tactics": [{"tactic": "simp only [prehaar, Compacts.coe_map, is_left_invariant_index K.isCompact _ hU]", "annotated_tactic": ["simp only [<a>prehaar</a>, <a>Compacts.coe_map</a>, <a>is_left_invariant_index</a> K.isCompact _ hU]", [{"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_map", "def_path": "Mathlib/Topology/Sets/Compacts.lean", "def_pos": [137, 9], "def_end_pos": [137, 16]}, {"full_name": "MeasureTheory.Measure.haar.is_left_invariant_index", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [286, 9], "def_end_pos": [286, 32]}]], "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)\ng : G\nK : Compacts G\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": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.singleton_product_singleton", "start": [232, 1], "end": [234, 76], "traced_tactics": [{"tactic": "simp only [product_singleton, Function.Embedding.coeFn_mk, map_singleton]", "annotated_tactic": ["simp only [<a>product_singleton</a>, <a>Function.Embedding.coeFn_mk</a>, <a>map_singleton</a>]", [{"full_name": "Finset.product_singleton", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [227, 9], "def_end_pos": [227, 26]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}, {"full_name": "Finset.map_singleton", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [226, 9], "def_end_pos": [226, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s' : Finset \u03b1\nt t' : Finset \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 {a} \u00d7\u02e2 {b} = {(a, b)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/BorelCantelli.lean", "full_name": "ProbabilityTheory.measure_limsup_eq_one", "start": [72, 1], "end": [103, 44], "traced_tactics": [{"tactic": "rw [measure_congr (eventuallyEq_set.2 (ae_mem_limsup_atTop_iff \u03bc <|\n  measurableSet_filtrationOfSet' hsm) : (limsup s atTop : Set \u03a9) =\u1d50[\u03bc]\n    {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n,\n      (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop atTop})]", "annotated_tactic": ["rw [<a>measure_congr</a> (<a>eventuallyEq_set</a>.2 (<a>ae_mem_limsup_atTop_iff</a> \u03bc <|\n    <a>measurableSet_filtrationOfSet'</a> hsm) : (<a>limsup</a> s <a>atTop</a> : <a>Set</a> \u03a9) =\u1d50[\u03bc]\n      {\u03c9 | <a>Tendsto</a> (fun n => \u2211 k in <a>Finset.range</a> n,\n        (\u03bc[(s (k + 1)).<a>indicator</a> (1 : \u03a9 \u2192 \u211d)|<a>filtrationOfSet</a> hsm k]) \u03c9) <a>atTop</a> <a>atTop</a>})]", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "Filter.eventuallyEq_set", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1455, 9], "def_end_pos": [1455, 25]}, {"full_name": "MeasureTheory.ae_mem_limsup_atTop_iff", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [376, 9], "def_end_pos": [376, 32]}, {"full_name": "MeasureTheory.measurableSet_filtrationOfSet'", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [253, 9], "def_end_pos": [253, 39]}, {"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": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"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": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.filtrationOfSet", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [242, 5], "def_end_pos": [242, 20]}, {"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]}]], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 1", "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 \u2191\u2191\u03bc\n      {\u03c9 |\n        Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n          atTop} =\n    1"}, {"tactic": "suffices {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n,\n    (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop atTop} =\u1d50[\u03bc] Set.univ by\n  rw [measure_congr this, measure_univ]", "annotated_tactic": ["suffices {\u03c9 | <a>Tendsto</a> (fun n => \u2211 k in <a>Finset.range</a> n,\n      (\u03bc[(s (k + 1)).<a>indicator</a> (1 : \u03a9 \u2192 \u211d)|<a>filtrationOfSet</a> hsm k]) \u03c9) <a>atTop</a> <a>atTop</a>} =\u1d50[\u03bc] <a>Set.univ</a> by\n    rw [<a>measure_congr</a> this, <a>measure_univ</a>]", [{"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": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.filtrationOfSet", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [242, 5], "def_end_pos": [242, 20]}, {"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": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}]], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 \u2191\u2191\u03bc\n      {\u03c9 |\n        Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n          atTop} =\n    1", "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1d50[\u03bc]\n    Set.univ"}, {"tactic": "have : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, (\u03bc[(s (n + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm n]) \u03c9 = _ :=\n  ae_all_iff.2 fun n => hs.condexp_indicator_filtrationOfSet_ae_eq hsm n.lt_succ_self", "annotated_tactic": ["have : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, (\u03bc[(s (n + 1)).<a>indicator</a> (1 : \u03a9 \u2192 \u211d)|<a>filtrationOfSet</a> hsm n]) \u03c9 = _ :=\n    <a>ae_all_iff</a>.2 fun n => hs.condexp_indicator_filtrationOfSet_ae_eq hsm n.lt_succ_self", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "MeasureTheory.filtrationOfSet", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [242, 5], "def_end_pos": [242, 20]}, {"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1d50[\u03bc]\n    Set.univ", "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1d50[\u03bc]\n    Set.univ"}, {"tactic": "filter_upwards [this] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [this] with \u03c9 h\u03c9", []], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1d50[\u03bc]\n    Set.univ", "state_after": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 setOf\n      (fun \u03c9 =>\n        Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n          atTop)\n      \u03c9 =\n    Set.univ \u03c9"}, {"tactic": "refine' eq_true (_ : Tendsto _ _ _)", "annotated_tactic": ["refine' <a>eq_true</a> (_ : <a>Tendsto</a> _ _ _)", [{"full_name": "eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [14, 9], "def_end_pos": [14, 16]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}]], "state_before": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 setOf\n      (fun \u03c9 =>\n        Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n          atTop)\n      \u03c9 =\n    Set.univ \u03c9", "state_after": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop"}, {"tactic": "simp_rw [h\u03c9]", "annotated_tactic": ["simp_rw [h\u03c9]", []], "state_before": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop", "state_after": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop"}, {"tactic": "have htends : Tendsto (fun n => \u2211 k in Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u221e) := by\n  rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]\n  exact ENNReal.tendsto_nat_tsum _", "annotated_tactic": ["have htends : <a>Tendsto</a> (fun n => \u2211 k in <a>Finset.range</a> n, \u03bc (s (k + 1))) <a>atTop</a> (\ud835\udcdd \u221e) := by\n    rw [\u2190 <a>ENNReal.tsum_add_one_eq_top</a> hs' (<a>measure_ne_top</a> _ _)]\n    exact <a>ENNReal.tendsto_nat_tsum</a> _", [{"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": "ENNReal.tsum_add_one_eq_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 28]}, {"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.tendsto_nat_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [929, 9], "def_end_pos": [929, 25]}]], "state_before": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop", "state_after": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop"}, {"tactic": "rw [ENNReal.tendsto_nhds_top_iff_nnreal] at htends", "annotated_tactic": ["rw [<a>ENNReal.tendsto_nhds_top_iff_nnreal</a>] at htends", [{"full_name": "ENNReal.tendsto_nhds_top_iff_nnreal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [160, 9], "def_end_pos": [160, 36]}]], "state_before": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop", "state_after": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop"}, {"tactic": "refine' tendsto_atTop_atTop_of_monotone' _ _", "annotated_tactic": ["refine' <a>tendsto_atTop_atTop_of_monotone'</a> _ _", [{"full_name": "Filter.tendsto_atTop_atTop_of_monotone'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1734, 9], "def_end_pos": [1734, 41]}]], "state_before": "case h\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop", "state_after": "case h.refine'_1\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 Monotone fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))\n\ncase h.refine'_2\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 \u00acBddAbove (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))"}, {"tactic": "rw [measure_congr this, measure_univ]", "annotated_tactic": ["rw [<a>measure_congr</a> this, <a>measure_univ</a>]", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}]], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  {\u03c9 |\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1d50[\u03bc]\n    Set.univ\n\u22a2 \u2191\u2191\u03bc\n      {\u03c9 |\n        Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n          atTop} =\n    1", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.tsum_add_one_eq_top</a> hs' (<a>measure_ne_top</a> _ _)]", [{"full_name": "ENNReal.tsum_add_one_eq_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 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": "\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)", "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd (\u2211' (n : \u2115), \u2191\u2191\u03bc (s (n + 1))))"}, {"tactic": "exact ENNReal.tendsto_nat_tsum _", "annotated_tactic": ["exact <a>ENNReal.tendsto_nat_tsum</a> _", [{"full_name": "ENNReal.tendsto_nat_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [929, 9], "def_end_pos": [929, 25]}]], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd (\u2211' (n : \u2115), \u2191\u2191\u03bc (s (n + 1))))", "state_after": "no goals"}, {"tactic": "refine' monotone_nat_of_le_succ fun n => _", "annotated_tactic": ["refine' <a>monotone_nat_of_le_succ</a> fun n => _", [{"full_name": "monotone_nat_of_le_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1025, 9], "def_end_pos": [1025, 32]}]], "state_before": "case h.refine'_1\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 Monotone fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))", "state_after": "case h.refine'_1\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nn : \u2115\n\u22a2 \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) \u2264\n    \u2211 x in Finset.range (n + 1), ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))"}, {"tactic": "rw [\u2190 sub_nonneg, Finset.sum_range_succ_sub_sum]", "annotated_tactic": ["rw [\u2190 <a>sub_nonneg</a>, <a>Finset.sum_range_succ_sub_sum</a>]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"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": "case h.refine'_1\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nn : \u2115\n\u22a2 \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) \u2264\n    \u2211 x in Finset.range (n + 1), ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))", "state_after": "case h.refine'_1\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nn : \u2115\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))"}, {"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 h.refine'_1\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nn : \u2115\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))", "state_after": "no goals"}, {"tactic": "rintro \u27e8B, hB\u27e9", "annotated_tactic": ["rintro \u27e8B, hB\u27e9", []], "state_before": "case h.refine'_2\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 \u00acBddAbove (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\n\u22a2 False"}, {"tactic": "refine' not_eventually.2 (frequently_of_forall fun n => _) (htends B.toNNReal)", "annotated_tactic": ["refine' <a>not_eventually</a>.2 (<a>frequently_of_forall</a> fun n => _) (htends B.toNNReal)", [{"full_name": "Filter.not_eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 23]}, {"full_name": "Filter.frequently_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1271, 9], "def_end_pos": [1271, 29]}]], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\n\u22a2 False", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\nn : \u2115\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))"}, {"tactic": "rw [mem_upperBounds] at hB", "annotated_tactic": ["rw [<a>mem_upperBounds</a>] at hB", [{"full_name": "mem_upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 24]}]], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\nn : \u2115\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))"}, {"tactic": "specialize hB (\u2211 k : \u2115 in Finset.range n, \u03bc (s (k + 1))).toReal _", "annotated_tactic": ["specialize hB (\u2211 k : \u2115 in <a>Finset.range</a> n, \u03bc (s (k + 1))).<a>toReal</a> _", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2208\n    Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))\n\ncase h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))"}, {"tactic": "refine' \u27e8n, _\u27e9", "annotated_tactic": ["refine' \u27e8n, _\u27e9", []], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2208\n    Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) n =\n    ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1)))"}, {"tactic": "rw [ENNReal.toReal_sum]", "annotated_tactic": ["rw [<a>ENNReal.toReal_sum</a>]", [{"full_name": "ENNReal.toReal_sum", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1288, 9], "def_end_pos": [1288, 19]}]], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) n =\n    ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1)))", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4"}, {"tactic": "exact fun _ _ => measure_ne_top _ _", "annotated_tactic": ["exact fun _ _ => <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 h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [not_lt, \u2190 ENNReal.toReal_le_toReal (ENNReal.sum_lt_top _).ne ENNReal.coe_ne_top]", "annotated_tactic": ["rw [<a>not_lt</a>, \u2190 <a>ENNReal.toReal_le_toReal</a> (<a>ENNReal.sum_lt_top</a> _).<a>ne</a> <a>ENNReal.coe_ne_top</a>]", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"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.sum_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 19]}, {"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_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))", "state_after": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 ENNReal.toReal (\u2211 a in Finset.range n, \u2191\u2191\u03bc (s (a + 1))) \u2264 ENNReal.toReal \u2191(Real.toNNReal B)\n\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4"}, {"tactic": "exact hB.trans (by simp)", "annotated_tactic": ["exact hB.trans (by simp)", []], "state_before": "case h.refine'_2.intro\n\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 ENNReal.toReal (\u2211 a in Finset.range n, \u2191\u2191\u03bc (s (a + 1))) \u2264 ENNReal.toReal \u2191(Real.toNNReal B)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 B \u2264 ENNReal.toReal \u2191(Real.toNNReal B)", "state_after": "no goals"}, {"tactic": "exact fun _ _ => measure_ne_top _ _", "annotated_tactic": ["exact fun _ _ => <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": "\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\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_pos", "start": [106, 9], "end": [107, 39], "traced_tactics": [{"tactic": "rw [pos_iff_ne_zero, encard_ne_zero]", "annotated_tactic": ["rw [<a>pos_iff_ne_zero</a>, <a>encard_ne_zero</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.encard_ne_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [103, 9], "def_end_pos": [103, 23]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 0 < encard s \u2194 Set.Nonempty s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_three", "start": [97, 1], "end": [104, 22], "traced_tactics": [{"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "n : \u2115\n\u22a2 ack 3 n = 2 ^ (n + 3) - 3", "state_after": "case zero\n\n\u22a2 ack 3 zero = 2 ^ (zero + 3) - 3\n\ncase succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 ack 3 (succ n) = 2 ^ (succ n + 3) - 3"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\n\u22a2 ack 3 zero = 2 ^ (zero + 3) - 3", "state_after": "no goals"}, {"tactic": "rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,\n    Nat.mul_sub_left_distrib, \u2190 Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]", "annotated_tactic": ["rw [<a>ack_succ_succ</a>, IH, <a>ack_two</a>, <a>Nat.succ_add</a>, <a>Nat.pow_succ</a> 2 (n + 3), <a>mul_comm</a> _ 2,\n        <a>Nat.mul_sub_left_distrib</a>, \u2190 <a>Nat.sub_add_comm</a>, <a>two_mul</a> 3, <a>Nat.add_sub_add_right</a>]", [{"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "ack_two", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"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": "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": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Nat.mul_sub_left_distrib", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [716, 19], "def_end_pos": [716, 39]}, {"full_name": "Nat.sub_add_comm", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [411, 19], "def_end_pos": [411, 31]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "Nat.add_sub_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [586, 19], "def_end_pos": [586, 36]}]], "state_before": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 ack 3 (succ n) = 2 ^ (succ n + 3) - 3", "state_after": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3)"}, {"tactic": "have H : 2 * 3 \u2264 2 * 2 ^ 3 := by norm_num", "annotated_tactic": ["have H : 2 * 3 \u2264 2 * 2 ^ 3 := by norm_num", []], "state_before": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3)", "state_after": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 \u2264 2 * 2 ^ 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3)"}, {"tactic": "apply H.trans", "annotated_tactic": ["apply H.trans", []], "state_before": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 \u2264 2 * 2 ^ 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3)", "state_after": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 \u2264 2 * 2 ^ 3\n\u22a2 2 * 2 ^ 3 \u2264 2 * 2 ^ (n + 3)"}, {"tactic": "simp [pow_le_pow]", "annotated_tactic": ["simp [<a>pow_le_pow</a>]", [{"full_name": "pow_le_pow", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [437, 9], "def_end_pos": [437, 19]}]], "state_before": "case succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 \u2264 2 * 2 ^ 3\n\u22a2 2 * 2 ^ 3 \u2264 2 * 2 ^ (n + 3)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "n : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.prod_mk", "start": [190, 1], "end": [193, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "Set.Finite.nullMeasurableSet_sUnion", "start": [381, 1], "end": [383, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurableSet_pi", "start": [976, 1], "end": [980, 75], "traced_tactics": [{"tactic": "cases' (pi s t).eq_empty_or_nonempty with h h", "annotated_tactic": ["cases' (<a>pi</a> s t).<a>eq_empty_or_nonempty</a> with h h", [{"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"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\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\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205", "state_after": "case inl\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.pi s t = \u2205\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205\n\ncase inr\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.Nonempty (Set.pi s t)\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case inl\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.pi s t = \u2205\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205", "state_after": "no goals"}, {"tactic": "simp [measurableSet_pi_of_nonempty hs, h, \u2190 not_nonempty_iff_eq_empty]", "annotated_tactic": ["simp [<a>measurableSet_pi_of_nonempty</a> hs, h, \u2190 <a>not_nonempty_iff_eq_empty</a>]", [{"full_name": "measurableSet_pi_of_nonempty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [966, 9], "def_end_pos": [966, 37]}, {"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 inr\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.Nonempty (Set.pi s t)\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205", "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_eq_zero_iff", "start": [349, 1], "end": [350, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.DominatedFinMeasAdditive.eq_zero", "start": [211, 1], "end": [214, 81], "traced_tactics": [{"tactic": "simp only [Measure.coe_zero, Pi.zero_apply]", "annotated_tactic": ["simp only [<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": "\u03b1 : Type u_1\nE : Type u_2\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2\u271d : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\u271d\nT\u271d T' : Set \u03b1 \u2192 \u03b2\u271d\nC\u271d C' : \u211d\n\u03b2 : Type u_8\ninst\u271d : NormedAddCommGroup \u03b2\nT : Set \u03b1 \u2192 \u03b2\nC : \u211d\nm : MeasurableSpace \u03b1\nhT : DominatedFinMeasAdditive 0 T C\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u21910 s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.totalDegree_add_eq_right_of_totalDegree_lt", "start": [670, 1], "end": [672, 61], "traced_tactics": [{"tactic": "rw [add_comm, totalDegree_add_eq_left_of_totalDegree_lt h]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>totalDegree_add_eq_left_of_totalDegree_lt</a> h]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MvPolynomial.totalDegree_add_eq_left_of_totalDegree_lt", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [647, 9], "def_end_pos": [647, 50]}]], "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 : totalDegree q < totalDegree p\n\u22a2 totalDegree (q + p) = totalDegree 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.swapRight_apply'", "start": [751, 1], "end": [753, 66], "traced_tactics": [{"tactic": "rw [swapRight_apply, Measure.map_apply measurable_swap hs]", "annotated_tactic": ["rw [<a>swapRight_apply</a>, <a>Measure.map_apply</a> <a>measurable_swap</a> hs]", [{"full_name": "ProbabilityTheory.kernel.swapRight_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [747, 9], "def_end_pos": [747, 24]}, {"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_swap", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [760, 9], "def_end_pos": [760, 24]}]], "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 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(swapRight \u03ba) a) s = \u2191\u2191(\u2191\u03ba a) {p | Prod.swap p \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 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.swap \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | Prod.swap p \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 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.swap \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | Prod.swap p \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousLinearMap.norm_compLp_le", "start": [1146, 1], "end": [1147, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.mem_uIcc'", "start": [1047, 1], "end": [1047, 92], "traced_tactics": [{"tactic": "simp [uIcc_eq_union]", "annotated_tactic": ["simp [<a>uIcc_eq_union</a>]", [{"full_name": "Finset.uIcc_eq_union", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1041, 9], "def_end_pos": [1041, 22]}]], "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 a \u2208 [[b, c]] \u2194 b \u2264 a \u2227 a \u2264 c \u2228 c \u2264 a \u2227 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.PointedMap.map_pt", "start": [368, 1], "end": [370, 23], "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_gt_nnreal", "start": [270, 1], "end": [313, 78], "traced_tactics": [{"tactic": "have fmeas : AEMeasurable f \u03bc := by\n  convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n  simp only [Real.toNNReal_coe]", "annotated_tactic": ["have fmeas : <a>AEMeasurable</a> f \u03bc := by\n    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n    simp only [<a>Real.toNNReal_coe</a>]", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 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\u22650\nfint : Integrable fun x => \u2191(f x)\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        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(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\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nfmeas : AEMeasurable f\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "lift \u03b5 to \u211d\u22650 using \u03b5pos.le", "annotated_tactic": ["lift \u03b5 to \u211d\u22650 using \u03b5pos.le", []], "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\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nfmeas : AEMeasurable f\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5", []], "state_before": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5\n\ncase 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "exact exists_between \u03b5pos", "annotated_tactic": ["exact <a>exists_between</a> \u03b5pos", [{"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\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5\n\ncase 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e :=\n  (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne", "annotated_tactic": ["have int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e :=\n    (<a>hasFiniteIntegral_iff_ofNNReal</a>.1 fint.hasFiniteIntegral).<a>ne</a>", [{"full_name": "MeasureTheory.hasFiniteIntegral_iff_ofNNReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [132, 9], "def_end_pos": [132, 39]}, {"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\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable \u03bc f fmeas\n    (ENNReal.coe_ne_zero.2 \u03b4pos.ne') with\n  \u27e8g, f_lt_g, gcont, gint\u27e9", "annotated_tactic": ["rcases <a>exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable</a> \u03bc f fmeas\n      (<a>ENNReal.coe_ne_zero</a>.2 \u03b4pos.ne') with\n    \u27e8g, f_lt_g, gcont, gint\u27e9", [{"full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [231, 9], "def_end_pos": [231, 67]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "state_before": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := ne_top_of_le_ne_top (by simpa) gint", "annotated_tactic": ["have gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := <a>ne_top_of_le_ne_top</a> (by simpa) gint", [{"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 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := ae_lt_top gcont.measurable gint_ne", "annotated_tactic": ["have g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := <a>ae_lt_top</a> gcont.measurable gint_ne", [{"full_name": "MeasureTheory.ae_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 18]}]], "state_before": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have Ig : (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) = \u222b\u207b a : \u03b1, g a \u2202\u03bc := by\n  apply lintegral_congr_ae\n  filter_upwards [g_lt_top] with _ hx\n  simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]", "annotated_tactic": ["have Ig : (\u222b\u207b a : \u03b1, <a>ENNReal.ofReal</a> (g a).<a>toReal</a> \u2202\u03bc) = \u222b\u207b a : \u03b1, g a \u2202\u03bc := by\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [g_lt_top] with _ hx\n    simp only [hx.ne, <a>ENNReal.ofReal_toReal</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"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 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "refine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9", "annotated_tactic": ["refine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9", []], "state_before": "case 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 Integrable fun x => ENNReal.toReal (g x)\n\ncase intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable", "annotated_tactic": ["convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable", []], "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\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 AEMeasurable f", "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\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nx\u271d : \u03b1\n\u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d)"}, {"tactic": "simp only [Real.toNNReal_coe]", "annotated_tactic": ["simp only [<a>Real.toNNReal_coe</a>]", [{"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 33]}]], "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\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nx\u271d : \u03b1\n\u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d)", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4", "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\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a"}, {"tactic": "filter_upwards [g_lt_top] with _ hx", "annotated_tactic": ["filter_upwards [g_lt_top] with _ hx", []], "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a", "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\na\u271d : \u03b1\nhx : g a\u271d < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d"}, {"tactic": "simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [hx.ne, <a>ENNReal.ofReal_toReal</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"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\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\na\u271d : \u03b1\nhx : g a\u271d < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d", "state_after": "no goals"}, {"tactic": "refine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9", []], "state_before": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 Integrable fun x => ENNReal.toReal (g x)", "state_after": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x)"}, {"tactic": "simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]", "annotated_tactic": ["simp only [<a>hasFiniteIntegral_iff_norm</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "MeasureTheory.hasFiniteIntegral_iff_norm", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [117, 9], "def_end_pos": [117, 35]}, {"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": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x)", "state_after": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4"}, {"tactic": "convert gint_ne.lt_top using 1", "annotated_tactic": ["convert gint_ne.lt_top using 1", []], "state_before": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"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": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) <\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.hfm\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.toReal (g x)\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.hfm\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc"}, {"tactic": "calc\n  ENNReal.toReal (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) =\n      ENNReal.toReal (\u222b\u207b a : \u03b1, g a \u2202\u03bc) :=\n    by congr 1\n  _ \u2264 ENNReal.toReal ((\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4) := by\n    apply ENNReal.toReal_mono _ gint\n    simpa using int_f_ne_top\n  _ = ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4 := by\n    rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]\n  _ < ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b5 := (add_lt_add_left h\u03b4\u03b5 _)\n  _ = (\u222b\u207b a : \u03b1, ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal + \u03b5 := by simp", "annotated_tactic": ["calc\n        <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, <a>ENNReal.ofReal</a> (g a).<a>toReal</a> \u2202\u03bc) =\n            <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, g a \u2202\u03bc) :=\n          by congr 1\n        _ \u2264 <a>ENNReal.toReal</a> ((\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4) := by\n          apply <a>ENNReal.toReal_mono</a> _ gint\n          simpa using int_f_ne_top\n        _ = <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4 := by\n          rw [<a>ENNReal.toReal_add</a> int_f_ne_top <a>ENNReal.coe_ne_top</a>, <a>ENNReal.coe_toReal</a>]\n        _ < <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b5 := (<a>add_lt_add_left</a> h\u03b4\u03b5 _)\n        _ = (\u222b\u207b a : \u03b1, <a>ENNReal.ofReal</a> \u2191(f a) \u2202\u03bc).<a>toReal</a> + \u03b5 := by simp", [{"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", "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": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 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": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "add_lt_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [120, 15], "def_end_pos": [120, 30]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "case intro.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) <\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5", "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "apply ENNReal.toReal_mono _ gint", "annotated_tactic": ["apply <a>ENNReal.toReal_mono</a> _ gint", [{"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}]], "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4"}, {"tactic": "simpa using int_f_ne_top", "annotated_tactic": ["simpa using int_f_ne_top", []], "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]", "annotated_tactic": ["rw [<a>ENNReal.toReal_add</a> int_f_ne_top <a>ENNReal.coe_ne_top</a>, <a>ENNReal.coe_toReal</a>]", [{"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 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": "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b4", "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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "apply Filter.eventually_of_forall fun x => _", "annotated_tactic": ["apply <a>Filter.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 intro.intro.intro.intro.intro.intro.refine'_2.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f 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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x"}, {"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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x", "state_after": "no goals"}, {"tactic": "exact fmeas.coe_nnreal_real.aestronglyMeasurable", "annotated_tactic": ["exact fmeas.coe_nnreal_real.aestronglyMeasurable", []], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.hfm\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc", "state_after": "no goals"}, {"tactic": "apply Filter.eventually_of_forall fun x => _", "annotated_tactic": ["apply <a>Filter.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 intro.intro.intro.intro.intro.intro.refine'_2.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x"}, {"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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x", "state_after": "no goals"}, {"tactic": "apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable", "annotated_tactic": ["apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable", []], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.hfm\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\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.natDegree_finSuccEquiv", "start": [505, 1], "end": [511, 9], "traced_tactics": [{"tactic": "by_cases c : f = 0", "annotated_tactic": ["by_cases c : f = 0", []], "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 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f", "state_after": "case pos\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\nc : f = 0\n\u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f\n\ncase neg\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\nc : \u00acf = 0\n\u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f"}, {"tactic": "rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]", "annotated_tactic": ["rw [c, (<a>finSuccEquiv</a> R n).<a>map_zero</a>, <a>Polynomial.natDegree_zero</a>, <a>degreeOf_zero</a>]", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "AlgEquiv.map_zero", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}, {"full_name": "Polynomial.natDegree_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [106, 9], "def_end_pos": [106, 23]}, {"full_name": "MvPolynomial.degreeOf_zero", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [509, 9], "def_end_pos": [509, 22]}]], "state_before": "case pos\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\nc : f = 0\n\u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f", "state_after": "no goals"}, {"tactic": "rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]", "annotated_tactic": ["rw [<a>Polynomial.natDegree</a>, <a>degree_finSuccEquiv</a> (by simpa only [<a>Ne.def</a>] )]", [{"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}, {"full_name": "MvPolynomial.degree_finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [492, 9], "def_end_pos": [492, 28]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "case neg\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\nc : \u00acf = 0\n\u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f", "state_after": "case neg\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\nc : \u00acf = 0\n\u22a2 WithBot.unbot' 0 \u2191(degreeOf 0 f) = degreeOf 0 f"}, {"tactic": "erw [WithBot.unbot'_coe]", "annotated_tactic": ["erw [<a>WithBot.unbot'_coe</a>]", [{"full_name": "WithBot.unbot'_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [126, 9], "def_end_pos": [126, 19]}]], "state_before": "case neg\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\nc : \u00acf = 0\n\u22a2 WithBot.unbot' 0 \u2191(degreeOf 0 f) = degreeOf 0 f", "state_after": "case neg\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\nc : \u00acf = 0\n\u22a2 \u2191(degreeOf 0 f) = degreeOf 0 f"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg\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\nc : \u00acf = 0\n\u22a2 \u2191(degreeOf 0 f) = degreeOf 0 f", "state_after": "no goals"}, {"tactic": "simpa only [Ne.def]", "annotated_tactic": ["simpa only [<a>Ne.def</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "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\nc : \u00acf = 0\n\u22a2 f \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.norm_indicatorConstLp_top", "start": [767, 1], "end": [771, 64], "traced_tactics": [{"tactic": "rw [Lp.norm_def, snorm_congr_ae indicatorConstLp_coeFn,\n  snorm_indicator_const' hs h\u03bcs_ne_zero ENNReal.top_ne_zero, ENNReal.top_toReal, _root_.div_zero,\n  ENNReal.rpow_zero, mul_one, ENNReal.coe_toReal, coe_nnnorm]", "annotated_tactic": ["rw [<a>Lp.norm_def</a>, <a>snorm_congr_ae</a> <a>indicatorConstLp_coeFn</a>,\n    <a>snorm_indicator_const'</a> hs h\u03bcs_ne_zero <a>ENNReal.top_ne_zero</a>, <a>ENNReal.top_toReal</a>, <a>_root_.div_zero</a>,\n    <a>ENNReal.rpow_zero</a>, <a>mul_one</a>, <a>ENNReal.coe_toReal</a>, <a>coe_nnnorm</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.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.indicatorConstLp_coeFn", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [748, 9], "def_end_pos": [748, 31]}, {"full_name": "MeasureTheory.snorm_indicator_const'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [615, 9], "def_end_pos": [615, 31]}, {"full_name": "ENNReal.top_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [337, 17], "def_end_pos": [337, 28]}, {"full_name": "ENNReal.top_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [227, 17], "def_end_pos": [227, 27]}, {"full_name": "div_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "ENNReal.rpow_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [381, 9], "def_end_pos": [381, 18]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"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": "\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\nh\u03bcs_ne_zero : \u2191\u2191\u03bc s \u2260 0\n\u22a2 \u2016indicatorConstLp \u22a4 hs h\u03bcs c\u2016 = \u2016c\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Partrec.merge'", "start": [74, 1], "end": [98, 18], "traced_tactics": [{"tactic": "let \u27e8k, hk, H\u27e9 := Nat.Partrec.merge' (bind_decode\u2082_iff.1 hf) (bind_decode\u2082_iff.1 hg)", "annotated_tactic": ["let \u27e8k, hk, H\u27e9 := <a>Nat.Partrec.merge'</a> (<a>bind_decode\u2082_iff</a>.1 hf) (<a>bind_decode\u2082_iff</a>.1 hg)", [{"full_name": "Nat.Partrec.merge'", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [27, 9], "def_end_pos": [27, 15]}, {"full_name": "Partrec.bind_decode\u2082_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [595, 9], "def_end_pos": [595, 25]}, {"full_name": "Partrec.bind_decode\u2082_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [595, 9], "def_end_pos": [595, 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\n\u22a2 \u2203 k, Partrec k \u2227 \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)", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\n\u22a2 \u2203 k, Partrec k \u2227 \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)"}, {"tactic": "let k' (a : \u03b1) := (k (encode a)).bind fun n => (decode (\u03b1 := \u03c3) n : Part \u03c3)", "annotated_tactic": ["let k' (a : \u03b1) := (k (<a>encode</a> a)).<a>bind</a> fun n => (<a>decode</a> (\u03b1 := \u03c3) n : <a>Part</a> \u03c3)", [{"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "Part.bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [428, 15], "def_end_pos": [428, 19]}, {"full_name": "Encodable.decode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [51, 3], "def_end_pos": [51, 9]}, {"full_name": "Part", "def_path": "Mathlib/Data/Part.lean", "def_pos": [52, 11], "def_end_pos": [52, 15]}]], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\n\u22a2 \u2203 k, Partrec k \u2227 \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)", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\n\u22a2 \u2203 k, Partrec k \u2227 \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)"}, {"tactic": "refine'\n  \u27e8k', ((nat_iff.2 hk).comp Computable.encode).bind (Computable.decode.ofOption.comp snd).to\u2082,\n    fun a => _\u27e9", "annotated_tactic": ["refine'\n    \u27e8k', ((<a>nat_iff</a>.2 hk).<a>comp</a> <a>Computable.encode</a>).<a>bind</a> (Computable.decode.ofOption.comp <a>snd</a>).<a>to\u2082</a>,\n      fun a => _\u27e9", [{"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]}, {"full_name": "Partrec.bind", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [455, 19], "def_end_pos": [455, 23]}, {"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]}]], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\n\u22a2 \u2203 k, Partrec k \u2227 \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)", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 (\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)"}, {"tactic": "suffices", "annotated_tactic": ["suffices", []], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 (\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)", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : ?m.108648\n\u22a2 (\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)\n\ncase this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 ?m.108648"}, {"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": "\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : ?m.108648\n\u22a2 (\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)\n\ncase this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 ?m.108648", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\n\u22a2 (f a).Dom \u2228 (g a).Dom \u2192 (k' a).Dom\n\ncase this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "intro x h'", "annotated_tactic": ["intro x h'", []], "state_before": "case this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a", "state_after": "case this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nh' : x \u2208 k' a\n\u22a2 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "simp only [exists_prop, mem_coe, mem_bind_iff, Option.mem_def] at h'", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_coe</a>, <a>mem_bind_iff</a>, <a>Option.mem_def</a>] at h'", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Part.mem_coe", "def_path": "Mathlib/Data/Part.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}, {"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\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nh' : x \u2208 k' a\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "case this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nh' : \u2203 a_1, a_1 \u2208 k (encode a) \u2227 decode a_1 = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "obtain \u27e8n, hn, hx\u27e9 := h'", "annotated_tactic": ["obtain \u27e8n, hn, hx\u27e9 := h'", []], "state_before": "case this\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nh' : \u2203 a_1, a_1 \u2208 k (encode a) \u2227 decode a_1 = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "case this.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "have := (H _).1 _ hn", "annotated_tactic": ["have := (H _).1 _ hn", []], "state_before": "case this.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "case this.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\nthis :\n  (n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)) \u2228\n    n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a)\n\u22a2 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "simp [mem_decode\u2082, encode_injective.eq_iff] at this", "annotated_tactic": ["simp [<a>mem_decode\u2082</a>, encode_injective.eq_iff] at this", [{"full_name": "Encodable.mem_decode\u2082", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 20]}]], "state_before": "case this.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\nthis :\n  (n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)) \u2228\n    n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a)\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "case this.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\nthis : (\u2203 a_1, a_1 \u2208 f a \u2227 encode a_1 = n) \u2228 \u2203 a_1, a_1 \u2208 g a \u2227 encode a_1 = n\n\u22a2 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "obtain \u27e8a', ha, rfl\u27e9 | \u27e8a', ha, rfl\u27e9 := this <;> simp only [encodek, Option.some_inj] at hx <;>\n  rw [hx] at ha", "annotated_tactic": ["obtain \u27e8a', ha, rfl\u27e9 | \u27e8a', ha, rfl\u27e9 := this <;> simp only [<a>encodek</a>, <a>Option.some_inj</a>] at hx <;>\n    rw [hx] at ha", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}, {"full_name": "Option.some_inj", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 17]}]], "state_before": "case this.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\nthis : (\u2203 a_1, a_1 \u2208 f a \u2227 encode a_1 = n) \u2228 \u2203 a_1, a_1 \u2208 g a \u2227 encode a_1 = n\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "case this.intro.intro.inl.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : x \u2208 f a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n\ncase this.intro.intro.inr.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : x \u2208 g a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a"}, {"tactic": "exact Or.inr ha", "annotated_tactic": ["exact <a>Or.inr</a> ha", [{"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 this.intro.intro.inr.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : x \u2208 g a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\n\u22a2 (f a).Dom \u2228 (g a).Dom \u2192 (k' a).Dom", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 (k' a).Dom"}, {"tactic": "rw [bind_dom]", "annotated_tactic": ["rw [<a>bind_dom</a>]", [{"full_name": "Part.bind_dom", "def_path": "Mathlib/Data/Part.lean", "def_pos": [665, 9], "def_end_pos": [665, 17]}]], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 (k' a).Dom", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom"}, {"tactic": "have hk : (k (encode a)).Dom :=\n  (H _).2.2 (by simpa only [encodek\u2082, bind_some, coe_some] using h)", "annotated_tactic": ["have hk : (k (<a>encode</a> a)).<a>Dom</a> :=\n      (H _).2.2 (by simpa only [<a>encodek\u2082</a>, <a>bind_some</a>, <a>coe_some</a>] using h)", [{"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "Part.Dom", "def_path": "Mathlib/Data/Part.lean", "def_pos": [54, 3], "def_end_pos": [54, 6]}, {"full_name": "Encodable.encodek\u2082", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 17]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "Part.coe_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [361, 9], "def_end_pos": [361, 17]}]], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\n\u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom"}, {"tactic": "exists hk", "annotated_tactic": ["exists hk", []], "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\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\n\u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom"}, {"tactic": "simp only [exists_prop, mem_map_iff, mem_coe, mem_bind_iff, Option.mem_def] at H", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_map_iff</a>, <a>mem_coe</a>, <a>mem_bind_iff</a>, <a>Option.mem_def</a>] at H", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Part.mem_map_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "Part.mem_coe", "def_path": "Mathlib/Data/Part.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}, {"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": "\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\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom", "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\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (\u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 f a_2 \u2227 encode a = x) \u2228\n            \u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 g a_2 \u2227 encode a = x) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom"}, {"tactic": "obtain \u27e8a', _, y, _, e\u27e9 | \u27e8a', _, y, _, e\u27e9 := (H _).1 _ \u27e8hk, rfl\u27e9 <;>\n  simp only [e.symm, encodek, coe_some, some_dom]", "annotated_tactic": ["obtain \u27e8a', _, y, _, e\u27e9 | \u27e8a', _, y, _, e\u27e9 := (H _).1 _ \u27e8hk, <a>rfl</a>\u27e9 <;>\n      simp only [e.symm, <a>encodek</a>, <a>coe_some</a>, <a>some_dom</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": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}, {"full_name": "Part.coe_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [361, 9], "def_end_pos": [361, 17]}, {"full_name": "Part.some_dom", "def_path": "Mathlib/Data/Part.lean", "def_pos": [140, 9], "def_end_pos": [140, 17]}]], "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\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (\u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 f a_2 \u2227 encode a = x) \u2228\n            \u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 g a_2 \u2227 encode a = x) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom", "state_after": "no goals"}, {"tactic": "simpa only [encodek\u2082, bind_some, coe_some] using h", "annotated_tactic": ["simpa only [<a>encodek\u2082</a>, <a>bind_some</a>, <a>coe_some</a>] using h", [{"full_name": "Encodable.encodek\u2082", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 17]}, {"full_name": "Part.bind_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [518, 9], "def_end_pos": [518, 18]}, {"full_name": "Part.coe_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [361, 9], "def_end_pos": [361, 17]}]], "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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 (Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)).Dom \u2228\n    (Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a)).Dom", "state_after": "no goals"}, {"tactic": "exact Or.inl ha", "annotated_tactic": ["exact <a>Or.inl</a> ha", [{"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 this.intro.intro.inl.intro.intro\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\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n  \u2200 (a : \u2115),\n    (\u2200 (x : \u2115),\n        x \u2208 k a \u2192\n          (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n            x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n      ((k a).Dom \u2194\n        (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n          (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : x \u2208 f a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.product_biUnion", "start": [133, 1], "end": [135, 73], "traced_tactics": [{"tactic": "classical simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion]", "annotated_tactic": ["classical simp_rw [<a>product_eq_biUnion</a>, <a>biUnion_biUnion</a>, <a>image_biUnion</a>]", [{"full_name": "Finset.product_eq_biUnion", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [117, 9], "def_end_pos": [117, 27]}, {"full_name": "Finset.biUnion_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3666, 9], "def_end_pos": [3666, 24]}, {"full_name": "Finset.image_biUnion", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [639, 9], "def_end_pos": [639, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u00d7 \u03b2 \u2192 Finset \u03b3\n\u22a2 Finset.biUnion (s \u00d7\u02e2 t) f = Finset.biUnion s fun a => Finset.biUnion t fun b => f (a, b)", "state_after": "no goals"}, {"tactic": "simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion]", "annotated_tactic": ["simp_rw [<a>product_eq_biUnion</a>, <a>biUnion_biUnion</a>, <a>image_biUnion</a>]", [{"full_name": "Finset.product_eq_biUnion", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [117, 9], "def_end_pos": [117, 27]}, {"full_name": "Finset.biUnion_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3666, 9], "def_end_pos": [3666, 24]}, {"full_name": "Finset.image_biUnion", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [639, 9], "def_end_pos": [639, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s' : Finset \u03b1\nt\u271d t' : Finset \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u00d7 \u03b2 \u2192 Finset \u03b3\n\u22a2 Finset.biUnion (s \u00d7\u02e2 t) f = Finset.biUnion s fun a => Finset.biUnion t fun b => f (a, b)", "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_le_of_uIcc_subset_uIcc", "start": [478, 1], "end": [481, 27], "traced_tactics": [{"tactic": "rw [\u2190 max_sub_min_eq_abs, \u2190 max_sub_min_eq_abs]", "annotated_tactic": ["rw [\u2190 <a>max_sub_min_eq_abs</a>, \u2190 <a>max_sub_min_eq_abs</a>]", [{"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": "max_sub_min_eq_abs", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [216, 9], "def_end_pos": [216, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nh : [[c, d]] \u2286 [[a, b]]\n\u22a2 |d - c| \u2264 |b - a|", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nh : [[c, d]] \u2286 [[a, b]]\n\u22a2 max c d - min c d \u2264 max a b - min a b"}, {"tactic": "rw [uIcc_subset_uIcc_iff_le] at h", "annotated_tactic": ["rw [<a>uIcc_subset_uIcc_iff_le</a>] at h", [{"full_name": "Set.uIcc_subset_uIcc_iff_le", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [243, 7], "def_end_pos": [243, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nh : [[c, d]] \u2286 [[a, b]]\n\u22a2 max c d - min c d \u2264 max a b - min a b", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nh : min a b \u2264 min c d \u2227 max c d \u2264 max a b\n\u22a2 max c d - min c d \u2264 max a b - min a b"}, {"tactic": "exact sub_le_sub h.2 h.1", "annotated_tactic": ["exact <a>sub_le_sub</a> h.2 h.1", [{"full_name": "sub_le_sub", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [850, 15], "def_end_pos": [850, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nh : min a b \u2264 min c d \u2227 max c d \u2264 max a b\n\u22a2 max c d - min c d \u2264 max a b - min a b", "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_sphere", "start": [523, 1], "end": [526, 42], "traced_tactics": [{"tactic": "rcases eq_or_ne r 0 with (rfl | h)", "annotated_tactic": ["rcases <a>eq_or_ne</a> r 0 with (rfl | h)", [{"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\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\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0", "state_after": "case inl\nE : 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\n\u22a2 \u2191\u2191\u03bc (sphere x 0) = 0\n\ncase inr\nE : 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\nh : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0"}, {"tactic": "rw [sphere_zero, measure_singleton]", "annotated_tactic": ["rw [<a>sphere_zero</a>, <a>measure_singleton</a>]", [{"full_name": "Metric.sphere_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2971, 17], "def_end_pos": [2971, 28]}, {"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 inl\nE : 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\n\u22a2 \u2191\u2191\u03bc (sphere x 0) = 0", "state_after": "no goals"}, {"tactic": "exact addHaar_sphere_of_ne_zero \u03bc x h", "annotated_tactic": ["exact <a>addHaar_sphere_of_ne_zero</a> \u03bc x h", [{"full_name": "MeasureTheory.Measure.addHaar_sphere_of_ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [515, 9], "def_end_pos": [515, 34]}]], "state_before": "case inr\nE : 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\nh : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.empty_pow", "start": [926, 1], "end": [927, 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": "Finset.empty_mul", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [352, 9], "def_end_pos": [352, 18]}]], "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\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/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_smul_left'", "start": [746, 1], "end": [749, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.monotone_iff", "start": [154, 1], "end": [158, 8], "traced_tactics": [{"tactic": "classical\nsimp only [monotone_iff_forall_covby, covby_iff, forall_exists_index, and_imp]\naesop", "annotated_tactic": ["classical\n  simp only [<a>monotone_iff_forall_covby</a>, <a>covby_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]\n  aesop", [{"full_name": "monotone_iff_forall_covby", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1278, 7], "def_end_pos": [1278, 32]}, {"full_name": "Finset.covby_iff", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [144, 7], "def_end_pos": [144, 16]}, {"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": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 Monotone f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi)", "state_after": "no goals"}, {"tactic": "simp only [monotone_iff_forall_covby, covby_iff, forall_exists_index, and_imp]", "annotated_tactic": ["simp only [<a>monotone_iff_forall_covby</a>, <a>covby_iff</a>, <a>forall_exists_index</a>, <a>and_imp</a>]", [{"full_name": "monotone_iff_forall_covby", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1278, 7], "def_end_pos": [1278, 32]}, {"full_name": "Finset.covby_iff", "def_path": "Mathlib/Data/Finset/Interval.lean", "def_pos": [144, 7], "def_end_pos": [144, 16]}, {"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": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 Monotone f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi)", "state_after": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (a b : Finset \u03b1) (x : \u03b1) (x_1 : \u00acx \u2208 a), b = cons x a x_1 \u2192 f a \u2264 f b) \u2194\n    \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi)"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\ns t : Finset \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf : Finset \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (a b : Finset \u03b1) (x : \u03b1) (x_1 : \u00acx \u2208 a), b = cons x a x_1 \u2192 f a \u2264 f b) \u2194\n    \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_le_of_forall_fin_meas_le'", "start": [1683, 1], "end": [1693, 88], "traced_tactics": [{"tactic": "let f' := hf_meas.mk f", "annotated_tactic": ["let f' := hf_meas.mk f", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C"}, {"tactic": "have hf' : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s \u2260 \u221e \u2192 \u222b\u207b x in s, f' x \u2202\u03bc \u2264 C := by\n  refine' fun s hs h\u03bcs => (le_of_eq _).trans (hf s hs h\u03bcs)\n  refine' lintegral_congr_ae (ae_restrict_of_ae (hf_meas.ae_eq_mk.mono fun x hx => _))\n  dsimp only\n  rw [hx]", "annotated_tactic": ["have hf' : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s \u2260 \u221e \u2192 \u222b\u207b x in s, f' x \u2202\u03bc \u2264 C := by\n    refine' fun s hs h\u03bcs => (<a>le_of_eq</a> _).<a>trans</a> (hf s hs h\u03bcs)\n    refine' <a>lintegral_congr_ae</a> (<a>ae_restrict_of_ae</a> (hf_meas.ae_eq_mk.mono fun x hx => _))\n    dsimp only\n    rw [hx]", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"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.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\nhf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C"}, {"tactic": "rw [lintegral_congr_ae hf_meas.ae_eq_mk]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> hf_meas.ae_eq_mk]", [{"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\nhf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\nhf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (a : \u03b1), AEMeasurable.mk f hf_meas a \u2202\u03bc \u2264 C"}, {"tactic": "exact lintegral_le_of_forall_fin_meas_le_of_measurable hm C hf_meas.measurable_mk hf'", "annotated_tactic": ["exact <a>lintegral_le_of_forall_fin_meas_le_of_measurable</a> hm C hf_meas.measurable_mk hf'", [{"full_name": "MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1643, 9], "def_end_pos": [1643, 57]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\nhf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (a : \u03b1), AEMeasurable.mk f hf_meas a \u2202\u03bc \u2264 C", "state_after": "no goals"}, {"tactic": "refine' fun s hs h\u03bcs => (le_of_eq _).trans (hf s hs h\u03bcs)", "annotated_tactic": ["refine' fun s hs h\u03bcs => (<a>le_of_eq</a> _).<a>trans</a> (hf s hs h\u03bcs)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "refine' lintegral_congr_ae (ae_restrict_of_ae (hf_meas.ae_eq_mk.mono fun x hx => _))", "annotated_tactic": ["refine' <a>lintegral_congr_ae</a> (<a>ae_restrict_of_ae</a> (hf_meas.ae_eq_mk.mono fun x hx => _))", [{"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.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2596, 9], "def_end_pos": [2596, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, 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 m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf_meas x\n\u22a2 (fun x => f' x) x = (fun x => f x) x"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf_meas x\n\u22a2 (fun x => f' x) x = (fun x => f x) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf_meas x\n\u22a2 AEMeasurable.mk f hf_meas x = f x"}, {"tactic": "rw [hx]", "annotated_tactic": ["rw [hx]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf_meas x\n\u22a2 AEMeasurable.mk f hf_meas x = f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean", "full_name": "ENNReal.aemeasurable_of_exist_almost_disjoint_supersets", "start": [113, 1], "end": [127, 41], "traced_tactics": [{"tactic": "obtain \u27e8s, s_count, s_dense, _, s_top\u27e9 :\n  \u2203 s : Set \u211d\u22650\u221e, s.Countable \u2227 Dense s \u2227 0 \u2209 s \u2227 \u221e \u2209 s :=\n  ENNReal.exists_countable_dense_no_zero_top", "annotated_tactic": ["obtain \u27e8s, s_count, s_dense, _, s_top\u27e9 :\n    \u2203 s : <a>Set</a> \u211d\u22650\u221e, s.Countable \u2227 <a>Dense</a> s \u2227 0 \u2209 s \u2227 \u221e \u2209 s :=\n    <a>ENNReal.exists_countable_dense_no_zero_top</a>", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Dense", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [616, 5], "def_end_pos": [616, 10]}, {"full_name": "ENNReal.exists_countable_dense_no_zero_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [682, 9], "def_end_pos": [682, 43]}]], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n\u22a2 AEMeasurable f", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\n\u22a2 AEMeasurable f"}, {"tactic": "have I : \u2200 x \u2208 s, x \u2260 \u221e := fun x xs hx => s_top (hx \u25b8 xs)", "annotated_tactic": ["have I : \u2200 x \u2208 s, x \u2260 \u221e := fun x xs hx => s_top (hx \u25b8 xs)", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\n\u22a2 AEMeasurable f", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\n\u22a2 AEMeasurable f"}, {"tactic": "apply MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets \u03bc s s_count s_dense _", "annotated_tactic": ["apply <a>MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets</a> \u03bc s s_count s_dense _", [{"full_name": "MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 70]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\n\u22a2 AEMeasurable f", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\n\u22a2 \u2200 (p : \u211d\u22650\u221e),\n    p \u2208 s \u2192\n      \u2200 (q : \u211d\u22650\u221e),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0"}, {"tactic": "rintro p hp q hq hpq", "annotated_tactic": ["rintro p hp q hq hpq", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\n\u22a2 \u2200 (p : \u211d\u22650\u221e),\n    p \u2208 s \u2192\n      \u2200 (q : \u211d\u22650\u221e),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\np : \u211d\u22650\u221e\nhp : p \u2208 s\nq : \u211d\u22650\u221e\nhq : q \u2208 s\nhpq : p < q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0"}, {"tactic": "lift p to \u211d\u22650 using I p hp", "annotated_tactic": ["lift p to \u211d\u22650 using I p hp", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\np : \u211d\u22650\u221e\nhp : p \u2208 s\nq : \u211d\u22650\u221e\nhq : q \u2208 s\nhpq : p < q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\nq : \u211d\u22650\u221e\nhq : q \u2208 s\np : \u211d\u22650\nhp : \u2191p \u2208 s\nhpq : \u2191p < q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0"}, {"tactic": "lift q to \u211d\u22650 using I q hq", "annotated_tactic": ["lift q to \u211d\u22650 using I q hq", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\nq : \u211d\u22650\u221e\nhq : q \u2208 s\np : \u211d\u22650\nhp : \u2191p \u2208 s\nhpq : \u2191p < q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\np : \u211d\u22650\nhp : \u2191p \u2208 s\nq : \u211d\u22650\nhq : \u2191q \u2208 s\nhpq : \u2191p < \u2191q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0"}, {"tactic": "exact h p q (ENNReal.coe_lt_coe.1 hpq)", "annotated_tactic": ["exact h p q (<a>ENNReal.coe_lt_coe</a>.1 hpq)", [{"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.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\np : \u211d\u22650\nhp : \u2191p \u2208 s\nq : \u211d\u22650\nhq : \u2191q \u2208 s\nhpq : \u2191p < \u2191q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.mem_powerset", "start": [34, 1], "end": [37, 18], "traced_tactics": [{"tactic": "cases s", "annotated_tactic": ["cases s", []], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\n\u22a2 s \u2208 powerset t \u2194 s \u2286 t", "state_after": "case mk\n\u03b1 : Type u_1\ns t\u271d t : Finset \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\n\u22a2 { val := val\u271d, nodup := nodup\u271d } \u2208 powerset t \u2194 { val := val\u271d, nodup := nodup\u271d } \u2286 t"}, {"tactic": "simp [powerset, mem_mk, mem_pmap, mk.injEq, mem_powerset, exists_prop, exists_eq_right,\n  \u2190 val_le_iff]", "annotated_tactic": ["simp [<a>powerset</a>, <a>mem_mk</a>, <a>mem_pmap</a>, mk.injEq, mem_powerset, <a>exists_prop</a>, <a>exists_eq_right</a>,\n    \u2190 <a>val_le_iff</a>]", [{"full_name": "Finset.powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.mem_mk", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 15]}, {"full_name": "Multiset.mem_pmap", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 17]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "exists_eq_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [462, 17], "def_end_pos": [462, 32]}, {"full_name": "Finset.val_le_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 19]}]], "state_before": "case mk\n\u03b1 : Type u_1\ns t\u271d t : Finset \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\n\u22a2 { val := val\u271d, nodup := nodup\u271d } \u2208 powerset t \u2194 { val := val\u271d, nodup := nodup\u271d } \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.leastGE_mono", "start": [67, 1], "end": [68, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_deriv_comp_smul_deriv'", "start": [1434, 1], "end": [1442, 43], "traced_tactics": [{"tactic": "rw [integral_comp_smul_deriv'' hf hff' hf' hg',\n  integral_eq_sub_of_hasDeriv_right hg hgg' (hg'.mono _).intervalIntegrable]", "annotated_tactic": ["rw [<a>integral_comp_smul_deriv''</a> hf hff' hf' hg',\n    <a>integral_eq_sub_of_hasDeriv_right</a> hg hgg' (hg'.mono _).<a>intervalIntegrable</a>]", [{"full_name": "intervalIntegral.integral_comp_smul_deriv''", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 35]}, {"full_name": "intervalIntegral.integral_eq_sub_of_hasDeriv_right", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 42]}, {"full_name": "ContinuousOn.intervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [354, 9], "def_end_pos": [354, 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\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' : \u211d \u2192 \u211d\ng g' : \u211d \u2192 E\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, f' x \u2022 (g' \u2218 f) x = (g \u2218 f) b - (g \u2218 f) a", "state_after": "\u03b9 : Type 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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' : \u211d \u2192 \u211d\ng g' : \u211d \u2192 E\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 [[f a, f b]] \u2286 f '' [[a, b]]"}, {"tactic": "exacts [rfl, intermediate_value_uIcc hf]", "annotated_tactic": ["exacts [<a>rfl</a>, <a>intermediate_value_uIcc</a> hf]", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "intermediate_value_uIcc", "def_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "def_pos": [541, 9], "def_end_pos": [541, 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\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' : \u211d \u2192 \u211d\ng g' : \u211d \u2192 E\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 g (f b) - g (f a) = (g \u2218 f) b - (g \u2218 f) a\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\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' : \u211d \u2192 \u211d\ng g' : \u211d \u2192 E\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 [[f a, f b]] \u2286 f '' [[a, b]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.symm_trans_self", "start": [281, 1], "end": [282, 74], "traced_tactics": [{"tactic": "simp [symm_trans_rev, self_trans_symm, -symm_symm]", "annotated_tactic": ["simp [<a>symm_trans_rev</a>, <a>self_trans_symm</a>, -<a>symm_symm</a>]", [{"full_name": "PEquiv.symm_trans_rev", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [266, 9], "def_end_pos": [266, 23]}, {"full_name": "PEquiv.self_trans_symm", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [270, 9], "def_end_pos": [270, 24]}, {"full_name": "PEquiv.symm_symm", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [135, 9], "def_end_pos": [135, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.symm (PEquiv.trans (PEquiv.symm f) f) = PEquiv.symm (ofSet {b | isSome (\u2191(PEquiv.symm f) b) = true})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_mono", "start": [2623, 1], "end": [2625, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.leastGE_eq_min", "start": [71, 1], "end": [86, 34], "traced_tactics": [{"tactic": "refine' le_antisymm (le_min (leastGE_le _) (leastGE_mono (h\u03c0n \u03c9) r \u03c9)) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>le_min</a> (<a>leastGE_le</a> _) (<a>leastGE_mono</a> (h\u03c0n \u03c9) r \u03c9)) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "le_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [47, 9], "def_end_pos": [47, 15]}, {"full_name": "MeasureTheory.leastGE_le", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [59, 9], "def_end_pos": [59, 19]}, {"full_name": "MeasureTheory.leastGE_mono", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [67, 9], "def_end_pos": [67, 21]}]], "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 leastGE f r (\u03c0 \u03c9) \u03c9 = min (\u03c0 \u03c9) (leastGE f r n \u03c9)", "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9"}, {"tactic": "by_cases hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9", "annotated_tactic": ["by_cases hle : \u03c0 \u03c9 \u2264 <a>leastGE</a> f r n \u03c9", [{"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}]], "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9", "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9\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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9"}, {"tactic": "rw [min_eq_left hle, leastGE]", "annotated_tactic": ["rw [<a>min_eq_left</a> hle, <a>leastGE</a>]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}]], "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9", "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9"}, {"tactic": "by_cases h : \u2203 j \u2208 Set.Icc 0 (\u03c0 \u03c9), f j \u03c9 \u2208 Set.Ici r", "annotated_tactic": ["by_cases h : \u2203 j \u2208 <a>Set.Icc</a> 0 (\u03c0 \u03c9), f j \u03c9 \u2208 <a>Set.Ici</a> r", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9", "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9\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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u00ac\u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9"}, {"tactic": "refine' hle.trans (Eq.le _)", "annotated_tactic": ["refine' hle.trans (<a>Eq.le</a> _)", [{"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9", "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 leastGE f r n \u03c9 = hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9"}, {"tactic": "rw [leastGE, \u2190 hitting_eq_hitting_of_exists (h\u03c0n \u03c9) h]", "annotated_tactic": ["rw [<a>leastGE</a>, \u2190 <a>hitting_eq_hitting_of_exists</a> (h\u03c0n \u03c9) h]", [{"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.hitting_eq_hitting_of_exists", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [195, 9], "def_end_pos": [195, 37]}]], "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 leastGE f r n \u03c9 = hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9", "state_after": "no goals"}, {"tactic": "simp only [hitting, if_neg h, le_rfl]", "annotated_tactic": ["simp only [<a>hitting</a>, <a>if_neg</a> h, <a>le_rfl</a>]", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"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": "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\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u00ac\u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9", "state_after": "no goals"}, {"tactic": "rw [min_eq_right (not_le.1 hle).le, leastGE, leastGE, \u2190\n  hitting_eq_hitting_of_exists (h\u03c0n \u03c9) _]", "annotated_tactic": ["rw [<a>min_eq_right</a> (<a>not_le</a>.1 hle).<a>le</a>, <a>leastGE</a>, <a>leastGE</a>, \u2190\n      <a>hitting_eq_hitting_of_exists</a> (h\u03c0n \u03c9) _]", [{"full_name": "min_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}, {"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]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.hitting_eq_hitting_of_exists", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [195, 9], "def_end_pos": [195, 37]}]], "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\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9", "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r"}, {"tactic": "rw [not_le, leastGE, hitting_lt_iff _ (h\u03c0n \u03c9)] at hle", "annotated_tactic": ["rw [<a>not_le</a>, <a>leastGE</a>, <a>hitting_lt_iff</a> _ (h\u03c0n \u03c9)] at hle", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.hitting_lt_iff", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [182, 9], "def_end_pos": [182, 23]}]], "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r", "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u2203 j, j \u2208 Set.Ico 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r"}, {"tactic": "exact\n  let \u27e8j, hj\u2081, hj\u2082\u27e9 := hle\n  \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.le\u27e9, hj\u2082\u27e9", "annotated_tactic": ["exact\n      let \u27e8j, hj\u2081, hj\u2082\u27e9 := hle\n      \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.<a>le</a>\u27e9, hj\u2082\u27e9", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "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\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u2203 j, j \u2208 Set.Ico 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.piecewise_mem_set_pi", "start": [2545, 1], "end": [2549, 42], "traced_tactics": [{"tactic": "classical\n  rw [\u2190 piecewise_coe]\n  exact Set.piecewise_mem_pi (\u2191s) hf hg", "annotated_tactic": ["classical\n    rw [\u2190 <a>piecewise_coe</a>]\n    exact <a>Set.piecewise_mem_pi</a> (\u2191s) hf hg", [{"full_name": "Finset.piecewise_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2506, 9], "def_end_pos": [2506, 22]}, {"full_name": "Set.piecewise_mem_pi", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1554, 9], "def_end_pos": [1554, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 piecewise s f g \u2208 Set.pi t t'", "state_after": "no goals"}, {"tactic": "rw [\u2190 piecewise_coe]", "annotated_tactic": ["rw [\u2190 <a>piecewise_coe</a>]", [{"full_name": "Finset.piecewise_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2506, 9], "def_end_pos": [2506, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 piecewise s f g \u2208 Set.pi t t'", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 Set.piecewise (\u2191s) f g \u2208 Set.pi t t'"}, {"tactic": "exact Set.piecewise_mem_pi (\u2191s) hf hg", "annotated_tactic": ["exact <a>Set.piecewise_mem_pi</a> (\u2191s) hf hg", [{"full_name": "Set.piecewise_mem_pi", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1554, 9], "def_end_pos": [1554, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 Set.piecewise (\u2191s) f g \u2208 Set.pi t t'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.iUnion_univ_pi_of_monotone", "start": [1563, 1], "end": [1566, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "volume_regionBetween_eq_lintegral'", "start": [531, 1], "end": [548, 49], "traced_tactics": [{"tactic": "rw [Measure.prod_apply]", "annotated_tactic": ["rw [<a>Measure.prod_apply</a>]", [{"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": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume (Prod.mk x \u207b\u00b9' regionBetween f g s) \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet (regionBetween f g s)"}, {"tactic": "dsimp only [regionBetween, preimage_setOf_eq]", "annotated_tactic": ["dsimp only [<a>regionBetween</a>, <a>preimage_setOf_eq</a>]", [{"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}, {"full_name": "Set.preimage_setOf_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [118, 9], "def_end_pos": [118, 26]}]], "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\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume (Prod.mk x \u207b\u00b9' regionBetween f g s) \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "rw [h, lintegral_indicator] <;> simp only [hs, Pi.sub_apply]", "annotated_tactic": ["rw [h, <a>lintegral_indicator</a>] <;> simp only [hs, <a>Pi.sub_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": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "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\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "funext x", "annotated_tactic": ["funext x", []], "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\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = indicator s (fun x => ofReal (g x - f x)) x"}, {"tactic": "rw [indicator_apply]", "annotated_tactic": ["rw [<a>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": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = indicator s (fun x => ofReal (g x - f x)) x", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = if x \u2208 s then ofReal (g x - f x) else 0"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = if x \u2208 s then ofReal (g x - f x) else 0", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0"}, {"tactic": "have hx : { a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x) } = Ioo (f x) (g x) := by simp [h, Ioo]", "annotated_tactic": ["have hx : { a | x \u2208 s \u2227 a \u2208 <a>Ioo</a> (f x) (g x) } = <a>Ioo</a> (f x) (g x) := by simp [h, <a>Ioo</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.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\nhx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x)\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x)"}, {"tactic": "simp only [hx, Real.volume_Ioo, sub_zero]", "annotated_tactic": ["simp only [hx, <a>Real.volume_Ioo</a>, <a>sub_zero</a>]", [{"full_name": "Real.volume_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\nhx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x)\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x)", "state_after": "no goals"}, {"tactic": "simp [h, Ioo]", "annotated_tactic": ["simp [h, <a>Ioo</a>]", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "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\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\n\u22a2 {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x)", "state_after": "no goals"}, {"tactic": "have hx : { a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x) } = \u2205 := by simp [h]", "annotated_tactic": ["have hx : { a | x \u2208 s \u2227 a \u2208 <a>Ioo</a> (f x) (g x) } = \u2205 := by simp [h]", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\nhx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0"}, {"tactic": "simp only [hx, measure_empty]", "annotated_tactic": ["simp only [hx, <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\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\nhx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "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\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205", "state_after": "no goals"}, {"tactic": "exact measurableSet_regionBetween hf hg hs", "annotated_tactic": ["exact <a>measurableSet_regionBetween</a> hf hg hs", [{"full_name": "measurableSet_regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [477, 9], "def_end_pos": [477, 36]}]], "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\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet (regionBetween f g s)", "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_singleton", "start": [264, 1], "end": [266, 87], "traced_tactics": [{"tactic": "simp [toMeasure_apply_eq_toOuterMeasure_apply _ _ h, toOuterMeasure_apply_singleton]", "annotated_tactic": ["simp [<a>toMeasure_apply_eq_toOuterMeasure_apply</a> _ _ h, <a>toOuterMeasure_apply_singleton</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_singleton", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 39]}]], "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\na : \u03b1\nh : MeasurableSet {a}\n\u22a2 \u2191\u2191(toMeasure p) {a} = \u2191p a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_univ_iff", "start": [2590, 1], "end": [2591, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.union_get_eq", "start": [856, 1], "end": [858, 29], "traced_tactics": [{"tactic": "simp [union_def]", "annotated_tactic": ["simp [<a>union_def</a>]", [{"full_name": "Part.union_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [708, 9], "def_end_pos": [708, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Union \u03b1\na b : Part \u03b1\nhab : (a \u222a b).Dom\n\u22a2 get (a \u222a b) hab = get a (_ : a.Dom) \u222a get b (_ : b.Dom)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Union \u03b1\na b : Part \u03b1\nhab : (a \u222a b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \u222a x) b) (_ : (Part.bind a fun y => map (fun x => y \u222a x) b).Dom) =\n    get a (_ : a.Dom) \u222a 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 : Union \u03b1\na b : Part \u03b1\nhab : (a \u222a b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \u222a x) b) (_ : (Part.bind a fun y => map (fun x => y \u222a x) b).Dom) =\n    get a (_ : a.Dom) \u222a get b (_ : b.Dom)", "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_one", "start": [518, 1], "end": [529, 17], "traced_tactics": [{"tactic": "split_ifs with hk", "annotated_tactic": ["split_ifs with hk", []], "state_before": "R : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\n\u22a2 \u2191(k - 1) = (if k = 0 then \u2191n else \u2191k) - 1", "state_after": "case pos\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : k = 0\n\u22a2 \u2191(k - 1) = \u2191n - 1\n\ncase neg\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1"}, {"tactic": "rw [hk, zero_sub, ZMod.cast_neg_one]", "annotated_tactic": ["rw [hk, <a>zero_sub</a>, <a>ZMod.cast_neg_one</a>]", [{"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "ZMod.cast_neg_one", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [512, 9], "def_end_pos": [512, 21]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : k = 0\n\u22a2 \u2191(k - 1) = \u2191n - 1", "state_after": "no goals"}, {"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "case neg\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1", "state_after": "case neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod Nat.zero\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1\n\ncase neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1"}, {"tactic": "rw [Int.cast_sub, Int.cast_one]", "annotated_tactic": ["rw [<a>Int.cast_sub</a>, <a>Int.cast_one</a>]", [{"full_name": "Int.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [77, 9], "def_end_pos": [77, 17]}]], "state_before": "case neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod Nat.zero\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1", "state_after": "no goals"}, {"tactic": "dsimp [ZMod, ZMod.cast, ZMod.val]", "annotated_tactic": ["dsimp [<a>ZMod</a>, <a>ZMod.cast</a>, <a>ZMod.val</a>]", [{"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod.cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [152, 12], "def_end_pos": [152, 16]}, {"full_name": "ZMod.val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1", "state_after": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191\u2191(k - 1) = \u2191\u2191k - 1"}, {"tactic": "rw [Fin.coe_sub_one, if_neg]", "annotated_tactic": ["rw [<a>Fin.coe_sub_one</a>, <a>if_neg</a>]", [{"full_name": "Fin.coe_sub_one", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1229, 9], "def_end_pos": [1229, 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.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191\u2191(k - 1) = \u2191\u2191k - 1", "state_after": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191(\u2191k - 1) = \u2191\u2191k - 1\n\ncase neg.succ.hnc\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u00ack = 0"}, {"tactic": "rw [Nat.cast_sub, Nat.cast_one]", "annotated_tactic": ["rw [<a>Nat.cast_sub</a>, <a>Nat.cast_one</a>]", [{"full_name": "Nat.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 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": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191(\u2191k - 1) = \u2191\u2191k - 1", "state_after": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 1 \u2264 \u2191k"}, {"tactic": "rwa [Fin.ext_iff, Fin.val_zero, \u2190 Ne, \u2190 Nat.one_le_iff_ne_zero] at hk", "annotated_tactic": ["rwa [<a>Fin.ext_iff</a>, <a>Fin.val_zero</a>, \u2190 <a>Ne</a>, \u2190 <a>Nat.one_le_iff_ne_zero</a>] at hk", [{"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.val_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "Nat.one_le_iff_ne_zero", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 27]}]], "state_before": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 1 \u2264 \u2191k", "state_after": "no goals"}, {"tactic": "exact hk", "annotated_tactic": ["exact hk", []], "state_before": "case neg.succ.hnc\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u00ack = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.of_to_int'", "start": [1352, 1], "end": [1356, 96], "traced_tactics": [{"tactic": "dsimp [ofInt', cast_zero]", "annotated_tactic": ["dsimp [<a>ofInt'</a>, <a>cast_zero</a>]", [{"full_name": "ZNum.ofInt'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [396, 5], "def_end_pos": [396, 11]}, {"full_name": "ZNum.cast_zero", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 18]}]], "state_before": "\u03b1 : Type u_1\n\u22a2 ofInt' \u21910 = 0", "state_after": "\u03b1 : Type u_1\n\u22a2 Num.toZNum (Num.ofNat' 0) = 0"}, {"tactic": "erw [Num.ofNat'_zero, Num.toZNum]", "annotated_tactic": ["erw [<a>Num.ofNat'_zero</a>, <a>Num.toZNum</a>]", [{"full_name": "Num.ofNat'_zero", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [236, 9], "def_end_pos": [236, 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\n\u22a2 Num.toZNum (Num.ofNat' 0) = 0", "state_after": "no goals"}, {"tactic": "rw [cast_pos, \u2190 PosNum.cast_to_nat, \u2190 Num.ofInt'_toZNum, PosNum.of_to_nat]", "annotated_tactic": ["rw [<a>cast_pos</a>, \u2190 <a>PosNum.cast_to_nat</a>, \u2190 <a>Num.ofInt'_toZNum</a>, <a>PosNum.of_to_nat</a>]", [{"full_name": "ZNum.cast_pos", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1038, 9], "def_end_pos": [1038, 17]}, {"full_name": "PosNum.cast_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}, {"full_name": "Num.ofInt'_toZNum", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 22]}, {"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\na : PosNum\n\u22a2 ofInt' \u2191(pos a) = pos a", "state_after": "\u03b1 : Type u_1\na : PosNum\n\u22a2 Num.toZNum (Num.pos a) = pos a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\na : PosNum\n\u22a2 Num.toZNum (Num.pos a) = pos a", "state_after": "no goals"}, {"tactic": "rw [cast_neg, ofInt'_neg, \u2190 PosNum.cast_to_nat, \u2190 Num.ofInt'_toZNum, PosNum.of_to_nat]", "annotated_tactic": ["rw [<a>cast_neg</a>, <a>ofInt'_neg</a>, \u2190 <a>PosNum.cast_to_nat</a>, \u2190 <a>Num.ofInt'_toZNum</a>, <a>PosNum.of_to_nat</a>]", [{"full_name": "ZNum.cast_neg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1043, 9], "def_end_pos": [1043, 17]}, {"full_name": "ZNum.ofInt'_neg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1345, 9], "def_end_pos": [1345, 19]}, {"full_name": "PosNum.cast_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}, {"full_name": "Num.ofInt'_toZNum", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 22]}, {"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\na : PosNum\n\u22a2 ofInt' \u2191(neg a) = neg a", "state_after": "\u03b1 : Type u_1\na : PosNum\n\u22a2 -Num.toZNum (Num.pos a) = neg a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\na : PosNum\n\u22a2 -Num.toZNum (Num.pos a) = neg a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essInf_const_top", "start": [177, 1], "end": [178, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Stream.next?_toList", "start": [427, 1], "end": [429, 45], "traced_tactics": [{"tactic": "cases s <;> simp [next?, toStream_toList']", "annotated_tactic": ["cases s <;> simp [<a>next?</a>, <a>toStream_toList'</a>]", [{"full_name": "Std.RBNode.Stream.next?", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [133, 5], "def_end_pos": [133, 10]}, {"full_name": "Std.RBNode.toStream_toList'", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [421, 9], "def_end_pos": [421, 25]}]], "state_before": "\u03b1 : Type u_1\ns : RBNode.Stream \u03b1\n\u22a2 Option.map\n      (fun x =>\n        match x with\n        | (a, b) => (a, toList b))\n      (next? s) =\n    List.next? (toList 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_val", "start": [1596, 1], "end": [1597, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "full_name": "Set.range_IciExtend", "start": [219, 1], "end": [220, 76], "traced_tactics": [{"tactic": "simp only [IciExtend, range_comp f, range_projIci, range_id', image_univ]", "annotated_tactic": ["simp only [<a>IciExtend</a>, <a>range_comp</a> f, <a>range_projIci</a>, <a>range_id'</a>, <a>image_univ</a>]", [{"full_name": "Set.IciExtend", "def_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "def_pos": [192, 5], "def_end_pos": [192, 14]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "Set.range_projIci", "def_path": "Mathlib/Data/Set/Intervals/ProjIcc.lean", "def_pos": [157, 9], "def_end_pos": [157, 22]}, {"full_name": "Set.range_id'", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [877, 9], "def_end_pos": [877, 18]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u2191(Ici a) \u2192 \u03b2\n\u22a2 range (IciExtend f) = range f", "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.merge", "start": [407, 1], "end": [408, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Action.lean", "full_name": "MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples", "start": [308, 1], "end": [321, 87], "traced_tactics": [{"tactic": "letI : MeasurableSpace (Multiplicative G) := (inferInstanceAs (MeasurableSpace G))", "annotated_tactic": ["letI : <a>MeasurableSpace</a> (<a>Multiplicative</a> G) := (<a>inferInstanceAs</a> (<a>MeasurableSpace</a> G))", [{"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}, {"full_name": "inferInstanceAs", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [100, 8], "def_end_pos": [100, 23]}, {"full_name": "MeasurableSpace", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [48, 20], "def_end_pos": [48, 35]}]], "state_before": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s", "state_after": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s"}, {"tactic": "letI : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n  \u27e8fun g => VAddInvariantMeasure.measure_preimage_vadd (Multiplicative.toAdd g)\u27e9", "annotated_tactic": ["letI : <a>SMulInvariantMeasure</a> (<a>Multiplicative</a> G) \u03b1 \u03bc :=\n    \u27e8fun g => <a>VAddInvariantMeasure.measure_preimage_vadd</a> (<a>Multiplicative.toAdd</a> g)\u27e9", [{"full_name": "MeasureTheory.SMulInvariantMeasure", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [45, 7], "def_end_pos": [45, 27]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}, {"full_name": "MeasureTheory.VAddInvariantMeasure.measure_preimage_vadd", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [37, 3], "def_end_pos": [37, 24]}, {"full_name": "Multiplicative.toAdd", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [81, 5], "def_end_pos": [81, 10]}]], "state_before": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s", "state_after": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis\u271d : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n  { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) }\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s"}, {"tactic": "letI : MeasurableSMul (Multiplicative G) \u03b1 :=\n  { measurable_const_smul := fun g => measurable_const_vadd (Multiplicative.toAdd g)\n    measurable_smul_const := fun a =>\n      @measurable_vadd_const (Multiplicative G) \u03b1 (inferInstanceAs (VAdd G \u03b1)) _ _\n        (inferInstanceAs (MeasurableVAdd G \u03b1)) a }", "annotated_tactic": ["letI : <a>MeasurableSMul</a> (<a>Multiplicative</a> G) \u03b1 :=\n    { measurable_const_smul := fun g => <a>measurable_const_vadd</a> (<a>Multiplicative.toAdd</a> g)\n      measurable_smul_const := fun a =>\n        @<a>measurable_vadd_const</a> (<a>Multiplicative</a> G) \u03b1 (<a>inferInstanceAs</a> (<a>VAdd</a> G \u03b1)) _ _\n          (<a>inferInstanceAs</a> (<a>MeasurableVAdd</a> G \u03b1)) a }", [{"full_name": "MeasurableSMul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [540, 7], "def_end_pos": [540, 21]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}, {"full_name": "MeasurableVAdd.measurable_const_vadd", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [531, 3], "def_end_pos": [531, 24]}, {"full_name": "Multiplicative.toAdd", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [81, 5], "def_end_pos": [81, 10]}, {"full_name": "MeasurableVAdd.measurable_vadd_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [532, 3], "def_end_pos": [532, 24]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}, {"full_name": "inferInstanceAs", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [100, 8], "def_end_pos": [100, 23]}, {"full_name": "VAdd", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [70, 7], "def_end_pos": [70, 11]}, {"full_name": "inferInstanceAs", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [100, 8], "def_end_pos": [100, 23]}, {"full_name": "MeasurableVAdd", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [529, 7], "def_end_pos": [529, 21]}]], "state_before": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis\u271d : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n  { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) }\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s", "state_after": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis\u271d\u00b9 : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis\u271d : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n  { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) }\nthis : MeasurableSMul (Multiplicative G) \u03b1 :=\n  { measurable_const_smul := fun g => measurable_const_vadd (\u2191Multiplicative.toAdd g),\n    measurable_smul_const := fun a => measurable_vadd_const a }\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s"}, {"tactic": "exact @smul_ae_eq_self_of_mem_zpowers (Multiplicative G) \u03b1 _ _ _ _ _ _ _ _ _ _ hs hy", "annotated_tactic": ["exact @<a>smul_ae_eq_self_of_mem_zpowers</a> (<a>Multiplicative</a> G) \u03b1 _ _ _ _ _ _ _ _ _ _ hs hy", [{"full_name": "MeasureTheory.smul_ae_eq_self_of_mem_zpowers", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [296, 9], "def_end_pos": [296, 39]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}]], "state_before": "G\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis\u271d\u00b9 : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis\u271d : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n  { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) }\nthis : MeasurableSMul (Multiplicative G) \u03b1 :=\n  { measurable_const_smul := fun g => measurable_const_vadd (\u2191Multiplicative.toAdd g),\n    measurable_smul_const := fun a => measurable_vadd_const a }\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.strongDownwardInductionOn_eq", "start": [733, 1], "end": [737, 31], "traced_tactics": [{"tactic": "dsimp only [strongDownwardInductionOn]", "annotated_tactic": ["dsimp only [<a>strongDownwardInductionOn</a>]", [{"full_name": "Finset.strongDownwardInductionOn", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [726, 5], "def_end_pos": [726, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInductionOn s H a) = H s fun {t} ht x => strongDownwardInductionOn t H ht", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInduction H s a) = H s fun {t} ht x => strongDownwardInduction H t ht"}, {"tactic": "rw [strongDownwardInduction]", "annotated_tactic": ["rw [<a>strongDownwardInduction</a>]", [{"full_name": "Finset.strongDownwardInduction", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [706, 5], "def_end_pos": [706, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInduction H s a) = H s fun {t} ht x => strongDownwardInduction H t ht", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "StronglyMeasurableAtFilter.eventually", "start": [47, 11], "end": [49, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithTop.image_coe_Ioc", "start": [118, 1], "end": [121, 78], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left\n    (Subset.trans Ioc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ioc</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>Ioc_subset_Iic_self</a> <| <a>Iic_subset_Iio</a>.2 <| <a>coe_lt_top</a> b)]", [{"full_name": "WithTop.preimage_coe_Ioc", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [67, 9], "def_end_pos": [67, 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.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.Ioc_subset_Iic_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [477, 9], "def_end_pos": [477, 28]}, {"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 '' Ioc a b = Ioc \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.evaln_prim", "start": [1086, 1], "end": [1145, 43], "traced_tactics": [{"tactic": "simp only [G, prod_ofNat_val, ofNat_nat, List.length_map, List.length_range,\n  Nat.pair_unpair, Option.some_inj]", "annotated_tactic": ["simp only [<a>G</a>, <a>prod_ofNat_val</a>, <a>ofNat_nat</a>, <a>List.length_map</a>, <a>List.length_range</a>,\n        <a>Nat.pair_unpair</a>, <a>Option.some_inj</a>]", [{"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.G", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [942, 13], "def_end_pos": [942, 14]}, {"full_name": "Denumerable.prod_ofNat_val", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [173, 9], "def_end_pos": [173, 23]}, {"full_name": "Denumerable.ofNat_nat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [123, 9], "def_end_pos": [123, 18]}, {"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": "List.length_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2074, 17], "def_end_pos": [2074, 29]}, {"full_name": "Nat.pair_unpair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [50, 9], "def_end_pos": [50, 20]}, {"full_name": "Option.some_inj", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 17]}]], "state_before": "x\u271d : Unit\np : \u2115\n\u22a2 Nat.Partrec.Code.G\n      (x\u271d,\n          List.map\n            (fun n =>\n              let a := ofNat (\u2115 \u00d7 Code) n;\n              List.map (evaln a.1 a.2) (List.range a.1))\n            (List.range p)).2 =\n    some\n      (let a := ofNat (\u2115 \u00d7 Code) p;\n      List.map (evaln a.1 a.2) (List.range a.1))", "state_after": "x\u271d : Unit\np : \u2115\n\u22a2 List.map\n      (fun n =>\n        Nat.rec Option.none\n          (fun n_1 n_ih =>\n            rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n              (fun cf cg x x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cf) n\n                let y \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) n\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) n\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) x)\n              (fun cf cg x x =>\n                Nat.rec\n                  (Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) (unpair n).1)\n                  (fun n_2 n_ih => do\n                    let i \u2190\n                      Nat.Partrec.Code.lup\n                          (List.map\n                            (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                            (List.range p))\n                          (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                  (unpair n).2)\n              (fun cf x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cf) n\n                Nat.rec (some (unpair n).2)\n                    (fun n_2 n_ih =>\n                      Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                    x)\n              (ofNat Code (unpair p).2))\n          (unpair p).1)\n      (List.range (unpair p).1) =\n    List.map (evaln (unpair p).1 (ofNat Code (unpair p).2)) (List.range (unpair p).1)"}, {"tactic": "refine List.map_congr fun n => ?_", "annotated_tactic": ["refine <a>List.map_congr</a> fun n => ?_", [{"full_name": "List.map_congr", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1716, 9], "def_end_pos": [1716, 18]}]], "state_before": "x\u271d : Unit\np : \u2115\n\u22a2 List.map\n      (fun n =>\n        Nat.rec Option.none\n          (fun n_1 n_ih =>\n            rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n              (fun cf cg x x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cf) n\n                let y \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) n\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) n\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) x)\n              (fun cf cg x x =>\n                Nat.rec\n                  (Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) (unpair n).1)\n                  (fun n_2 n_ih => do\n                    let i \u2190\n                      Nat.Partrec.Code.lup\n                          (List.map\n                            (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                            (List.range p))\n                          (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                  (unpair n).2)\n              (fun cf x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cf) n\n                Nat.rec (some (unpair n).2)\n                    (fun n_2 n_ih =>\n                      Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                    x)\n              (ofNat Code (unpair p).2))\n          (unpair p).1)\n      (List.range (unpair p).1) =\n    List.map (evaln (unpair p).1 (ofNat Code (unpair p).2)) (List.range (unpair p).1)", "state_after": "x\u271d : Unit\np n : \u2115\n\u22a2 n \u2208 List.range (unpair p).1 \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        (unpair p).1 =\n      evaln (unpair p).1 (ofNat Code (unpair p).2) n"}, {"tactic": "have : List.range p = List.range (Nat.pair p.unpair.1 (encode (ofNat Code p.unpair.2))) := by\n  simp", "annotated_tactic": ["have : <a>List.range</a> p = <a>List.range</a> (<a>Nat.pair</a> p.unpair.1 (<a>encode</a> (<a>ofNat</a> <a>Code</a> p.unpair.2))) := by\n        simp", [{"full_name": "List.range", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [572, 5], "def_end_pos": [572, 10]}, {"full_name": "List.range", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [572, 5], "def_end_pos": [572, 10]}, {"full_name": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}]], "state_before": "x\u271d : Unit\np n : \u2115\n\u22a2 n \u2208 List.range (unpair p).1 \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        (unpair p).1 =\n      evaln (unpair p).1 (ofNat Code (unpair p).2) n", "state_after": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\n\u22a2 n \u2208 List.range (unpair p).1 \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        (unpair p).1 =\n      evaln (unpair p).1 (ofNat Code (unpair p).2) n"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\n\u22a2 n \u2208 List.range (unpair p).1 \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range p))\n                  ((unpair p).1, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range p))\n                    ((unpair p).1, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range p))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        (unpair p).1 =\n      evaln (unpair p).1 (ofNat Code (unpair p).2) n", "state_after": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\n\u22a2 n \u2208 List.range (unpair p).1 \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                  ((unpair p).1, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                  ((unpair p).1, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                      ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        (unpair p).1 =\n      evaln (unpair p).1 (ofNat Code (unpair p).2) n"}, {"tactic": "generalize p.unpair.1 = k", "annotated_tactic": ["generalize p.unpair.1 = k", []], "state_before": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\n\u22a2 n \u2208 List.range (unpair p).1 \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                  ((unpair p).1, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                  ((unpair p).1, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                      ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                    ((unpair p).1, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        (unpair p).1 =\n      evaln (unpair p).1 (ofNat Code (unpair p).2) n", "state_after": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : \u2115\n\u22a2 n \u2208 List.range k \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                  (k, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                  (k, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                      (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        k =\n      evaln k (ofNat Code (unpair p).2) n"}, {"tactic": "generalize ofNat Code p.unpair.2 = c", "annotated_tactic": ["generalize <a>ofNat</a> <a>Code</a> p.unpair.2 = c", [{"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"full_name": "Nat.Partrec.Code", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [76, 11], "def_end_pos": [76, 15]}]], "state_before": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : \u2115\n\u22a2 n \u2208 List.range k \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                  (k, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                  (k, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                      (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                    (k, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))\n                      (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            (ofNat Code (unpair p).2))\n        k =\n      evaln k (ofNat Code (unpair p).2) n", "state_after": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : \u2115\nc : Code\n\u22a2 n \u2208 List.range k \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range (Nat.pair k (encode c))))\n                        (n_1, c) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode c))))\n                      (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            c)\n        k =\n      evaln k c n"}, {"tactic": "intro nk", "annotated_tactic": ["intro nk", []], "state_before": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : \u2115\nc : Code\n\u22a2 n \u2208 List.range k \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) (unpair n).1)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                          (List.range (Nat.pair k (encode c))))\n                        (n_1, c) (Nat.pair (unpair n).1 n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode c))))\n                      (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n                (unpair n).2)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cf) n\n              Nat.rec (some (unpair n).2)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                  x)\n            c)\n        k =\n      evaln k c n", "state_after": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : \u2115\nc : Code\nnk : n \u2208 List.range k\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair k (encode c))))\n                (k, cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair k (encode c))))\n                (k, cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      k =\n    evaln k c n"}, {"tactic": "cases' k with k'", "annotated_tactic": ["cases' k with k'", []], "state_before": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : \u2115\nc : Code\nnk : n \u2208 List.range k\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair k (encode c))))\n                (k, cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair k (encode c))))\n                (k, cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair k (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair k (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      k =\n    evaln k c n", "state_after": "case zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nnk : n \u2208 List.range Nat.zero\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair Nat.zero (encode c))))\n                (Nat.zero, cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair Nat.zero (encode c))))\n                (Nat.zero, cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair Nat.zero (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair Nat.zero (encode c))))\n                    (Nat.zero, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair Nat.zero (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      Nat.zero =\n    evaln Nat.zero c n\n\ncase succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (Nat.succ k') (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (Nat.succ k', cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      (Nat.succ k') =\n    evaln (Nat.succ k') c n"}, {"tactic": "let k := k' + 1", "annotated_tactic": ["let k := k' + 1", []], "state_before": "case succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (Nat.succ k') (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (Nat.succ k', cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      (Nat.succ k') =\n    evaln (Nat.succ k') c n", "state_after": "case succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\nk : \u2115 := k' + 1\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (Nat.succ k') (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (Nat.succ k', cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      (Nat.succ k') =\n    evaln (Nat.succ k') c n"}, {"tactic": "simp only [show k'.succ = k from rfl]", "annotated_tactic": ["simp only [show k'.succ = k from <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 succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\nk : \u2115 := k' + 1\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair (Nat.succ k') (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (Nat.succ k', cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      (Nat.succ k') =\n    evaln (Nat.succ k') c n", "state_after": "case succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\nk : \u2115 := k' + 1\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).1)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).1 n_1)\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).2)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n"}, {"tactic": "simp [Nat.lt_succ_iff] at nk", "annotated_tactic": ["simp [<a>Nat.lt_succ_iff</a>] at nk", [{"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 succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\nk : \u2115 := k' + 1\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).1)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).1 n_1)\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).2)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n", "state_after": "case succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).1)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).1 n_1)\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).2)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n"}, {"tactic": "have hg :\n  \u2200 {k' c' n},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      lup ((List.range (Nat.pair k (encode c))).map fun n =>\n        (List.range n.unpair.1).map (evaln n.unpair.1 (ofNat Code n.unpair.2))) (k', c') n =\n      evaln k' c' n := by\n  intro k\u2081 c\u2081 n\u2081 hl\n  simp [lup, List.get?_range hl, evaln_map, Bind.bind]", "annotated_tactic": ["have hg :\n        \u2200 {k' c' n},\n          <a>Nat.pair</a> k' (<a>encode</a> c') < <a>Nat.pair</a> k (<a>encode</a> c) \u2192\n            <a>lup</a> ((<a>List.range</a> (<a>Nat.pair</a> k (<a>encode</a> c))).<a>map</a> fun n =>\n              (<a>List.range</a> n.unpair.1).<a>map</a> (<a>evaln</a> n.unpair.1 (<a>ofNat</a> <a>Code</a> n.unpair.2))) (k', c') n =\n            <a>evaln</a> k' c' n := by\n        intro k\u2081 c\u2081 n\u2081 hl\n        simp [<a>lup</a>, <a>List.get?_range</a> hl, <a>evaln_map</a>, <a>Bind.bind</a>]", [{"full_name": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.lup", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [931, 13], "def_end_pos": [931, 16]}, {"full_name": "List.range", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [572, 5], "def_end_pos": [572, 10]}, {"full_name": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}, {"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.range", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [572, 5], "def_end_pos": [572, 10]}, {"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": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}, {"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.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.lup", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [931, 13], "def_end_pos": [931, 16]}, {"full_name": "List.get?_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2094, 9], "def_end_pos": [2094, 19]}, {"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.evaln_map", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [1075, 17], "def_end_pos": [1075, 26]}, {"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 succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).1)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).1 n_1)\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).2)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n", "state_after": "case succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode c))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).1)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).1 n_1)\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).2)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n"}, {"tactic": "cases' c with cf cg cf cg cf cg cf <;>\n  simp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]", "annotated_tactic": ["cases' c with cf cg cf cg cf cg cf <;>\n        simp [<a>evaln</a>, nk, <a>Bind.bind</a>, <a>Functor.map</a>, <a>Seq.seq</a>, <a>pure</a>]", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"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]}, {"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]}]], "state_before": "case succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode c))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).1)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).1 n_1)\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).2)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n", "state_after": "case succ.pair\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Option.bind\n        (Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n          (k' + 1, cg) n)\n        fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)\n\ncase succ.comp\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cg) n)\n      fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x\n\ncase succ.prec\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 Nat.rec\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cf) (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2 =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2\n\ncase succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "x\u271d : Unit\np n : \u2115\n\u22a2 List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))", "state_after": "no goals"}, {"tactic": "simp [evaln]", "annotated_tactic": ["simp [<a>evaln</a>]", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "case zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nnk : n \u2208 List.range Nat.zero\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cg) n\n            Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair Nat.zero (encode c))))\n                (Nat.zero, cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                  (List.range (Nat.pair Nat.zero (encode c))))\n                (Nat.zero, cf) (unpair n).1)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                        (List.range (Nat.pair Nat.zero (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).1 n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair Nat.zero (encode c))))\n                    (Nat.zero, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))\n              (unpair n).2)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cf) n\n            Nat.rec (some (unpair n).2)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n                      (List.range (Nat.pair Nat.zero (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n                x)\n          c)\n      Nat.zero =\n    evaln Nat.zero c n", "state_after": "no goals"}, {"tactic": "intro k\u2081 c\u2081 n\u2081 hl", "annotated_tactic": ["intro k\u2081 c\u2081 n\u2081 hl", []], "state_before": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\n\u22a2 \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode c))))\n          (k', c') n =\n        evaln k' c' n", "state_after": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nk\u2081 : \u2115\nc\u2081 : Code\nn\u2081 : \u2115\nhl : Nat.pair k\u2081 (encode c\u2081) < Nat.pair k (encode c)\n\u22a2 Nat.Partrec.Code.lup\n      (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n        (List.range (Nat.pair k (encode c))))\n      (k\u2081, c\u2081) n\u2081 =\n    evaln k\u2081 c\u2081 n\u2081"}, {"tactic": "simp [lup, List.get?_range hl, evaln_map, Bind.bind]", "annotated_tactic": ["simp [<a>lup</a>, <a>List.get?_range</a> hl, <a>evaln_map</a>, <a>Bind.bind</a>]", [{"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.lup", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [931, 13], "def_end_pos": [931, 16]}, {"full_name": "List.get?_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2094, 9], "def_end_pos": [2094, 19]}, {"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.evaln_map", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [1075, 17], "def_end_pos": [1075, 26]}, {"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": "x\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nk\u2081 : \u2115\nc\u2081 : Code\nn\u2081 : \u2115\nhl : Nat.pair k\u2081 (encode c\u2081) < Nat.pair k (encode c)\n\u22a2 Nat.Partrec.Code.lup\n      (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n        (List.range (Nat.pair k (encode c))))\n      (k\u2081, c\u2081) n\u2081 =\n    evaln k\u2081 c\u2081 n\u2081", "state_after": "no goals"}, {"tactic": "cases' encode_lt_pair cf cg with lf lg", "annotated_tactic": ["cases' <a>encode_lt_pair</a> cf cg with lf lg", [{"full_name": "Nat.Partrec.Code.encode_lt_pair", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [202, 9], "def_end_pos": [202, 23]}]], "state_before": "case succ.pair\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Option.bind\n        (Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n          (k' + 1, cg) n)\n        fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)", "state_after": "case succ.pair.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Option.bind\n        (Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n          (k' + 1, cg) n)\n        fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)"}, {"tactic": "rw [hg (Nat.pair_lt_pair_right _ lf), hg (Nat.pair_lt_pair_right _ lg)]", "annotated_tactic": ["rw [hg (<a>Nat.pair_lt_pair_right</a> _ lf), hg (<a>Nat.pair_lt_pair_right</a> _ lg)]", [{"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}, {"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "case succ.pair.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Option.bind\n        (Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n          (k' + 1, cg) n)\n        fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)", "state_after": "case succ.pair.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind (evaln k cf n) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)"}, {"tactic": "cases evaln k cf n", "annotated_tactic": ["cases <a>evaln</a> k cf n", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "case succ.pair.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind (evaln k cf n) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)", "state_after": "case succ.pair.intro.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind Option.none fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair Option.none) fun y => Option.map y (evaln (k' + 1) cg n)\n\ncase succ.pair.intro.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (some val\u271d)) fun y => Option.map y (evaln (k' + 1) cg n)"}, {"tactic": "cases evaln k cg n <;> rfl", "annotated_tactic": ["cases <a>evaln</a> k cg n <;> rfl", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "case succ.pair.intro.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (some val\u271d)) fun y => Option.map y (evaln (k' + 1) cg n)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.pair.intro.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind Option.none fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair Option.none) fun y => Option.map y (evaln (k' + 1) cg n)", "state_after": "no goals"}, {"tactic": "cases' encode_lt_comp cf cg with lf lg", "annotated_tactic": ["cases' <a>encode_lt_comp</a> cf cg with lf lg", [{"full_name": "Nat.Partrec.Code.encode_lt_comp", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [211, 9], "def_end_pos": [211, 23]}]], "state_before": "case succ.comp\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cg) n)\n      fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x", "state_after": "case succ.comp.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cg) n)\n      fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x"}, {"tactic": "rw [hg (Nat.pair_lt_pair_right _ lg)]", "annotated_tactic": ["rw [hg (<a>Nat.pair_lt_pair_right</a> _ lg)]", [{"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "case succ.comp.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cg) n)\n      fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x", "state_after": "case succ.comp.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind (evaln k cg n) fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x"}, {"tactic": "cases evaln k cg n", "annotated_tactic": ["cases <a>evaln</a> k cg n", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "case succ.comp.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind (evaln k cg n) fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x", "state_after": "case succ.comp.intro.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind Option.none fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind Option.none fun x => evaln (k' + 1) cf x\n\ncase succ.comp.intro.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (some val\u271d) fun x => evaln (k' + 1) cf x"}, {"tactic": "simp [hg (Nat.pair_lt_pair_right _ lf)]", "annotated_tactic": ["simp [hg (<a>Nat.pair_lt_pair_right</a> _ lf)]", [{"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "case succ.comp.intro.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (some val\u271d) fun x => evaln (k' + 1) cf x", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.comp.intro.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind Option.none fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind Option.none fun x => evaln (k' + 1) cf x", "state_after": "no goals"}, {"tactic": "cases' encode_lt_prec cf cg with lf lg", "annotated_tactic": ["cases' <a>encode_lt_prec</a> cf cg with lf lg", [{"full_name": "Nat.Partrec.Code.encode_lt_prec", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [217, 9], "def_end_pos": [217, 23]}]], "state_before": "case succ.prec\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 Nat.rec\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cf) (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2 =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2", "state_after": "case succ.prec.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cf) (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2 =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2"}, {"tactic": "rw [hg (Nat.pair_lt_pair_right _ lf)]", "annotated_tactic": ["rw [hg (<a>Nat.pair_lt_pair_right</a> _ lf)]", [{"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "case succ.prec.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cf) (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2 =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2", "state_after": "case succ.prec.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec (evaln k cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2 =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2"}, {"tactic": "cases n.unpair.2", "annotated_tactic": ["cases n.unpair.2", []], "state_before": "case succ.prec.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec (evaln k cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2 =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (unpair n).2", "state_after": "case succ.prec.intro.zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec (evaln k cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      Nat.zero =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      Nat.zero\n\ncase succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 Nat.rec (evaln k cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (Nat.succ n\u271d) =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (Nat.succ n\u271d)"}, {"tactic": "simp only [decode_eq_ofNat, Option.some.injEq]", "annotated_tactic": ["simp only [<a>decode_eq_ofNat</a>, Option.some.injEq]", [{"full_name": "Denumerable.decode_eq_ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [55, 9], "def_end_pos": [55, 24]}]], "state_before": "case succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 Nat.rec (evaln k cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (Nat.succ n\u271d) =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      (Nat.succ n\u271d)", "state_after": "case succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k', prec cf cg) (Nat.pair (unpair n).1 n\u271d))\n      fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>\n      evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))"}, {"tactic": "rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]", "annotated_tactic": ["rw [hg (<a>Nat.pair_lt_pair_left</a> _ k'.lt_succ_self)]", [{"full_name": "Nat.pair_lt_pair_left", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [127, 9], "def_end_pos": [127, 26]}]], "state_before": "case succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k', prec cf cg) (Nat.pair (unpair n).1 n\u271d))\n      fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>\n      evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))", "state_after": "case succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>\n      evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))"}, {"tactic": "cases evaln k' _ _", "annotated_tactic": ["cases <a>evaln</a> k' _ _", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "case succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>\n      evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))", "state_after": "case succ.prec.intro.succ.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind Option.none fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind Option.none fun i => evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))\n\ncase succ.prec.intro.succ.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d val\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind (some val\u271d) fun i => evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))"}, {"tactic": "simp [hg (Nat.pair_lt_pair_right _ lg)]", "annotated_tactic": ["simp [hg (<a>Nat.pair_lt_pair_right</a> _ lg)]", [{"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "case succ.prec.intro.succ.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d val\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind (some val\u271d) fun i => evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.prec.intro.zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec (evaln k cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).1 n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      Nat.zero =\n    Nat.rec (evaln (k' + 1) cf (unpair n).1)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n      Nat.zero", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.prec.intro.succ.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind Option.none fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =\n    Option.bind Option.none fun i => evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i))", "state_after": "no goals"}, {"tactic": "have lf := encode_lt_rfind' cf", "annotated_tactic": ["have lf := <a>encode_lt_rfind'</a> cf", [{"full_name": "Nat.Partrec.Code.encode_lt_rfind'", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [223, 9], "def_end_pos": [223, 25]}]], "state_before": "case succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "case succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))"}, {"tactic": "rw [hg (Nat.pair_lt_pair_right _ lf)]", "annotated_tactic": ["rw [hg (<a>Nat.pair_lt_pair_right</a> _ lf)]", [{"full_name": "Nat.pair_lt_pair_right", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "case succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n          (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "case succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind (evaln k cf n) fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))"}, {"tactic": "cases' evaln k cf n with x", "annotated_tactic": ["cases' <a>evaln</a> k cf n with x", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "case succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind (evaln k cf n) fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "case succ.rfind'.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind Option.none fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind Option.none fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))\n\ncase succ.rfind'.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : \u2115\n\u22a2 (Option.bind (some x) fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (some x) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))"}, {"tactic": "simp only [decode_eq_ofNat, Option.some.injEq, Option.some_bind]", "annotated_tactic": ["simp only [<a>decode_eq_ofNat</a>, Option.some.injEq, <a>Option.some_bind</a>]", [{"full_name": "Denumerable.decode_eq_ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [55, 9], "def_end_pos": [55, 24]}, {"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]}]], "state_before": "case succ.rfind'.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : \u2115\n\u22a2 (Option.bind (some x) fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind (some x) fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "case succ.rfind'.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : \u2115\n\u22a2 Nat.rec (some (unpair n).2)\n      (fun n_1 n_ih =>\n        Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n          (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n      x =\n    if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))"}, {"tactic": "cases x <;> simp [Nat.succ_ne_zero]", "annotated_tactic": ["cases x <;> simp [<a>Nat.succ_ne_zero</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]}]], "state_before": "case succ.rfind'.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : \u2115\n\u22a2 Nat.rec (some (unpair n).2)\n      (fun n_1 n_ih =>\n        Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n          (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n      x =\n    if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "case succ.rfind'.some.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nn\u271d : \u2115\n\u22a2 Nat.Partrec.Code.lup\n      (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n        (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n      (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) =\n    evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))"}, {"tactic": "rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]", "annotated_tactic": ["rw [hg (<a>Nat.pair_lt_pair_left</a> _ k'.lt_succ_self)]", [{"full_name": "Nat.pair_lt_pair_left", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [127, 9], "def_end_pos": [127, 26]}]], "state_before": "case succ.rfind'.some.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nn\u271d : \u2115\n\u22a2 Nat.Partrec.Code.lup\n      (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n        (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n      (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) =\n    evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ.rfind'.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind Option.none fun x =>\n      Nat.rec (some (unpair n).2)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))\n        x) =\n    Option.bind Option.none fun x =>\n      if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))", "state_after": "no goals"}, {"tactic": "simp [evaln_map]", "annotated_tactic": ["simp [<a>evaln_map</a>]", [{"full_name": "_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.evaln_map", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [1075, 17], "def_end_pos": [1075, 26]}]], "state_before": "this :\n  Primrec\u2082 fun x n =>\n    let a := ofNat (\u2115 \u00d7 Code) n;\n    List.map (evaln a.1 a.2) (List.range a.1)\nx\u271d : (\u2115 \u00d7 Code) \u00d7 \u2115\nk : \u2115\nc : Code\nn : \u2115\n\u22a2 (Option.bind\n      (List.get?\n        (let a := ofNat (\u2115 \u00d7 Code) (encode ((k, c), n).1);\n        List.map (evaln a.1 a.2) (List.range a.1))\n        ((k, c), n).2)\n      fun b => (((k, c), n), b).2) =\n    evaln ((k, c), n).1.1 ((k, c), n).1.2 ((k, c), n).2", "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_Ioo_add_int_cast", "start": [245, 1], "end": [248, 46], "traced_tactics": [{"tactic": "simpa only [zsmul_one, Int.cast_add, Int.cast_one, \u2190 add_assoc] using\n  pairwise_disjoint_Ioo_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_Ioo_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_Ioo_add_zsmul", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [199, 3], "def_end_pos": [199, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedRing \u03b1\na : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioo (a + \u2191n) (a + \u2191n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_union", "start": [981, 1], "end": [984, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_normalize_of_tendsto", "start": [476, 1], "end": [481, 77], "traced_tactics": [{"tactic": "rw [ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds,\n  tendsto_iff_forall_testAgainstNN_tendsto]", "annotated_tactic": ["rw [<a>ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds</a>,\n    <a>tendsto_iff_forall_testAgainstNN_tendsto</a>]", [{"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]}, {"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": "\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\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))", "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\n\u22a2 \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))"}, {"tactic": "exact fun f => tendsto_normalize_testAgainstNN_of_tendsto \u03bcs_lim nonzero f", "annotated_tactic": ["exact fun f => <a>tendsto_normalize_testAgainstNN_of_tendsto</a> \u03bcs_lim nonzero f", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_normalize_testAgainstNN_of_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [438, 9], "def_end_pos": [438, 51]}]], "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\n\u22a2 \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))", "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_sub_eq_lintegral", "start": [1386, 1], "end": [1391, 31], "traced_tactics": [{"tactic": "simp_rw [ofReal_norm_eq_lintegral, \u2190 edist_eq_coe_nnnorm]", "annotated_tactic": ["simp_rw [<a>ofReal_norm_eq_lintegral</a>, \u2190 <a>edist_eq_coe_nnnorm</a>]", [{"full_name": "MeasureTheory.L1.ofReal_norm_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1377, 9], "def_end_pos": [1377, 33]}, {"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 g : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 ENNReal.ofReal \u2016f - g\u2016 = \u222b\u207b (x : \u03b1), \u2191\u2016\u2191\u2191f x - \u2191\u2191g 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 g : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 \u222b\u207b (x : \u03b1), edist (\u2191\u2191(f - g) x) 0 \u2202\u03bc = \u222b\u207b (x : \u03b1), edist (\u2191\u2191f x - \u2191\u2191g x) 0 \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\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 : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 \u222b\u207b (x : \u03b1), edist (\u2191\u2191(f - g) x) 0 \u2202\u03bc = \u222b\u207b (x : \u03b1), edist (\u2191\u2191f x - \u2191\u2191g x) 0 \u2202\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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf g : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 (fun a => edist (\u2191\u2191(f - g) a) 0) =\u1d50[\u03bc] fun a => edist (\u2191\u2191f a - \u2191\u2191g a) 0"}, {"tactic": "filter_upwards [Lp.coeFn_sub f g] with _ ha", "annotated_tactic": ["filter_upwards [<a>Lp.coeFn_sub</a> f g] with _ ha", [{"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 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 g : { x // x \u2208 Lp \u03b2 1 }\n\u22a2 (fun a => edist (\u2191\u2191(f - g) a) 0) =\u1d50[\u03bc] fun a => edist (\u2191\u2191f a - \u2191\u2191g a) 0", "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 g : { x // x \u2208 Lp \u03b2 1 }\na\u271d : \u03b1\nha : \u2191\u2191(f - g) a\u271d = (\u2191\u2191f - \u2191\u2191g) a\u271d\n\u22a2 edist (\u2191\u2191(f - g) a\u271d) 0 = edist (\u2191\u2191f a\u271d - \u2191\u2191g a\u271d) 0"}, {"tactic": "simp only [ha, Pi.sub_apply]", "annotated_tactic": ["simp only [ha, <a>Pi.sub_apply</a>]", [{"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\u03b1 : Type u_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 : { x // x \u2208 Lp \u03b2 1 }\na\u271d : \u03b1\nha : \u2191\u2191(f - g) a\u271d = (\u2191\u2191f - \u2191\u2191g) a\u271d\n\u22a2 edist (\u2191\u2191(f - g) a\u271d) 0 = edist (\u2191\u2191f a\u271d - \u2191\u2191g a\u271d) 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.toDual_max'", "start": [1546, 1], "end": [1551, 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\nhs : Finset.Nonempty s\n\u22a2 \u2191toDual (max' s hs) = min' (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 (max' s hs)) = \u2191(min' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s)))"}, {"tactic": "simp only [max'_eq_sup', id_eq, toDual_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>toDual_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.toDual_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1112, 9], "def_end_pos": [1112, 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\nhs : Finset.Nonempty s\n\u22a2 \u2191(\u2191toDual (max' s hs)) = \u2191(min' (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 inf s (WithTop.some \u2218 fun x => \u2191toDual x) = inf s ((WithTop.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 inf s (WithTop.some \u2218 fun x => \u2191toDual x) = inf s ((WithTop.some \u2218 fun x => x) \u2218 \u2191toDual)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.cgf_zero_measure", "start": [126, 1], "end": [127, 46], "traced_tactics": [{"tactic": "simp only [cgf, log_zero, mgf_zero_measure]", "annotated_tactic": ["simp only [<a>cgf</a>, <a>log_zero</a>, <a>mgf_zero_measure</a>]", [{"full_name": "ProbabilityTheory.cgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [108, 5], "def_end_pos": [108, 8]}, {"full_name": "Real.log_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 17]}, {"full_name": "ProbabilityTheory.mgf_zero_measure", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [122, 9], "def_end_pos": [122, 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\nt : \u211d\n\u22a2 cgf X 0 t = 0", "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_add_withDensity_rnDeriv_eq", "start": [930, 1], "end": [950, 20], "traced_tactics": [{"tactic": "conv_rhs =>\n  rw [\u2190 toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure]", "annotated_tactic": ["conv_rhs =>\n    rw [\u2190 <a>toSignedMeasure_toJordanDecomposition</a> s, <a>JordanDecomposition.toSignedMeasure</a>]", [{"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.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\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 singularPart s \u03bc + withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 singularPart s \u03bc + withDensity\u1d65 \u03bc (rnDeriv s \u03bc) =\n    toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart"}, {"tactic": "rw [singularPart, rnDeriv,\n  withDensity\u1d65_sub' (integrable_toReal_of_lintegral_ne_top _ _)\n    (integrable_toReal_of_lintegral_ne_top _ _),\n  withDensity\u1d65_toReal, withDensity\u1d65_toReal, sub_eq_add_neg, sub_eq_add_neg,\n  add_comm (s.toJordanDecomposition.posPart.singularPart \u03bc).toSignedMeasure, \u2190 add_assoc,\n  add_assoc (-(s.toJordanDecomposition.negPart.singularPart \u03bc).toSignedMeasure),\n  \u2190 toSignedMeasure_add, add_comm, \u2190 add_assoc, \u2190 neg_add, \u2190 toSignedMeasure_add, add_comm,\n  \u2190 sub_eq_add_neg]", "annotated_tactic": ["rw [<a>singularPart</a>, <a>rnDeriv</a>,\n    <a>withDensity\u1d65_sub'</a> (<a>integrable_toReal_of_lintegral_ne_top</a> _ _)\n      (<a>integrable_toReal_of_lintegral_ne_top</a> _ _),\n    <a>withDensity\u1d65_toReal</a>, <a>withDensity\u1d65_toReal</a>, <a>sub_eq_add_neg</a>, <a>sub_eq_add_neg</a>,\n    <a>add_comm</a> (s.toJordanDecomposition.posPart.singularPart \u03bc).<a>toSignedMeasure</a>, \u2190 <a>add_assoc</a>,\n    <a>add_assoc</a> (-(s.toJordanDecomposition.negPart.singularPart \u03bc).<a>toSignedMeasure</a>),\n    \u2190 <a>toSignedMeasure_add</a>, <a>add_comm</a>, \u2190 <a>add_assoc</a>, \u2190 <a>neg_add</a>, \u2190 <a>toSignedMeasure_add</a>, <a>add_comm</a>,\n    \u2190 <a>sub_eq_add_neg</a>]", [{"full_name": "MeasureTheory.SignedMeasure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [839, 5], "def_end_pos": [839, 17]}, {"full_name": "MeasureTheory.SignedMeasure.rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [894, 5], "def_end_pos": [894, 12]}, {"full_name": "MeasureTheory.withDensity\u1d65_sub'", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [107, 9], "def_end_pos": [107, 26]}, {"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]}, {"full_name": "MeasureTheory.withDensity\u1d65_toReal", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [166, 9], "def_end_pos": [166, 28]}, {"full_name": "MeasureTheory.withDensity\u1d65_toReal", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [166, 9], "def_end_pos": [166, 28]}, {"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_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [410, 5], "def_end_pos": [410, 20]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [410, 5], "def_end_pos": [410, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_add", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [465, 9], "def_end_pos": [465, 28]}, {"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": "neg_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [481, 15], "def_end_pos": [481, 22]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_add", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [465, 9], "def_end_pos": [465, 28]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 singularPart s \u03bc + withDensity\u1d65 \u03bc (rnDeriv s \u03bc) =\n    toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 toSignedMeasure\n        (Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n          withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x) -\n      toSignedMeasure\n        ((withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) +\n          Measure.singularPart (toJordanDecomposition s).negPart \u03bc) =\n    toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4"}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 toSignedMeasure\n        (Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n          withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x) -\n      toSignedMeasure\n        ((withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) +\n          Measure.singularPart (toJordanDecomposition s).negPart \u03bc) =\n    toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4", "state_after": "case h.e'_3.h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 (toJordanDecomposition s).posPart =\n    Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n      withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\ncase h.e'_3.h.e'_6.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 (toJordanDecomposition s).negPart =\n    (withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) +\n      Measure.singularPart (toJordanDecomposition s).negPart \u03bc\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4"}, {"tactic": "all_goals\n  first\n  | exact (lintegral_rnDeriv_lt_top _ _).ne\n  | measurability", "annotated_tactic": ["all_goals\n    first\n    | exact (<a>lintegral_rnDeriv_lt_top</a> _ _).<a>ne</a>\n    | measurability", [{"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 hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\ncase hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc", "annotated_tactic": ["exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc", []], "state_before": "case h.e'_3.h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 (toJordanDecomposition s).posPart =\n    Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n      withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x", "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": "case h.e'_3.h.e'_6.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 (toJordanDecomposition s).negPart =\n    (withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) +\n      Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case h.e'_3.h.e'_6.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 (toJordanDecomposition s).negPart =\n    Measure.singularPart (toJordanDecomposition s).negPart \u03bc +\n      withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x"}, {"tactic": "exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc", "annotated_tactic": ["exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc", []], "state_before": "case h.e'_3.h.e'_6.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 (toJordanDecomposition s).negPart =\n    Measure.singularPart (toJordanDecomposition s).negPart \u03bc +\n      withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x", "state_after": "no goals"}, {"tactic": "first\n| exact (lintegral_rnDeriv_lt_top _ _).ne\n| measurability", "annotated_tactic": ["first\n    | exact (<a>lintegral_rnDeriv_lt_top</a> _ _).<a>ne</a>\n    | measurability", [{"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "measurability", "annotated_tactic": ["measurability", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t : SignedMeasure \u03b1\ninst\u271d : HaveLebesgueDecomposition s \u03bc\n\u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x", "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_Ico", "start": [1093, 1], "end": [1096, 38], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_map, coe_Ico, coe_Ico]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_map</a>, <a>coe_Ico</a>, <a>coe_Ico</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_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]}]], "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) (Ico a b) = Ico (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.Ico a b = Set.Ico (c + a) (c + b)"}, {"tactic": "exact Set.image_const_add_Ico _ _ _", "annotated_tactic": ["exact <a>Set.image_const_add_Ico</a> _ _ _", [{"full_name": "Set.image_const_add_Ico", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [129, 9], "def_end_pos": [129, 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.Ico a b = Set.Ico (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed", "start": [329, 1], "end": [338, 35], "traced_tactics": [{"tactic": "apply measure_of_cont_bdd_of_tendsto_indicator \u03bc F_closed.measurableSet\n  (fun n => thickenedIndicator (\u03b4s_pos n) F) fun n \u03c9 => thickenedIndicator_le_one (\u03b4s_pos n) F \u03c9", "annotated_tactic": ["apply <a>measure_of_cont_bdd_of_tendsto_indicator</a> \u03bc F_closed.measurableSet\n    (fun n => <a>thickenedIndicator</a> (\u03b4s_pos n) F) fun n \u03c9 => <a>thickenedIndicator_le_one</a> (\u03b4s_pos n) F \u03c9", [{"full_name": "MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [312, 9], "def_end_pos": [312, 49]}, {"full_name": "thickenedIndicator", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [163, 5], "def_end_pos": [163, 23]}, {"full_name": "thickenedIndicator_le_one", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [187, 9], "def_end_pos": [187, 34]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nF : Set \u03a9\nF_closed : IsClosed F\n\u03b4s : \u2115 \u2192 \u211d\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc F))", "state_after": "\u03a9\u271d : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nF : Set \u03a9\nF_closed : IsClosed F\n\u03b4s : \u2115 \u2192 \u211d\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator F fun x => 1))"}, {"tactic": "have key := thickenedIndicator_tendsto_indicator_closure \u03b4s_pos \u03b4s_lim F", "annotated_tactic": ["have key := <a>thickenedIndicator_tendsto_indicator_closure</a> \u03b4s_pos \u03b4s_lim F", [{"full_name": "thickenedIndicator_tendsto_indicator_closure", "def_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "def_pos": [236, 9], "def_end_pos": [236, 53]}]], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nF : Set \u03a9\nF_closed : IsClosed F\n\u03b4s : \u2115 \u2192 \u211d\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator F fun x => 1))", "state_after": "\u03a9\u271d : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nF : Set \u03a9\nF_closed : IsClosed F\n\u03b4s : \u2115 \u2192 \u211d\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey : Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator (closure F) fun x => 1))\n\u22a2 Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator F fun x => 1))"}, {"tactic": "rwa [F_closed.closure_eq] at key", "annotated_tactic": ["rwa [F_closed.closure_eq] at key", []], "state_before": "\u03a9\u271d : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nF : Set \u03a9\nF_closed : IsClosed F\n\u03b4s : \u2115 \u2192 \u211d\n\u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n\n\u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0)\nkey : Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator (closure F) fun x => 1))\n\u22a2 Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator F fun x => 1))", "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_smul_deriv_Ioi", "start": [805, 1], "end": [829, 94], "traced_tactics": [{"tactic": "rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2", "annotated_tactic": ["rw [<a>integrableOn_Ici_iff_integrableOn_Ioi</a>] at hg2", [{"full_name": "integrableOn_Ici_iff_integrableOn_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [723, 9], "def_end_pos": [723, 46]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u"}, {"tactic": "have t2 := intervalIntegral_tendsto_integral_Ioi _ hg2 tendsto_id", "annotated_tactic": ["have t2 := <a>intervalIntegral_tendsto_integral_Ioi</a> _ hg2 <a>tendsto_id</a>", [{"full_name": "MeasureTheory.intervalIntegral_tendsto_integral_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [642, 9], "def_end_pos": [642, 46]}, {"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\u271d : \u211d \u2192 E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\nt2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x))\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u"}, {"tactic": "have : Ioi (f a) \u2286 f '' Ici a :=\n  Ioi_subset_Ici_self.trans <|\n    IsPreconnected.intermediate_value_Ici isPreconnected_Ici left_mem_Ici\n      (le_principal_iff.mpr <| Ici_mem_atTop _) hf hft", "annotated_tactic": ["have : <a>Ioi</a> (f a) \u2286 f '' <a>Ici</a> a :=\n    Ioi_subset_Ici_self.trans <|\n      <a>IsPreconnected.intermediate_value_Ici</a> <a>isPreconnected_Ici</a> <a>left_mem_Ici</a>\n        (le_principal_iff.mpr <| <a>Ici_mem_atTop</a> _) hf hft", [{"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": "IsPreconnected.intermediate_value_Ici", "def_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "def_pos": [151, 9], "def_end_pos": [151, 46]}, {"full_name": "isPreconnected_Ici", "def_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "def_pos": [434, 9], "def_end_pos": [434, 27]}, {"full_name": "Set.left_mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 21]}, {"full_name": "Filter.Ici_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [57, 9], "def_end_pos": [57, 22]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\nt2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x))\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\nt2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x))\nthis : Ioi (f a) \u2286 f '' Ici a\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u"}, {"tactic": "have t1 := (intervalIntegral_tendsto_integral_Ioi _ (hg1.mono_set this) tendsto_id).comp hft", "annotated_tactic": ["have t1 := (<a>intervalIntegral_tendsto_integral_Ioi</a> _ (hg1.mono_set this) <a>tendsto_id</a>).<a>comp</a> hft", [{"full_name": "MeasureTheory.intervalIntegral_tendsto_integral_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [642, 9], "def_end_pos": [642, 46]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3028, 9], "def_end_pos": [3028, 19]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\nt2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x))\nthis : Ioi (f a) \u2286 f '' Ici a\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\nt2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x))\nthis : Ioi (f a) \u2286 f '' Ici a\nt1 : Tendsto ((fun i => \u222b (x : \u211d) in f a..id i, g x) \u2218 f) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi (f a), g x))\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u"}, {"tactic": "exact tendsto_nhds_unique (Tendsto.congr' (eventuallyEq_of_mem (Ioi_mem_atTop a) eq) t2) t1", "annotated_tactic": ["exact <a>tendsto_nhds_unique</a> (<a>Tendsto.congr'</a> (<a>eventuallyEq_of_mem</a> (<a>Ioi_mem_atTop</a> a) eq) t2) t1", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}, {"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}, {"full_name": "Filter.eventuallyEq_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1473, 9], "def_end_pos": [1473, 28]}, {"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ioi a)\neq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u\nt2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x))\nthis : Ioi (f a) \u2286 f '' Ici a\nt1 : Tendsto ((fun i => \u222b (x : \u211d) in f a..id i, g x) \u2218 f) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi (f a), g x))\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u", "state_after": "no goals"}, {"tactic": "have i1 : Ioo (min a b) (max a b) \u2286 Ioi a := by\n  rw [min_eq_left hb.le]\n  exact Ioo_subset_Ioi_self", "annotated_tactic": ["have i1 : <a>Ioo</a> (<a>min</a> a b) (<a>max</a> a b) \u2286 <a>Ioi</a> a := by\n      rw [<a>min_eq_left</a> hb.le]\n      exact <a>Ioo_subset_Ioi_self</a>", [{"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": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "Set.Ioo_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [529, 9], "def_end_pos": [529, 28]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u"}, {"tactic": "have i2 : [[a, b]] \u2286 Ici a := by rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self", "annotated_tactic": ["have i2 : [[a, b]] \u2286 <a>Ici</a> a := by rw [<a>uIcc_of_le</a> hb.le]; exact <a>Icc_subset_Ici_self</a>", [{"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 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": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u"}, {"tactic": "refine'\n  intervalIntegral.integral_comp_smul_deriv''' (hf.mono i2)\n    (fun x hx => hff' x <| mem_of_mem_of_subset hx i1) (hg_cont.mono <| image_subset _ _)\n    (hg1.mono_set <| image_subset _ _) (hg2.mono_set i2)", "annotated_tactic": ["refine'\n      <a>intervalIntegral.integral_comp_smul_deriv'''</a> (hf.mono i2)\n        (fun x hx => hff' x <| <a>mem_of_mem_of_subset</a> hx i1) (hg_cont.mono <| <a>image_subset</a> _ _)\n        (hg1.mono_set <| <a>image_subset</a> _ _) (hg2.mono_set i2)", [{"full_name": "intervalIntegral.integral_comp_smul_deriv'''", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1354, 9], "def_end_pos": [1354, 36]}, {"full_name": "Set.mem_of_mem_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 29]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u", "state_after": "case refine'_1\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 Ioo (min a b) (max a b) \u2286 Ioi a\n\ncase refine'_2\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 [[a, b]] \u2286 Ici a"}, {"tactic": "rw [min_eq_left hb.le]", "annotated_tactic": ["rw [<a>min_eq_left</a> hb.le]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\n\u22a2 Ioo (min a b) (max a b) \u2286 Ioi a", "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\n\u22a2 Ioo a (max a b) \u2286 Ioi a"}, {"tactic": "exact Ioo_subset_Ioi_self", "annotated_tactic": ["exact <a>Ioo_subset_Ioi_self</a>", [{"full_name": "Set.Ioo_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [529, 9], "def_end_pos": [529, 28]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\n\u22a2 Ioo a (max a b) \u2286 Ioi a", "state_after": "no goals"}, {"tactic": "rw [uIcc_of_le hb.le]", "annotated_tactic": ["rw [<a>uIcc_of_le</a> hb.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": "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\n\u22a2 [[a, b]] \u2286 Ici a", "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\n\u22a2 Icc a b \u2286 Ici a"}, {"tactic": "exact Icc_subset_Ici_self", "annotated_tactic": ["exact <a>Icc_subset_Ici_self</a>", [{"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\n\u22a2 Icc a b \u2286 Ici a", "state_after": "no goals"}, {"tactic": "rw [min_eq_left hb.le]", "annotated_tactic": ["rw [<a>min_eq_left</a> hb.le]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "state_before": "case refine'_1\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 Ioo (min a b) (max a b) \u2286 Ioi a", "state_after": "case refine'_1\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 Ioo a (max a b) \u2286 Ioi a"}, {"tactic": "exact Ioo_subset_Ioi_self", "annotated_tactic": ["exact <a>Ioo_subset_Ioi_self</a>", [{"full_name": "Set.Ioo_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [529, 9], "def_end_pos": [529, 28]}]], "state_before": "case refine'_1\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 Ioo a (max a b) \u2286 Ioi a", "state_after": "no goals"}, {"tactic": "rw [uIcc_of_le hb.le]", "annotated_tactic": ["rw [<a>uIcc_of_le</a> hb.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 refine'_2\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 [[a, b]] \u2286 Ici a", "state_after": "case refine'_2\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 Icc a b \u2286 Ici a"}, {"tactic": "exact Icc_subset_Ici_self", "annotated_tactic": ["exact <a>Icc_subset_Ici_self</a>", [{"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "state_before": "case refine'_2\nE : 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' : \u211d \u2192 \u211d\ng : \u211d \u2192 E\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 => f' x \u2022 (g \u2218 f) x) (Ici a)\nb : \u211d\nhb : a < b\ni1 : Ioo (min a b) (max a b) \u2286 Ioi a\ni2 : [[a, b]] \u2286 Ici a\n\u22a2 Icc a b \u2286 Ici a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "LipschitzWith.continuous_compLp", "start": [1074, 1], "end": [1076, 42], "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_combine", "start": [130, 1], "end": [137, 8], "traced_tactics": [{"tactic": "unfold combine", "annotated_tactic": ["unfold <a>combine</a>", [{"full_name": "Std.PairingHeapImp.Heap.combine", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [51, 19], "def_end_pos": [51, 31]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 size (combine le s) = size s", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 size\n      (match s with\n      | h\u2081@h_1:(node a child h\u2082@h:(node a_1 child_1 s)) => merge le (merge le h\u2081 h\u2082) (combine le s)\n      | h => h) =\n    size s"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\n\u22a2 size\n      (match s with\n      | h\u2081@h_1:(node a child h\u2082@h:(node a_1 child_1 s)) => merge le (merge le h\u2081 h\u2082) (combine le s)\n      | h => h) =\n    size s", "state_after": "case h_1\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nx\u271d : Heap \u03b1\na\u271d\u00b9 : \u03b1\nchild\u271d\u00b9 : Heap \u03b1\na\u271d : \u03b1\nchild\u271d s\u271d : Heap \u03b1\n\u22a2 size (merge le (merge le (node a\u271d\u00b9 child\u271d\u00b9 (node a\u271d child\u271d s\u271d)) (node a\u271d child\u271d s\u271d)) (combine le s\u271d)) =\n    size (node a\u271d\u00b9 child\u271d\u00b9 (node a\u271d child\u271d s\u271d))\n\ncase h_2\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns x\u271d\u00b9 : Heap \u03b1\nx\u271d : \u2200 (a : \u03b1) (child : Heap \u03b1) (a_1 : \u03b1) (child_1 s_1 : Heap \u03b1), s = node a child (node a_1 child_1 s_1) \u2192 False\n\u22a2 size s = size s"}, {"tactic": "rename_i a\u2081 c\u2081 a\u2082 c\u2082 s", "annotated_tactic": ["rename_i a\u2081 c\u2081 a\u2082 c\u2082 s", []], "state_before": "case h_1\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nx\u271d : Heap \u03b1\na\u271d\u00b9 : \u03b1\nchild\u271d\u00b9 : Heap \u03b1\na\u271d : \u03b1\nchild\u271d s\u271d : Heap \u03b1\n\u22a2 size (merge le (merge le (node a\u271d\u00b9 child\u271d\u00b9 (node a\u271d child\u271d s\u271d)) (node a\u271d child\u271d s\u271d)) (combine le s\u271d)) =\n    size (node a\u271d\u00b9 child\u271d\u00b9 (node a\u271d child\u271d s\u271d))", "state_after": "case h_1\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nx\u271d : Heap \u03b1\na\u2081 : \u03b1\nc\u2081 : Heap \u03b1\na\u2082 : \u03b1\nc\u2082 s : Heap \u03b1\n\u22a2 size (merge le (merge le (node a\u2081 c\u2081 (node a\u2082 c\u2082 s)) (node a\u2082 c\u2082 s)) (combine le s)) =\n    size (node a\u2081 c\u2081 (node a\u2082 c\u2082 s))"}, {"tactic": "rw [size_merge le (noSibling_merge _ _ _) (noSibling_combine _ _),\n  size_merge_node, size_combine le s]", "annotated_tactic": ["rw [<a>size_merge</a> le (<a>noSibling_merge</a> _ _ _) (<a>noSibling_combine</a> _ _),\n      <a>size_merge_node</a>, size_combine le s]", [{"full_name": "Std.PairingHeapImp.Heap.size_merge", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [124, 9], "def_end_pos": [124, 24]}, {"full_name": "Std.PairingHeapImp.Heap.noSibling_merge", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [91, 9], "def_end_pos": [91, 29]}, {"full_name": "Std.PairingHeapImp.Heap.noSibling_combine", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [96, 9], "def_end_pos": [96, 31]}, {"full_name": "Std.PairingHeapImp.Heap.size_merge_node", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [120, 9], "def_end_pos": [120, 29]}]], "state_before": "case h_1\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nx\u271d : Heap \u03b1\na\u2081 : \u03b1\nc\u2081 : Heap \u03b1\na\u2082 : \u03b1\nc\u2082 s : Heap \u03b1\n\u22a2 size (merge le (merge le (node a\u2081 c\u2081 (node a\u2082 c\u2082 s)) (node a\u2082 c\u2082 s)) (combine le s)) =\n    size (node a\u2081 c\u2081 (node a\u2082 c\u2082 s))", "state_after": "case h_1\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nx\u271d : Heap \u03b1\na\u2081 : \u03b1\nc\u2081 : Heap \u03b1\na\u2082 : \u03b1\nc\u2082 s : Heap \u03b1\n\u22a2 size c\u2081 + size c\u2082 + 2 + size s = size (node a\u2081 c\u2081 (node a\u2082 c\u2082 s))"}, {"tactic": "simp_arith [size]", "annotated_tactic": ["simp_arith [<a>size</a>]", [{"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 h_1\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nx\u271d : Heap \u03b1\na\u2081 : \u03b1\nc\u2081 : Heap \u03b1\na\u2082 : \u03b1\nc\u2082 s : Heap \u03b1\n\u22a2 size c\u2081 + size c\u2082 + 2 + size s = size (node a\u2081 c\u2081 (node a\u2082 c\u2082 s))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h_2\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns x\u271d\u00b9 : Heap \u03b1\nx\u271d : \u2200 (a : \u03b1) (child : Heap \u03b1) (a_1 : \u03b1) (child_1 s_1 : Heap \u03b1), s = node a child (node a_1 child_1 s_1) \u2192 False\n\u22a2 size s = size s", "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.foldl_eq", "start": [59, 9], "end": [61, 45], "traced_tactics": [{"tactic": "simp [List.foldl_eq_foldlM, foldl, Id.run]", "annotated_tactic": ["simp [<a>List.foldl_eq_foldlM</a>, <a>foldl</a>, <a>Id.run</a>]", [{"full_name": "List.foldl_eq_foldlM", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [205, 9], "def_end_pos": [205, 24]}, {"full_name": "Std.AssocList.foldl", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [56, 15], "def_end_pos": [56, 20]}, {"full_name": "Id.run", "def_path": "lake-packages/lean4/src/lean/Init/Control/Id.lean", "def_pos": [27, 15], "def_end_pos": [27, 18]}]], "state_before": "\u03b4 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nf : \u03b4 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b4\ninit : \u03b4\nl : AssocList \u03b1 \u03b2\n\u22a2 foldl f init l =\n    List.foldl\n      (fun d x =>\n        match x with\n        | (a, b) => f d a b)\n      init (toList l)", "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_add_le_left", "start": [1001, 11], "end": [1003, 36], "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 : -b + a \u2264 c\n\u22a2 a \u2264 b + c", "state_after": "a b c : Int\nh : a + -b \u2264 c\n\u22a2 a \u2264 b + c"}, {"tactic": "exact Int.le_add_of_sub_left_le h", "annotated_tactic": ["exact <a>Int.le_add_of_sub_left_le</a> h", [{"full_name": "Int.le_add_of_sub_left_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [985, 19], "def_end_pos": [985, 40]}]], "state_before": "a b c : Int\nh : a + -b \u2264 c\n\u22a2 a \u2264 b + c", "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_of_inter_eq", "start": [2342, 1], "end": [2343, 67], "traced_tactics": [{"tactic": "rw [\u2190 ite_inter, \u2190 h, ite_same]", "annotated_tactic": ["rw [\u2190 <a>ite_inter</a>, \u2190 h, <a>ite_same</a>]", [{"full_name": "Set.ite_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2338, 9], "def_end_pos": [2338, 18]}, {"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\nh : s\u2081 \u2229 s = s\u2082 \u2229 s\n\u22a2 Set.ite t s\u2081 s\u2082 \u2229 s = s\u2081 \u2229 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Option/Basic.lean", "full_name": "Option.isNone_iff_eq_none", "start": [24, 1], "end": [25, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.evaln_complete", "start": [858, 1], "end": [924, 33], "traced_tactics": [{"tactic": "rsuffices \u27e8k, h\u27e9 : \u2203 k, x \u2208 evaln (k + 1) c n", "annotated_tactic": ["rsuffices \u27e8k, h\u27e9 : \u2203 k, x \u2208 <a>evaln</a> (k + 1) c n", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}]], "state_before": "c : Code\nn x : \u2115\nh : x \u2208 eval c n\n\u22a2 \u2203 k, x \u2208 evaln k c n", "state_after": "case intro\nc : Code\nn x : \u2115\nh\u271d : x \u2208 eval c n\nk : \u2115\nh : x \u2208 evaln (k + 1) c n\n\u22a2 \u2203 k, x \u2208 evaln k c n\n\nc : Code\nn x : \u2115\nh : x \u2208 eval c n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) c n"}, {"tactic": "induction c generalizing n x <;> simp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2", "annotated_tactic": ["induction c generalizing n x <;> simp [<a>eval</a>, <a>evaln</a>, <a>pure</a>, <a>PFun.pure</a>, <a>Seq.seq</a>, <a>Bind.bind</a>] at h \u22a2", [{"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [620, 5], "def_end_pos": [620, 9]}, {"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"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": "Seq.seq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2698, 3], "def_end_pos": [2698, 6]}, {"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": "c : Code\nn x : \u2115\nh : x \u2208 eval c n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) c n", "state_after": "case zero\nn x : \u2115\nh : x = 0\n\u22a2 (\u2203 x, n \u2264 x) \u2227 0 = x\n\ncase succ\nn x : \u2115\nh : x = Nat.succ n\n\u22a2 (\u2203 x, n \u2264 x) \u2227 Nat.succ n = x\n\ncase left\nn x : \u2115\nh : x = (unpair n).1\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).1 = x\n\ncase right\nn x : \u2115\nh : x = (unpair n).2\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).2 = x\n\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x"}, {"tactic": "iterate 4 exact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9", "annotated_tactic": ["iterate 4 exact \u27e8\u27e8_, <a>le_rfl</a>\u27e9, h.symm\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case zero\nn x : \u2115\nh : x = 0\n\u22a2 (\u2203 x, n \u2264 x) \u2227 0 = x\n\ncase succ\nn x : \u2115\nh : x = Nat.succ n\n\u22a2 (\u2203 x, n \u2264 x) \u2227 Nat.succ n = x\n\ncase left\nn x : \u2115\nh : x = (unpair n).1\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).1 = x\n\ncase right\nn x : \u2115\nh : x = (unpair n).2\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).2 = x\n\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x", "state_after": "case pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x"}, {"tactic": "case pair cf cg hf hg =>\n  rcases h with \u27e8x, hx, y, hy, rfl\u27e9\n  rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9; rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  refine'\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n      evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081, _,\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9", "annotated_tactic": ["case pair cf cg hf hg =>\n      rcases h with \u27e8x, hx, y, hy, rfl\u27e9\n      rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9; rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9\n      refine' \u27e8<a>max</a> k\u2081 k\u2082, _\u27e9\n      refine'\n        \u27e8<a>le_max_of_le_left</a> <| <a>Nat.le_of_lt_succ</a> <| <a>evaln_bound</a> hk\u2081, _,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_left</a> _ _) hk\u2081, _,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_right</a> _ _) hk\u2082, <a>rfl</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": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"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": "Nat.Partrec.Code.evaln_bound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [775, 9], "def_end_pos": [775, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 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 pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x", "state_after": "case comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x"}, {"tactic": "case comp cf cg hf hg =>\n  rcases h with \u27e8y, hy, hx\u27e9\n  rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9; rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  exact\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n      evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9", "annotated_tactic": ["case comp cf cg hf hg =>\n      rcases h with \u27e8y, hy, hx\u27e9\n      rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9; rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9\n      refine' \u27e8<a>max</a> k\u2081 k\u2082, _\u27e9\n      exact\n        \u27e8<a>le_max_of_le_left</a> <| <a>Nat.le_of_lt_succ</a> <| <a>evaln_bound</a> hk\u2081, _,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_left</a> _ _) hk\u2081,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_right</a> _ _) hk\u2082\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": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"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": "Nat.Partrec.Code.evaln_bound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [775, 9], "def_end_pos": [775, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x", "state_after": "case prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x"}, {"tactic": "exact \u27e8k + 1, h\u27e9", "annotated_tactic": ["exact \u27e8k + 1, h\u27e9", []], "state_before": "case intro\nc : Code\nn x : \u2115\nh\u271d : x \u2208 eval c n\nk : \u2115\nh : x \u2208 evaln (k + 1) c n\n\u22a2 \u2203 k, x \u2208 evaln k c n", "state_after": "no goals"}, {"tactic": "exact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9", "annotated_tactic": ["exact \u27e8\u27e8_, <a>le_rfl</a>\u27e9, h.symm\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case right\nn x : \u2115\nh : x = (unpair n).2\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).2 = x\n\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x", "state_after": "case pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\n\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\n\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x\n\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x"}, {"tactic": "rcases h with \u27e8x, hx, y, hy, rfl\u27e9", "annotated_tactic": ["rcases h with \u27e8x, hx, y, hy, rfl\u27e9", []], "state_before": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh : \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x", "state_after": "case intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y"}, {"tactic": "rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9", "annotated_tactic": ["rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9", []], "state_before": "case intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y", "state_after": "case intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y"}, {"tactic": "rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9", "annotated_tactic": ["rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9", []], "state_before": "case intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y", "state_after": "case intro.intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\nk\u2082 : \u2115\nhk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y"}, {"tactic": "refine' \u27e8max k\u2081 k\u2082, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>max</a> k\u2081 k\u2082, _\u27e9", [{"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": "case intro.intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\nk\u2082 : \u2115\nhk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y", "state_after": "case intro.intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\nk\u2082 : \u2115\nhk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n\n\u22a2 n \u2264 max k\u2081 k\u2082 \u2227\n    \u2203 a,\n      evaln (max k\u2081 k\u2082 + 1) cf n = some a \u2227 \u2203 a_1, evaln (max k\u2081 k\u2082 + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y"}, {"tactic": "refine'\n  \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081, _,\n    evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9", "annotated_tactic": ["refine'\n        \u27e8<a>le_max_of_le_left</a> <| <a>Nat.le_of_lt_succ</a> <| <a>evaln_bound</a> hk\u2081, _,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_left</a> _ _) hk\u2081, _,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_right</a> _ _) hk\u2082, <a>rfl</a>\u27e9", [{"full_name": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"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": "Nat.Partrec.Code.evaln_bound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [775, 9], "def_end_pos": [775, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 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.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\nk\u2082 : \u2115\nhk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n\n\u22a2 n \u2264 max k\u2081 k\u2082 \u2227\n    \u2203 a,\n      evaln (max k\u2081 k\u2082 + 1) cf n = some a \u2227 \u2203 a_1, evaln (max k\u2081 k\u2082 + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8y, hy, hx\u27e9", "annotated_tactic": ["rcases h with \u27e8y, hy, hx\u27e9", []], "state_before": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh : \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x", "state_after": "case intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x"}, {"tactic": "rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9", "annotated_tactic": ["rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9", []], "state_before": "case intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x", "state_after": "case intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x"}, {"tactic": "rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9", "annotated_tactic": ["rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9", []], "state_before": "case intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x", "state_after": "case intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x"}, {"tactic": "refine' \u27e8max k\u2081 k\u2082, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>max</a> k\u2081 k\u2082, _\u27e9", [{"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": "case intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x", "state_after": "case intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y\n\u22a2 n \u2264 max k\u2081 k\u2082 \u2227 \u2203 a, evaln (max k\u2081 k\u2082 + 1) cg n = some a \u2227 evaln (max k\u2081 k\u2082 + 1) cf a = some x"}, {"tactic": "exact\n  \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n    evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9", "annotated_tactic": ["exact\n        \u27e8<a>le_max_of_le_left</a> <| <a>Nat.le_of_lt_succ</a> <| <a>evaln_bound</a> hk\u2081, _,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_left</a> _ _) hk\u2081,\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_right</a> _ _) hk\u2082\u27e9", [{"full_name": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"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": "Nat.Partrec.Code.evaln_bound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [775, 9], "def_end_pos": [775, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y\n\u22a2 n \u2264 max k\u2081 k\u2082 \u2227 \u2203 a, evaln (max k\u2081 k\u2082 + 1) cg n = some a \u2227 evaln (max k\u2081 k\u2082 + 1) cf a = some x", "state_after": "no goals"}, {"tactic": "revert h", "annotated_tactic": ["revert h", []], "state_before": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval cf (unpair n).1) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).1 (Nat.pair y i)))\n      (unpair n).2\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) cf (unpair n).1)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n          (unpair n).2 =\n        some x", "state_after": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\n\u22a2 x \u2208\n      Nat.rec (eval cf (unpair n).1) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).1 (Nat.pair y i)))\n        (unpair n).2 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf (unpair n).1)\n            (fun n_1 n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n                evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n            (unpair n).2 =\n          some x"}, {"tactic": "generalize n.unpair.1 = n\u2081", "annotated_tactic": ["generalize n.unpair.1 = n\u2081", []], "state_before": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\n\u22a2 x \u2208\n      Nat.rec (eval cf (unpair n).1) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).1 (Nat.pair y i)))\n        (unpair n).2 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf (unpair n).1)\n            (fun n_1 n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>\n                evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))\n            (unpair n).2 =\n          some x", "state_after": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x n\u2081 : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) (unpair n).2 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            (unpair n).2 =\n          some x"}, {"tactic": "generalize n.unpair.2 = n\u2082", "annotated_tactic": ["generalize n.unpair.2 = n\u2082", []], "state_before": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x n\u2081 : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) (unpair n).2 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            (unpair n).2 =\n          some x", "state_after": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x n\u2081 n\u2082 : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) n\u2082 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            n\u2082 =\n          some x"}, {"tactic": "induction' n\u2082 with m IH generalizing x n <;> simp", "annotated_tactic": ["induction' n\u2082 with m IH generalizing x n <;> simp", []], "state_before": "cf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x n\u2081 n\u2082 : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) n\u2082 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            n\u2082 =\n          some x", "state_after": "case zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\n\u22a2 x \u2208 eval cf n\u2081 \u2192 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x\n\ncase succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x : \u2115\n\u22a2 \u2200 (x_1 : \u2115),\n    x_1 \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m x_1)) \u2192\n        \u2203 k,\n          n \u2264 k \u2227\n            \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\n\u22a2 x \u2208 eval cf n\u2081 \u2192 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x", "state_after": "case zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\nh : x \u2208 eval cf n\u2081\n\u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x"}, {"tactic": "rcases hf h with \u27e8k, hk\u27e9", "annotated_tactic": ["rcases hf h with \u27e8k, hk\u27e9", []], "state_before": "case zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\nh : x \u2208 eval cf n\u2081\n\u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x", "state_after": "case zero.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\nh : x \u2208 eval cf n\u2081\nk : \u2115\nhk : x \u2208 evaln (k + 1) cf n\u2081\n\u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x"}, {"tactic": "exact \u27e8_, le_max_left _ _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u27e9", "annotated_tactic": ["exact \u27e8_, <a>le_max_left</a> _ _, <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_right</a> _ _) hk\u27e9", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case zero.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\nh : x \u2208 eval cf n\u2081\nk : \u2115\nhk : x \u2208 evaln (k + 1) cf n\u2081\n\u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x", "state_after": "no goals"}, {"tactic": "intro y hy hx", "annotated_tactic": ["intro y hy hx", []], "state_before": "case succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x : \u2115\n\u22a2 \u2200 (x_1 : \u2115),\n    x_1 \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m x_1)) \u2192\n        \u2203 k,\n          n \u2264 k \u2227\n            \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x", "state_after": "case succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x"}, {"tactic": "rcases IH hy with \u27e8k\u2081, nk\u2081, hk\u2081\u27e9", "annotated_tactic": ["rcases IH hy with \u27e8k\u2081, nk\u2081, hk\u2081\u27e9", []], "state_before": "case succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x", "state_after": "case succ.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : ?m.1146745 \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x"}, {"tactic": "rcases hg hx with \u27e8k\u2082, hk\u2082\u27e9", "annotated_tactic": ["rcases hg hx with \u27e8k\u2082, hk\u2082\u27e9", []], "state_before": "case succ.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : ?m.1146745 \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x", "state_after": "case succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : ?m.1146745 \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x"}, {"tactic": "refine'\n  \u27e8(max k\u2081 k\u2082).succ,\n    Nat.le_succ_of_le <| le_max_of_le_left <|\n      le_trans (le_max_left _ (Nat.pair n\u2081 m)) nk\u2081, y,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) _,\n    evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_right _ _) hk\u2082\u27e9", "annotated_tactic": ["refine'\n          \u27e8(<a>max</a> k\u2081 k\u2082).<a>succ</a>,\n            <a>Nat.le_succ_of_le</a> <| <a>le_max_of_le_left</a> <|\n              <a>le_trans</a> (<a>le_max_left</a> _ (<a>Nat.pair</a> n\u2081 m)) nk\u2081, y,\n            <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_left</a> _ _) _,\n            <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>Nat.le_succ_of_le</a> <| <a>le_max_right</a> _ _) hk\u2082\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": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.le_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1602, 9], "def_end_pos": [1602, 26]}, {"full_name": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"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": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1602, 9], "def_end_pos": [1602, 26]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : ?m.1146745 \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x", "state_after": "case succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 y \u2208 evaln (Nat.succ k\u2081) (prec cf cg) (Nat.pair n\u2081 m)"}, {"tactic": "simp only [evaln._eq_8, bind, unpaired, unpair_pair, Option.mem_def, Option.bind_eq_some,\n  Option.guard_eq_some', exists_and_left, exists_const]", "annotated_tactic": ["simp only [evaln._eq_8, <a>bind</a>, <a>unpaired</a>, <a>unpair_pair</a>, <a>Option.mem_def</a>, <a>Option.bind_eq_some</a>,\n          <a>Option.guard_eq_some'</a>, <a>exists_and_left</a>, <a>exists_const</a>]", [{"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": "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]}, {"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.bind_eq_some", "def_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "def_pos": [93, 17], "def_end_pos": [93, 29]}, {"full_name": "Option.guard_eq_some'", "def_path": "Mathlib/Data/Option/Basic.lean", "def_pos": [329, 9], "def_end_pos": [329, 23]}, {"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_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [369, 17], "def_end_pos": [369, 29]}]], "state_before": "case succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 y \u2208 evaln (Nat.succ k\u2081) (prec cf cg) (Nat.pair n\u2081 m)", "state_after": "case succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 Nat.pair n\u2081 m \u2264 k\u2081 \u2227\n    Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n        (fun n n_ih =>\n          Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n        m =\n      some y"}, {"tactic": "exact \u27e8le_trans (le_max_right _ _) nk\u2081, hk\u2081\u27e9", "annotated_tactic": ["exact \u27e8<a>le_trans</a> (<a>le_max_right</a> _ _) nk\u2081, hk\u2081\u27e9", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 Nat.pair n\u2081 m \u2264 k\u2081 \u2227\n    Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n        (fun n n_ih =>\n          Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n        m =\n      some y", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9", "annotated_tactic": ["rcases h with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9", []], "state_before": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).2 = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x", "state_after": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) =\n            some (y + (unpair n).2)"}, {"tactic": "suffices \u2203 k, y + n.unpair.2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair n.unpair.1 n.unpair.2) by\n  simpa [evaln, Bind.bind]", "annotated_tactic": ["suffices \u2203 k, y + n.unpair.2 \u2208 <a>evaln</a> (k + 1) (<a>rfind'</a> cf) (<a>Nat.pair</a> n.unpair.1 n.unpair.2) by\n        simpa [<a>evaln</a>, <a>Bind.bind</a>]", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"full_name": "Nat.Partrec.Code.rfind'", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [84, 5], "def_end_pos": [84, 11]}, {"full_name": "Nat.pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [37, 5], "def_end_pos": [37, 9]}, {"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"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 intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) =\n            some (y + (unpair n).2)", "state_after": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2)"}, {"tactic": "revert hy\u2081 hy\u2082", "annotated_tactic": ["revert hy\u2081 hy\u2082", []], "state_before": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2)", "state_after": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2)) \u2192\n    (\u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2192\n      \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2)"}, {"tactic": "generalize n.unpair.2 = m", "annotated_tactic": ["generalize n.unpair.2 = m", []], "state_before": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2)) \u2192\n    (\u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2192\n      \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2)", "state_after": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m : \u2115\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n    (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n      \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)"}, {"tactic": "intro hy\u2081 hy\u2082", "annotated_tactic": ["intro hy\u2081 hy\u2082", []], "state_before": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m : \u2115\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n    (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n      \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)", "state_after": "case intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)"}, {"tactic": "induction' y with y IH generalizing m <;> simp [evaln, Bind.bind]", "annotated_tactic": ["induction' y with y IH generalizing m <;> simp [<a>evaln</a>, <a>Bind.bind</a>]", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"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 intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)", "state_after": "case intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.zero + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m\n\ncase intro.intro.intro.succ\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)"}, {"tactic": "simpa [evaln, Bind.bind]", "annotated_tactic": ["simpa [<a>evaln</a>, <a>Bind.bind</a>]", [{"full_name": "Nat.Partrec.Code.evaln", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [736, 5], "def_end_pos": [736, 10]}, {"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": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0\nthis : \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2)\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) =\n            some (y + (unpair n).2)", "state_after": "no goals"}, {"tactic": "simp at hy\u2081", "annotated_tactic": ["simp at hy\u2081", []], "state_before": "case intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.zero + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m", "state_after": "case intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m"}, {"tactic": "rcases hf hy\u2081 with \u27e8k, hk\u27e9", "annotated_tactic": ["rcases hf hy\u2081 with \u27e8k, hk\u27e9", []], "state_before": "case intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m", "state_after": "case intro.intro.intro.zero.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m)\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m"}, {"tactic": "exact \u27e8_, Nat.le_of_lt_succ <| evaln_bound hk, _, hk, by simp; rfl\u27e9", "annotated_tactic": ["exact \u27e8_, <a>Nat.le_of_lt_succ</a> <| <a>evaln_bound</a> hk, _, hk, by simp; rfl\u27e9", [{"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": "Nat.Partrec.Code.evaln_bound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [775, 9], "def_end_pos": [775, 20]}]], "state_before": "case intro.intro.intro.zero.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m)\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m)\n\u22a2 (if 0 = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m", "state_after": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m)\n\u22a2 pure m = some m"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m)\n\u22a2 pure m = some m", "state_after": "no goals"}, {"tactic": "rcases hy\u2082 (Nat.succ_pos _) with \u27e8a, ha, a0\u27e9", "annotated_tactic": ["rcases hy\u2082 (<a>Nat.succ_pos</a> _) with \u27e8a, ha, a0\u27e9", [{"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 intro.intro.intro.succ\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)", "state_after": "case intro.intro.intro.succ.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)"}, {"tactic": "rcases hf ha with \u27e8k\u2081, hk\u2081\u27e9", "annotated_tactic": ["rcases hf ha with \u27e8k\u2081, hk\u2081\u27e9", []], "state_before": "case intro.intro.intro.succ.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)", "state_after": "case intro.intro.intro.succ.intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)"}, {"tactic": "rcases IH m.succ (by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081)\n    fun {i} hi => by\n    simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using\n      hy\u2082 (Nat.succ_lt_succ hi) with\n  \u27e8k\u2082, hk\u2082\u27e9", "annotated_tactic": ["rcases IH m.succ (by simpa [<a>Nat.succ_eq_add_one</a>, <a>add_comm</a>, <a>add_left_comm</a>] using hy\u2081)\n            fun {i} hi => by\n            simpa [<a>Nat.succ_eq_add_one</a>, <a>add_comm</a>, <a>add_left_comm</a>] using\n              hy\u2082 (<a>Nat.succ_lt_succ</a> hi) with\n          \u27e8k\u2082, hk\u2082\u27e9", [{"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]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"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]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Nat.succ_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 21]}]], "state_before": "case intro.intro.intro.succ.intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)", "state_after": "case intro.intro.intro.succ.intro.intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)"}, {"tactic": "use (max k\u2081 k\u2082).succ", "annotated_tactic": ["use (<a>max</a> k\u2081 k\u2082).<a>succ</a>", [{"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": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "case intro.intro.intro.succ.intro.intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).1 m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m)", "state_after": "case h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 Nat.pair (unpair n).1 m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227\n    \u2203 a,\n      evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n        (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n          some (Nat.succ y + m)"}, {"tactic": "rw [zero_add] at hk\u2081", "annotated_tactic": ["rw [<a>zero_add</a>] at hk\u2081", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 Nat.pair (unpair n).1 m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227\n    \u2203 a,\n      evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n        (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n          some (Nat.succ y + m)", "state_after": "case h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 Nat.pair (unpair n).1 m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227\n    \u2203 a,\n      evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n        (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n          some (Nat.succ y + m)"}, {"tactic": "use Nat.le_succ_of_le <| le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081", "annotated_tactic": ["use <a>Nat.le_succ_of_le</a> <| <a>le_max_of_le_left</a> <| <a>Nat.le_of_lt_succ</a> <| <a>evaln_bound</a> hk\u2081", [{"full_name": "Nat.le_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1602, 9], "def_end_pos": [1602, 26]}, {"full_name": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"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": "Nat.Partrec.Code.evaln_bound", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [775, 9], "def_end_pos": [775, 20]}]], "state_before": "case h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 Nat.pair (unpair n).1 m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227\n    \u2203 a,\n      evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n        (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n          some (Nat.succ y + m)", "state_after": "case right\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 \u2203 a,\n    evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n      (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n        some (Nat.succ y + m)"}, {"tactic": "use a", "annotated_tactic": ["use a", []], "state_before": "case right\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 \u2203 a,\n    evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n      (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n        some (Nat.succ y + m)", "state_after": "case h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n    (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n      some (Nat.succ y + m)"}, {"tactic": "use evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_left _ _) hk\u2081", "annotated_tactic": ["use <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>Nat.le_succ_of_le</a> <| <a>le_max_left</a> _ _) hk\u2081", [{"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1602, 9], "def_end_pos": [1602, 26]}, {"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\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227\n    (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n      some (Nat.succ y + m)", "state_after": "case right\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n    some (Nat.succ y + m)"}, {"tactic": "simpa [Nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using\n  evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082", "annotated_tactic": ["simpa [<a>Nat.succ_eq_add_one</a>, a0, -<a>max_eq_left</a>, -<a>max_eq_right</a>, <a>add_comm</a>, <a>add_left_comm</a>] using\n          <a>evaln_mono</a> (<a>Nat.succ_le_succ</a> <| <a>le_max_right</a> _ _) hk\u2082", [{"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]}, {"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Nat.Partrec.Code.evaln_mono", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [783, 9], "def_end_pos": [783, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case right\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m))\n\u22a2 (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =\n    some (Nat.succ y + m)", "state_after": "no goals"}, {"tactic": "simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081", "annotated_tactic": ["simpa [<a>Nat.succ_eq_add_one</a>, <a>add_comm</a>, <a>add_left_comm</a>] using hy\u2081", [{"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]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}]], "state_before": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + Nat.succ m))", "state_after": "no goals"}, {"tactic": "simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using\n  hy\u2082 (Nat.succ_lt_succ hi)", "annotated_tactic": ["simpa [<a>Nat.succ_eq_add_one</a>, <a>add_comm</a>, <a>add_left_comm</a>] using\n              hy\u2082 (<a>Nat.succ_lt_succ</a> hi)", [{"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]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Nat.succ_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 21]}]], "state_before": "cf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m))\ni : \u2115\nhi : i < y\n\u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (i + Nat.succ m)) \u2227 \u00aca = 0", "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", "start": [1121, 1], "end": [1144, 35], "traced_tactics": [{"tactic": "obtain rfl | a_lt_b := hab.eq_or_lt", "annotated_tactic": ["obtain rfl | a_lt_b := hab.eq_or_lt", []], "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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo 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": "case inl\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 : \u211d\nhab : a \u2264 a\nhcont : ContinuousOn g (Icc a a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a a)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 g' x \u2264 \u03c6 x\n\u22a2 g a - g a \u2264 \u222b (y : \u211d) in a..a, \u03c6 y\n\ncase 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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y"}, {"tactic": "set s := {t | g b - g t \u2264 \u222b u in t..b, \u03c6 u} \u2229 Icc a b", "annotated_tactic": ["set s := {t | g b - g t \u2264 \u222b u in t..b, \u03c6 u} \u2229 <a>Icc</a> a b", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y", "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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y"}, {"tactic": "have s_closed : IsClosed s := by\n  have : ContinuousOn (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (Icc a b) := by\n    rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2\n    exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int)\n  simp only [inter_comm]\n  exact this.preimage_closed_of_closed isClosed_Icc isClosed_le_prod", "annotated_tactic": ["have s_closed : <a>IsClosed</a> s := by\n    have : <a>ContinuousOn</a> (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (<a>Icc</a> a b) := by\n      rw [\u2190 <a>uIcc_of_le</a> hab] at hcont \u03c6int \u22a2\n      exact (continuousOn_const.sub hcont).<a>prod</a> (<a>continuousOn_primitive_interval_left</a> \u03c6int)\n    simp only [<a>inter_comm</a>]\n    exact this.preimage_closed_of_closed <a>isClosed_Icc</a> <a>isClosed_le_prod</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": "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.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 26]}, {"full_name": "intervalIntegral.continuousOn_primitive_interval_left", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 45]}, {"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": "isClosed_le_prod", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 25]}]], "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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y", "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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y"}, {"tactic": "have A : closure (Ioc a b) \u2286 s := by\n  apply s_closed.closure_subset_iff.2\n  intro t ht\n  refine' \u27e8_, \u27e8ht.1.le, ht.2\u27e9\u27e9\n  exact\n    sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n      (fun x hx => hderiv x \u27e8ht.1.trans_le hx.1, hx.2\u27e9)\n      (\u03c6int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => h\u03c6g x \u27e8ht.1.trans_le hx.1, hx.2\u27e9", "annotated_tactic": ["have A : <a>closure</a> (<a>Ioc</a> a b) \u2286 s := by\n    apply s_closed.closure_subset_iff.2\n    intro t ht\n    refine' \u27e8_, \u27e8ht.1.<a>le</a>, ht.2\u27e9\u27e9\n    exact\n      <a>sub_le_integral_of_hasDeriv_right_of_le_Ico</a> ht.2 (hcont.mono (<a>Icc_subset_Icc</a> ht.1.<a>le</a> <a>le_rfl</a>))\n        (fun x hx => hderiv x \u27e8ht.1.<a>trans_le</a> hx.1, hx.2\u27e9)\n        (\u03c6int.mono_set (<a>Icc_subset_Icc</a> ht.1.<a>le</a> <a>le_rfl</a>)) fun x hx => h\u03c6g x \u27e8ht.1.<a>trans_le</a> hx.1, hx.2\u27e9", [{"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 52]}, {"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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y", "state_after": "case inr\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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nA : closure (Ioc a b) \u2286 s\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y"}, {"tactic": "rw [closure_Ioc a_lt_b.ne] at A", "annotated_tactic": ["rw [<a>closure_Ioc</a> a_lt_b.ne] at A", [{"full_name": "closure_Ioc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2336, 9], "def_end_pos": [2336, 20]}]], "state_before": "case inr\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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nA : closure (Ioc a b) \u2286 s\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y", "state_after": "case inr\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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nA : Icc a b \u2286 s\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y"}, {"tactic": "exact (A (left_mem_Icc.2 hab)).1", "annotated_tactic": ["exact (A (<a>left_mem_Icc</a>.2 hab)).1", [{"full_name": "Set.left_mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [191, 9], "def_end_pos": [191, 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\u271d : 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nA : Icc a b \u2286 s\n\u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\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 : \u211d\nhab : a \u2264 a\nhcont : ContinuousOn g (Icc a a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a a)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 g' x \u2264 \u03c6 x\n\u22a2 g a - g a \u2264 \u222b (y : \u211d) in a..a, \u03c6 y", "state_after": "no goals"}, {"tactic": "have : ContinuousOn (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (Icc a b) := by\n  rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2\n  exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int)", "annotated_tactic": ["have : <a>ContinuousOn</a> (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (<a>Icc</a> a b) := by\n      rw [\u2190 <a>uIcc_of_le</a> hab] at hcont \u03c6int \u22a2\n      exact (continuousOn_const.sub hcont).<a>prod</a> (<a>continuousOn_primitive_interval_left</a> \u03c6int)", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"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.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 26]}, {"full_name": "intervalIntegral.continuousOn_primitive_interval_left", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 45]}]], "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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b)\n\u22a2 IsClosed (Icc a b \u2229 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u})"}, {"tactic": "exact this.preimage_closed_of_closed isClosed_Icc isClosed_le_prod", "annotated_tactic": ["exact this.preimage_closed_of_closed <a>isClosed_Icc</a> <a>isClosed_le_prod</a>", [{"full_name": "isClosed_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [214, 9], "def_end_pos": [214, 21]}, {"full_name": "isClosed_le_prod", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 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\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\nthis : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b)\n\u22a2 IsClosed (Icc a b \u2229 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u})", "state_after": "no goals"}, {"tactic": "rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2", "annotated_tactic": ["rw [\u2190 <a>uIcc_of_le</a> hab] at hcont \u03c6int \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\n\u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 [[a, b]]\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\n\u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) [[a, b]]"}, {"tactic": "exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int)", "annotated_tactic": ["exact (continuousOn_const.sub hcont).<a>prod</a> (<a>continuousOn_primitive_interval_left</a> \u03c6int)", [{"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_left", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 45]}]], "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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 [[a, b]]\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\n\u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) [[a, b]]", "state_after": "no goals"}, {"tactic": "apply s_closed.closure_subset_iff.2", "annotated_tactic": ["apply s_closed.closure_subset_iff.2", []], "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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 closure (Ioc a b) \u2286 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 Ioc a b \u2286 s"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\n\u22a2 Ioc a b \u2286 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 Ioc a b\n\u22a2 t \u2208 s"}, {"tactic": "refine' \u27e8_, \u27e8ht.1.le, ht.2\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8_, \u27e8ht.1.<a>le</a>, ht.2\u27e9\u27e9", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 Ioc a b\n\u22a2 t \u2208 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 Ioc a b\n\u22a2 t \u2208 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u}"}, {"tactic": "exact\n  sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n    (fun x hx => hderiv x \u27e8ht.1.trans_le hx.1, hx.2\u27e9)\n    (\u03c6int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => h\u03c6g x \u27e8ht.1.trans_le hx.1, hx.2\u27e9", "annotated_tactic": ["exact\n      <a>sub_le_integral_of_hasDeriv_right_of_le_Ico</a> ht.2 (hcont.mono (<a>Icc_subset_Icc</a> ht.1.<a>le</a> <a>le_rfl</a>))\n        (fun x hx => hderiv x \u27e8ht.1.<a>trans_le</a> hx.1, hx.2\u27e9)\n        (\u03c6int.mono_set (<a>Icc_subset_Icc</a> ht.1.<a>le</a> <a>le_rfl</a>)) fun x hx => h\u03c6g x \u27e8ht.1.<a>trans_le</a> hx.1, hx.2\u27e9", [{"full_name": "intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 52]}, {"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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"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": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x\n\u03c6int : IntegrableOn \u03c6 (Icc a b)\nh\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x\na_lt_b : a < b\ns : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b\ns_closed : IsClosed s\nt : \u211d\nht : t \u2208 Ioc a b\n\u22a2 t \u2208 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "Measurable.congr_ae", "start": [475, 1], "end": [477, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.degrees_rename_of_injective", "start": [257, 1], "end": [263, 42], "traced_tactics": [{"tactic": "classical\nsimp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,\n  support_rename_of_injective h, Finset.sup_image]\nrefine' Finset.sup_congr rfl fun x _ => _\nexact (Finsupp.toMultiset_map _ _).symm", "annotated_tactic": ["classical\n  simp only [<a>degrees</a>, <a>Multiset.map_finset_sup</a> p.support <a>Finsupp.toMultiset</a> f h,\n    <a>support_rename_of_injective</a> h, <a>Finset.sup_image</a>]\n  refine' <a>Finset.sup_congr</a> <a>rfl</a> fun x _ => _\n  exact (<a>Finsupp.toMultiset_map</a> _ _).<a>symm</a>", [{"full_name": "MvPolynomial.degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [87, 5], "def_end_pos": [87, 12]}, {"full_name": "Multiset.map_finset_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1829, 9], "def_end_pos": [1829, 23]}, {"full_name": "Finsupp.toMultiset", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [34, 5], "def_end_pos": [34, 15]}, {"full_name": "MvPolynomial.support_rename_of_injective", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [341, 9], "def_end_pos": [341, 36]}, {"full_name": "Finset.sup_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}, {"full_name": "Finset.sup_congr", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [85, 9], "def_end_pos": [85, 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": "Finsupp.toMultiset_map", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [72, 9], "def_end_pos": [72, 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": "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 p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\n\u22a2 degrees (\u2191(rename f) p) = Multiset.map f (degrees p)", "state_after": "no goals"}, {"tactic": "simp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,\n  support_rename_of_injective h, Finset.sup_image]", "annotated_tactic": ["simp only [<a>degrees</a>, <a>Multiset.map_finset_sup</a> p.support <a>Finsupp.toMultiset</a> f h,\n    <a>support_rename_of_injective</a> h, <a>Finset.sup_image</a>]", [{"full_name": "MvPolynomial.degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [87, 5], "def_end_pos": [87, 12]}, {"full_name": "Multiset.map_finset_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1829, 9], "def_end_pos": [1829, 23]}, {"full_name": "Finsupp.toMultiset", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [34, 5], "def_end_pos": [34, 15]}, {"full_name": "MvPolynomial.support_rename_of_injective", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [341, 9], "def_end_pos": [341, 36]}, {"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 : 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 p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\n\u22a2 degrees (\u2191(rename f) p) = Multiset.map f (degrees 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 : CommSemiring R\np\u271d q p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\n\u22a2 Finset.sup (support p) ((fun s => \u2191toMultiset s) \u2218 Finsupp.mapDomain f) =\n    Finset.sup (support p) (Multiset.map f \u2218 \u2191toMultiset)"}, {"tactic": "refine' Finset.sup_congr rfl fun x _ => _", "annotated_tactic": ["refine' <a>Finset.sup_congr</a> <a>rfl</a> fun x _ => _", [{"full_name": "Finset.sup_congr", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [85, 9], "def_end_pos": [85, 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": "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 p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\n\u22a2 Finset.sup (support p) ((fun s => \u2191toMultiset s) \u2218 Finsupp.mapDomain f) =\n    Finset.sup (support p) (Multiset.map f \u2218 \u2191toMultiset)", "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\u271d q p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\nx : \u03c3 \u2192\u2080 \u2115\nx\u271d : x \u2208 support p\n\u22a2 ((fun s => \u2191toMultiset s) \u2218 Finsupp.mapDomain f) x = (Multiset.map f \u2218 \u2191toMultiset) x"}, {"tactic": "exact (Finsupp.toMultiset_map _ _).symm", "annotated_tactic": ["exact (<a>Finsupp.toMultiset_map</a> _ _).<a>symm</a>", [{"full_name": "Finsupp.toMultiset_map", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [72, 9], "def_end_pos": [72, 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": "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 p : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\nh : Injective f\nx : \u03c3 \u2192\u2080 \u2115\nx\u271d : x \u2208 support p\n\u22a2 ((fun s => \u2191toMultiset s) \u2218 Finsupp.mapDomain f) x = (Multiset.map f \u2218 \u2191toMultiset) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max'_eq_sup'", "start": [1472, 1], "end": [1473, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.foldr_induction", "start": [155, 1], "end": [161, 16], "traced_tactics": [{"tactic": "have := SatisfiesM_foldrM (m := Id) (as := as) (f := f) motive h0", "annotated_tactic": ["have := <a>SatisfiesM_foldrM</a> (m := <a>Id</a>) (as := as) (f := f) motive h0", [{"full_name": "Array.SatisfiesM_foldrM", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [139, 9], "def_end_pos": [139, 26]}, {"full_name": "Id", "def_path": "lake-packages/lean4/src/lean/Init/Control/Id.lean", "def_pos": [13, 5], "def_end_pos": [13, 7]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nmotive : Nat \u2192 \u03b2 \u2192 Prop\ninit : \u03b2\nh0 : motive (size as) init\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nhf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b)\n\u22a2 motive 0 (foldr f init as (size as))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nmotive : Nat \u2192 \u03b2 \u2192 Prop\ninit : \u03b2\nh0 : motive (size as) init\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nhf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b)\nthis :\n  (\u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 SatisfiesM (motive i.val) (f as[i] b)) \u2192\n    SatisfiesM (motive 0) (foldrM f init as (size as))\n\u22a2 motive 0 (foldr f init as (size as))"}, {"tactic": "simp [SatisfiesM_Id_eq] at this", "annotated_tactic": ["simp [<a>SatisfiesM_Id_eq</a>] at this", [{"full_name": "SatisfiesM_Id_eq", "def_path": "lake-packages/std/Std/Classes/LawfulMonad.lean", "def_pos": [184, 17], "def_end_pos": [184, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nmotive : Nat \u2192 \u03b2 \u2192 Prop\ninit : \u03b2\nh0 : motive (size as) init\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nhf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b)\nthis :\n  (\u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 SatisfiesM (motive i.val) (f as[i] b)) \u2192\n    SatisfiesM (motive 0) (foldrM f init as (size as))\n\u22a2 motive 0 (foldr f init as (size as))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nmotive : Nat \u2192 \u03b2 \u2192 Prop\ninit : \u03b2\nh0 : motive (size as) init\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nhf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b)\nthis :\n  (\u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i.val] b)) \u2192\n    motive 0 (foldrM f init as (size as))\n\u22a2 motive 0 (foldr f init as (size as))"}, {"tactic": "exact this hf", "annotated_tactic": ["exact this hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nas : Array \u03b1\nmotive : Nat \u2192 \u03b2 \u2192 Prop\ninit : \u03b2\nh0 : motive (size as) init\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nhf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b)\nthis :\n  (\u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i.val] b)) \u2192\n    motive 0 (foldrM f init as (size as))\n\u22a2 motive 0 (foldr f init as (size as))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "comap_subtype_coe_apply", "start": [4154, 1], "end": [4156, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "IndexedPartition.index_out'", "start": [457, 1], "end": [458, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integrableOn_Ioi_deriv_of_nonneg", "start": [699, 1], "end": [721, 25], "traced_tactics": [{"tactic": "refine integrableOn_Ioi_of_intervalIntegral_norm_tendsto (l - g a) a (fun x => ?_) tendsto_id ?_", "annotated_tactic": ["refine <a>integrableOn_Ioi_of_intervalIntegral_norm_tendsto</a> (l - g a) a (fun x => ?_) <a>tendsto_id</a> ?_", [{"full_name": "MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [583, 9], "def_end_pos": [583, 58]}, {"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 f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\n\u22a2 IntegrableOn g' (Ioi a)", "state_after": "case refine_1\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 IntegrableOn g' (Ioc a (id x))\n\ncase refine_2\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun i => \u222b (x : \u211d) in a..id i, \u2016g' x\u2016) atTop (\ud835\udcdd (l - g a))"}, {"tactic": "apply Tendsto.congr' _ (hg.sub_const _)", "annotated_tactic": ["apply <a>Tendsto.congr'</a> _ (hg.sub_const _)", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}]], "state_before": "case refine_2\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun i => \u222b (x : \u211d) in a..id i, \u2016g' x\u2016) atTop (\ud835\udcdd (l - g a))", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\n\u22a2 (fun k => g k - g a) =\u1da0[atTop] fun i => \u222b (x : \u211d) in a..id i, \u2016g' x\u2016"}, {"tactic": "filter_upwards [Ioi_mem_atTop a] with x hx", "annotated_tactic": ["filter_upwards [<a>Ioi_mem_atTop</a> a] with x hx", [{"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\n\u22a2 (fun k => g k - g a) =\u1da0[atTop] fun i => \u222b (x : \u211d) in a..id i, \u2016g' x\u2016", "state_after": "case h\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\n\u22a2 g x - g a = \u222b (x : \u211d) in a..id x, \u2016g' x\u2016"}, {"tactic": "have h'x : a \u2264 id x := le_of_lt hx", "annotated_tactic": ["have h'x : a \u2264 <a>id</a> x := <a>le_of_lt</a> hx", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 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 h\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\n\u22a2 g x - g a = \u222b (x : \u211d) in a..id x, \u2016g' x\u2016", "state_after": "case h\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 g x - g a = \u222b (x : \u211d) in a..id x, \u2016g' x\u2016"}, {"tactic": "calc\n  g x - g a = \u222b y in a..id x, g' y := by\n    symm\n    apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x\n      (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1\n    rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]\n    exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)\n      (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1\n  _ = \u222b y in a..id x, \u2016g' y\u2016 := by\n    simp_rw [intervalIntegral.integral_of_le h'x]\n    refine' set_integral_congr measurableSet_Ioc fun y hy => _\n    dsimp\n    rw [abs_of_nonneg]\n    exact g'pos _ hy.1", "annotated_tactic": ["calc\n    g x - g a = \u222b y in a..<a>id</a> x, g' y := by\n      symm\n      apply <a>intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le</a> h'x\n        (hcont.mono <a>Icc_subset_Ici_self</a>) fun y hy => hderiv y hy.1\n      rw [<a>intervalIntegrable_iff_integrable_Ioc_of_le</a> h'x]\n      exact <a>intervalIntegral.integrableOn_deriv_of_nonneg</a> (hcont.mono <a>Icc_subset_Ici_self</a>)\n        (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1\n    _ = \u222b y in a..<a>id</a> x, \u2016g' y\u2016 := by\n      simp_rw [<a>intervalIntegral.integral_of_le</a> h'x]\n      refine' <a>set_integral_congr</a> <a>measurableSet_Ioc</a> fun y hy => _\n      dsimp\n      rw [<a>abs_of_nonneg</a>]\n      exact g'pos _ hy.1", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 44]}, {"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": "intervalIntegrable_iff_integrable_Ioc_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [91, 9], "def_end_pos": [91, 52]}, {"full_name": "intervalIntegral.integrableOn_deriv_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1290, 9], "def_end_pos": [1290, 37]}, {"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": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"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.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"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 h\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 g x - g a = \u222b (x : \u211d) in a..id x, \u2016g' x\u2016", "state_after": "no goals"}, {"tactic": "exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)\n  (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1", "annotated_tactic": ["exact <a>intervalIntegral.integrableOn_deriv_of_nonneg</a> (hcont.mono <a>Icc_subset_Ici_self</a>)\n      (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1", [{"full_name": "intervalIntegral.integrableOn_deriv_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1290, 9], "def_end_pos": [1290, 37]}, {"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "state_before": "case refine_1\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\n\u22a2 IntegrableOn g' (Ioc a (id x))", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 g x - g a = \u222b (y : \u211d) in a..id x, g' y", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 \u222b (y : \u211d) in a..id x, g' y = g x - g a"}, {"tactic": "apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x\n  (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1", "annotated_tactic": ["apply <a>intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le</a> h'x\n        (hcont.mono <a>Icc_subset_Ici_self</a>) fun y hy => hderiv y hy.1", [{"full_name": "intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 44]}, {"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 \u222b (y : \u211d) in a..id x, g' y = g x - g a", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 IntervalIntegrable (fun y => g' y) volume a (id x)"}, {"tactic": "rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]", "annotated_tactic": ["rw [<a>intervalIntegrable_iff_integrable_Ioc_of_le</a> h'x]", [{"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": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 IntervalIntegrable (fun y => g' y) volume a (id x)", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 IntegrableOn (fun y => g' y) (Ioc a (id x))"}, {"tactic": "exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)\n  (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1", "annotated_tactic": ["exact <a>intervalIntegral.integrableOn_deriv_of_nonneg</a> (hcont.mono <a>Icc_subset_Ici_self</a>)\n        (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1", [{"full_name": "intervalIntegral.integrableOn_deriv_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1290, 9], "def_end_pos": [1290, 37]}, {"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 IntegrableOn (fun y => g' y) (Ioc a (id x))", "state_after": "no goals"}, {"tactic": "simp_rw [intervalIntegral.integral_of_le h'x]", "annotated_tactic": ["simp_rw [<a>intervalIntegral.integral_of_le</a> h'x]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 \u222b (y : \u211d) in a..id x, g' y = \u222b (y : \u211d) in a..id x, \u2016g' y\u2016", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 \u222b (y : \u211d) in Ioc a (id x), g' y = \u222b (y : \u211d) in Ioc a (id x), \u2016g' y\u2016"}, {"tactic": "refine' set_integral_congr measurableSet_Ioc fun y hy => _", "annotated_tactic": ["refine' <a>set_integral_congr</a> <a>measurableSet_Ioc</a> fun y hy => _", [{"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_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\n\u22a2 \u222b (y : \u211d) in Ioc a (id x), g' y = \u222b (y : \u211d) in Ioc a (id x), \u2016g' y\u2016", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\ny : \u211d\nhy : y \u2208 Ioc a (id x)\n\u22a2 g' y = \u2016g' y\u2016"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\ny : \u211d\nhy : y \u2208 Ioc a (id x)\n\u22a2 g' y = \u2016g' y\u2016", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\ny : \u211d\nhy : y \u2208 Ioc a (id x)\n\u22a2 g' y = |g' y|"}, {"tactic": "rw [abs_of_nonneg]", "annotated_tactic": ["rw [<a>abs_of_nonneg</a>]", [{"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\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\ny : \u211d\nhy : y \u2208 Ioc a (id x)\n\u22a2 g' y = |g' y|", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\ny : \u211d\nhy : y \u2208 Ioc a (id x)\n\u22a2 0 \u2264 g' y"}, {"tactic": "exact g'pos _ hy.1", "annotated_tactic": ["exact g'pos _ hy.1", []], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousOn g (Ici a)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\ng'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x\nhg : Tendsto g atTop (\ud835\udcdd l)\nx : \u211d\nhx : x \u2208 Ioi a\nh'x : a \u2264 id x\ny : \u211d\nhy : y \u2208 Ioc a (id x)\n\u22a2 0 \u2264 g' y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Filtration.lean", "full_name": "MeasureTheory.measurableSet_filtrationOfSet'", "start": [253, 1], "end": [255, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasureTheory.Measure.ext_of_Iic", "start": [819, 1], "end": [830, 34], "traced_tactics": [{"tactic": "refine' ext_of_Ioc_finite \u03bc \u03bd _ fun a b hlt => _", "annotated_tactic": ["refine' <a>ext_of_Ioc_finite</a> \u03bc \u03bd _ fun a b hlt => _", [{"full_name": "MeasureTheory.Measure.ext_of_Ioc_finite", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [754, 9], "def_end_pos": [754, 26]}]], "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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\n\u22a2 \u03bc = \u03bd", "state_after": "case refine'_1\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 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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\n\u22a2 \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\n\ncase refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)"}, {"tactic": "rw [\u2190 Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic,\n  measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic, h a, h b]", "annotated_tactic": ["rw [\u2190 <a>Iic_diff_Iic</a>, <a>measure_diff</a> (<a>Iic_subset_Iic</a>.2 hlt.le) <a>measurableSet_Iic</a>,\n    <a>measure_diff</a> (<a>Iic_subset_Iic</a>.2 hlt.le) <a>measurableSet_Iic</a>, h a, h b]", [{"full_name": "Set.Iic_diff_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1110, 9], "def_end_pos": [1110, 21]}, {"full_name": "MeasureTheory.measure_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [252, 9], "def_end_pos": [252, 21]}, {"full_name": "Set.Iic_subset_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [419, 9], "def_end_pos": [419, 23]}, {"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_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [252, 9], "def_end_pos": [252, 21]}, {"full_name": "Set.Iic_subset_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [419, 9], "def_end_pos": [419, 23]}, {"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}]], "state_before": "case refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)", "state_after": "case refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bd (Iic a) \u2260 \u22a4\n\ncase refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Iic a) \u2260 \u22a4"}, {"tactic": "rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, -, hst\u27e9", "annotated_tactic": ["rcases <a>exists_countable_dense_bot_top</a> \u03b1 with \u27e8s, hsc, hsd, -, hst\u27e9", [{"full_name": "exists_countable_dense_bot_top", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [609, 9], "def_end_pos": [609, 39]}]], "state_before": "case refine'_1\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 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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\n\u22a2 \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ", "state_after": "case refine'_1.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 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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nhst : \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s\n\u22a2 \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ"}, {"tactic": "simp only [\u2190 biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]", "annotated_tactic": ["simp only [\u2190 <a>biSup_measure_Iic</a> hsc (hsd.exists_ge' hst) this, h]", [{"full_name": "MeasureTheory.biSup_measure_Iic", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2739, 9], "def_end_pos": [2739, 26]}]], "state_before": "case refine'_1.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 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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nhst : \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s\nthis : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ", "state_after": "no goals"}, {"tactic": "rw [\u2190 h a]", "annotated_tactic": ["rw [\u2190 h a]", []], "state_before": "case refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bd (Iic a) \u2260 \u22a4", "state_after": "case refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Iic a) \u2260 \u22a4"}, {"tactic": "exact (measure_lt_top \u03bc _).ne", "annotated_tactic": ["exact (<a>measure_lt_top</a> \u03bc _).<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 refine'_2\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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a)\na b : \u03b1\nhlt : a < b\n\u22a2 \u2191\u2191\u03bc (Iic a) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.computable_iff_re_compl_re", "start": [244, 1], "end": [258, 29], "traced_tactics": [{"tactic": "obtain \u27e8k, pk, hk\u27e9 :=\n  Partrec.merge (h\u2081.map (Computable.const true).to\u2082) (h\u2082.map (Computable.const false).to\u2082)\n  (by\n    intro a x hx y hy\n    simp at hx hy\n    cases hy.1 hx.1)", "annotated_tactic": ["obtain \u27e8k, pk, hk\u27e9 :=\n        <a>Partrec.merge</a> (h\u2081.map (<a>Computable.const</a> <a>true</a>).<a>to\u2082</a>) (h\u2082.map (<a>Computable.const</a> <a>false</a>).<a>to\u2082</a>)\n        (by\n          intro a x hx y hy\n          simp at hx hy\n          cases hy.1 hx.1)", [{"full_name": "Partrec.merge", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [101, 9], "def_end_pos": [101, 14]}, {"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [284, 9], "def_end_pos": [284, 14]}, {"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": "Computable.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [294, 9], "def_end_pos": [294, 12]}, {"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [284, 9], "def_end_pos": [284, 14]}, {"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": "Computable.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [294, 9], "def_end_pos": [294, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\n\u22a2 Computable fun a => decide (p a)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\n\u22a2 Computable fun a => decide (p a)"}, {"tactic": "intro a x hx y hy", "annotated_tactic": ["intro a x hx y hy", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\n\u22a2 \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2192\n      \u2200 (y : Bool), y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) \u2192 x = y", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\na : \u03b1\nx : Bool\nhx : x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ())\ny : Bool\nhy : y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\n\u22a2 x = y"}, {"tactic": "simp at hx hy", "annotated_tactic": ["simp at hx hy", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\na : \u03b1\nx : Bool\nhx : x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ())\ny : Bool\nhy : y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\n\u22a2 x = y", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\na : \u03b1\nx y : Bool\nhx : p a \u2227 true = x\nhy : \u00acp a \u2227 false = y\n\u22a2 x = y"}, {"tactic": "cases hy.1 hx.1", "annotated_tactic": ["cases hy.1 hx.1", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\na : \u03b1\nx y : Bool\nhx : p a \u2227 true = x\nhy : \u00acp a \u2227 false = y\n\u22a2 x = y", "state_after": "no goals"}, {"tactic": "refine' Partrec.of_eq pk fun n => Part.eq_some_iff.2 _", "annotated_tactic": ["refine' <a>Partrec.of_eq</a> pk fun n => <a>Part.eq_some_iff</a>.2 _", [{"full_name": "Partrec.of_eq", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [430, 9], "def_end_pos": [430, 14]}, {"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\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\n\u22a2 Computable fun a => decide (p a)", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 (fun a => decide (p a)) n \u2208 k n"}, {"tactic": "rw [hk]", "annotated_tactic": ["rw [hk]", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 (fun a => decide (p a)) n \u2208 k n", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 (fun a => decide (p a)) n \u2208 Part.map (fun b => true) (Part.assert (p n) fun x => Part.some ()) \u2228\n    (fun a => decide (p a)) n \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) n) fun x => Part.some ())"}, {"tactic": "simp only [Part.mem_map_iff, Part.mem_assert_iff, Part.mem_some_iff, exists_prop, and_true,\n  Bool.true_eq_decide_iff, and_self, exists_const, Bool.false_eq_decide_iff]", "annotated_tactic": ["simp only [<a>Part.mem_map_iff</a>, <a>Part.mem_assert_iff</a>, <a>Part.mem_some_iff</a>, <a>exists_prop</a>, <a>and_true</a>,\n          <a>Bool.true_eq_decide_iff</a>, <a>and_self</a>, <a>exists_const</a>, <a>Bool.false_eq_decide_iff</a>]", [{"full_name": "Part.mem_map_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [445, 9], "def_end_pos": [445, 20]}, {"full_name": "Part.mem_assert_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [466, 9], "def_end_pos": [466, 23]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "and_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [83, 17], "def_end_pos": [83, 25]}, {"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": "and_self", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [82, 17], "def_end_pos": [82, 25]}, {"full_name": "exists_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [369, 17], "def_end_pos": [369, 29]}, {"full_name": "Bool.false_eq_decide_iff", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 28]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 (fun a => decide (p a)) n \u2208 Part.map (fun b => true) (Part.assert (p n) fun x => Part.some ()) \u2228\n    (fun a => decide (p a)) n \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) n) fun x => Part.some ())", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 p n \u2228 \u00acp n"}, {"tactic": "apply Decidable.em", "annotated_tactic": ["apply <a>Decidable.em</a>", [{"full_name": "Decidable.em", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [734, 9], "def_end_pos": [734, 11]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n  \u2200 (a : \u03b1) (x : Bool),\n    x \u2208 k a \u2194\n      x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n        x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 p n \u2228 \u00acp n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'", "start": [787, 1], "end": [790, 7], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.laverage_union", "start": [162, 1], "end": [165, 96], "traced_tactics": [{"tactic": "rw [restrict_union\u2080 hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]", "annotated_tactic": ["rw [<a>restrict_union\u2080</a> hd ht, <a>laverage_add_measure</a>, <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.laverage_add_measure", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [142, 9], "def_end_pos": [142, 29]}, {"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 t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n    \u2191\u2191\u03bc s / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) * \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc + \u2191\u2191\u03bc t / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) * \u2a0d\u207b (x : \u03b1) in t, f x \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.all_eq_not_any_not", "start": [990, 1], "end": [991, 78], "traced_tactics": [{"tactic": "rw [Bool.eq_iff_iff]", "annotated_tactic": ["rw [<a>Bool.eq_iff_iff</a>]", [{"full_name": "Bool.eq_iff_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [812, 9], "def_end_pos": [812, 24]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 all l p = !any l fun c => !p c", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 all l p = true \u2194 (!any l fun c => !p c) = true"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 all l p = true \u2194 (!any l fun c => !p c) = true", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true) \u2194 (any l fun c => !p c) = false"}, {"tactic": "rw [\u2190 Bool.not_eq_true, List.any_eq_true]", "annotated_tactic": ["rw [\u2190 <a>Bool.not_eq_true</a>, <a>List.any_eq_true</a>]", [{"full_name": "Bool.not_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [136, 17], "def_end_pos": [136, 33]}, {"full_name": "List.any_eq_true", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [988, 17], "def_end_pos": [988, 28]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true) \u2194 (any l fun c => !p c) = false", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true) \u2194 \u00ac\u2203 x, x \u2208 l \u2227 (!p x) = true"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true) \u2194 \u00ac\u2203 x, x \u2208 l \u2227 (!p x) = true", "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_zero'_apply", "start": [1536, 1], "end": [1538, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurableSet_of_continuousAt", "start": [335, 1], "end": [337, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_eq_three", "start": [1108, 1], "end": [1112, 14], "traced_tactics": [{"tactic": "rw [\u2190encard_eq_three, ncard_def, \u2190Nat.cast_inj (R := \u2115\u221e), Nat.cast_ofNat]", "annotated_tactic": ["rw [\u2190<a>encard_eq_three</a>, <a>ncard_def</a>, \u2190<a>Nat.cast_inj</a> (R := \u2115\u221e), <a>Nat.cast_ofNat</a>]", [{"full_name": "Set.encard_eq_three", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [324, 9], "def_end_pos": [324, 24]}, {"full_name": "Set.ncard_def", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [475, 9], "def_end_pos": [475, 18]}, {"full_name": "Nat.cast_inj", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "Nat.cast_ofNat", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [66, 28], "def_end_pos": [66, 42]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 ncard s = 3 \u2194 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z}", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 \u2191(\u2191ENat.toNat (encard s)) = 3 \u2194 encard s = 3"}, {"tactic": "refine' \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 \u2191(\u2191ENat.toNat (encard s)) = 3 \u2194 encard s = 3", "state_after": "case refine'_1\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : \u2191(\u2191ENat.toNat (encard s)) = 3\n\u22a2 encard s = 3\n\ncase refine'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 3\n\u22a2 \u2191(\u2191ENat.toNat (encard s)) = 3"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 3\n\u22a2 \u2191(\u2191ENat.toNat (encard s)) = 3", "state_after": "case refine'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 3\n\u22a2 \u2191(\u2191ENat.toNat 3) = 3"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : encard s = 3\n\u22a2 \u2191(\u2191ENat.toNat 3) = 3", "state_after": "no goals"}, {"tactic": "rwa [ENat.coe_toNat] at h", "annotated_tactic": ["rwa [<a>ENat.coe_toNat</a>] at h", [{"full_name": "ENat.coe_toNat", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [161, 11], "def_end_pos": [161, 20]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : \u2191(\u2191ENat.toNat (encard s)) = 3\n\u22a2 encard s = 3", "state_after": "case refine'_1\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : \u2191(\u2191ENat.toNat (encard s)) = 3\n\u22a2 encard s \u2260 \u22a4"}, {"tactic": "rintro h'", "annotated_tactic": ["rintro h'", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : \u2191(\u2191ENat.toNat (encard s)) = 3\n\u22a2 encard s \u2260 \u22a4", "state_after": "case refine'_1\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : \u2191(\u2191ENat.toNat (encard s)) = 3\nh' : encard s = \u22a4\n\u22a2 False"}, {"tactic": "simp [h'] at h", "annotated_tactic": ["simp [h'] at h", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\ns t : Set \u03b1\nh : \u2191(\u2191ENat.toNat (encard s)) = 3\nh' : encard s = \u22a4\n\u22a2 False", "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.modify_get?_length", "start": [870, 9], "end": [871, 53], "traced_tactics": [{"tactic": "cases l <;> rfl", "annotated_tactic": ["cases l <;> rfl", []], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nl : List \u03b1\n\u22a2 length (modifyHead f l) = length l", "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_assoc", "start": [262, 11], "end": [282, 44], "traced_tactics": [{"tactic": "rw [Int.add_comm, \u2190 aux1, Int.add_comm k, aux1, Int.add_comm b]", "annotated_tactic": ["rw [<a>Int.add_comm</a>, \u2190 aux1, <a>Int.add_comm</a> k, aux1, <a>Int.add_comm</a> b]", [{"full_name": "Int.add_comm", "def_path": 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{"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.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}]], "state_before": "a : Int\nn k : Nat\n\u22a2 a + \u2191n + \u2191k = a + (\u2191n + \u2191k)", "state_after": "no goals"}, {"tactic": "rw [Int.add_comm, \u2190 aux2, Int.add_comm n, \u2190 aux2, Int.add_comm -[m+1]]", "annotated_tactic": ["rw [<a>Int.add_comm</a>, \u2190 aux2, <a>Int.add_comm</a> n, \u2190 aux2, <a>Int.add_comm</a> -[m+1]]", [{"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.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}, {"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": "m n k : Nat\n\u22a2 -[m+1] + \u2191n + -[k+1] = -[m+1] + (\u2191n + -[k+1])", "state_after": "no goals"}, {"tactic": "rw [Int.add_comm, Int.add_comm m, Int.add_comm m, \u2190 aux2, Int.add_comm -[k+1]]", "annotated_tactic": ["rw [<a>Int.add_comm</a>, <a>Int.add_comm</a> m, <a>Int.add_comm</a> m, \u2190 aux2, <a>Int.add_comm</a> -[k+1]]", [{"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.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}, {"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.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}]], "state_before": "m n k : Nat\n\u22a2 \u2191m + -[n+1] + -[k+1] = \u2191m + (-[n+1] + -[k+1])", "state_after": "no goals"}, {"tactic": "simp [add_succ, Nat.add_comm, Nat.add_left_comm, neg_ofNat_succ]", "annotated_tactic": ["simp [<a>add_succ</a>, <a>Nat.add_comm</a>, <a>Nat.add_left_comm</a>, <a>neg_ofNat_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.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_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": "Int.neg_ofNat_succ", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [40, 23], "def_end_pos": [40, 37]}]], "state_before": "m n k : Nat\n\u22a2 -[m+1] + -[n+1] + -[k+1] = -[m+1] + (-[n+1] + -[k+1])", "state_after": "no goals"}, {"tactic": "simp [Nat.add_assoc]", "annotated_tactic": ["simp [<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": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 \u2191m + \u2191n + \u2191k = \u2191m + (\u2191n + \u2191k)", "state_after": "no goals"}, {"tactic": "simp [subNatNat_add]", "annotated_tactic": ["simp [<a>subNatNat_add</a>]", [{"full_name": "Int.subNatNat_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [236, 9], "def_end_pos": [236, 22]}]], "state_before": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 \u2191m + \u2191n + -[k+1] = \u2191m + (\u2191n + -[k+1])", "state_after": "no goals"}, {"tactic": "simp [add_succ]", "annotated_tactic": ["simp [<a>add_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]}]], "state_before": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 -[m+1] + -[n+1] + \u2191k = -[m+1] + (-[n+1] + \u2191k)", "state_after": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 subNatNat k (succ (succ (m + n))) = -[m+1] + subNatNat k (succ n)"}, {"tactic": "rw [Int.add_comm, subNatNat_add_negSucc]", "annotated_tactic": ["rw [<a>Int.add_comm</a>, <a>subNatNat_add_negSucc</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.subNatNat_add_negSucc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [245, 9], "def_end_pos": [245, 30]}]], "state_before": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 subNatNat k (succ (succ (m + n))) = -[m+1] + subNatNat k (succ n)", "state_after": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 subNatNat k (succ (succ (m + n))) = subNatNat k (succ n + succ m)"}, {"tactic": "simp [add_succ, succ_add, Nat.add_comm]", "annotated_tactic": ["simp [<a>add_succ</a>, <a>succ_add</a>, <a>Nat.add_comm</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_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}, {"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": "x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int\nm n k : Nat\n\u22a2 subNatNat k (succ (succ (m + n))) = subNatNat k (succ n + succ 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.infix_cons_iff", "start": [1865, 1], "end": [1874, 27], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 l\u2081 <:+: a :: l\u2082 \u2194 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082", "state_after": "case mp\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 l\u2081 <:+: a :: l\u2082 \u2192 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082\n\ncase mpr\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 \u2192 l\u2081 <:+: a :: l\u2082"}, {"tactic": "rintro \u27e8\u27e8hd, tl\u27e9, t, hl\u2083\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8hd, tl\u27e9, t, hl\u2083\u27e9", []], "state_before": "case mp\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 l\u2081 <:+: a :: l\u2082 \u2192 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082", "state_after": "case mp.intro.nil.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 t : List \u03b1\u271d\nhl\u2083 : [] ++ l\u2081 ++ t = a :: l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082\n\ncase mp.intro.cons.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhead\u271d : \u03b1\u271d\ntail\u271d t : List \u03b1\u271d\nhl\u2083 : head\u271d :: tail\u271d ++ l\u2081 ++ t = a :: l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082"}, {"tactic": "exact Or.inl \u27e8t, hl\u2083\u27e9", "annotated_tactic": ["exact <a>Or.inl</a> \u27e8t, hl\u2083\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 mp.intro.nil.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 t : List \u03b1\u271d\nhl\u2083 : [] ++ l\u2081 ++ t = a :: l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082", "state_after": "no goals"}, {"tactic": "simp only [cons_append] at hl\u2083", "annotated_tactic": ["simp only [<a>cons_append</a>] at hl\u2083", [{"full_name": "List.cons_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [99, 17], "def_end_pos": [99, 28]}]], "state_before": "case mp.intro.cons.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhead\u271d : \u03b1\u271d\ntail\u271d t : List \u03b1\u271d\nhl\u2083 : head\u271d :: tail\u271d ++ l\u2081 ++ t = a :: l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082", "state_after": "case mp.intro.cons.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhead\u271d : \u03b1\u271d\ntail\u271d t : List \u03b1\u271d\nhl\u2083 : head\u271d :: (tail\u271d ++ l\u2081 ++ t) = a :: l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082"}, {"tactic": "injection hl\u2083 with _ hl\u2084", "annotated_tactic": ["injection hl\u2083 with _ hl\u2084", []], "state_before": "case mp.intro.cons.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhead\u271d : \u03b1\u271d\ntail\u271d t : List \u03b1\u271d\nhl\u2083 : head\u271d :: (tail\u271d ++ l\u2081 ++ t) = a :: l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082", "state_after": "case mp.intro.cons.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhead\u271d : \u03b1\u271d\ntail\u271d t : List \u03b1\u271d\nhead_eq\u271d : head\u271d = a\nhl\u2084 : tail\u271d ++ l\u2081 ++ t = l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082"}, {"tactic": "exact Or.inr \u27e8_, t, hl\u2084\u27e9", "annotated_tactic": ["exact <a>Or.inr</a> \u27e8_, t, hl\u2084\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 mp.intro.cons.intro\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhead\u271d : \u03b1\u271d\ntail\u271d t : List \u03b1\u271d\nhead_eq\u271d : head\u271d = a\nhl\u2084 : tail\u271d ++ l\u2081 ++ t = l\u2082\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082", "state_after": "no goals"}, {"tactic": "rintro (h | hl\u2081)", "annotated_tactic": ["rintro (h | hl\u2081)", []], "state_before": "case mpr\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 \u2192 l\u2081 <:+: a :: l\u2082", "state_after": "case mpr.inl\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nh : l\u2081 <+: a :: l\u2082\n\u22a2 l\u2081 <:+: a :: l\u2082\n\ncase mpr.inr\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhl\u2081 : l\u2081 <:+: l\u2082\n\u22a2 l\u2081 <:+: a :: l\u2082"}, {"tactic": "exact h.isInfix", "annotated_tactic": ["exact h.isInfix", []], "state_before": "case mpr.inl\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nh : l\u2081 <+: a :: l\u2082\n\u22a2 l\u2081 <:+: a :: l\u2082", "state_after": "no goals"}, {"tactic": "exact infix_cons hl\u2081", "annotated_tactic": ["exact <a>infix_cons</a> hl\u2081", [{"full_name": "List.infix_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1760, 9], "def_end_pos": [1760, 19]}]], "state_before": "case mpr.inr\n\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\nhl\u2081 : l\u2081 <:+: l\u2082\n\u22a2 l\u2081 <:+: a :: l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Polynomial.lean", "full_name": "MvPolynomial.eval_polynomial_eval_finSuccEquiv", "start": [30, 1], "end": [40, 41], "traced_tactics": [{"tactic": "simp only [finSuccEquiv_apply, coe_eval\u2082Hom, polynomial_eval_eval\u2082, eval_eval\u2082]", "annotated_tactic": ["simp only [<a>finSuccEquiv_apply</a>, <a>coe_eval\u2082Hom</a>, <a>polynomial_eval_eval\u2082</a>, <a>eval_eval\u2082</a>]", [{"full_name": "MvPolynomial.finSuccEquiv_apply", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [332, 9], "def_end_pos": [332, 27]}, {"full_name": "MvPolynomial.coe_eval\u2082Hom", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1051, 9], "def_end_pos": [1051, 21]}, {"full_name": "MvPolynomial.polynomial_eval_eval\u2082", "def_path": "Mathlib/Data/MvPolynomial/Polynomial.lean", "def_pos": [19, 9], "def_end_pos": [19, 30]}, {"full_name": "MvPolynomial.eval_eval\u2082", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1210, 9], "def_end_pos": [1210, 19]}]], "state_before": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 \u2191(eval x) (Polynomial.eval q (\u2191(finSuccEquiv R n) f)) = \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f", "state_after": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 eval\u2082 (RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C)))\n      (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f"}, {"tactic": "conv in RingHom.comp _ _ =>\n{ refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_\n  simp }", "annotated_tactic": ["conv in <a>RingHom.comp</a> _ _ =>\n  { refine @<a>RingHom.ext</a> _ _ _ _ _ (<a>RingHom.id</a> _) fun r => ?_\n    simp }", [{"full_name": "RingHom.comp", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [657, 5], "def_end_pos": [657, 9]}, {"full_name": "RingHom.ext", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [531, 9], "def_end_pos": [531, 12]}, {"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_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 eval\u2082 (RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C)))\n      (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f", "state_after": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 eval\u2082 (RingHom.id R)\n      (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f"}, {"tactic": "simp only [eval\u2082_id]", "annotated_tactic": ["simp only [<a>eval\u2082_id</a>]", [{"full_name": "MvPolynomial.eval\u2082_id", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 17]}]], "state_before": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 eval\u2082 (RingHom.id R)\n      (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f", "state_after": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 \u2191(eval fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 \u2191(eval fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f", "state_after": "case e_a.e_f\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) = fun i =>\n    Fin.cases (\u2191(eval x) q) x i"}, {"tactic": "funext i", "annotated_tactic": ["funext i", []], "state_before": "case e_a.e_f\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) = fun i =>\n    Fin.cases (\u2191(eval x) q) x i", "state_after": "case e_a.e_f.h\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i)) = Fin.cases (\u2191(eval x) q) x i"}, {"tactic": "refine Fin.cases (by simp) (by simp) i", "annotated_tactic": ["refine <a>Fin.cases</a> (by simp) (by simp) i", [{"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": "case e_a.e_f.h\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i)) = Fin.cases (\u2191(eval x) q) x i", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) 0)) = Fin.cases (\u2191(eval x) q) x 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2200 (i : Fin n),\n    \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) (Fin.succ i))) =\n      Fin.cases (\u2191(eval x) q) x (Fin.succ i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.rename_monomial", "start": [98, 1], "end": [104, 39], "traced_tactics": [{"tactic": "rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),\n  Finsupp.prod_mapDomain_index]", "annotated_tactic": ["rw [<a>rename</a>, <a>aeval_monomial</a>, <a>monomial_eq</a> (s := <a>Finsupp.mapDomain</a> f d),\n    <a>Finsupp.prod_mapDomain_index</a>]", [{"full_name": "MvPolynomial.rename", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}, {"full_name": "MvPolynomial.aeval_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1563, 9], "def_end_pos": [1563, 23]}, {"full_name": "MvPolynomial.monomial_eq", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 20]}, {"full_name": "Finsupp.mapDomain", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [448, 5], "def_end_pos": [448, 14]}, {"full_name": "Finsupp.prod_mapDomain_index", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [581, 9], "def_end_pos": [581, 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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2191(rename f) (\u2191(monomial d) r) = \u2191(monomial (Finsupp.mapDomain f d)) r", "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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (\u2191(algebraMap R (MvPolynomial \u03c4 R)) r * Finsupp.prod d fun i k => (X \u2218 f) i ^ k) =\n    \u2191C r * Finsupp.prod d fun a m => X (f a) ^ m\n\ncase h_zero\n\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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4), X b ^ 0 = 1\n\ncase h_add\n\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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4) (m\u2081 m\u2082 : \u2115), X b ^ (m\u2081 + m\u2082) = X b ^ m\u2081 * X b ^ m\u2082"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (\u2191(algebraMap R (MvPolynomial \u03c4 R)) r * Finsupp.prod d fun i k => (X \u2218 f) i ^ k) =\n    \u2191C r * Finsupp.prod d fun a m => X (f a) ^ m", "state_after": "no goals"}, {"tactic": "exact fun n => pow_zero _", "annotated_tactic": ["exact fun n => <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": "case h_zero\n\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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4), X b ^ 0 = 1", "state_after": "no goals"}, {"tactic": "exact fun n i\u2081 i\u2082 => pow_add _ _ _", "annotated_tactic": ["exact fun n i\u2081 i\u2082 => <a>pow_add</a> _ _ _", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "case h_add\n\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\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4) (m\u2081 m\u2082 : \u2115), X b ^ (m\u2081 + m\u2082) = X b ^ m\u2081 * X b ^ m\u2082", "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.HeapNode.WF.realSize_eq", "start": [21, 9], "end": [26, 40], "traced_tactics": [{"tactic": "rw [realSize, realSize_eq c, Nat.pow_succ, Nat.mul_succ]", "annotated_tactic": ["rw [<a>realSize</a>, realSize_eq c, <a>Nat.pow_succ</a>, <a>Nat.mul_succ</a>]", [{"full_name": "Std.BinomialHeap.Imp.HeapNode.realSize", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [39, 13], "def_end_pos": [39, 30]}, {"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_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na a\u271d\u00b2 : \u03b1\na\u271d\u00b9 a\u271d : HeapNode \u03b1\nw\u271d : Nat\nleft\u271d : \u2200 [inst : TotalBLE le], le a a\u271d\u00b2 = true\nc : WF le a\u271d\u00b2 a\u271d\u00b9 w\u271d\ns : WF le a a\u271d w\u271d\n\u22a2 realSize (node a\u271d\u00b2 a\u271d\u00b9 a\u271d) + 1 = 2 ^ (w\u271d + 1)", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na a\u271d\u00b2 : \u03b1\na\u271d\u00b9 a\u271d : HeapNode \u03b1\nw\u271d : Nat\nleft\u271d : \u2200 [inst : TotalBLE le], le a a\u271d\u00b2 = true\nc : WF le a\u271d\u00b2 a\u271d\u00b9 w\u271d\ns : WF le a a\u271d w\u271d\n\u22a2 2 ^ w\u271d + realSize a\u271d + 1 = 2 ^ w\u271d * 1 + 2 ^ w\u271d"}, {"tactic": "simp [Nat.add_assoc, realSize_eq s]", "annotated_tactic": ["simp [<a>Nat.add_assoc</a>, realSize_eq s]", [{"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\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na a\u271d\u00b2 : \u03b1\na\u271d\u00b9 a\u271d : HeapNode \u03b1\nw\u271d : Nat\nleft\u271d : \u2200 [inst : TotalBLE le], le a a\u271d\u00b2 = true\nc : WF le a\u271d\u00b2 a\u271d\u00b9 w\u271d\ns : WF le a a\u271d w\u271d\n\u22a2 2 ^ w\u271d + realSize a\u271d + 1 = 2 ^ w\u271d * 1 + 2 ^ w\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.AEStronglyMeasurable'.const_inner", "start": [98, 1], "end": [106, 10], "traced_tactics": [{"tactic": "rcases hfm with \u27e8f', hf'_meas, hf_ae\u27e9", "annotated_tactic": ["rcases hfm with \u27e8f', hf'_meas, hf_ae\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nhfm : AEStronglyMeasurable' m f \u03bc\nc : \u03b2\n\u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc"}, {"tactic": "refine'\n  \u27e8fun x => (inner c (f' x) : \ud835\udd5c), (@stronglyMeasurable_const _ _ m _ c).inner hf'_meas,\n    hf_ae.mono fun x hx => _\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun x => (<a>inner</a> c (f' x) : \ud835\udd5c), (@<a>stronglyMeasurable_const</a> _ _ m _ c).<a>inner</a> hf'_meas,\n      hf_ae.mono fun x hx => _\u27e9", [{"full_name": "Inner.inner", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [79, 3], "def_end_pos": [79, 8]}, {"full_name": "MeasureTheory.stronglyMeasurable_const", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 33]}, {"full_name": "MeasureTheory.StronglyMeasurable.inner", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Inner.lean", "def_pos": [26, 19], "def_end_pos": [26, 24]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 (fun x => inner c (f x)) x = (fun x => inner c (f' x)) x"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 (fun x => inner c (f x)) x = (fun x => inner c (f' x)) x", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 inner c (f x) = inner c (f' x)"}, {"tactic": "rw [hx]", "annotated_tactic": ["rw [hx]", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 inner c (f x) = inner c (f' x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Mem.lean", "full_name": "Vector.mem_map_succ_iff", "start": [85, 1], "end": [87, 67], "traced_tactics": [{"tactic": "rw [mem_succ_iff, head_map, tail_map, mem_map_iff, @eq_comm _ b]", "annotated_tactic": ["rw [<a>mem_succ_iff</a>, <a>head_map</a>, <a>tail_map</a>, <a>mem_map_iff</a>, @<a>eq_comm</a> _ b]", [{"full_name": "Vector.mem_succ_iff", "def_path": "Mathlib/Data/Vector/Mem.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Vector.head_map", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "Vector.tail_map", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Vector.mem_map_iff", "def_path": "Mathlib/Data/Vector/Mem.lean", "def_pos": [76, 9], "def_end_pos": [76, 20]}, {"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\nn : \u2115\na a' : \u03b1\nb : \u03b2\nv : Vector \u03b1 (n + 1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 b \u2208 toList (map f v) \u2194 f (head v) = b \u2228 \u2203 a, a \u2208 toList (tail v) \u2227 f a = b", "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_left_iff", "start": [101, 1], "end": [102, 68], "traced_tactics": [{"tactic": "rw [\u2190 sum_cond, sum_left, Bool.forall_bool, cond, cond, and_comm]", "annotated_tactic": ["rw [\u2190 <a>sum_cond</a>, <a>sum_left</a>, <a>Bool.forall_bool</a>, <a>cond</a>, <a>cond</a>, <a>and_comm</a>]", [{"full_name": "MeasureTheory.Measure.sum_cond", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2055, 9], "def_end_pos": [2055, 17]}, {"full_name": "MeasureTheory.Measure.MutuallySingular.sum_left", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [86, 9], "def_end_pos": [86, 17]}, {"full_name": "Bool.forall_bool", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}, {"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]}, {"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u22a2 \u03bc\u2081 + \u03bc\u2082 \u27c2\u2098 \u03bd \u2194 \u03bc\u2081 \u27c2\u2098 \u03bd \u2227 \u03bc\u2082 \u27c2\u2098 \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Nat.rfind_zero_none", "start": [118, 1], "end": [121, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_exchange'", "start": [640, 1], "end": [642, 84], "traced_tactics": [{"tactic": "rw [\u2190 ncard_exchange ha hb, \u2190 singleton_union, \u2190 singleton_union, union_diff_distrib,\n  @diff_singleton_eq_self _ b {a} fun h \u21a6 ha (by rwa [\u2190 mem_singleton_iff.mp h])]", "annotated_tactic": ["rw [\u2190 <a>ncard_exchange</a> ha hb, \u2190 <a>singleton_union</a>, \u2190 <a>singleton_union</a>, <a>union_diff_distrib</a>,\n    @<a>diff_singleton_eq_self</a> _ b {a} fun h \u21a6 ha (by rwa [\u2190 mem_singleton_iff.mp h])]", [{"full_name": "Set.ncard_exchange", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [636, 9], "def_end_pos": [636, 23]}, {"full_name": "Set.singleton_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 24]}, {"full_name": "Set.singleton_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 24]}, {"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_singleton_eq_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2068, 9], "def_end_pos": [2068, 31]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 ncard (insert a s \\ {b}) = ncard s", "state_after": "no goals"}, {"tactic": "rwa [\u2190 mem_singleton_iff.mp h]", "annotated_tactic": ["rwa [\u2190 mem_singleton_iff.mp h]", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\nhb : b \u2208 s\nh : b \u2208 {a}\n\u22a2 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Set.Infinite.not_bddBelow", "start": [482, 1], "end": [483, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.totalDegree_multiset_prod", "start": [718, 1], "end": [722, 32], "traced_tactics": [{"tactic": "refine' Quotient.inductionOn s fun l => _", "annotated_tactic": ["refine' <a>Quotient.inductionOn</a> s fun l => _", [{"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": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns : Multiset (MvPolynomial \u03c3 R)\n\u22a2 totalDegree (Multiset.prod s) \u2264 Multiset.sum (Multiset.map totalDegree s)", "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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns : Multiset (MvPolynomial \u03c3 R)\nl : List (MvPolynomial \u03c3 R)\n\u22a2 totalDegree (Multiset.prod (Quotient.mk (List.isSetoid (MvPolynomial \u03c3 R)) l)) \u2264\n    Multiset.sum (Multiset.map totalDegree (Quotient.mk (List.isSetoid (MvPolynomial \u03c3 R)) l))"}, {"tactic": "rw [Multiset.quot_mk_to_coe, Multiset.coe_prod, Multiset.coe_map, Multiset.coe_sum]", "annotated_tactic": ["rw [<a>Multiset.quot_mk_to_coe</a>, <a>Multiset.coe_prod</a>, <a>Multiset.coe_map</a>, <a>Multiset.coe_sum</a>]", [{"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.coe_prod", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}, {"full_name": "Multiset.coe_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1166, 9], "def_end_pos": [1166, 16]}, {"full_name": "Multiset.coe_sum", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [63, 3], "def_end_pos": [63, 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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns : Multiset (MvPolynomial \u03c3 R)\nl : List (MvPolynomial \u03c3 R)\n\u22a2 totalDegree (Multiset.prod (Quotient.mk (List.isSetoid (MvPolynomial \u03c3 R)) l)) \u2264\n    Multiset.sum (Multiset.map totalDegree (Quotient.mk (List.isSetoid (MvPolynomial \u03c3 R)) l))", "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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns : Multiset (MvPolynomial \u03c3 R)\nl : List (MvPolynomial \u03c3 R)\n\u22a2 totalDegree (List.prod l) \u2264 List.sum (List.map totalDegree l)"}, {"tactic": "exact totalDegree_list_prod l", "annotated_tactic": ["exact <a>totalDegree_list_prod</a> l", [{"full_name": "MvPolynomial.totalDegree_list_prod", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [710, 9], "def_end_pos": [710, 30]}]], "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\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\ns : Multiset (MvPolynomial \u03c3 R)\nl : List (MvPolynomial \u03c3 R)\n\u22a2 totalDegree (List.prod l) \u2264 List.sum (List.map totalDegree l)", "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.size_tail", "start": [149, 1], "end": [153, 48], "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": [75, 15], "def_end_pos": [75, 24]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\n\u22a2 size (tail le s) = size s - 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\n\u22a2 size (Option.getD (tail? le s) nil) = size s - 1"}, {"tactic": "match eq : s.tail? le with\n| none => cases s with cases eq | nil => rfl\n| some tl => simp [Heap.size_tail? h 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.size_tail?</a> h 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.PairingHeapImp.Heap.nil", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [23, 5], "def_end_pos": [23, 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.PairingHeapImp.Heap.size_tail?", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [143, 9], "def_end_pos": [143, 24]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\n\u22a2 size (Option.getD (tail? le s) nil) = size 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.PairingHeapImp.Heap.nil", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [23, 5], "def_end_pos": [23, 8]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\neq : tail? le s = none\n\u22a2 size (Option.getD none nil) = size 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\nh : NoSibling nil\n\u22a2 size (Option.getD none nil) = size nil - 1", "state_after": "no goals"}, {"tactic": "simp [Heap.size_tail? h eq]", "annotated_tactic": ["simp [<a>Heap.size_tail?</a> h eq]", [{"full_name": "Std.PairingHeapImp.Heap.size_tail?", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [143, 9], "def_end_pos": [143, 24]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\ntl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 size (Option.getD (some tl) nil) = size s - 1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\ntl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 size (Option.getD (some tl) nil) = size tl + 1 - 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\ns : Heap \u03b1\nh : NoSibling s\ntl : Heap \u03b1\neq : tail? le s = some tl\n\u22a2 size (Option.getD (some tl) nil) = size tl + 1 - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/DFA.lean", "full_name": "DFA.evalFrom_append_singleton", "start": [64, 1], "end": [66, 75], "traced_tactics": [{"tactic": "simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]", "annotated_tactic": ["simp only [<a>evalFrom</a>, <a>List.foldl_append</a>, <a>List.foldl_cons</a>, <a>List.foldl_nil</a>]", [{"full_name": "DFA.evalFrom", "def_path": "Mathlib/Computability/DFA.lean", "def_pos": [49, 5], "def_end_pos": [49, 13]}, {"full_name": "List.foldl_append", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [226, 17], "def_end_pos": [226, 29]}, {"full_name": "List.foldl_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [234, 17], "def_end_pos": [234, 27]}, {"full_name": "List.foldl_nil", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [232, 17], "def_end_pos": [232, 26]}]], "state_before": "\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ns : \u03c3\nx : List \u03b1\na : \u03b1\n\u22a2 evalFrom M s (x ++ [a]) = step M (evalFrom M s x) a", "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.mod_eq_zero_of_dvd", "start": [679, 1], "end": [680, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to1.trTape_mk'", "start": [1746, 1], "end": [1747, 53], "traced_tactics": [{"tactic": "simp only [trTape, Tape.mk'_left, Tape.mk'_right\u2080]", "annotated_tactic": ["simp only [<a>trTape</a>, <a>Tape.mk'_left</a>, <a>Tape.mk'_right\u2080</a>]", [{"full_name": "Turing.TM1to1.trTape", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1742, 5], "def_end_pos": [1742, 11]}, {"full_name": "Turing.Tape.mk'_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}, {"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\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\nL R : ListBlank \u0393\n\u22a2 trTape enc0 (Tape.mk' L R) = trTape' enc0 L R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.ite_diff_self", "start": [2285, 1], "end": [2286, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.denseRange", "start": [786, 11], "end": [788, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.eq_mk_of_mem", "start": [62, 1], "end": [63, 14], "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", "start": [752, 1], "end": [760, 29], "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\nf g : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 setToL1S T (f + g) = setToL1S T f + 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\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\nf g : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) =\n    SimpleFunc.setToSimpleFunc T (toSimpleFunc f) + SimpleFunc.setToSimpleFunc T (toSimpleFunc g)"}, {"tactic": "rw [\u2190 SimpleFunc.setToSimpleFunc_add T h_add (SimpleFunc.integrable f)\n    (SimpleFunc.integrable g)]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.setToSimpleFunc_add</a> T h_add (<a>SimpleFunc.integrable</a> f)\n      (<a>SimpleFunc.integrable</a> g)]", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_add", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [449, 9], "def_end_pos": [449, 28]}, {"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.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\nf g : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) =\n    SimpleFunc.setToSimpleFunc T (toSimpleFunc f) + SimpleFunc.setToSimpleFunc T (toSimpleFunc 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\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\nf g : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) = SimpleFunc.setToSimpleFunc T (toSimpleFunc f + toSimpleFunc g)"}, {"tactic": "exact\n  SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _)\n    (add_toSimpleFunc f g)", "annotated_tactic": ["exact\n    <a>SimpleFunc.setToSimpleFunc_congr</a> T h_zero h_add (<a>SimpleFunc.integrable</a> _)\n      (<a>add_toSimpleFunc</a> f g)", [{"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]}, {"full_name": "MeasureTheory.Lp.simpleFunc.add_toSimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [645, 9], "def_end_pos": [645, 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\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\nf g : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) = SimpleFunc.setToSimpleFunc T (toSimpleFunc f + toSimpleFunc g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "full_name": "MeasureTheory.SimpleFunc.edist_approxOn_mono", "start": [167, 1], "end": [171, 65], "traced_tactics": [{"tactic": "dsimp only [approxOn, coe_comp, Function.comp]", "annotated_tactic": ["dsimp only [<a>approxOn</a>, <a>coe_comp</a>, <a>Function.comp</a>]", [{"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "MeasureTheory.SimpleFunc.coe_comp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [337, 9], "def_end_pos": [337, 17]}, {"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\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhf : Measurable f\ns : Set \u03b1\ny\u2080 : \u03b1\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nm n : \u2115\nh : m \u2264 n\n\u22a2 edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist (\u2191(approxOn f hf s y\u2080 h\u2080 m) x) (f x)", "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\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhf : Measurable f\ns : Set \u03b1\ny\u2080 : \u03b1\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nm n : \u2115\nh : m \u2264 n\n\u22a2 edist (\u2191(nearestPt (fun k => Nat.rec y\u2080 (fun n n_ih => \u2191(denseSeq (\u2191s) n)) k) n) (f x)) (f x) \u2264\n    edist (\u2191(nearestPt (fun k => Nat.rec y\u2080 (fun n n_ih => \u2191(denseSeq (\u2191s) n)) k) m) (f x)) (f x)"}, {"tactic": "exact edist_nearestPt_le _ _ ((nearestPtInd_le _ _ _).trans h)", "annotated_tactic": ["exact <a>edist_nearestPt_le</a> _ _ ((<a>nearestPtInd_le</a> _ _ _).<a>trans</a> h)", [{"full_name": "MeasureTheory.SimpleFunc.edist_nearestPt_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [101, 9], "def_end_pos": [101, 27]}, {"full_name": "MeasureTheory.SimpleFunc.nearestPtInd_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [94, 9], "def_end_pos": [94, 24]}, {"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\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhf : Measurable f\ns : Set \u03b1\ny\u2080 : \u03b1\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nm n : \u2115\nh : m \u2264 n\n\u22a2 edist (\u2191(nearestPt (fun k => Nat.rec y\u2080 (fun n n_ih => \u2191(denseSeq (\u2191s) n)) k) n) (f x)) (f x) \u2264\n    edist (\u2191(nearestPt (fun k => Nat.rec y\u2080 (fun n n_ih => \u2191(denseSeq (\u2191s) n)) k) m) (f x)) (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Action.lean", "full_name": "MeasureTheory.measure_eq_zero_iff_eq_empty_of_smulInvariant", "start": [287, 1], "end": [290, 78], "traced_tactics": [{"tactic": "rw [\u2190 not_iff_not, \u2190 Ne.def, \u2190 pos_iff_ne_zero,\n  measure_pos_iff_nonempty_of_smulInvariant G h\u03bc hU, nonempty_iff_ne_empty]", "annotated_tactic": ["rw [\u2190 <a>not_iff_not</a>, \u2190 <a>Ne.def</a>, \u2190 <a>pos_iff_ne_zero</a>,\n    <a>measure_pos_iff_nonempty_of_smulInvariant</a> G h\u03bc hU, <a>nonempty_iff_ne_empty</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"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": "MeasureTheory.measure_pos_iff_nonempty_of_smulInvariant", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [279, 9], "def_end_pos": [279, 50]}, {"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": "G : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : MulAction G \u03b1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ContinuousConstSMul G \u03b1\ninst\u271d\u00b9 : MulAction.IsMinimal G \u03b1\nK U : Set \u03b1\ninst\u271d : Regular \u03bc\nh\u03bc : \u03bc \u2260 0\nhU : IsOpen U\n\u22a2 \u2191\u2191\u03bc U = 0 \u2194 U = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.mul_X_modMonomial", "start": [188, 1], "end": [190, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.hasCondCdf_ae", "start": [533, 1], "end": [536, 29], "traced_tactics": [{"tactic": "filter_upwards [monotone_preCdf \u03c1, preCdf_le_one \u03c1, tendsto_preCdf_atTop_one \u03c1,\n  tendsto_preCdf_atBot_zero \u03c1, inf_gt_preCdf \u03c1] with a h1 h2 h3 h4 h5", "annotated_tactic": ["filter_upwards [<a>monotone_preCdf</a> \u03c1, <a>preCdf_le_one</a> \u03c1, <a>tendsto_preCdf_atTop_one</a> \u03c1,\n    <a>tendsto_preCdf_atBot_zero</a> \u03c1, <a>inf_gt_preCdf</a> \u03c1] with a h1 h2 h3 h4 h5", [{"full_name": "ProbabilityTheory.monotone_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [318, 9], "def_end_pos": [318, 24]}, {"full_name": "ProbabilityTheory.preCdf_le_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [349, 9], "def_end_pos": [349, 22]}, {"full_name": "ProbabilityTheory.tendsto_preCdf_atTop_one", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [373, 9], "def_end_pos": [373, 33]}, {"full_name": "ProbabilityTheory.tendsto_preCdf_atBot_zero", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [443, 9], "def_end_pos": [443, 34]}, {"full_name": "ProbabilityTheory.inf_gt_preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [511, 9], "def_end_pos": [511, 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\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, HasCondCdf \u03c1 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\na : \u03b1\nh1 : Monotone fun r => preCdf \u03c1 r a\nh2 : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh3 : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\nh4 : Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\nh5 : \u2200 (t : \u211a), \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t a\n\u22a2 HasCondCdf \u03c1 a"}, {"tactic": "exact \u27e8h1, h2, h3, h4, h5\u27e9", "annotated_tactic": ["exact \u27e8h1, h2, h3, h4, h5\u27e9", []], "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\nh1 : Monotone fun r => preCdf \u03c1 r a\nh2 : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh3 : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\nh4 : Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\nh5 : \u2200 (t : \u211a), \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t a\n\u22a2 HasCondCdf \u03c1 a", "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.finset_sum", "start": [317, 1], "end": [326, 34], "traced_tactics": [{"tactic": "induction' I using Finset.induction with i I hi_nmem_I h_ind h", "annotated_tactic": ["induction' I using <a>Finset.induction</a> with i I hi_nmem_I h_ind h", [{"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\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u2211 i in I, \u03bas i)", "state_after": "case empty\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u2211 i in \u2205, \u03bas i)\n\ncase insert\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\n\u22a2 IsSFiniteKernel (\u2211 i in insert i I, \u03bas i)"}, {"tactic": "rw [Finset.sum_empty]", "annotated_tactic": ["rw [<a>Finset.sum_empty</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case empty\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u2211 i in \u2205, \u03bas i)", "state_after": "case empty\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel 0"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case empty\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel 0", "state_after": "no goals"}, {"tactic": "rw [Finset.sum_insert hi_nmem_I]", "annotated_tactic": ["rw [<a>Finset.sum_insert</a> hi_nmem_I]", [{"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\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\n\u22a2 IsSFiniteKernel (\u2211 i in insert i I, \u03bas i)", "state_after": "case insert\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)"}, {"tactic": "haveI : IsSFiniteKernel (\u03bas i) := h i (Finset.mem_insert_self _ _)", "annotated_tactic": ["haveI : <a>IsSFiniteKernel</a> (\u03bas i) := h i (<a>Finset.mem_insert_self</a> _ _)", [{"full_name": "ProbabilityTheory.IsSFiniteKernel", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [278, 7], "def_end_pos": [278, 47]}, {"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\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)", "state_after": "case insert\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\nthis : IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)"}, {"tactic": "have : IsSFiniteKernel (\u2211 x : \u03b9 in I, \u03bas x) :=\n  h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)", "annotated_tactic": ["have : <a>IsSFiniteKernel</a> (\u2211 x : \u03b9 in I, \u03bas x) :=\n      h_ind fun i hiI => h i (<a>Finset.mem_insert_of_mem</a> hiI)", [{"full_name": "ProbabilityTheory.IsSFiniteKernel", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [278, 7], "def_end_pos": [278, 47]}, {"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\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\nthis : IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)", "state_after": "case insert\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\nthis\u271d : IsSFiniteKernel (\u03bas i)\nthis : IsSFiniteKernel (\u2211 x in I, \u03bas x)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)"}, {"tactic": "exact IsSFiniteKernel.add _ _", "annotated_tactic": ["exact <a>IsSFiniteKernel.add</a> _ _", [{"full_name": "ProbabilityTheory.kernel.IsSFiniteKernel.add", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [311, 10], "def_end_pos": [311, 29]}]], "state_before": "case insert\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 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\nthis\u271d : IsSFiniteKernel (\u03bas i)\nthis : IsSFiniteKernel (\u2211 x in I, \u03bas x)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_product_right", "start": [425, 1], "end": [427, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Mem.lean", "full_name": "Vector.mem_of_mem_tail", "start": [70, 1], "end": [73, 43], "traced_tactics": [{"tactic": "induction' n with n _", "annotated_tactic": ["induction' n with n _", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\na a' : \u03b1\nv : Vector \u03b1 n\nha : a \u2208 toList (tail v)\n\u22a2 a \u2208 toList v", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\na a' : \u03b1\nv\u271d : Vector \u03b1 n\nha\u271d : a \u2208 toList (tail v\u271d)\nv : Vector \u03b1 Nat.zero\nha : a \u2208 toList (tail v)\n\u22a2 a \u2208 toList v\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d : \u2115\na a' : \u03b1\nv\u271d : Vector \u03b1 n\u271d\nha\u271d : a \u2208 toList (tail v\u271d)\nn : \u2115\nn_ih\u271d : \u2200 (v : Vector \u03b1 n), a \u2208 toList (tail v) \u2192 a \u2208 toList v\nv : Vector \u03b1 (Nat.succ n)\nha : a \u2208 toList (tail v)\n\u22a2 a \u2208 toList v"}, {"tactic": "exact False.elim (Vector.not_mem_zero a v.tail ha)", "annotated_tactic": ["exact <a>False.elim</a> (<a>Vector.not_mem_zero</a> a v.tail ha)", [{"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": "Vector.not_mem_zero", "def_path": "Mathlib/Data/Vector/Mem.lean", "def_pos": [44, 9], "def_end_pos": [44, 21]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\na a' : \u03b1\nv\u271d : Vector \u03b1 n\nha\u271d : a \u2208 toList (tail v\u271d)\nv : Vector \u03b1 Nat.zero\nha : a \u2208 toList (tail v)\n\u22a2 a \u2208 toList v", "state_after": "no goals"}, {"tactic": "exact (mem_succ_iff a v).2 (Or.inr ha)", "annotated_tactic": ["exact (<a>mem_succ_iff</a> a v).2 (<a>Or.inr</a> ha)", [{"full_name": "Vector.mem_succ_iff", "def_path": "Mathlib/Data/Vector/Mem.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"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 succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d : \u2115\na a' : \u03b1\nv\u271d : Vector \u03b1 n\u271d\nha\u271d : a \u2208 toList (tail v\u271d)\nn : \u2115\nn_ih\u271d : \u2200 (v : Vector \u03b1 n), a \u2208 toList (tail v) \u2192 a \u2208 toList v\nv : Vector \u03b1 (Nat.succ n)\nha : a \u2208 toList (tail v)\n\u22a2 a \u2208 toList v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.extend_iUnion", "start": [1412, 1], "end": [1419, 29], "traced_tactics": [{"tactic": "cases nonempty_encodable \u03b2", "annotated_tactic": ["cases <a>nonempty_encodable</a> \u03b2", [{"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\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\n\u22a2 extend m (\u22c3 i, f i) = \u2211' (i : \u03b2), extend m (f i)", "state_after": "case intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m (\u22c3 i, f i) = \u2211' (i : \u03b2), extend m (f i)"}, {"tactic": "rw [\u2190 Encodable.iUnion_decode\u2082, \u2190 tsum_iUnion_decode\u2082]", "annotated_tactic": ["rw [\u2190 <a>Encodable.iUnion_decode\u2082</a>, \u2190 <a>tsum_iUnion_decode\u2082</a>]", [{"full_name": "Encodable.iUnion_decode\u2082", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [35, 9], "def_end_pos": [35, 23]}, {"full_name": "tsum_iUnion_decode\u2082", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 28]}]], "state_before": "case intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m (\u22c3 i, f i) = \u2211' (i : \u03b2), extend m (f i)", "state_after": "case intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m (\u22c3 i, \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) = \u2211' (i : \u2115), extend m (\u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b)\n\ncase intro.m0\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m \u2205 = 0"}, {"tactic": "exact\n  extend_iUnion_nat PU (fun n => Encodable.iUnion_decode\u2082_cases P0 hm)\n    (mU _ (Encodable.iUnion_decode\u2082_disjoint_on hd))", "annotated_tactic": ["exact\n      <a>extend_iUnion_nat</a> PU (fun n => <a>Encodable.iUnion_decode\u2082_cases</a> P0 hm)\n        (mU _ (<a>Encodable.iUnion_decode\u2082_disjoint_on</a> hd))", [{"full_name": "MeasureTheory.extend_iUnion_nat", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1375, 9], "def_end_pos": [1375, 26]}, {"full_name": "Encodable.iUnion_decode\u2082_cases", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [41, 9], "def_end_pos": [41, 29]}, {"full_name": "Encodable.iUnion_decode\u2082_disjoint_on", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [52, 9], "def_end_pos": [52, 35]}]], "state_before": "case intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m (\u22c3 i, \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) = \u2211' (i : \u2115), extend m (\u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b)", "state_after": "no goals"}, {"tactic": "exact extend_empty P0 m0", "annotated_tactic": ["exact <a>extend_empty</a> P0 m0", [{"full_name": "MeasureTheory.extend_empty", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1371, 9], "def_end_pos": [1371, 21]}]], "state_before": "case intro.m0\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i)\nmU :\n  \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n    Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m \u2205 = 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.fmod_add_fdiv", "start": [282, 1], "end": [297, 72], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a\u271d : Nat\n\u22a2 0 + 0 = 0", "state_after": "no goals"}, {"tactic": "show subNatNat (m % succ n) n + (\u2191(succ n * (m / succ n)) + n + 1) = (m + 1)", "annotated_tactic": ["show <a>subNatNat</a> (m % <a>succ</a> n) n + (\u2191(<a>succ</a> n * (m / <a>succ</a> n)) + n + 1) = (m + 1)", [{"full_name": "Int.subNatNat", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [86, 5], "def_end_pos": [86, 14]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "m n : Nat\n\u22a2 fmod \u2191(succ m) -[n+1] + -[n+1] * fdiv \u2191(succ m) -[n+1] = \u2191(succ m)", "state_after": "m n : Nat\n\u22a2 subNatNat (m % succ n) n + (\u2191(succ n * (m / succ n)) + \u2191n + 1) = \u2191m + 1"}, {"tactic": "rw [Int.add_comm _ n, \u2190 Int.add_assoc, \u2190 Int.add_assoc,\n  Int.subNatNat_eq_coe, Int.sub_add_cancel]", "annotated_tactic": ["rw [<a>Int.add_comm</a> _ n, \u2190 <a>Int.add_assoc</a>, \u2190 <a>Int.add_assoc</a>,\n      <a>Int.subNatNat_eq_coe</a>, <a>Int.sub_add_cancel</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.add_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [262, 19], "def_end_pos": [262, 28]}, {"full_name": "Int.add_assoc", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [262, 19], "def_end_pos": [262, 28]}, {"full_name": "Int.subNatNat_eq_coe", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [505, 19], "def_end_pos": [505, 35]}, {"full_name": "Int.sub_add_cancel", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [486, 27], "def_end_pos": [486, 41]}]], "state_before": "m n : Nat\n\u22a2 subNatNat (m % succ n) n + (\u2191(succ n * (m / succ n)) + \u2191n + 1) = \u2191m + 1", "state_after": "m n : Nat\n\u22a2 \u2191(m % succ n) + \u2191(succ n * (m / succ n)) + 1 = \u2191m + 1"}, {"tactic": "rw [fmod_zero]", "annotated_tactic": ["rw [<a>fmod_zero</a>]", [{"full_name": "Int.fmod_zero", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [258, 17], "def_end_pos": [258, 26]}]], "state_before": "a\u271d : Nat\n\u22a2 fmod -[a\u271d+1] 0 + 0 * fdiv -[a\u271d+1] 0 = -[a\u271d+1]", "state_after": "a\u271d : Nat\n\u22a2 -[a\u271d+1] + 0 * fdiv -[a\u271d+1] 0 = -[a\u271d+1]"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "a\u271d : Nat\n\u22a2 -[a\u271d+1] + 0 * fdiv -[a\u271d+1] 0 = -[a\u271d+1]", "state_after": "no goals"}, {"tactic": "show subNatNat .. - (\u2191(succ n * (m / succ n)) + \u2191(succ n)) = -\u2191(succ m)", "annotated_tactic": ["show <a>subNatNat</a> .. - (\u2191(<a>succ</a> n * (m / <a>succ</a> n)) + \u2191(<a>succ</a> n)) = -\u2191(<a>succ</a> m)", [{"full_name": "Int.subNatNat", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [86, 5], "def_end_pos": [86, 14]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "m n : Nat\n\u22a2 fmod -[m+1] \u2191(succ n) + \u2191(succ n) * fdiv -[m+1] \u2191(succ n) = -[m+1]", "state_after": "m n : Nat\n\u22a2 subNatNat (succ n) (succ (m % succ n)) - (\u2191(succ n * (m / succ n)) + \u2191(succ n)) = -\u2191(succ m)"}, {"tactic": "rw [Int.subNatNat_eq_coe, \u2190 Int.sub_sub, \u2190 Int.neg_sub, Int.sub_sub, Int.sub_sub_self]", "annotated_tactic": ["rw [<a>Int.subNatNat_eq_coe</a>, \u2190 <a>Int.sub_sub</a>, \u2190 <a>Int.neg_sub</a>, <a>Int.sub_sub</a>, <a>Int.sub_sub_self</a>]", [{"full_name": "Int.subNatNat_eq_coe", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [505, 19], "def_end_pos": [505, 35]}, {"full_name": "Int.sub_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [361, 19], "def_end_pos": [361, 26]}, {"full_name": "Int.neg_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [364, 19], "def_end_pos": [364, 26]}, {"full_name": "Int.sub_sub", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [361, 19], "def_end_pos": [361, 26]}, {"full_name": "Int.sub_sub_self", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [367, 19], "def_end_pos": [367, 31]}]], "state_before": "m n : Nat\n\u22a2 subNatNat (succ n) (succ (m % succ n)) - (\u2191(succ n * (m / succ n)) + \u2191(succ n)) = -\u2191(succ m)", "state_after": "m n : Nat\n\u22a2 -(\u2191(succ (m % succ n)) + \u2191(succ n * (m / succ n))) = -\u2191(succ m)"}, {"tactic": "show -(\u2191(succ m % succ n) : Int) + -\u2191(succ n * (succ m / succ n)) = -\u2191(succ m)", "annotated_tactic": ["show -(\u2191(<a>succ</a> m % <a>succ</a> n) : <a>Int</a>) + -\u2191(<a>succ</a> n * (<a>succ</a> m / <a>succ</a> n)) = -\u2191(<a>succ</a> m)", [{"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Int", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [40, 11], "def_end_pos": [40, 14]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"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.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "m n : Nat\n\u22a2 fmod -[m+1] -[n+1] + -[n+1] * fdiv -[m+1] -[n+1] = -[m+1]", "state_after": "m n : Nat\n\u22a2 -\u2191(succ m % succ n) + -\u2191(succ n * (succ m / succ n)) = -\u2191(succ m)"}, {"tactic": "rw [\u2190 Int.neg_add]", "annotated_tactic": ["rw [\u2190 <a>Int.neg_add</a>]", [{"full_name": "Int.neg_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [334, 33], "def_end_pos": [334, 40]}]], "state_before": "m n : Nat\n\u22a2 -\u2191(succ m % succ n) + -\u2191(succ n * (succ m / succ n)) = -\u2191(succ m)", "state_after": "m n : Nat\n\u22a2 -(\u2191(succ m % succ n) + \u2191(succ n * (succ m / succ n))) = -\u2191(succ m)"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_preserving_finTwoArrow", "start": [859, 1], "end": [862, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.mem\u2112p_iff", "start": [351, 1], "end": [354, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.liftOn_ofMem", "start": [119, 1], "end": [121, 48], "traced_tactics": [{"tactic": "revert h", "annotated_tactic": ["revert h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (a : \u03b1), a \u2208 q \u2192 \u2200 (b : \u03b1), b \u2208 q \u2192 f a = f b\na : \u03b1\naq : a \u2208 q\n\u22a2 liftOn q f h = f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\naq : a \u2208 q\n\u22a2 \u2200 (h : \u2200 (a : \u03b1), a \u2208 q \u2192 \u2200 (b : \u03b1), b \u2208 q \u2192 f a = f b), liftOn q f h = f a"}, {"tactic": "rw [eq_mk_of_mem aq]", "annotated_tactic": ["rw [<a>eq_mk_of_mem</a> aq]", [{"full_name": "Semiquot.eq_mk_of_mem", "def_path": "Mathlib/Data/Semiquot.lean", "def_pos": [62, 9], "def_end_pos": [62, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\naq : a \u2208 q\n\u22a2 \u2200 (h : \u2200 (a : \u03b1), a \u2208 q \u2192 \u2200 (b : \u03b1), b \u2208 q \u2192 f a = f b), liftOn q f h = f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\naq : a \u2208 q\n\u22a2 \u2200 (h : \u2200 (a_1 : \u03b1), a_1 \u2208 mk aq \u2192 \u2200 (b : \u03b1), b \u2208 mk aq \u2192 f a_1 = f b), liftOn (mk aq) f h = f a"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\naq : a \u2208 q\n\u22a2 \u2200 (h : \u2200 (a_1 : \u03b1), a_1 \u2208 mk aq \u2192 \u2200 (b : \u03b1), b \u2208 mk aq \u2192 f a_1 = f b), liftOn (mk aq) f h = f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\naq : a \u2208 q\nh\u271d : \u2200 (a_1 : \u03b1), a_1 \u2208 mk aq \u2192 \u2200 (b : \u03b1), b \u2208 mk aq \u2192 f a_1 = f b\n\u22a2 liftOn (mk aq) f h\u271d = f a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq : Semiquot \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\naq : a \u2208 q\nh\u271d : \u2200 (a_1 : \u03b1), a_1 \u2208 mk aq \u2192 \u2200 (b : \u03b1), b \u2208 mk aq \u2192 f a_1 = f b\n\u22a2 liftOn (mk aq) f h\u271d = f a", "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": [222, 1], "end": [229, 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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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": [116, 9], "def_end_pos": [116, 30]}]], "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\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2079 : NormedSpace \u211d F'\ninst\u271d\u2078 : CompleteSpace F'\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \u211d G'\ninst\u271d\u2074 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm)\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nc : \ud835\udd5c\nx : F\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/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.mod_one", "start": [346, 9], "end": [347, 57], "traced_tactics": [{"tactic": "simp [mod_def, Int.div_one, Int.one_mul, Int.sub_self]", "annotated_tactic": ["simp [<a>mod_def</a>, <a>Int.div_one</a>, <a>Int.one_mul</a>, <a>Int.sub_self</a>]", [{"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.div_one", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [120, 27], "def_end_pos": [120, 34]}, {"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.sub_self", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [343, 19], "def_end_pos": [343, 27]}]], "state_before": "a : Int\n\u22a2 mod a 1 = 0", "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_zero_left", "start": [890, 1], "end": [892, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iSup_directed", "start": [1154, 1], "end": [1179, 46], "traced_tactics": [{"tactic": "simp_rw [\u2190 iSup_apply]", "annotated_tactic": ["simp_rw [\u2190 <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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 b, f b a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc"}, {"tactic": "let p : \u03b1 \u2192 (\u03b2 \u2192 ENNReal) \u2192 Prop := fun x f' => Directed LE.le f'", "annotated_tactic": ["let p : \u03b1 \u2192 (\u03b2 \u2192 <a>ENNReal</a>) \u2192 Prop := fun x f' => <a>Directed</a> <a>LE.le</a> f'", [{"full_name": "ENNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [83, 5], "def_end_pos": [83, 12]}, {"full_name": "Directed", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [43, 5], "def_end_pos": [43, 13]}, {"full_name": "LE.le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1075, 3], "def_end_pos": [1075, 5]}]], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc"}, {"tactic": "have hp : \u2200\u1d50 x \u2202\u03bc, p x fun i => f i x := by\n  filter_upwards [] with x i j\n  obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j\n  exact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9", "annotated_tactic": ["have hp : \u2200\u1d50 x \u2202\u03bc, p x fun i => f i x := by\n    filter_upwards [] with x i j\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j\n    exact \u27e8z, hz\u2081 x, hz\u2082 x\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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc"}, {"tactic": "convert lintegral_iSup_directed_of_measurable (aeSeq.measurable hf p) h_ae_seq_directed using 1", "annotated_tactic": ["convert <a>lintegral_iSup_directed_of_measurable</a> (<a>aeSeq.measurable</a> hf p) h_ae_seq_directed using 1", [{"full_name": "MeasureTheory.lintegral_iSup_directed_of_measurable", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1132, 9], "def_end_pos": [1132, 46]}, {"full_name": "aeSeq.measurable", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [95, 9], "def_end_pos": [95, 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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc", "state_after": "case h.e'_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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 b, aeSeq hf p b a \u2202?m.1158777\n\ncase 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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202?m.1158777\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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 Measure \u03b1"}, {"tactic": "filter_upwards [] with x i j", "annotated_tactic": ["filter_upwards [] with x i j", []], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i 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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nx : \u03b1\ni j : \u03b2\n\u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z"}, {"tactic": "obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j", "annotated_tactic": ["obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j", []], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nx : \u03b1\ni j : \u03b2\n\u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z", "state_after": "case h.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nx : \u03b1\ni j z : \u03b2\nhz\u2081 : f i \u2264 f z\nhz\u2082 : f j \u2264 f z\n\u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z"}, {"tactic": "exact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9", "annotated_tactic": ["exact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9", []], "state_before": "case h.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nx : \u03b1\ni j z : \u03b2\nhz\u2081 : f i \u2264 f z\nhz\u2082 : f j \u2264 f z\n\u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z", "state_after": "no goals"}, {"tactic": "intro b\u2081 b\u2082", "annotated_tactic": ["intro b\u2081 b\u2082", []], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n\u22a2 Directed LE.le (aeSeq hf p)", "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 : \u03b2\n\u22a2 \u2203 z, aeSeq hf p b\u2081 \u2264 aeSeq hf p z \u2227 aeSeq hf p b\u2082 \u2264 aeSeq hf p z"}, {"tactic": "obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed b\u2081 b\u2082", "annotated_tactic": ["obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed b\u2081 b\u2082", []], "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 : \u03b2\n\u22a2 \u2203 z, aeSeq hf p b\u2081 \u2264 aeSeq hf p z \u2227 aeSeq hf p b\u2082 \u2264 aeSeq hf p z", "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\n\u22a2 \u2203 z, aeSeq hf p b\u2081 \u2264 aeSeq hf p z \u2227 aeSeq hf p b\u2082 \u2264 aeSeq hf p z"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case intro.intro.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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\n\u22a2 aeSeq hf p b\u2082 \u2264 aeSeq hf p z", "state_after": "case intro.intro.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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x"}, {"tactic": "by_cases hx : x \u2208 aeSeqSet hf p", "annotated_tactic": ["by_cases hx : x \u2208 <a>aeSeqSet</a> hf p", [{"full_name": "aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [36, 5], "def_end_pos": [36, 13]}]], "state_before": "case intro.intro.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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x", "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x\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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x"}, {"tactic": "repeat' rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]", "annotated_tactic": ["repeat' rw [<a>aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet</a> hf hx]", [{"full_name": "aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [64, 9], "def_end_pos": [64, 37]}]], "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x", "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 f z x"}, {"tactic": "apply_rules [hz\u2081, hz\u2082]", "annotated_tactic": ["apply_rules [hz\u2081, hz\u2082]", []], "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 f z x", "state_after": "no goals"}, {"tactic": "rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]", "annotated_tactic": ["rw [<a>aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet</a> hf hx]", [{"full_name": "aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [64, 9], "def_end_pos": [64, 37]}]], "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 aeSeq hf p z x", "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 f z x"}, {"tactic": "simp only [aeSeq, hx, if_false]", "annotated_tactic": ["simp only [<a>aeSeq</a>, hx, <a>if_false</a>]", [{"full_name": "aeSeq", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [42, 19], "def_end_pos": [42, 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 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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x", "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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e)"}, {"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\u03b1 : Type u_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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e)", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 iSup_apply]", "annotated_tactic": ["simp_rw [\u2190 <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'_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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 b, aeSeq hf p b a \u2202?m.1158777", "state_after": "case h.e'_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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc =\n    \u222b\u207b (a : \u03b1), iSup (fun i => aeSeq hf (fun x f' => Directed LE.le f') i) a \u2202?m.1158777"}, {"tactic": "rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> (<a>aeSeq.iSup</a> hf hp).<a>symm</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": "aeSeq.iSup", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [130, 9], "def_end_pos": [130, 13]}, {"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.e'_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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc =\n    \u222b\u207b (a : \u03b1), iSup (fun i => aeSeq hf (fun x f' => Directed LE.le f') i) a \u2202?m.1158777", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc", "state_after": "case h.e'_3.e_s\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 (fun b => \u222b\u207b (a : \u03b1), f b a \u2202\u03bc) = fun b => \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc"}, {"tactic": "ext1 b", "annotated_tactic": ["ext1 b", []], "state_before": "case h.e'_3.e_s\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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 (fun b => \u222b\u207b (a : \u03b1), f b a \u2202\u03bc) = fun b => \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc", "state_after": "case h.e'_3.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae]", "annotated_tactic": ["rw [<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.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc", "state_after": "case h.e'_3.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 (fun a => f b a) =\u1d50[\u03bc] fun a => aeSeq hf p b a"}, {"tactic": "apply EventuallyEq.symm", "annotated_tactic": ["apply <a>EventuallyEq.symm</a>", [{"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.e'_3.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\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 (fun a => f b a) =\u1d50[\u03bc] fun a => aeSeq hf p b a", "state_after": "case h.e'_3.e_s.h.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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 (fun a => aeSeq hf p b a) =\u1d50[\u03bc] fun a => f b a"}, {"tactic": "refine' aeSeq.aeSeq_n_eq_fun_n_ae hf hp _", "annotated_tactic": ["refine' <a>aeSeq.aeSeq_n_eq_fun_n_ae</a> hf hp _", [{"full_name": "aeSeq.aeSeq_n_eq_fun_n_ae", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [125, 9], "def_end_pos": [125, 28]}]], "state_before": "case h.e'_3.e_s.h.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 : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 (fun a => aeSeq hf p b a) =\u1d50[\u03bc] fun a => f b a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Set.Finite.toFinset_smul_set", "start": [2279, 1], "end": [2281, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_smul'", "start": [199, 1], "end": [203, 42], "traced_tactics": [{"tactic": "convert variance_smul (algebraMap A \u211d c) X \u03bc using 1", "annotated_tactic": ["convert <a>variance_smul</a> (<a>algebraMap</a> A \u211d c) X \u03bc using 1", [{"full_name": "ProbabilityTheory.variance_smul", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [194, 9], "def_end_pos": [194, 22]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 variance (c \u2022 X) \u03bc = c ^ 2 \u2022 variance X \u03bc", "state_after": "case h.e'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 variance (c \u2022 X) \u03bc = variance (\u2191(algebraMap A \u211d) c \u2022 X) \u03bc\n\ncase h.e'_3\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 c ^ 2 \u2022 variance X \u03bc = \u2191(algebraMap A \u211d) c ^ 2 * variance X \u03bc"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.e'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 variance (c \u2022 X) \u03bc = variance (\u2191(algebraMap A \u211d) c \u2022 X) \u03bc", "state_after": "case h.e'_2.e_X\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 c \u2022 X = \u2191(algebraMap A \u211d) c \u2022 X"}, {"tactic": "simp only [algebraMap_smul]", "annotated_tactic": ["simp only [<a>algebraMap_smul</a>]", [{"full_name": "algebraMap_smul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [835, 9], "def_end_pos": [835, 24]}]], "state_before": "case h.e'_2.e_X\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 c \u2022 X = \u2191(algebraMap A \u211d) c \u2022 X", "state_after": "no goals"}, {"tactic": "simp only [Algebra.smul_def, map_pow]", "annotated_tactic": ["simp only [<a>Algebra.smul_def</a>, <a>map_pow</a>]", [{"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "map_pow", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [435, 9], "def_end_pos": [435, 16]}]], "state_before": "case h.e'_3\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nA : Type u_2\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A \u211d\nc : A\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 c ^ 2 \u2022 variance X \u03bc = \u2191(algebraMap A \u211d) c ^ 2 * variance X \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.setToL1SCLM_congr_measure", "start": [913, 1], "end": [916, 79], "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": [320, 1], "end": [323, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.range_restrictPreimage", "start": [570, 1], "end": [573, 47], "traced_tactics": [{"tactic": "delta Set.restrictPreimage", "annotated_tactic": ["delta <a>Set.restrictPreimage</a>", [{"full_name": "Set.restrictPreimage", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [565, 5], "def_end_pos": [565, 21]}]], "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 range (restrictPreimage t f) = Subtype.val \u207b\u00b9' range f", "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 range (MapsTo.restrict f (f \u207b\u00b9' t) t (_ : MapsTo f (f \u207b\u00b9' t) t)) = Subtype.val \u207b\u00b9' range f"}, {"tactic": "rw [MapsTo.range_restrict, Set.image_preimage_eq_inter_range, Set.preimage_inter,\n  Subtype.coe_preimage_self, Set.univ_inter]", "annotated_tactic": ["rw [<a>MapsTo.range_restrict</a>, <a>Set.image_preimage_eq_inter_range</a>, <a>Set.preimage_inter</a>,\n    <a>Subtype.coe_preimage_self</a>, <a>Set.univ_inter</a>]", [{"full_name": "Set.MapsTo.range_restrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [385, 9], "def_end_pos": [385, 30]}, {"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.preimage_inter", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [87, 9], "def_end_pos": [87, 23]}, {"full_name": "Subtype.coe_preimage_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1430, 9], "def_end_pos": [1430, 26]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}]], "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 range (MapsTo.restrict f (f \u207b\u00b9' t) t (_ : MapsTo f (f \u207b\u00b9' t) t)) = Subtype.val \u207b\u00b9' range f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Basic.lean", "full_name": "WType.elim_injective", "start": [93, 1], "end": [98, 75], "traced_tactics": [{"tactic": "obtain \u27e8rfl, h\u27e9 := Sigma.mk.inj_iff.mp (f\u03b3_injective h)", "annotated_tactic": ["obtain \u27e8rfl, h\u27e9 := Sigma.mk.inj_iff.mp (f\u03b3_injective h)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\na\u2082 : \u03b1\nf\u2082 : \u03b2 a\u2082 \u2192 WType fun a => \u03b2 a\nh : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2082 f\u2082)\n\u22a2 mk a\u2081 f\u2081 = mk a\u2082 f\u2082", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\nh\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082)\nh : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b)\n\u22a2 mk a\u2081 f\u2081 = mk a\u2081 f\u2082"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\nh\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082)\nh : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b)\n\u22a2 mk a\u2081 f\u2081 = mk a\u2081 f\u2082", "state_after": "case intro.e_f.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\nh\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082)\nh : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b)\nx : \u03b2 a\u2081\n\u22a2 f\u2081 x = f\u2082 x"}, {"tactic": "exact elim_injective \u03b3 f\u03b3 f\u03b3_injective (congr_fun (eq_of_heq h) x : _)", "annotated_tactic": ["exact elim_injective \u03b3 f\u03b3 f\u03b3_injective (<a>congr_fun</a> (<a>eq_of_heq</a> h) x : _)", [{"full_name": "congr_fun", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [42, 7], "def_end_pos": [42, 16]}, {"full_name": "eq_of_heq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}]], "state_before": "case intro.e_f.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\nh\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082)\nh : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b)\nx : \u03b2 a\u2081\n\u22a2 f\u2081 x = f\u2082 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.fundamentalInterior_smul", "start": [628, 1], "end": [630, 98], "traced_tactics": [{"tactic": "simp_rw [fundamentalInterior, smul_set_sdiff, smul_set_iUnion, smul_comm g (_ : G) (_ : Set \u03b1)]", "annotated_tactic": ["simp_rw [<a>fundamentalInterior</a>, <a>smul_set_sdiff</a>, <a>smul_set_iUnion</a>, <a>smul_comm</a> g (_ : G) (_ : <a>Set</a> \u03b1)]", [{"full_name": 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fundamentalInterior G (g \u2022 s) = g \u2022 fundamentalInterior G s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.limsup_measure_le_of_le_liminf_measure_compl", "start": [159, 1], "end": [163, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.SimpleFunc.norm_integral_le_integral_norm", "start": [442, 1], "end": [445, 59], "traced_tactics": [{"tactic": "refine' (norm_setToSimpleFunc_le_integral_norm _ (fun s _ _ => _) hf).trans (one_mul _).le", "annotated_tactic": ["refine' (<a>norm_setToSimpleFunc_le_integral_norm</a> _ (fun s _ _ => _) hf).<a>trans</a> (<a>one_mul</a> _).<a>le</a>", [{"full_name": "MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_integral_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [432, 9], "def_end_pos": [432, 46]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "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 E\nhf : Integrable \u2191f\n\u22a2 \u2016integral \u03bc f\u2016 \u2264 integral \u03bc (map norm 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\nf : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\ns : Set \u03b1\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2016weightedSMul \u03bc s\u2016 \u2264 1 * ENNReal.toReal (\u2191\u2191\u03bc s)"}, {"tactic": "exact (norm_weightedSMul_le s).trans (one_mul _).symm.le", "annotated_tactic": ["exact (<a>norm_weightedSMul_le</a> s).<a>trans</a> (<a>one_mul</a> _).symm.le", [{"full_name": "MeasureTheory.norm_weightedSMul_le", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [233, 9], "def_end_pos": [233, 29]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"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\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type 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[<a>comp\u2082_eq_pair</a>, <a>pair_eq_mk</a>, <a>comp_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.comp\u2082_eq_pair", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [372, 9], "def_end_pos": [372, 22]}, {"full_name": "MeasureTheory.AEEqFun.pair_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [344, 9], "def_end_pos": [344, 19]}, {"full_name": "MeasureTheory.AEEqFun.comp_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [282, 9], "def_end_pos": [282, 16]}]], "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 comp\u2082 g hg f\u2081 f\u2082 = mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)", "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 mk (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) =\n    mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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 mk (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) =\n    mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)", "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.size_tail?_lt", "start": [159, 1], "end": [163, 53], "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 size s' < size 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 size s' < size 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 size s' < size 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 size s' < size s"}, {"tactic": "match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact size_deleteMin_lt 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_lt</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.size_deleteMin_lt", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [155, 9], "def_end_pos": [155, 31]}]], "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 size s' < size s", "state_after": "no goals"}, {"tactic": "exact size_deleteMin_lt eq\u2082", "annotated_tactic": ["exact <a>size_deleteMin_lt</a> eq\u2082", [{"full_name": "Std.PairingHeapImp.Heap.size_deleteMin_lt", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [155, 9], "def_end_pos": [155, 31]}]], "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 size ((fun x => x.snd) (a, tl)) < size s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Multiset.toFinset_eq_empty", "start": [3244, 1], "end": [3245, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upperCrossingTime_succ", "start": [168, 1], "end": [170, 25], "traced_tactics": [{"tactic": "rw [upperCrossingTime]", "annotated_tactic": ["rw [<a>upperCrossingTime</a>]", [{"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : OrderBot \u03b9\ninst\u271d : InfSet \u03b9\na b : \u211d\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nN : \u03b9\nn m : \u2115\n\u03c9 : \u03a9\n\u22a2 upperCrossingTime a b f N (n + 1) \u03c9 =\n    hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n \u03c9) N \u03c9) N \u03c9", "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", "start": [815, 1], "end": [817, 80], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico h\u2081 h\u2082]", "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</a> h\u2081 h\u2082]", [{"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", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1547, 9], "def_end_pos": [1547, 22]}]], "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 d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)", "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_fun_fst", "start": [539, 1], "end": [541, 25], "traced_tactics": [{"tactic": "rw [\u2190 integral_prod_swap]", "annotated_tactic": ["rw [\u2190 <a>integral_prod_swap</a>]", [{"full_name": "MeasureTheory.integral_prod_swap", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [345, 9], "def_end_pos": [345, 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\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 \u2192 E\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2202Measure.prod \u03bc \u03bd = ENNReal.toReal (\u2191\u2191\u03bd univ) \u2022 \u222b (x : \u03b1), f x \u2202\u03bc", "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\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 \u2192 E\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z).1 \u2202Measure.prod \u03bd \u03bc = ENNReal.toReal (\u2191\u2191\u03bd univ) \u2022 \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "apply integral_fun_snd", "annotated_tactic": ["apply <a>integral_fun_snd</a>", [{"full_name": "MeasureTheory.integral_fun_snd", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [536, 9], "def_end_pos": [536, 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\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 \u2192 E\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z).1 \u2202Measure.prod \u03bd \u03bc = ENNReal.toReal (\u2191\u2191\u03bd univ) \u2022 \u222b (x : \u03b1), f x \u2202\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.finset_sum_apply", "start": [103, 1], "end": [104, 84], "traced_tactics": [{"tactic": "rw [coe_finset_sum, Finset.sum_apply]", "annotated_tactic": ["rw [<a>coe_finset_sum</a>, <a>Finset.sum_apply</a>]", [{"full_name": "ProbabilityTheory.kernel.coe_finset_sum", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 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\nI : Finset \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\n\u22a2 \u2191(\u2211 i in I, \u03ba i) a = \u2211 i in I, \u2191(\u03ba i) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sign.lean", "full_name": "Int.sign_eq_sign", "start": [480, 1], "end": [481, 73], "traced_tactics": [{"tactic": "obtain (n | _) | _ := n <;> simp [sign, Int.sign_neg, negSucc_lt_zero]", "annotated_tactic": ["obtain (n | _) | _ := n <;> simp [<a>sign</a>, <a>Int.sign_neg</a>, <a>negSucc_lt_zero</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": "Int.sign_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [211, 17], "def_end_pos": [211, 25]}, {"full_name": "Int.negSucc_lt_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [738, 9], "def_end_pos": [738, 24]}]], "state_before": "\u03b1 : Type u_1\nn : \u2124\n\u22a2 sign n = \u2191(\u2191SignType.sign n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "start": [1457, 9], "end": [1471, 70], "traced_tactics": [{"tactic": "let f_norm_diff i x := \u2016f (i + 1) x - f i x\u2016", "annotated_tactic": ["let f_norm_diff i x := \u2016f (i + 1) x - f i x\u2016", []], "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\n\u22a2 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", "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\n\u22a2 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"}, {"tactic": "have hgf_norm_diff :\n  \u2200 n,\n    (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) =\n      \u2211 i in Finset.range (n + 1), f_norm_diff i :=\n  fun n => funext fun x => by simp", "annotated_tactic": ["have hgf_norm_diff :\n    \u2200 n,\n      (fun x => \u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) =\n        \u2211 i in <a>Finset.range</a> (n + 1), f_norm_diff i :=\n    fun n => <a>funext</a> fun x => by simp", [{"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": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\n\u22a2 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", "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 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"}, {"tactic": "rw [hgf_norm_diff]", "annotated_tactic": ["rw [hgf_norm_diff]", []], "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 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", "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 snorm' (\u2211 i in Finset.range (n + 1), f_norm_diff i) p \u03bc \u2264 \u2211' (i : \u2115), B i"}, {"tactic": "refine' (snorm'_sum_le (fun i _ => ((hf (i + 1)).sub (hf i)).norm) hp1).trans _", "annotated_tactic": ["refine' (<a>snorm'_sum_le</a> (fun i _ => ((hf (i + 1)).<a>sub</a> (hf i)).<a>norm</a>) hp1).<a>trans</a> _", [{"full_name": "MeasureTheory.snorm'_sum_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1218, 9], "def_end_pos": [1218, 22]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 3], "def_end_pos": [1322, 14]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.norm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1479, 19], "def_end_pos": [1479, 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\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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 snorm' (\u2211 i in Finset.range (n + 1), f_norm_diff i) p \u03bc \u2264 \u2211' (i : \u2115), B i", "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 \u2211 i in Finset.range (n + 1), snorm' (fun x => \u2016(f (i + 1) - f i) x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i"}, {"tactic": "simp_rw [snorm'_norm]", "annotated_tactic": ["simp_rw [<a>snorm'_norm</a>]", [{"full_name": "MeasureTheory.snorm'_norm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [497, 9], "def_end_pos": [497, 20]}]], "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 \u2211 i in Finset.range (n + 1), snorm' (fun x => \u2016(f (i + 1) - f i) x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i", "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 \u2211 x in Finset.range (n + 1), snorm' (fun a => (f (x + 1) - f x) a) p \u03bc \u2264 \u2211' (i : \u2115), B i"}, {"tactic": "refine' (Finset.sum_le_sum _).trans (sum_le_tsum _ (fun m _ => zero_le _) ENNReal.summable)", "annotated_tactic": ["refine' (<a>Finset.sum_le_sum</a> _).<a>trans</a> (<a>sum_le_tsum</a> _ (fun m _ => <a>zero_le</a> _) <a>ENNReal.summable</a>)", [{"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "sum_le_tsum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [95, 9], "def_end_pos": [95, 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.summable", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [777, 19], "def_end_pos": [777, 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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 \u2211 x in Finset.range (n + 1), snorm' (fun a => (f (x + 1) - f x) a) p \u03bc \u2264 \u2211' (i : \u2115), B i", "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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range (n + 1) \u2192 snorm' (fun a => (f (i + 1) - f i) a) p \u03bc \u2264 B i"}, {"tactic": "exact fun m _ => (h_cau m (m + 1) m (Nat.le_succ m) (le_refl m)).le", "annotated_tactic": ["exact fun m _ => (h_cau m (m + 1) m (<a>Nat.le_succ</a> m) (<a>le_refl</a> m)).<a>le</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]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"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\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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nhgf_norm_diff :\n  \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range (n + 1) \u2192 snorm' (fun a => (f (i + 1) - f i) a) p \u03bc \u2264 B i", "state_after": "no goals"}, {"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\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\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nn\u271d : \u2115\nf_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016\nn : \u2115\nx : \u03b1\n\u22a2 \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016 = Finset.sum (Finset.range (n + 1)) (fun i => f_norm_diff i) 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_empty", "start": [430, 1], "end": [437, 68], "traced_tactics": [{"tactic": "let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5", "annotated_tactic": ["let eval : (<a>Compacts</a> G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5", [{"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 \u22a5 = 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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\n\u22a2 chaar K\u2080 \u22a5 = 0"}, {"tactic": "have : Continuous eval := continuous_apply \u22a5", "annotated_tactic": ["have : <a>Continuous</a> eval := <a>continuous_apply</a> \u22a5", [{"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 \u22a5\n\u22a2 chaar K\u2080 \u22a5 = 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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u22a5 = 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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u22a5 = 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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {0})"}, {"tactic": "rintro _ \u27e8U, _, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, _, 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 \u22a5\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\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 \u22a5\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2208 {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}"}, {"tactic": "apply prehaar_empty", "annotated_tactic": ["apply <a>prehaar_empty</a>", [{"full_name": "MeasureTheory.Measure.haar.prehaar_empty", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2208 {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}", "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 \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\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\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 IsClosed {0}", "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_smul_left", "start": [1105, 1], "end": [1113, 58], "traced_tactics": [{"tactic": "suffices setToL1 (hT.smul c) = c \u2022 setToL1 hT by rw [this, ContinuousLinearMap.smul_apply]", "annotated_tactic": ["suffices <a>setToL1</a> (hT.smul c) = c \u2022 <a>setToL1</a> hT by rw [this, <a>ContinuousLinearMap.smul_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": "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\nc : \u211d\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f = c \u2022 \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\nc : \u211d\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) = c \u2022 setToL1 hT"}, {"tactic": "refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc (hT.smul c)) _ _ _ _ _", "annotated_tactic": ["refine' <a>ContinuousLinearMap.extend_unique</a> (<a>setToL1SCLM</a> \u03b1 E \u03bc (hT.smul c)) _ _ _ _ _", [{"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\nc : \u211d\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) = c \u2022 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\nc : \u211d\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d) =\n    setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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\nc : \u211d\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d) =\n    setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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\nc : \u211d\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 =\n    \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f"}, {"tactic": "suffices c \u2022 setToL1 hT f = setToL1SCLM \u03b1 E \u03bc (hT.smul c) f by rw [\u2190 this]; congr", "annotated_tactic": ["suffices c \u2022 <a>setToL1</a> hT f = <a>setToL1SCLM</a> \u03b1 E \u03bc (hT.smul c) 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\nc : \u211d\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 =\n    \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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\nc : \u211d\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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left c hT]", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>, <a>setToL1SCLM_smul_left</a> c 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.SimpleFunc.setToL1SCLM_smul_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [932, 9], "def_end_pos": [932, 30]}]], "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\nc : \u211d\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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) 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\nc : \u211d\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) = c \u2022 setToL1 hT\n\u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) 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\nc : \u211d\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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f\n\u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f =\n    \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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\nc : \u211d\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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) 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\nc : \u211d\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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) 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/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.OuterMeasure.toMeasure_top", "start": [1067, 1], "end": [1072, 63], "traced_tactics": [{"tactic": "rw [OuterMeasure.top_caratheodory]", "annotated_tactic": ["rw [<a>OuterMeasure.top_caratheodory</a>]", [{"full_name": "MeasureTheory.OuterMeasure.top_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 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\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 inst\u271d \u2264 OuterMeasure.caratheodory \u22a4", "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\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 inst\u271d \u2264 \u22a4"}, {"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": "\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 inst\u271d \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "cases' s.eq_empty_or_nonempty with h h <;>\n  simp [h, toMeasure_apply \u22a4 _ hs, OuterMeasure.top_apply]", "annotated_tactic": ["cases' s.eq_empty_or_nonempty with h h <;>\n      simp [h, <a>toMeasure_apply</a> \u22a4 _ hs, <a>OuterMeasure.top_apply</a>]", [{"full_name": "MeasureTheory.toMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [674, 9], "def_end_pos": [674, 24]}, {"full_name": "MeasureTheory.OuterMeasure.top_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [619, 9], "def_end_pos": [619, 18]}]], "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\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s", "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_Icc", "start": [89, 1], "end": [90, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.mem_ite_univ_left", "start": [2231, 1], "end": [2233, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "Set.Finite.nullMeasurableSet_biUnion", "start": [371, 1], "end": [373, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.measure_iUnion\u2080", "start": [282, 1], "end": [288, 74], "traced_tactics": [{"tactic": "rcases exists_subordinate_pairwise_disjoint h hd with \u27e8t, _ht_sub, ht_eq, htm, htd\u27e9", "annotated_tactic": ["rcases <a>exists_subordinate_pairwise_disjoint</a> h hd with \u27e8t, _ht_sub, ht_eq, htm, htd\u27e9", [{"full_name": "MeasureTheory.exists_subordinate_pairwise_disjoint", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [258, 9], "def_end_pos": [258, 45]}]], "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\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on f)\nh : \u2200 (i : \u03b9), NullMeasurableSet (f i)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)", "state_after": "case intro.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 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on f)\nh : \u2200 (i : \u03b9), NullMeasurableSet (f i)\nt : \u03b9 \u2192 Set \u03b1\n_ht_sub : \u2200 (i : \u03b9), t i \u2286 f i\nht_eq : \u2200 (i : \u03b9), f i =\u1d50[\u03bc] t i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nhtd : Pairwise (Disjoint on t)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)"}, {"tactic": "calc\n  \u03bc (\u22c3 i, f i) = \u03bc (\u22c3 i, t i) := measure_congr (EventuallyEq.countable_iUnion ht_eq)\n  _ = \u2211' i, \u03bc (t i) := (measure_iUnion htd htm)\n  _ = \u2211' i, \u03bc (f i) := tsum_congr fun i => measure_congr (ht_eq _).symm", "annotated_tactic": ["calc\n    \u03bc (\u22c3 i, f i) = \u03bc (\u22c3 i, t i) := <a>measure_congr</a> (<a>EventuallyEq.countable_iUnion</a> ht_eq)\n    _ = \u2211' i, \u03bc (t i) := (<a>measure_iUnion</a> htd htm)\n    _ = \u2211' i, \u03bc (f i) := <a>tsum_congr</a> fun i => <a>measure_congr</a> (ht_eq _).<a>symm</a>", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "EventuallyEq.countable_iUnion", "def_path": "Mathlib/Order/Filter/CountableInter.lean", "def_pos": [79, 9], "def_end_pos": [79, 38]}, {"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}, {"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case intro.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 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on f)\nh : \u2200 (i : \u03b9), NullMeasurableSet (f i)\nt : \u03b9 \u2192 Set \u03b1\n_ht_sub : \u2200 (i : \u03b9), t i \u2286 f i\nht_eq : \u2200 (i : \u03b9), f i =\u1d50[\u03bc] t i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nhtd : Pairwise (Disjoint on t)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Expand.lean", "full_name": "MvPolynomial.expand_one", "start": [61, 1], "end": [63, 41], "traced_tactics": [{"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "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\n\u22a2 expand 1 = AlgHom.id R (MvPolynomial \u03c3 R)", "state_after": "case hf\n\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 : \u03c3\n\u22a2 \u2191(expand 1) (X f) = \u2191(AlgHom.id R (MvPolynomial \u03c3 R)) (X f)"}, {"tactic": "rw [expand_one_apply, AlgHom.id_apply]", "annotated_tactic": ["rw [<a>expand_one_apply</a>, <a>AlgHom.id_apply</a>]", [{"full_name": "MvPolynomial.expand_one_apply", "def_path": "Mathlib/Data/MvPolynomial/Expand.lean", "def_pos": [55, 9], "def_end_pos": [55, 25]}, {"full_name": "AlgHom.id_apply", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "case hf\n\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 : \u03c3\n\u22a2 \u2191(expand 1) (X f) = \u2191(AlgHom.id R (MvPolynomial \u03c3 R)) (X f)", "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", "start": [734, 1], "end": [736, 77], "traced_tactics": [{"tactic": "split <;> [exact find?_insert_of_eq t \u2039_\u203a; exact find?_insert_of_ne t \u2039_\u203a]", "annotated_tactic": ["split <;> [exact <a>find?_insert_of_eq</a> t \u2039_\u203a; exact <a>find?_insert_of_ne</a> t \u2039_\u203a]", [{"full_name": "Std.RBSet.find?_insert_of_eq", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [722, 9], "def_end_pos": [722, 27]}, {"full_name": "Std.RBSet.find?_insert_of_ne", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [726, 9], "def_end_pos": [726, 27]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nt : RBSet \u03b1 cmp\nv v' : \u03b1\n\u22a2 find? (insert t v) v' = if cmp v' v = Ordering.eq then some v else find? t v'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "full_name": "aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet", "start": [64, 1], "end": [66, 85], "traced_tactics": [{"tactic": "simp only [aeSeq_eq_mk_of_mem_aeSeqSet hf hx i, mk_eq_fun_of_mem_aeSeqSet hf hx i]", "annotated_tactic": ["simp only [<a>aeSeq_eq_mk_of_mem_aeSeqSet</a> hf hx i, <a>mk_eq_fun_of_mem_aeSeqSet</a> hf hx i]", [{"full_name": "aeSeq.aeSeq_eq_mk_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [59, 9], "def_end_pos": [59, 36]}, {"full_name": "aeSeq.mk_eq_fun_of_mem_aeSeqSet", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "def_pos": [50, 9], "def_end_pos": [50, 34]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 aeSeq hf p i x = f i x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto", "start": [384, 1], "end": [389, 94], "traced_tactics": [{"tactic": "apply FiniteMeasure.limsup_measure_closed_le_of_tendsto\n  ((ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds L).mp \u03bcs_lim) F_closed", "annotated_tactic": ["apply <a>FiniteMeasure.limsup_measure_closed_le_of_tendsto</a>\n    ((<a>ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds</a> L).<a>mp</a> \u03bcs_lim) F_closed", [{"full_name": "MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [345, 9], "def_end_pos": [345, 58]}, {"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]}, {"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": "\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)\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": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Basic.lean", "full_name": "List.sections_eq_nil_of_isEmpty", "start": [964, 1], "end": [969, 65], "traced_tactics": [{"tactic": "simp only [any, foldr, Bool.or_eq_true] at h", "annotated_tactic": ["simp only [<a>any</a>, <a>foldr</a>, <a>Bool.or_eq_true</a>] at h", [{"full_name": "List.any", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [522, 15], "def_end_pos": [522, 18]}, {"full_name": "List.foldr", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [514, 19], "def_end_pos": [514, 24]}, {"full_name": "Bool.or_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [107, 17], "def_end_pos": [107, 32]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nh : any (l :: L) isEmpty = true\n\u22a2 sections (l :: L) = []", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nh : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true\n\u22a2 sections (l :: L) = []"}, {"tactic": "match l, h with\n| [], .inl rfl => simp; induction sections L <;> simp [*]\n| l, .inr h => simp [sections, sections_eq_nil_of_isEmpty h]", "annotated_tactic": ["match l, h with\n    | [], .inl <a>rfl</a> => simp; induction <a>sections</a> L <;> simp [*]\n    | l, .inr h => simp [<a>sections</a>, sections_eq_nil_of_isEmpty h]", [{"full_name": "Bool", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}, {"full_name": "List.sections", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [949, 13], "def_end_pos": [949, 21]}, {"full_name": "List.sections", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [949, 13], "def_end_pos": [949, 21]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nh : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true\n\u22a2 sections (l :: L) = []", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nh : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true\n\u22a2 sections ([] :: L) = []", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nh : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true\n\u22a2 (List.bind (sections L) fun s => []) = []"}, {"tactic": "induction sections L <;> simp [*]", "annotated_tactic": ["induction <a>sections</a> L <;> simp [*]", [{"full_name": "List.sections", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [949, 13], "def_end_pos": [949, 21]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nL : List (List \u03b1)\nh : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true\n\u22a2 (List.bind (sections L) fun s => []) = []", "state_after": "no goals"}, {"tactic": "simp [sections, sections_eq_nil_of_isEmpty h]", "annotated_tactic": ["simp [<a>sections</a>, sections_eq_nil_of_isEmpty h]", [{"full_name": "List.sections", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [949, 13], "def_end_pos": [949, 21]}]], "state_before": "\u03b1 : Type u_1\nl\u271d : List \u03b1\nL : List (List \u03b1)\nh\u271d : isEmpty l\u271d = true \u2228 foldr (fun a r => isEmpty a || r) false L = true\nl : List \u03b1\nh : foldr (fun a r => isEmpty a || r) false L = true\n\u22a2 sections (l :: L) = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.fin_succ", "start": [1278, 1], "end": [1279, 47], "traced_tactics": [{"tactic": "simp [succ.comp fin_val]", "annotated_tactic": ["simp [succ.comp <a>fin_val</a>]", [{"full_name": "Primrec.fin_val", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1274, 9], "def_end_pos": [1274, 16]}]], "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\n\u22a2 Primrec fun a => \u2191(Fin.succ a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.not_mem_erase", "start": [1891, 1], "end": [1892, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.Iic_add_bij", "start": [411, 1], "end": [412, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.lt_add_right_iff_pos", "start": [330, 11], "end": [331, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.subNatNat_add_add", "start": [121, 1], "end": [128, 30], "traced_tactics": [{"tactic": "apply subNatNat_elim m n (fun m n i => subNatNat (m + k) (n + k) = i)", "annotated_tactic": ["apply <a>subNatNat_elim</a> m n (fun m n i => <a>subNatNat</a> (m + k) (n + k) = i)", [{"full_name": "Int.subNatNat_elim", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [101, 9], "def_end_pos": [101, 23]}, {"full_name": "Int.subNatNat", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [86, 5], "def_end_pos": [86, 14]}]], "state_before": "m n k : Nat\n\u22a2 subNatNat (m + k) (n + k) = subNatNat m n", "state_after": "case hp\nm n k : Nat\n\u22a2 \u2200 (i n : Nat), subNatNat (n + i + k) (n + k) = \u2191i\n\ncase hn\nm n k : Nat\n\u22a2 \u2200 (i m : Nat), subNatNat (m + k) (m + i + 1 + k) = -[i+1]"}, {"tactic": "intro i j", "annotated_tactic": ["intro i j", []], "state_before": "case hp\nm n k : Nat\n\u22a2 \u2200 (i n : Nat), subNatNat (n + i + k) (n + k) = \u2191i", "state_after": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + i + k) (j + k) = \u2191i"}, {"tactic": "rw [Nat.add_assoc, Nat.add_comm i k, \u2190 Nat.add_assoc]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a>, <a>Nat.add_comm</a> i k, \u2190 <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_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_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 hp\nm n k i j : Nat\n\u22a2 subNatNat (j + i + k) (j + k) = \u2191i", "state_after": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + k + i) (j + k) = \u2191i"}, {"tactic": "exact subNatNat_add_left", "annotated_tactic": ["exact <a>subNatNat_add_left</a>", [{"full_name": "Int.subNatNat_add_left", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [114, 9], "def_end_pos": [114, 27]}]], "state_before": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + k + i) (j + k) = \u2191i", "state_after": "no goals"}, {"tactic": "intro i j", "annotated_tactic": ["intro i j", []], "state_before": "case hn\nm n k : Nat\n\u22a2 \u2200 (i m : Nat), subNatNat (m + k) (m + i + 1 + k) = -[i+1]", "state_after": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + i + 1 + k) = -[i+1]"}, {"tactic": "rw [Nat.add_assoc j i 1, Nat.add_comm j (i+1), Nat.add_assoc, Nat.add_comm (i+1) (j+k)]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a> j i 1, <a>Nat.add_comm</a> j (i+1), <a>Nat.add_assoc</a>, <a>Nat.add_comm</a> (i+1) (j+k)]", [{"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]}, {"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": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + i + 1 + k) = -[i+1]", "state_after": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + k + (i + 1)) = -[i+1]"}, {"tactic": "exact subNatNat_add_right", "annotated_tactic": ["exact <a>subNatNat_add_right</a>", [{"full_name": "Int.subNatNat_add_right", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [118, 9], "def_end_pos": [118, 28]}]], "state_before": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + k + (i + 1)) = -[i+1]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.kstar_eq_iSup_pow", "start": [247, 1], "end": [254, 23], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext 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 : List \u03b1\nl : Language \u03b1\n\u22a2 l\u2217 = \u2a06 i, l ^ i", "state_after": "case h\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 x \u2208 l\u2217 \u2194 x \u2208 \u2a06 i, l ^ i"}, {"tactic": "simp only [mem_kstar, mem_iSup, mem_pow]", "annotated_tactic": ["simp only [<a>mem_kstar</a>, <a>mem_iSup</a>, <a>mem_pow</a>]", [{"full_name": "Language.mem_kstar", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [121, 9], "def_end_pos": [121, 18]}, {"full_name": "Language.mem_iSup", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [205, 9], "def_end_pos": [205, 17]}, {"full_name": "Language.mem_pow", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [229, 9], "def_end_pos": [229, 16]}]], "state_before": "case h\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 x \u2208 l\u2217 \u2194 x \u2208 \u2a06 i, l ^ i", "state_after": "case h\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 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l) \u2194 \u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\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 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l) \u2194 \u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case h.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 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l) \u2192 \u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\ncase h.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 : List \u03b1\nl : Language \u03b1\nx : List \u03b1\n\u22a2 (\u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192 \u2203 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l"}, {"tactic": "rintro \u27e8S, rfl, hS\u27e9", "annotated_tactic": ["rintro \u27e8S, rfl, hS\u27e9", []], "state_before": "case h.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 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l) \u2192 \u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l", "state_after": "case h.mp.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 : List \u03b1\nl : Language \u03b1\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\u22a2 \u2203 i S_1, join S = join S_1 \u2227 length S_1 = i \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l"}, {"tactic": "exact \u27e8_, S, rfl, rfl, hS\u27e9", "annotated_tactic": ["exact \u27e8_, S, <a>rfl</a>, <a>rfl</a>, hS\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.mp.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 : List \u03b1\nl : Language \u03b1\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\u22a2 \u2203 i S_1, join S = join S_1 \u2227 length S_1 = i \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l", "state_after": "no goals"}, {"tactic": "rintro \u27e8_, S, rfl, rfl, hS\u27e9", "annotated_tactic": ["rintro \u27e8_, S, rfl, rfl, hS\u27e9", []], "state_before": "case h.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 : List \u03b1\nl : Language \u03b1\nx : List \u03b1\n\u22a2 (\u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192 \u2203 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l", "state_after": "case h.mpr.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 b x : List \u03b1\nl : Language \u03b1\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\u22a2 \u2203 L, join S = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l"}, {"tactic": "exact \u27e8S, rfl, hS\u27e9", "annotated_tactic": ["exact \u27e8S, <a>rfl</a>, hS\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.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 : List \u03b1\nl : Language \u03b1\nS : List (List \u03b1)\nhS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l\n\u22a2 \u2203 L, join S = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l", "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.neg_right_iff", "start": [1255, 1], "end": [1257, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Decidable.Partrec.const'", "start": [446, 1], "end": [447, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.Ico_eq_image_ssubsets", "start": [90, 1], "end": [97, 54], "traced_tactics": [{"tactic": "ext u", "annotated_tactic": ["ext u", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 Ico s t = image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s))", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 u \u2208 Ico s t \u2194 u \u2208 image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s))"}, {"tactic": "simp_rw [mem_Ico, mem_image, mem_ssubsets]", "annotated_tactic": ["simp_rw [<a>mem_Ico</a>, <a>mem_image</a>, <a>mem_ssubsets</a>]", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [331, 9], "def_end_pos": [331, 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_ssubsets", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 u \u2208 Ico s t \u2194 u \u2208 image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s))", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u < t \u2194 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u < t \u2194 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u", "state_after": "case a.mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u < t \u2192 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u\n\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 (\u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u) \u2192 s \u2264 u \u2227 u < t"}, {"tactic": "rintro \u27e8hs, ht\u27e9", "annotated_tactic": ["rintro \u27e8hs, ht\u27e9", []], "state_before": "case a.mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u < t \u2192 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u", "state_after": "case a.mp.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhs : s \u2264 u\nht : u < t\n\u22a2 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u"}, {"tactic": "exact \u27e8u \\ s, sdiff_lt_sdiff_right ht hs, sup_sdiff_cancel_right hs\u27e9", "annotated_tactic": ["exact \u27e8u \\ s, <a>sdiff_lt_sdiff_right</a> ht hs, <a>sup_sdiff_cancel_right</a> hs\u27e9", [{"full_name": "sdiff_lt_sdiff_right", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [329, 9], "def_end_pos": [329, 29]}, {"full_name": "sup_sdiff_cancel_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}]], "state_before": "case a.mp.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhs : s \u2264 u\nht : u < t\n\u22a2 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u", "state_after": "no goals"}, {"tactic": "rintro \u27e8v, hv, rfl\u27e9", "annotated_tactic": ["rintro \u27e8v, hv, rfl\u27e9", []], "state_before": "case a.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 (\u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u) \u2192 s \u2264 u \u2227 u < t", "state_after": "case a.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nv : Finset \u03b1\nhv : v \u2282 t \\ s\n\u22a2 s \u2264 s \u222a v \u2227 s \u222a v < t"}, {"tactic": "exact \u27e8le_sup_left, sup_lt_of_lt_sdiff_left hv h\u27e9", "annotated_tactic": ["exact \u27e8<a>le_sup_left</a>, <a>sup_lt_of_lt_sdiff_left</a> hv h\u27e9", [{"full_name": "le_sup_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}, {"full_name": "sup_lt_of_lt_sdiff_left", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [471, 9], "def_end_pos": [471, 32]}]], "state_before": "case a.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nv : Finset \u03b1\nhv : v \u2282 t \\ s\n\u22a2 s \u2264 s \u222a v \u2227 s \u222a v < t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.disj_sum_strictMono_right", "start": [108, 1], "end": [110, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.induction_on", "start": [451, 1], "end": [453, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "ENNReal.measurable_of_measurable_nnreal_prod", "start": [2045, 1], "end": [2051, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "parallelepiped_eq_sum_segment", "start": [114, 1], "end": [124, 15], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "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 = \u2211 i : \u03b9, segment \u211d 0 (v i)", "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\nx\u271d : E\n\u22a2 x\u271d \u2208 parallelepiped v \u2194 x\u271d \u2208 \u2211 i : \u03b9, segment \u211d 0 (v i)"}, {"tactic": "simp only [mem_parallelepiped_iff, Set.mem_finset_sum, Finset.mem_univ, forall_true_left,\n  segment_eq_image, smul_zero, zero_add, \u2190 Set.pi_univ_Icc, Set.mem_univ_pi]", "annotated_tactic": ["simp only [<a>mem_parallelepiped_iff</a>, <a>Set.mem_finset_sum</a>, <a>Finset.mem_univ</a>, <a>forall_true_left</a>,\n    <a>segment_eq_image</a>, <a>smul_zero</a>, <a>zero_add</a>, \u2190 <a>Set.pi_univ_Icc</a>, <a>Set.mem_univ_pi</a>]", [{"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_finset_sum", "def_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "segment_eq_image", "def_path": "Mathlib/Analysis/Convex/Segment.lean", "def_pos": [191, 9], "def_end_pos": [191, 25]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [732, 9], "def_end_pos": [732, 18]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"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": "Set.mem_univ_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [675, 9], "def_end_pos": [675, 20]}]], "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\nx\u271d : E\n\u22a2 x\u271d \u2208 parallelepiped v \u2194 x\u271d \u2208 \u2211 i : \u03b9, segment \u211d 0 (v i)", "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\nx\u271d : E\n\u22a2 (\u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i) \u2194 \u2203 g h, \u2211 i : \u03b9, g i = x\u271d"}, {"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\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\nx\u271d : E\n\u22a2 (\u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i) \u2194 \u2203 g h, \u2211 i : \u03b9, g i = x\u271d", "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\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\nx\u271d : E\n\u22a2 (\u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i) \u2192 \u2203 g h, \u2211 i : \u03b9, g i = x\u271d\n\ncase h.mpr\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\nx\u271d : E\n\u22a2 (\u2203 g h, \u2211 i : \u03b9, g i = x\u271d) \u2192 \u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i"}, {"tactic": "rintro \u27e8g, hg, rfl\u27e9", "annotated_tactic": ["rintro \u27e8g, hg, rfl\u27e9", []], "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\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\nx\u271d : E\n\u22a2 (\u2203 g h, \u2211 i : \u03b9, g i = x\u271d) \u2192 \u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i", "state_after": "case h.mpr.intro.intro\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 g : \u03b9 \u2192 E\nhg : \u2200 {i : \u03b9}, g i \u2208 (fun a => a \u2022 v i) '' Icc 0 1\n\u22a2 \u2203 t h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i"}, {"tactic": "choose t ht hg using @hg", "annotated_tactic": ["choose t ht hg using @hg", []], "state_before": "case h.mpr.intro.intro\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 g : \u03b9 \u2192 E\nhg : \u2200 {i : \u03b9}, g i \u2208 (fun a => a \u2022 v i) '' Icc 0 1\n\u22a2 \u2203 t h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i", "state_after": "case h.mpr.intro.intro\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 g : \u03b9 \u2192 E\nt : {i : \u03b9} \u2192 \u211d\nht : \u2200 {i : \u03b9}, t \u2208 Icc 0 1\nhg : \u2200 {i : \u03b9}, (fun a => a \u2022 v i) t = g i\n\u22a2 \u2203 t h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i"}, {"tactic": "refine \u27e8@t, @ht, ?_\u27e9", "annotated_tactic": ["refine \u27e8@t, @ht, ?_\u27e9", []], "state_before": "case h.mpr.intro.intro\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 g : \u03b9 \u2192 E\nt : {i : \u03b9} \u2192 \u211d\nht : \u2200 {i : \u03b9}, t \u2208 Icc 0 1\nhg : \u2200 {i : \u03b9}, (fun a => a \u2022 v i) t = g i\n\u22a2 \u2203 t h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i", "state_after": "case h.mpr.intro.intro\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 g : \u03b9 \u2192 E\nt : {i : \u03b9} \u2192 \u211d\nht : \u2200 {i : \u03b9}, t \u2208 Icc 0 1\nhg : \u2200 {i : \u03b9}, (fun a => a \u2022 v i) t = g i\n\u22a2 \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t \u2022 v i"}, {"tactic": "simp_rw [hg]", "annotated_tactic": ["simp_rw [hg]", []], "state_before": "case h.mpr.intro.intro\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 g : \u03b9 \u2192 E\nt : {i : \u03b9} \u2192 \u211d\nht : \u2200 {i : \u03b9}, t \u2208 Icc 0 1\nhg : \u2200 {i : \u03b9}, (fun a => a \u2022 v i) t = g i\n\u22a2 \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t \u2022 v i", "state_after": "no goals"}, {"tactic": "rintro \u27e8t, ht, rfl\u27e9", "annotated_tactic": ["rintro \u27e8t, ht, rfl\u27e9", []], "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\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\nx\u271d : E\n\u22a2 (\u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i) \u2192 \u2203 g h, \u2211 i : \u03b9, g i = x\u271d", "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\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\nt : \u03b9 \u2192 \u211d\nht : \u2200 (i : \u03b9), t i \u2208 Icc (OfNat.ofNat 0 i) (OfNat.ofNat 1 i)\n\u22a2 \u2203 g h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i"}, {"tactic": "exact \u27e8t \u2022 v, fun {i} => \u27e8t i, ht _, by simp\u27e9, rfl\u27e9", "annotated_tactic": ["exact \u27e8t \u2022 v, fun {i} => \u27e8t i, ht _, by simp\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 h.mp.intro.intro\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\nt : \u03b9 \u2192 \u211d\nht : \u2200 (i : \u03b9), t i \u2208 Icc (OfNat.ofNat 0 i) (OfNat.ofNat 1 i)\n\u22a2 \u2203 g h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nt : \u03b9 \u2192 \u211d\nht : \u2200 (i : \u03b9), t i \u2208 Icc (OfNat.ofNat 0 i) (OfNat.ofNat 1 i)\ni : \u03b9\n\u22a2 (fun a => a \u2022 v i) (t i) = (t \u2022 v) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg", "start": [281, 1], "end": [292, 20], "traced_tactics": [{"tactic": "rcases hf.1 with \u27e8f', hf'_meas, hf_ae\u27e9", "annotated_tactic": ["rcases hf.1 with \u27e8f', hf'_meas, hf_ae\u27e9", []], "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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f", "state_after": "case 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 0 \u2264\u1d50[\u03bc] f"}, {"tactic": "have hf'_integrable : Integrable f' \u03bc := Integrable.congr hf hf_ae", "annotated_tactic": ["have hf'_integrable : <a>Integrable</a> f' \u03bc := <a>Integrable.congr</a> hf hf_ae", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"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\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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 0 \u2264\u1d50[\u03bc] f", "state_after": "case 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\n\u22a2 0 \u2264\u1d50[\u03bc] f"}, {"tactic": "have hf'_zero : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 0 \u2264 \u222b x in s, f' x \u2202\u03bc := by\n  intro s hs h's\n  rw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]\n  exact hf_zero s hs h's", "annotated_tactic": ["have hf'_zero : \u2200 s, <a>MeasurableSet</a> s \u2192 \u03bc s < \u221e \u2192 0 \u2264 \u222b x in s, f' x \u2202\u03bc := by\n    intro s hs h's\n    rw [<a>set_integral_congr_ae</a> hs (hf_ae.mono fun x hx _ => hx.symm)]\n    exact hf_zero s hs h's", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"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 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\n\u22a2 0 \u2264\u1d50[\u03bc] f", "state_after": "case 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\nhf'_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f"}, {"tactic": "exact\n  (ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable hf'_meas hf'_integrable\n        hf'_zero).trans\n    hf_ae.symm.le", "annotated_tactic": ["exact\n    (<a>ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable</a> hf'_meas hf'_integrable\n          hf'_zero).<a>trans</a>\n      hf_ae.symm.le", [{"full_name": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [242, 9], "def_end_pos": [242, 70]}, {"full_name": "Filter.EventuallyLE.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 27]}]], "state_before": "case 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\nhf'_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f", "state_after": "no goals"}, {"tactic": "intro s hs h's", "annotated_tactic": ["intro s hs h's", []], "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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' 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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc"}, {"tactic": "rw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]", "annotated_tactic": ["rw [<a>set_integral_congr_ae</a> hs (hf_ae.mono fun x hx _ => hx.symm)]", [{"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": "\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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f' 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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "exact hf_zero s hs h's", "annotated_tactic": ["exact hf_zero s hs h's", []], "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 : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc", "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": [188, 1], "end": [198, 99], "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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : Tendsto U fltr (\ud835\udcdd u)\n\u22a2 ProgMeasurable f u", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : Tendsto U fltr (\ud835\udcdd u)\ni : \u03b9\n\u22a2 StronglyMeasurable fun p => u (\u2191p.1) p.2"}, {"tactic": "apply @stronglyMeasurable_of_tendsto (Set.Iic i \u00d7 \u03a9) \u03b2 \u03b3\n  (MeasurableSpace.prod _ (f i)) _ _ fltr _ _ _ _ fun l => h l i", "annotated_tactic": ["apply @<a>stronglyMeasurable_of_tendsto</a> (<a>Set.Iic</a> i \u00d7 \u03a9) \u03b2 \u03b3\n    (<a>MeasurableSpace.prod</a> _ (f i)) _ _ fltr _ _ _ _ fun l => h l i", [{"full_name": "stronglyMeasurable_of_tendsto", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [733, 9], "def_end_pos": [733, 45]}, {"full_name": "Set.Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [64, 5], "def_end_pos": [64, 8]}, {"full_name": "MeasurableSpace.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [681, 5], "def_end_pos": [681, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : Tendsto U fltr (\ud835\udcdd u)\ni : \u03b9\n\u22a2 StronglyMeasurable fun p => u (\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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : Tendsto U fltr (\ud835\udcdd u)\ni : \u03b9\n\u22a2 Tendsto (fun l p => U l (\u2191p.1) p.2) fltr (\ud835\udcdd fun p => u (\u2191p.1) p.2)"}, {"tactic": "rw [tendsto_pi_nhds] at h_tendsto \u22a2", "annotated_tactic": ["rw [<a>tendsto_pi_nhds</a>] at h_tendsto \u22a2", [{"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\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : Tendsto U fltr (\ud835\udcdd u)\ni : \u03b9\n\u22a2 Tendsto (fun l p => U l (\u2191p.1) p.2) fltr (\ud835\udcdd fun p => u (\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\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : \u2200 (x : \u03b9), Tendsto (fun i => U i x) fltr (\ud835\udcdd (u x))\ni : \u03b9\n\u22a2 \u2200 (x : \u2191(Set.Iic i) \u00d7 \u03a9), Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2))"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : \u2200 (x : \u03b9), Tendsto (fun i => U i x) fltr (\ud835\udcdd (u x))\ni : \u03b9\n\u22a2 \u2200 (x : \u2191(Set.Iic i) \u00d7 \u03a9), Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2))", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : \u2200 (x : \u03b9), Tendsto (fun i => U i x) fltr (\ud835\udcdd (u x))\ni : \u03b9\nx : \u2191(Set.Iic i) \u00d7 \u03a9\n\u22a2 Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2))"}, {"tactic": "specialize h_tendsto x.fst", "annotated_tactic": ["specialize h_tendsto x.fst", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\nh_tendsto : \u2200 (x : \u03b9), Tendsto (fun i => U i x) fltr (\ud835\udcdd (u x))\ni : \u03b9\nx : \u2191(Set.Iic i) \u00d7 \u03a9\n\u22a2 Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2))", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\ni : \u03b9\nx : \u2191(Set.Iic i) \u00d7 \u03a9\nh_tendsto : Tendsto (fun i_1 => U i_1 \u2191x.1) fltr (\ud835\udcdd (u \u2191x.1))\n\u22a2 Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2))"}, {"tactic": "rw [tendsto_nhds] at h_tendsto \u22a2", "annotated_tactic": ["rw [<a>tendsto_nhds</a>] at h_tendsto \u22a2", [{"full_name": "tendsto_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1038, 9], "def_end_pos": [1038, 21]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\ni : \u03b9\nx : \u2191(Set.Iic i) \u00d7 \u03a9\nh_tendsto : Tendsto (fun i_1 => U i_1 \u2191x.1) fltr (\ud835\udcdd (u \u2191x.1))\n\u22a2 Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2))", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\ni : \u03b9\nx : \u2191(Set.Iic i) \u00d7 \u03a9\nh_tendsto : \u2200 (s : Set (\u03a9 \u2192 \u03b2)), IsOpen s \u2192 u \u2191x.1 \u2208 s \u2192 (fun i_1 => U i_1 \u2191x.1) \u207b\u00b9' s \u2208 fltr\n\u22a2 \u2200 (s : Set \u03b2), IsOpen s \u2192 u (\u2191x.1) x.2 \u2208 s \u2192 (fun i_1 => U i_1 (\u2191x.1) x.2) \u207b\u00b9' s \u2208 fltr"}, {"tactic": "exact fun s hs h_mem => h_tendsto {g | g x.snd \u2208 s} (hs.preimage (continuous_apply x.snd)) h_mem", "annotated_tactic": ["exact fun s hs h_mem => h_tendsto {g | g x.snd \u2208 s} (hs.preimage (<a>continuous_apply</a> x.snd)) h_mem", [{"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b9\nu v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b9\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\nfltr : Filter \u03b3\ninst\u271d\u00b9 : NeBot fltr\ninst\u271d : IsCountablyGenerated fltr\nU : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2\nh : \u2200 (l : \u03b3), ProgMeasurable f (U l)\ni : \u03b9\nx : \u2191(Set.Iic i) \u00d7 \u03a9\nh_tendsto : \u2200 (s : Set (\u03a9 \u2192 \u03b2)), IsOpen s \u2192 u \u2191x.1 \u2208 s \u2192 (fun i_1 => U i_1 \u2191x.1) \u207b\u00b9' s \u2208 fltr\n\u22a2 \u2200 (s : Set \u03b2), IsOpen s \u2192 u (\u2191x.1) x.2 \u2208 s \u2192 (fun i_1 => U i_1 (\u2191x.1) x.2) \u207b\u00b9' s \u2208 fltr", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.append_get_eq", "start": [816, 1], "end": [818, 30], "traced_tactics": [{"tactic": "simp [append_def]", "annotated_tactic": ["simp [<a>append_def</a>]", [{"full_name": "Part.append_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [706, 9], "def_end_pos": [706, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Append \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 : Append \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 : Append \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/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Basic.lean", "full_name": "Rat.normalize.den_nz", "start": [55, 1], "end": [57, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_lt_iff_sq_lt", "start": [51, 1], "end": [53, 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_lt_iff_mul_self_lt", "annotated_tactic": ["exact <a>natAbs_lt_iff_mul_self_lt</a>", [{"full_name": "Int.natAbs_lt_iff_mul_self_lt", "def_path": "Mathlib/Data/Int/Order/Lemmas.lean", "def_pos": [36, 9], "def_end_pos": [36, 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/Density.lean", "full_name": "MeasureTheory.pdf.integral_fun_mul_eq_integral", "start": [166, 1], "end": [201, 57], "traced_tactics": [{"tactic": "by_cases hpdf : Integrable (fun x => f x * (pdf X \u2119 \u03bc x).toReal) \u03bc", "annotated_tactic": ["by_cases hpdf : <a>Integrable</a> (fun x => f x * (<a>pdf</a> X \u2119 \u03bc x).<a>toReal</a>) \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\n\u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119\n\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : \u00acIntegrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119"}, {"tactic": "rw [\u2190 integral_map (HasPDF.measurable X \u2119 \u03bc).aemeasurable hf.aestronglyMeasurable,\n  map_eq_withDensity_pdf X \u2119 \u03bc, integral_eq_lintegral_pos_part_sub_lintegral_neg_part hpdf,\n  integral_eq_lintegral_pos_part_sub_lintegral_neg_part,\n  lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf X \u2119 \u03bc) hf.neg.ennreal_ofReal,\n  lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf X \u2119 \u03bc) hf.ennreal_ofReal]", "annotated_tactic": ["rw [\u2190 <a>integral_map</a> (<a>HasPDF.measurable</a> X \u2119 \u03bc).<a>aemeasurable</a> hf.aestronglyMeasurable,\n      <a>map_eq_withDensity_pdf</a> X \u2119 \u03bc, <a>integral_eq_lintegral_pos_part_sub_lintegral_neg_part</a> hpdf,\n      <a>integral_eq_lintegral_pos_part_sub_lintegral_neg_part</a>,\n      <a>lintegral_withDensity_eq_lintegral_mul</a> _ (<a>measurable_pdf</a> X \u2119 \u03bc) hf.neg.ennreal_ofReal,\n      <a>lintegral_withDensity_eq_lintegral_mul</a> _ (<a>measurable_pdf</a> X \u2119 \u03bc) hf.ennreal_ofReal]", [{"full_name": "MeasureTheory.integral_map", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1610, 9], "def_end_pos": [1610, 21]}, {"full_name": "MeasureTheory.HasPDF.measurable", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [74, 9], "def_end_pos": [74, 26]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "MeasureTheory.map_eq_withDensity_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [116, 9], "def_end_pos": [116, 31]}, {"full_name": "MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 62]}, {"full_name": "MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 62]}, {"full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [275, 9], "def_end_pos": [275, 47]}, {"full_name": "MeasureTheory.measurable_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [109, 9], "def_end_pos": [109, 23]}, {"full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [275, 9], "def_end_pos": [275, 47]}, {"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\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 ENNReal.toReal (\u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc) -\n      ENNReal.toReal (\u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc) -\n      ENNReal.toReal (\u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc)\n\ncase pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 Integrable fun y => f y"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 ENNReal.toReal (\u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc) -\n      ENNReal.toReal (\u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc) -\n      ENNReal.toReal (\u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc)", "state_after": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc\n\ncase pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc"}, {"tactic": "have : \u2200 x, ENNReal.ofReal (f x * (pdf X \u2119 \u03bc x).toReal) =\n    ENNReal.ofReal (pdf X \u2119 \u03bc x).toReal * ENNReal.ofReal (f x) := fun x \u21a6 by\n  rw [mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]", "annotated_tactic": ["have : \u2200 x, <a>ENNReal.ofReal</a> (f x * (<a>pdf</a> X \u2119 \u03bc x).<a>toReal</a>) =\n            <a>ENNReal.ofReal</a> (<a>pdf</a> X \u2119 \u03bc x).<a>toReal</a> * <a>ENNReal.ofReal</a> (f x) := fun x \u21a6 by\n          rw [<a>mul_comm</a>, <a>ENNReal.ofReal_mul</a> <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"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.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"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": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc", "state_after": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc"}, {"tactic": "simp_rw [this]", "annotated_tactic": ["simp_rw [this]", []], "state_before": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc", "state_after": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 a)) * ENNReal.ofReal (f a) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc"}, {"tactic": "exact lintegral_congr_ae (Filter.EventuallyEq.mul ofReal_toReal_ae_eq (ae_eq_refl _))", "annotated_tactic": ["exact <a>lintegral_congr_ae</a> (<a>Filter.EventuallyEq.mul</a> <a>ofReal_toReal_ae_eq</a> (<a>ae_eq_refl</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": "Filter.EventuallyEq.mul", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1531, 9], "def_end_pos": [1531, 25]}, {"full_name": "MeasureTheory.pdf.ofReal_toReal_ae_eq", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [149, 16], "def_end_pos": [149, 35]}, {"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.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 a)) * ENNReal.ofReal (f a) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>ENNReal.ofReal_mul</a> <a>ENNReal.toReal_nonneg</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_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nx : E\n\u22a2 ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) = ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x)", "state_after": "no goals"}, {"tactic": "have :\n  \u2200 x,\n    ENNReal.ofReal (-(f x * (pdf X \u2119 \u03bc x).toReal)) =\n      ENNReal.ofReal (pdf X \u2119 \u03bc x).toReal * ENNReal.ofReal (-f x) := by\n  intro x\n  rw [neg_mul_eq_neg_mul, mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]", "annotated_tactic": ["have :\n          \u2200 x,\n            <a>ENNReal.ofReal</a> (-(f x * (<a>pdf</a> X \u2119 \u03bc x).<a>toReal</a>)) =\n              <a>ENNReal.ofReal</a> (<a>pdf</a> X \u2119 \u03bc x).<a>toReal</a> * <a>ENNReal.ofReal</a> (-f x) := by\n          intro x\n          rw [<a>neg_mul_eq_neg_mul</a>, <a>mul_comm</a>, <a>ENNReal.ofReal_mul</a> <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "MeasureTheory.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"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.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"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": "neg_mul_eq_neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [301, 9], "def_end_pos": [301, 27]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc", "state_after": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc"}, {"tactic": "simp_rw [this]", "annotated_tactic": ["simp_rw [this]", []], "state_before": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc", "state_after": "case pos.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 a)) * ENNReal.ofReal (-f a) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc"}, {"tactic": "exact lintegral_congr_ae (Filter.EventuallyEq.mul ofReal_toReal_ae_eq (ae_eq_refl _))", "annotated_tactic": ["exact <a>lintegral_congr_ae</a> (<a>Filter.EventuallyEq.mul</a> <a>ofReal_toReal_ae_eq</a> (<a>ae_eq_refl</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": "Filter.EventuallyEq.mul", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1531, 9], "def_end_pos": [1531, 25]}, {"full_name": "MeasureTheory.pdf.ofReal_toReal_ae_eq", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [149, 16], "def_end_pos": [149, 35]}, {"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.e_a.e_a\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis :\n  \u2200 (x : E),\n    ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 a)) * ENNReal.ofReal (-f a) \u2202\u03bc =\n    \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u2200 (x : E),\n    ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n      ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nx : E\n\u22a2 ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n    ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)"}, {"tactic": "rw [neg_mul_eq_neg_mul, mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]", "annotated_tactic": ["rw [<a>neg_mul_eq_neg_mul</a>, <a>mul_comm</a>, <a>ENNReal.ofReal_mul</a> <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "neg_mul_eq_neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [301, 9], "def_end_pos": [301, 27]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 19]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nx : E\n\u22a2 ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =\n    ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x)", "state_after": "no goals"}, {"tactic": "refine' \u27e8hf.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8hf.aestronglyMeasurable, _\u27e9", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 Integrable fun y => f y", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 HasFiniteIntegral fun y => f y"}, {"tactic": "rw [HasFiniteIntegral,\n  lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf _ _ _)\n    hf.nnnorm.coe_nnreal_ennreal]", "annotated_tactic": ["rw [<a>HasFiniteIntegral</a>,\n        <a>lintegral_withDensity_eq_lintegral_mul</a> _ (<a>measurable_pdf</a> _ _ _)\n          hf.nnnorm.coe_nnreal_ennreal]", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [275, 9], "def_end_pos": [275, 47]}, {"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\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 HasFiniteIntegral fun y => f y", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b\u207b (a : E), (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) a \u2202\u03bc < \u22a4"}, {"tactic": "have : (fun x => (pdf X \u2119 \u03bc * fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) x) =\u1d50[\u03bc]\n    fun x => \u2016f x * (pdf X \u2119 \u03bc x).toReal\u2016\u208a := by\n  simp_rw [\u2190 smul_eq_mul, nnnorm_smul, ENNReal.coe_mul]\n  rw [smul_eq_mul, mul_comm]\n  refine' Filter.EventuallyEq.mul (ae_eq_refl _) (ae_eq_trans ofReal_toReal_ae_eq.symm _)\n  simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]", "annotated_tactic": ["have : (fun x => (<a>pdf</a> X \u2119 \u03bc * fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) x) =\u1d50[\u03bc]\n          fun x => \u2016f x * (<a>pdf</a> X \u2119 \u03bc x).<a>toReal</a>\u2016\u208a := by\n        simp_rw [\u2190 <a>smul_eq_mul</a>, <a>nnnorm_smul</a>, <a>ENNReal.coe_mul</a>]\n        rw [<a>smul_eq_mul</a>, <a>mul_comm</a>]\n        refine' <a>Filter.EventuallyEq.mul</a> (<a>ae_eq_refl</a> _) (<a>ae_eq_trans</a> ofReal_toReal_ae_eq.symm _)\n        simp only [<a>Real.ennnorm_eq_ofReal</a> <a>ENNReal.toReal_nonneg</a>, <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.pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [82, 5], "def_end_pos": [82, 8]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "nnnorm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"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": "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": "MeasureTheory.ae_eq_trans", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [444, 9], "def_end_pos": [444, 20]}, {"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]}, {"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\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 \u222b\u207b (a : E), (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) a \u2202\u03bc < \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis : (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a\n\u22a2 \u222b\u207b (a : E), (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) a \u2202\u03bc < \u22a4"}, {"tactic": "rw [lintegral_congr_ae this]", "annotated_tactic": ["rw [<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": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis : (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a\n\u22a2 \u222b\u207b (a : E), (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) a \u2202\u03bc < \u22a4", "state_after": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis : (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a\n\u22a2 \u222b\u207b (a : E), \u2191\u2016f a * ENNReal.toReal (pdf X \u2119 a)\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "exact hpdf.2", "annotated_tactic": ["exact hpdf.2", []], "state_before": "case pos\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\nthis : (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a\n\u22a2 \u222b\u207b (a : E), \u2191\u2016f a * ENNReal.toReal (pdf X \u2119 a)\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 smul_eq_mul, nnnorm_smul, ENNReal.coe_mul]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_eq_mul</a>, <a>nnnorm_smul</a>, <a>ENNReal.coe_mul</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "nnnorm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => (pdf X \u2119 \u2022 fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x\u2016\u208a * \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a"}, {"tactic": "rw [smul_eq_mul, mul_comm]", "annotated_tactic": ["rw [<a>smul_eq_mul</a>, <a>mul_comm</a>]", [{"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]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => (pdf X \u2119 \u2022 fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x\u2016\u208a * \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => ((fun x => \u2191\u2016f x\u2016\u208a) * pdf X \u2119) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x\u2016\u208a * \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a"}, {"tactic": "refine' Filter.EventuallyEq.mul (ae_eq_refl _) (ae_eq_trans ofReal_toReal_ae_eq.symm _)", "annotated_tactic": ["refine' <a>Filter.EventuallyEq.mul</a> (<a>ae_eq_refl</a> _) (<a>ae_eq_trans</a> ofReal_toReal_ae_eq.symm _)", [{"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": "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\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => ((fun x => \u2191\u2016f x\u2016\u208a) * pdf X \u2119) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x\u2016\u208a * \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x))) =\u1da0[ae \u03bc] fun x => \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a"}, {"tactic": "simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]", "annotated_tactic": ["simp only [<a>Real.ennnorm_eq_ofReal</a> <a>ENNReal.toReal_nonneg</a>, <a>ae_eq_refl</a>]", [{"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]}, {"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\u00b2 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\ninst\u271d\u00b9 : IsFiniteMeasure \u2119\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119\nf : E \u2192 \u211d\nhf : Measurable f\nhpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x)\n\u22a2 (fun x => ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x))) =\u1da0[ae \u03bc] fun x => \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [integral_undef hpdf, integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a> hpdf, <a>integral_undef</a>]", [{"full_name": 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"state_after": "no goals"}, {"tactic": "rcases h with \u27e8f, hf\u27e9", "annotated_tactic": ["rcases h with \u27e8f, hf\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.Nonempty (Set.pi s t)\n\u22a2 MeasurableSet (Set.pi s t) \u2194 \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t 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\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\n\u22a2 MeasurableSet (Set.pi s t) \u2194 \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)"}, {"tactic": "refine' \u27e8fun hst i hi => _, MeasurableSet.pi hs\u27e9", "annotated_tactic": ["refine' \u27e8fun hst i hi => _, <a>MeasurableSet.pi</a> hs\u27e9", [{"full_name": "MeasurableSet.pi", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [955, 19], "def_end_pos": [955, 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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\n\u22a2 MeasurableSet (Set.pi s t) \u2194 \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t 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\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\nhst : MeasurableSet (Set.pi s t)\ni : \u03b4\nhi : i \u2208 s\n\u22a2 MeasurableSet (t i)"}, {"tactic": "convert measurable_update f (a := i) hst", "annotated_tactic": ["convert <a>measurable_update</a> f (a := i) hst", [{"full_name": "measurable_update", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [928, 9], "def_end_pos": [928, 26]}]], "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\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\nhst : MeasurableSet (Set.pi s t)\ni : \u03b4\nhi : i \u2208 s\n\u22a2 MeasurableSet (t i)", "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\nhst : MeasurableSet (Set.pi s t)\ni : \u03b4\nhi : i \u2208 s\n\u22a2 t i = update f i \u207b\u00b9' Set.pi s t"}, {"tactic": "rw [update_preimage_pi hi]", "annotated_tactic": ["rw [<a>update_preimage_pi</a> hi]", [{"full_name": "Set.update_preimage_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [902, 9], "def_end_pos": [902, 27]}]], "state_before": "case h.e'_3\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\nhst : MeasurableSet (Set.pi s t)\ni : \u03b4\nhi : i \u2208 s\n\u22a2 t i = update f i \u207b\u00b9' Set.pi s t", "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\nhst : MeasurableSet (Set.pi s t)\ni : \u03b4\nhi : i \u2208 s\n\u22a2 \u2200 (j : \u03b4), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j"}, {"tactic": "exact fun j hj _ => hf j hj", "annotated_tactic": ["exact fun j hj _ => hf j hj", []], "state_before": "case h.e'_3\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\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nf : (i : \u03b4) \u2192 \u03c0 i\nhf : f \u2208 Set.pi s t\nhst : MeasurableSet (Set.pi s t)\ni : \u03b4\nhi : i \u2208 s\n\u22a2 \u2200 (j : \u03b4), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.bindOnSupport_pure", "start": [270, 1], "end": [271, 55], "traced_tactics": [{"tactic": "simp only [PMF.bind_pure, PMF.bindOnSupport_eq_bind]", "annotated_tactic": ["simp only [<a>PMF.bind_pure</a>, <a>PMF.bindOnSupport_eq_bind</a>]", [{"full_name": "PMF.bind_pure", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [139, 9], "def_end_pos": [139, 18]}, {"full_name": "PMF.bindOnSupport_eq_bind", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "def_pos": [245, 9], "def_end_pos": [245, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2\np : PMF \u03b1\n\u22a2 (bindOnSupport p fun a x => pure a) = p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/HittingTime.lean", "full_name": "MeasureTheory.hitting_le_iff_of_lt", "start": [172, 1], "end": [179, 61], "traced_tactics": [{"tactic": "by_cases h_exists : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s", "annotated_tactic": ["by_cases h_exists : \u2203 j \u2208 <a>Set.Icc</a> n m, u j \u03c9 \u2208 s", [{"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\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s"}, {"tactic": "rw [hitting_le_iff_of_exists h_exists]", "annotated_tactic": ["rw [<a>hitting_le_iff_of_exists</a> h_exists]", [{"full_name": "MeasureTheory.hitting_le_iff_of_exists", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [157, 9], "def_end_pos": [157, 33]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s", "state_after": "no goals"}, {"tactic": "simp_rw [hitting, if_neg h_exists]", "annotated_tactic": ["simp_rw [<a>hitting</a>, <a>if_neg</a> h_exists]", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"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\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s"}, {"tactic": "push_neg at h_exists", "annotated_tactic": ["push_neg at h_exists", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n\u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s"}, {"tactic": "simp only [not_le.mpr hi, Set.mem_Icc, false_iff_iff, not_exists, not_and, and_imp]", "annotated_tactic": ["simp only [not_le.mpr hi, <a>Set.mem_Icc</a>, <a>false_iff_iff</a>, <a>not_exists</a>, <a>not_and</a>, <a>and_imp</a>]", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "false_iff_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 22]}, {"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": "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\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n\u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n\u22a2 \u2200 (x : \u03b9), n \u2264 x \u2192 x \u2264 i \u2192 \u00acu x \u03c9 \u2208 s"}, {"tactic": "exact fun k hkn hki => h_exists k \u27e8hkn, hki.trans hi.le\u27e9", "annotated_tactic": ["exact fun k hkn hki => h_exists k \u27e8hkn, hki.trans hi.le\u27e9", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n\u22a2 \u2200 (x : \u03b9), n \u2264 x \u2192 x \u2264 i \u2192 \u00acu x \u03c9 \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Zip.lean", "full_name": "Vector.prod_mul_prod_eq_prod_zipWith", "start": [48, 1], "end": [51, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.getElem_ofFn", "start": [300, 9], "end": [302, 49], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : Nat\n\u03b1 : Type u_1\nf : Fin n \u2192 \u03b1\ni : Nat\nh : i < size (ofFn f)\n\u22a2 0 \u2264 n", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : Nat\n\u03b1 : Type u_1\nf : Fin n \u2192 \u03b1\ni : Nat\nh : i < size (ofFn f)\n\u22a2 0 = size (mkEmpty n)", "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_Ici_self", "start": [428, 1], "end": [429, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "exists_lt_ack_of_nat_primrec", "start": [313, 1], "end": [378, 87], "traced_tactics": [{"tactic": "induction' hf with f g hf hg IHf IHg f g hf hg IHf IHg f g hf hg IHf IHg", "annotated_tactic": ["induction' hf with f g hf hg IHf IHg f g hf hg IHf IHg f g hf hg IHf IHg", []], "state_before": "f : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\n\u22a2 \u2203 m, \u2200 (n : \u2115), f n < ack m n", "state_after": "case zero\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun x => 0) n < ack m n\n\ncase succ\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), succ n < ack m n\n\ncase left\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).1) n < ack m n\n\ncase right\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).2) n < ack m n\n\ncase pair\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\n\ncase comp\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\n\ncase prec\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n"}, {"tactic": "all_goals cases' IHf with a ha; cases' IHg with b hb", "annotated_tactic": ["all_goals cases' IHf with a ha; cases' IHg with b hb", []], "state_before": "case pair\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\n\ncase comp\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\n\ncase prec\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n", "state_after": "case pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\n\ncase comp.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\n\ncase prec.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n"}, {"tactic": "exact \u27e80, ack_pos 0\u27e9", "annotated_tactic": ["exact \u27e80, <a>ack_pos</a> 0\u27e9", [{"full_name": "ack_pos", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [107, 9], "def_end_pos": [107, 16]}]], "state_before": "case zero\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun x => 0) n < ack m n", "state_after": "no goals"}, {"tactic": "refine' \u27e81, fun n => _\u27e9", "annotated_tactic": ["refine' \u27e81, fun n => _\u27e9", []], "state_before": "case succ\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), succ n < ack m n", "state_after": "case succ\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 succ n < ack 1 n"}, {"tactic": "rw [succ_eq_one_add]", "annotated_tactic": ["rw [<a>succ_eq_one_add</a>]", [{"full_name": "Nat.succ_eq_one_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [241, 9], "def_end_pos": [241, 24]}]], "state_before": "case succ\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 succ n < ack 1 n", "state_after": "case succ\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 1 + n < ack 1 n"}, {"tactic": "apply add_lt_ack", "annotated_tactic": ["apply <a>add_lt_ack</a>", [{"full_name": "add_lt_ack", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [175, 9], "def_end_pos": [175, 19]}]], "state_before": "case succ\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 1 + n < ack 1 n", "state_after": "no goals"}, {"tactic": "refine' \u27e80, fun n => _\u27e9", "annotated_tactic": ["refine' \u27e80, fun n => _\u27e9", []], "state_before": "case left\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).1) n < ack m n", "state_after": "case left\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).1) n < ack 0 n"}, {"tactic": "rw [ack_zero, lt_succ_iff]", "annotated_tactic": ["rw [<a>ack_zero</a>, <a>lt_succ_iff</a>]", [{"full_name": "ack_zero", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [70, 9], "def_end_pos": [70, 17]}, {"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 left\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).1) n < ack 0 n", "state_after": "case left\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).1) n \u2264 n"}, {"tactic": "exact unpair_left_le n", "annotated_tactic": ["exact <a>unpair_left_le</a> n", [{"full_name": "Nat.unpair_left_le", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [110, 9], "def_end_pos": [110, 23]}]], "state_before": "case left\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).1) n \u2264 n", "state_after": "no goals"}, {"tactic": "refine' \u27e80, fun n => _\u27e9", "annotated_tactic": ["refine' \u27e80, fun n => _\u27e9", []], "state_before": "case right\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).2) n < ack m n", "state_after": "case right\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).2) n < ack 0 n"}, {"tactic": "rw [ack_zero, lt_succ_iff]", "annotated_tactic": ["rw [<a>ack_zero</a>, <a>lt_succ_iff</a>]", [{"full_name": "ack_zero", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [70, 9], "def_end_pos": [70, 17]}, {"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 right\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).2) n < ack 0 n", "state_after": "case right\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).2) n \u2264 n"}, {"tactic": "exact unpair_right_le n", "annotated_tactic": ["exact <a>unpair_right_le</a> n", [{"full_name": "Nat.unpair_right_le", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [123, 9], "def_end_pos": [123, 24]}]], "state_before": "case right\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).2) n \u2264 n", "state_after": "no goals"}, {"tactic": "cases' IHf with a ha", "annotated_tactic": ["cases' IHf with a ha", []], "state_before": "case prec\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n", "state_after": "case prec.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n"}, {"tactic": "cases' IHg with b hb", "annotated_tactic": ["cases' IHg with b hb", []], "state_before": "case prec.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n", "state_after": "case prec.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n"}, {"tactic": "refine'\n  \u27e8max a b + 3, fun n =>\n    (pair_lt_max_add_one_sq _ _).trans_le <|\n      (pow_le_pow_of_le_left (add_le_add_right _ _) 2).trans <|\n        ack_add_one_sq_lt_ack_add_three _ _\u27e9", "annotated_tactic": ["refine'\n      \u27e8<a>max</a> a b + 3, fun n =>\n        (<a>pair_lt_max_add_one_sq</a> _ _).<a>trans_le</a> <|\n          (<a>pow_le_pow_of_le_left</a> (<a>add_le_add_right</a> _ _) 2).<a>trans</a> <|\n            <a>ack_add_one_sq_lt_ack_add_three</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": "Nat.pair_lt_max_add_one_sq", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [152, 9], "def_end_pos": [152, 31]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "Nat.pow_le_pow_of_le_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [478, 9], "def_end_pos": [478, 30]}, {"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": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "ack_add_one_sq_lt_ack_add_three", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [277, 9], "def_end_pos": [277, 40]}]], "state_before": "case pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n", "state_after": "case pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nn : \u2115\n\u22a2 max (f n) (g n) \u2264 ack (max a b) n"}, {"tactic": "rw [max_ack_left]", "annotated_tactic": ["rw [<a>max_ack_left</a>]", [{"full_name": "max_ack_left", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [245, 9], "def_end_pos": [245, 21]}]], "state_before": "case pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nn : \u2115\n\u22a2 max (f n) (g n) \u2264 ack (max a b) n", "state_after": "case pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nn : \u2115\n\u22a2 max (f n) (g n) \u2264 max (ack a n) (ack b n)"}, {"tactic": "exact max_le_max (ha n).le (hb n).le", "annotated_tactic": ["exact <a>max_le_max</a> (ha n).<a>le</a> (hb n).<a>le</a>", [{"full_name": "max_le_max", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [73, 9], "def_end_pos": [73, 19]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nn : \u2115\n\u22a2 max (f n) (g n) \u2264 max (ack a n) (ack b n)", "state_after": "no goals"}, {"tactic": "exact\n  \u27e8max a b + 2, fun n =>\n    (ha _).trans <| (ack_strictMono_right a <| hb n).trans <| ack_ack_lt_ack_max_add_two a b n\u27e9", "annotated_tactic": ["exact\n      \u27e8<a>max</a> a b + 2, fun n =>\n        (ha _).<a>trans</a> <| (<a>ack_strictMono_right</a> a <| hb n).<a>trans</a> <| <a>ack_ack_lt_ack_max_add_two</a> a b n\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.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]}, {"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "ack_ack_lt_ack_max_add_two", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [288, 9], "def_end_pos": [288, 35]}]], "state_before": "case comp.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n", "state_after": "no goals"}, {"tactic": "exact \u27e8max a b + 9, fun n => this.trans_le <| ack_mono_right _ <| unpair_add_le n\u27e9", "annotated_tactic": ["exact \u27e8<a>max</a> a b + 9, fun n => this.trans_le <| <a>ack_mono_right</a> _ <| <a>unpair_add_le</a> n\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": "ack_mono_right", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [148, 9], "def_end_pos": [148, 23]}, {"full_name": "Nat.unpair_add_le", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [175, 9], "def_end_pos": [175, 22]}]], "state_before": "case prec.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nthis : \u2200 {m n : \u2115}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n", "state_after": "no goals"}, {"tactic": "intro m n", "annotated_tactic": ["intro m n", []], "state_before": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2200 {m n : \u2115}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)", "state_after": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)", "state_after": "case zero\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm : \u2115\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) zero < ack (max a b + 9) (m + zero)\n\ncase succ\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) (succ n) < ack (max a b + 9) (m + succ n)"}, {"tactic": "apply (ha m).trans (ack_strictMono_left m <| (le_max_left a b).trans_lt _)", "annotated_tactic": ["apply (ha m).<a>trans</a> (<a>ack_strictMono_left</a> m <| (<a>le_max_left</a> a b).<a>trans_lt</a> _)", [{"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "ack_strictMono_left", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [218, 9], "def_end_pos": [218, 28]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case zero\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm : \u2115\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) zero < ack (max a b + 9) (m + zero)", "state_after": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm : \u2115\n\u22a2 max a b < max a b + 9"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm : \u2115\n\u22a2 max a b < max a b + 9", "state_after": "no goals"}, {"tactic": "simp only [ge_iff_le]", "annotated_tactic": ["simp only [<a>ge_iff_le</a>]", [{"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}]], "state_before": "case succ\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) (succ n) < ack (max a b + 9) (m + succ n)", "state_after": "case succ\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 g (pair m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) < ack (max a b + 9) (m + succ n)"}, {"tactic": "apply (hb _).trans ((ack_pair_lt _ _ _).trans_le _)", "annotated_tactic": ["apply (hb _).<a>trans</a> ((<a>ack_pair_lt</a> _ _ _).<a>trans_le</a> _)", [{"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "ack_pair_lt", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [307, 9], "def_end_pos": [307, 20]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case succ\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 g (pair m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) < ack (max a b + 9) (m + succ n)", "state_after": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "cases' lt_or_le _ m with h\u2081 h\u2081", "annotated_tactic": ["cases' <a>lt_or_le</a> _ m with h\u2081 h\u2081", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : ?m.127778 < m\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)\n\ncase inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 ?m.127778\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "rw [max_eq_right h\u2081]", "annotated_tactic": ["rw [<a>max_eq_right</a> h\u2081]", [{"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}]], "state_before": "case inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4) (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "apply (ack_pair_lt _ _ _).le.trans", "annotated_tactic": ["apply (<a>ack_pair_lt</a> _ _ _).le.trans", [{"full_name": "ack_pair_lt", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [307, 9], "def_end_pos": [307, 20]}]], "state_before": "case inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4) (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "cases' lt_or_le _ n with h\u2082 h\u2082", "annotated_tactic": ["cases' <a>lt_or_le</a> _ n with h\u2082 h\u2082", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr.inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : ?m.128145 < n\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)\n\ncase inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 ?m.128145\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "rw [max_eq_right h\u2082]", "annotated_tactic": ["rw [<a>max_eq_right</a> h\u2082]", [{"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}]], "state_before": "case inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "apply (ack_strictMono_right _ IH).le.trans", "annotated_tactic": ["apply (<a>ack_strictMono_right</a> _ IH).le.trans", [{"full_name": "ack_strictMono_right", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [136, 9], "def_end_pos": [136, 29]}]], "state_before": "case inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (ack (max a b + 9) (m + n)) \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "rw [add_succ m, add_succ _ 8, succ_eq_add_one, succ_eq_add_one,\n    ack_succ_succ (_ + 8), add_assoc]", "annotated_tactic": ["rw [<a>add_succ</a> m, <a>add_succ</a> _ 8, <a>succ_eq_add_one</a>, <a>succ_eq_add_one</a>,\n            <a>ack_succ_succ</a> (_ + 8), <a>add_assoc</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.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_eq_add_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 24]}, {"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]}, {"full_name": "ack_succ_succ", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "case inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (ack (max a b + 9) (m + n)) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + (4 + 4)) (ack (max a b + 8 + 1) (m + n)) \u2264 ack (max a b + 8) (ack (max a b + 8 + 1) (m + n))"}, {"tactic": "exact ack_mono_left _ (Nat.add_le_add (le_max_right a b) le_rfl)", "annotated_tactic": ["exact <a>ack_mono_left</a> _ (<a>Nat.add_le_add</a> (<a>le_max_right</a> a b) <a>le_rfl</a>)", [{"full_name": "ack_mono_left", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}, {"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": "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 inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + (4 + 4)) (ack (max a b + 8 + 1) (m + n)) \u2264 ack (max a b + 8) (ack (max a b + 8 + 1) (m + n))", "state_after": "no goals"}, {"tactic": "rw [max_eq_left h\u2081.le]", "annotated_tactic": ["rw [<a>max_eq_left</a> h\u2081.le]", [{"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}]], "state_before": "case inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : ?m.127778 < m\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n\u22a2 ack (b + 4) m \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "exact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num)\n                 (self_le_add_right m _)", "annotated_tactic": ["exact <a>ack_le_ack</a> (<a>Nat.add_le_add</a> (<a>le_max_right</a> a b) <| by norm_num)\n                           (<a>self_le_add_right</a> m _)", [{"full_name": "ack_le_ack", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [249, 9], "def_end_pos": [249, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "self_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [167, 3], "def_end_pos": [167, 14]}]], "state_before": "case inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n\u22a2 ack (b + 4) m \u2264 ack (max a b + 9) (m + succ n)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n\u22a2 4 \u2264 9", "state_after": "no goals"}, {"tactic": "rw [max_eq_left h\u2082.le, add_assoc]", "annotated_tactic": ["rw [<a>max_eq_left</a> h\u2082.le, <a>add_assoc</a>]", [{"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "case inr.inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : ?m.128145 < n\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)", "state_after": "case inr.inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n\u22a2 ack (b + (4 + 4)) n \u2264 ack (max a b + 9) (m + succ n)"}, {"tactic": "exact\n  ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num)\n    ((le_succ n).trans <| self_le_add_left _ _)", "annotated_tactic": ["exact\n            <a>ack_le_ack</a> (<a>Nat.add_le_add</a> (<a>le_max_right</a> a b) <| by norm_num)\n              ((<a>le_succ</a> n).<a>trans</a> <| <a>self_le_add_left</a> _ _)", [{"full_name": "ack_le_ack", "def_path": "Mathlib/Computability/Ackermann.lean", "def_pos": [249, 9], "def_end_pos": [249, 19]}, {"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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 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": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "self_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [173, 3], "def_end_pos": [173, 14]}]], "state_before": "case inr.inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n\u22a2 ack (b + (4 + 4)) n \u2264 ack (max a b + 9) (m + succ n)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n\u22a2 4 + 4 \u2264 9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.truncation_nonneg", "start": [128, 1], "end": [129, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_fintype", "start": [1509, 1], "end": [1511, 66], "traced_tactics": [{"tactic": "rw [\u2190 lintegral_finset, Finset.coe_univ, Measure.restrict_univ]", "annotated_tactic": ["rw [\u2190 <a>lintegral_finset</a>, <a>Finset.coe_univ</a>, <a>Measure.restrict_univ</a>]", [{"full_name": "MeasureTheory.lintegral_finset", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1504, 9], "def_end_pos": [1504, 25]}, {"full_name": "Finset.coe_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 17]}, {"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u2211 x : \u03b1, f x * \u2191\u2191\u03bc {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Count.lean", "full_name": "MeasureTheory.Measure.empty_of_count_eq_zero'", "start": [119, 1], "end": [123, 50], "traced_tactics": [{"tactic": "have hs : s.Finite := by\n  rw [\u2190 count_apply_lt_top' s_mble, hsc]\n  exact WithTop.zero_lt_top", "annotated_tactic": ["have hs : s.Finite := by\n    rw [\u2190 <a>count_apply_lt_top'</a> s_mble, hsc]\n    exact <a>WithTop.zero_lt_top</a>", [{"full_name": "MeasureTheory.Measure.count_apply_lt_top'", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [102, 9], "def_end_pos": [102, 28]}, {"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16305\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\ns_mble : MeasurableSet s\nhsc : \u2191\u2191count s = 0\n\u22a2 s = \u2205", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16305\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\ns_mble : MeasurableSet s\nhsc : \u2191\u2191count s = 0\nhs : Set.Finite s\n\u22a2 s = \u2205"}, {"tactic": "simpa [count_apply_finite' hs s_mble] using hsc", "annotated_tactic": ["simpa [<a>count_apply_finite'</a> hs s_mble] using hsc", [{"full_name": "MeasureTheory.Measure.count_apply_finite'", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [63, 9], "def_end_pos": [63, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16305\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\ns_mble : MeasurableSet s\nhsc : \u2191\u2191count s = 0\nhs : Set.Finite s\n\u22a2 s = \u2205", "state_after": "no goals"}, {"tactic": "rw [\u2190 count_apply_lt_top' s_mble, hsc]", "annotated_tactic": ["rw [\u2190 <a>count_apply_lt_top'</a> s_mble, hsc]", [{"full_name": "MeasureTheory.Measure.count_apply_lt_top'", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [102, 9], "def_end_pos": [102, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16305\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\ns_mble : MeasurableSet s\nhsc : \u2191\u2191count s = 0\n\u22a2 Set.Finite s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16305\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\ns_mble : MeasurableSet s\nhsc : \u2191\u2191count s = 0\n\u22a2 0 < \u22a4"}, {"tactic": "exact WithTop.zero_lt_top", "annotated_tactic": ["exact <a>WithTop.zero_lt_top</a>", [{"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16305\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\ns_mble : MeasurableSet s\nhsc : \u2191\u2191count s = 0\n\u22a2 0 < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.exists_eq_insert_iff_ncard", "start": [1004, 1], "end": [1015, 25], "traced_tactics": [{"tactic": "cases' t.finite_or_infinite with ht ht", "annotated_tactic": ["cases' t.finite_or_infinite with ht ht", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t\n\ncase inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Infinite t\n\u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t"}, {"tactic": "simp only [ht.ncard, exists_prop, add_eq_zero, and_false, iff_false, not_exists, not_and]", "annotated_tactic": ["simp only [ht.ncard, <a>exists_prop</a>, <a>add_eq_zero</a>, <a>and_false</a>, <a>iff_false</a>, <a>not_exists</a>, <a>not_and</a>]", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "add_eq_zero", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [566, 3], "def_end_pos": [566, 14]}, {"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": "iff_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [95, 17], "def_end_pos": [95, 26]}, {"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]}]], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Infinite t\n\u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t", "state_after": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Infinite t\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u00acinsert x s = t"}, {"tactic": "rintro x - rfl", "annotated_tactic": ["rintro x - rfl", []], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Infinite t\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u00acinsert x s = t", "state_after": "case inr\n\u03b1 : Type u_1\ns : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nht : Set.Infinite (insert x s)\n\u22a2 False"}, {"tactic": "exact ht (hs.insert x)", "annotated_tactic": ["exact ht (hs.insert x)", []], "state_before": "case inr\n\u03b1 : Type u_1\ns : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nx : \u03b1\nht : Set.Infinite (insert x s)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [ncard_eq_toFinset_card _ hs, ncard_eq_toFinset_card _ ht,\n  \u2190@Finite.toFinset_subset_toFinset _ _ _ hs ht, \u2190Finset.exists_eq_insert_iff]", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a> _ hs, <a>ncard_eq_toFinset_card</a> _ ht,\n      \u2190@<a>Finite.toFinset_subset_toFinset</a> _ _ _ hs ht, \u2190<a>Finset.exists_eq_insert_iff</a>]", [{"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.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_subset_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [218, 19], "def_end_pos": [218, 43]}, {"full_name": "Finset.exists_eq_insert_iff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [527, 9], "def_end_pos": [527, 29]}]], "state_before": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (\u2203 a x, insert a s = t) \u2194 \u2203 a x, insert a (Finite.toFinset hs) = Finite.toFinset ht"}, {"tactic": "convert Iff.rfl using 2", "annotated_tactic": ["convert <a>Iff.rfl</a> using 2", [{"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 inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (\u2203 a x, insert a s = t) \u2194 \u2203 a x, insert a (Finite.toFinset hs) = Finite.toFinset ht", "state_after": "case h.e'_2.h.e'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (fun a => \u2203 x, insert a (Finite.toFinset hs) = Finite.toFinset ht) = fun a => \u2203 x, insert a s = t"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (fun a => \u2203 x, insert a (Finite.toFinset hs) = Finite.toFinset ht) = fun a => \u2203 x, insert a s = t", "state_after": "case h.e'_2.h.e'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (fun a => \u00aca \u2208 s \u2227 insert a (Finite.toFinset hs) = Finite.toFinset ht) = fun a => \u00aca \u2208 s \u2227 insert a s = t"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_2.h.e'_2\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\n\u22a2 (fun a => \u00aca \u2208 s \u2227 insert a (Finite.toFinset hs) = Finite.toFinset ht) = fun a => \u00aca \u2208 s \u2227 insert a s = t", "state_after": "case h.e'_2.h.e'_2.h.a\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\nx : \u03b1\n\u22a2 \u00acx \u2208 s \u2227 insert x (Finite.toFinset hs) = Finite.toFinset ht \u2194 \u00acx \u2208 s \u2227 insert x s = t"}, {"tactic": "simp [Finset.ext_iff, Set.ext_iff]", "annotated_tactic": ["simp [<a>Finset.ext_iff</a>, <a>Set.ext_iff</a>]", [{"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}, {"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.h.e'_2.h.a\n\u03b1 : Type u_1\ns t : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nht : Set.Finite t\nx : \u03b1\n\u22a2 \u00acx \u2208 s \u2227 insert x (Finite.toFinset hs) = Finite.toFinset ht \u2194 \u00acx \u2208 s \u2227 insert x s = 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_Ico_left", "start": [230, 1], "end": [232, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "full_name": "MeasureTheory.sigmaFinite_trim_bot_iff", "start": [126, 1], "end": [130, 58], "traced_tactics": [{"tactic": "rw [sigmaFinite_bot_iff]", "annotated_tactic": ["rw [<a>sigmaFinite_bot_iff</a>]", [{"full_name": "MeasureTheory.sigmaFinite_bot_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3706, 9], "def_end_pos": [3706, 28]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 SigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 IsFiniteMeasure (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc"}, {"tactic": "refine' \u27e8fun h => \u27e8_\u27e9, fun h => \u27e8_\u27e9\u27e9 <;> have h_univ := h.measure_univ_lt_top", "annotated_tactic": ["refine' \u27e8fun h => \u27e8_\u27e9, fun h => \u27e8_\u27e9\u27e9 <;> have h_univ := h.measure_univ_lt_top", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 IsFiniteMeasure (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\nh_univ : \u2191\u2191(Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) Set.univ < \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ < \u22a4\n\ncase refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure \u03bc\nh_univ : \u2191\u2191\u03bc Set.univ < \u22a4\n\u22a2 \u2191\u2191(Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) Set.univ < \u22a4"}, {"tactic": "rwa [trim_measurableSet_eq bot_le MeasurableSet.univ] at h_univ", "annotated_tactic": ["rwa [<a>trim_measurableSet_eq</a> <a>bot_le</a> <a>MeasurableSet.univ</a>] at h_univ", [{"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure (Measure.trim \u03bc (_ : \u22a5 \u2264 m0))\nh_univ : \u2191\u2191(Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) Set.univ < \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ < \u22a4", "state_after": "no goals"}, {"tactic": "rwa [trim_measurableSet_eq bot_le MeasurableSet.univ]", "annotated_tactic": ["rwa [<a>trim_measurableSet_eq</a> <a>bot_le</a> <a>MeasurableSet.univ</a>]", [{"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure \u03bc\nh_univ : \u2191\u2191\u03bc Set.univ < \u22a4\n\u22a2 \u2191\u2191(Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) Set.univ < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "ComputablePred.halting_problem_re", "start": [232, 1], "end": [233, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aevalTower_comp_toAlgHom", "start": [1657, 1], "end": [1659, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.kernel.integral_integral_add", "start": [186, 1], "end": [191, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "full_name": "Besicovitch.SatelliteConfig.exists_normalized", "start": [500, 1], "end": [526, 74], "traced_tactics": [{"tactic": "let c' : Fin N.succ \u2192 E := fun i => if \u2016a.c i\u2016 \u2264 2 then a.c i else (2 / \u2016a.c i\u2016) \u2022 a.c i", "annotated_tactic": ["let c' : <a>Fin</a> N.succ \u2192 E := fun i => if \u2016a.c i\u2016 \u2264 2 then a.c i else (2 / \u2016a.c i\u2016) \u2022 a.c i", [{"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\n\u22a2 \u2203 c', (\u2200 (n : Fin (Nat.succ N)), \u2016c' n\u2016 \u2264 2) \u2227 \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\n\u22a2 \u2203 c', (\u2200 (n : Fin (Nat.succ N)), \u2016c' n\u2016 \u2264 2) \u2227 \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "refine' \u27e8c', fun n => norm_c'_le n, fun i j inej => _\u27e9", "annotated_tactic": ["refine' \u27e8c', fun n => norm_c'_le n, fun i j inej => _\u27e9", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\n\u22a2 \u2203 c', (\u2200 (n : Fin (Nat.succ N)), \u2016c' n\u2016 \u2264 2) \u2227 \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "wlog hij : \u2016a.c i\u2016 \u2264 \u2016a.c j\u2016 generalizing i j", "annotated_tactic": ["wlog hij : \u2016a.c i\u2016 \u2264 \u2016a.c j\u2016 generalizing i j", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nthis : \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\nhij : \u00ac\u2016c a i\u2016 \u2264 \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\n\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "rcases le_or_lt \u2016a.c j\u2016 2 with (Hj | Hj)", "annotated_tactic": ["rcases <a>le_or_lt</a> \u2016a.c j\u2016 2 with (Hj | Hj)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : \u2016c a j\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\n\ncase inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\n\u22a2 \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\n\u22a2 \u2016c' i\u2016 \u2264 2"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\n\u22a2 \u2016c' i\u2016 \u2264 2", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\n\u22a2 \u2016if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\u2016 \u2264 2"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\n\u22a2 \u2016if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\u2016 \u2264 2", "state_after": "case pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\nh : \u2016c a i\u2016 \u2264 2\n\u22a2 \u2016c a i\u2016 \u2264 2\n\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\nh : \u00ac\u2016c a i\u2016 \u2264 2\n\u22a2 \u2016(2 / \u2016c a i\u2016) \u2022 c a i\u2016 \u2264 2"}, {"tactic": "by_cases hi : \u2016a.c i\u2016 = 0 <;> field_simp [norm_smul, hi]", "annotated_tactic": ["by_cases hi : \u2016a.c i\u2016 = 0 <;> field_simp [<a>norm_smul</a>, hi]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}]], "state_before": "case neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\nh : \u00ac\u2016c a i\u2016 \u2264 2\n\u22a2 \u2016(2 / \u2016c a i\u2016) \u2022 c a i\u2016 \u2264 2", "state_after": "case pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\nh : \u00ac\u2016c a i\u2016 \u2264 2\nhi : \u2016c a i\u2016 = 0\n\u22a2 0 \u2264 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\nh : \u00ac\u2016c a i\u2016 \u2264 2\nhi : \u2016c a i\u2016 = 0\n\u22a2 0 \u2264 2", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\ni : Fin (Nat.succ N)\nh : \u2016c a i\u2016 \u2264 2\n\u22a2 \u2016c a i\u2016 \u2264 2", "state_after": "no goals"}, {"tactic": "rw [norm_sub_rev]", "annotated_tactic": ["rw [<a>norm_sub_rev</a>]", [{"full_name": "norm_sub_rev", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [431, 3], "def_end_pos": [431, 14]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nthis : \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\nhij : \u00ac\u2016c a i\u2016 \u2264 \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nthis : \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\nhij : \u00ac\u2016c a i\u2016 \u2264 \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' j - c' i\u2016"}, {"tactic": "exact this j i inej.symm (le_of_not_le hij)", "annotated_tactic": ["exact this j i inej.symm (<a>le_of_not_le</a> hij)", [{"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\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nthis : \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\nhij : \u00ac\u2016c a i\u2016 \u2264 \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' j - c' i\u2016", "state_after": "no goals"}, {"tactic": "simp_rw [Hj, hij.trans Hj, if_true]", "annotated_tactic": ["simp_rw [Hj, hij.trans Hj, <a>if_true</a>]", [{"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : \u2016c a j\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : \u2016c a j\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c a i - c a j\u2016"}, {"tactic": "exact exists_normalized_aux1 a lastr h\u03c4 \u03b4 h\u03b41 h\u03b42 i j inej", "annotated_tactic": ["exact <a>exists_normalized_aux1</a> a lastr h\u03c4 \u03b4 h\u03b41 h\u03b42 i j inej", [{"full_name": "Besicovitch.SatelliteConfig.exists_normalized_aux1", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [355, 9], "def_end_pos": [355, 31]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : \u2016c a j\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c a i - c a j\u2016", "state_after": "no goals"}, {"tactic": "have H'j : \u2016a.c j\u2016 \u2264 2 \u2194 False := by simpa only [not_le, iff_false_iff] using Hj", "annotated_tactic": ["have H'j : \u2016a.c j\u2016 \u2264 2 \u2194 <a>False</a> := by simpa only [<a>not_le</a>, <a>iff_false_iff</a>] using Hj", [{"full_name": "False", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [189, 11], "def_end_pos": [189, 16]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "rcases le_or_lt \u2016a.c i\u2016 2 with (Hi | Hi)", "annotated_tactic": ["rcases <a>le_or_lt</a> \u2016a.c i\u2016 2 with (Hi | Hi)", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : \u2016c a i\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016\n\ncase inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "simpa only [not_le, iff_false_iff] using Hj", "annotated_tactic": ["simpa only [<a>not_le</a>, <a>iff_false_iff</a>] using Hj", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\n\u22a2 \u2016c a j\u2016 \u2264 2 \u2194 False", "state_after": "no goals"}, {"tactic": "simp_rw [Hi, if_true, H'j, if_false]", "annotated_tactic": ["simp_rw [Hi, <a>if_true</a>, H'j, <a>if_false</a>]", [{"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 inr.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : \u2016c a i\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : \u2016c a i\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c a i - (2 / \u2016c a j\u2016) \u2022 c a j\u2016"}, {"tactic": "exact exists_normalized_aux2 a lastc lastr h\u03c4 \u03b4 h\u03b41 h\u03b42 i j inej Hi Hj", "annotated_tactic": ["exact <a>exists_normalized_aux2</a> a lastc lastr h\u03c4 \u03b4 h\u03b41 h\u03b42 i j inej Hi Hj", [{"full_name": "Besicovitch.SatelliteConfig.exists_normalized_aux2", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [387, 9], "def_end_pos": [387, 31]}]], "state_before": "case inr.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : \u2016c a i\u2016 \u2264 2\n\u22a2 1 - \u03b4 \u2264 \u2016c a i - (2 / \u2016c a j\u2016) \u2022 c a j\u2016", "state_after": "no goals"}, {"tactic": "have H'i : \u2016a.c i\u2016 \u2264 2 \u2194 False := by simpa only [not_le, iff_false_iff] using Hi", "annotated_tactic": ["have H'i : \u2016a.c i\u2016 \u2264 2 \u2194 <a>False</a> := by simpa only [<a>not_le</a>, <a>iff_false_iff</a>] using Hi", [{"full_name": "False", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [189, 11], "def_end_pos": [189, 16]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\nH'i : \u2016c a i\u2016 \u2264 2 \u2194 False\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016"}, {"tactic": "simp_rw [H'i, if_false, H'j, if_false]", "annotated_tactic": ["simp_rw [H'i, <a>if_false</a>, H'j, <a>if_false</a>]", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\nH'i : \u2016c a i\u2016 \u2264 2 \u2194 False\n\u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016", "state_after": "case inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\nH'i : \u2016c a i\u2016 \u2264 2 \u2194 False\n\u22a2 1 - \u03b4 \u2264 \u2016(2 / \u2016c a i\u2016) \u2022 c a i - (2 / \u2016c a j\u2016) \u2022 c a j\u2016"}, {"tactic": "exact exists_normalized_aux3 a lastc lastr h\u03c4 \u03b4 h\u03b41 i j inej Hi hij", "annotated_tactic": ["exact <a>exists_normalized_aux3</a> a lastc lastr h\u03c4 \u03b4 h\u03b41 i j inej Hi hij", [{"full_name": "Besicovitch.SatelliteConfig.exists_normalized_aux3", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [443, 9], "def_end_pos": [443, 31]}]], "state_before": "case inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\nH'i : \u2016c a i\u2016 \u2264 2 \u2194 False\n\u22a2 1 - \u03b4 \u2264 \u2016(2 / \u2016c a i\u2016) \u2022 c a i - (2 / \u2016c a j\u2016) \u2022 c a j\u2016", "state_after": "no goals"}, {"tactic": "simpa only [not_le, iff_false_iff] using Hi", "annotated_tactic": ["simpa only [<a>not_le</a>, <a>iff_false_iff</a>] using Hi", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\nc' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\nnorm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2\ni j : Fin (Nat.succ N)\ninej : i \u2260 j\nhij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016\nHj : 2 < \u2016c a j\u2016\nH'j : \u2016c a j\u2016 \u2264 2 \u2194 False\nHi : 2 < \u2016c a i\u2016\n\u22a2 \u2016c a i\u2016 \u2264 2 \u2194 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_compl_eq", "start": [448, 1], "end": [449, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.idv", "start": [354, 1], "end": [355, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.set_biUnion_finset_image", "start": [2146, 1], "end": [2148, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.indep_iSup_directed_limsup", "start": [83, 1], "end": [94, 20], "traced_tactics": [{"tactic": "refine' indep_iSup_of_directed_le _ _ _ _", "annotated_tactic": ["refine' <a>indep_iSup_of_directed_le</a> _ _ _ _", [{"full_name": "ProbabilityTheory.indep_iSup_of_directed_le", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [419, 9], "def_end_pos": [419, 34]}]], "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)\n\u22a2 Indep (\u2a06 a, \u2a06 n \u2208 ns a, s n) (limsup s f)", "state_after": "case refine'_1\n\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)\n\u22a2 \u2200 (i : \u03b1), Indep (\u2a06 n \u2208 ns i, s n) (limsup s f)\n\ncase refine'_2\n\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)\n\u22a2 \u2200 (i : \u03b1), \u2a06 n \u2208 ns i, s n \u2264 m0\n\ncase refine'_3\n\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)\n\u22a2 limsup s f \u2264 m0\n\ncase refine'_4\n\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)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => \u2a06 n \u2208 ns a, s n"}, {"tactic": "exact fun a => indep_biSup_limsup h_le h_indep hf (hnsp a)", "annotated_tactic": ["exact fun a => <a>indep_biSup_limsup</a> h_le h_indep hf (hnsp a)", [{"full_name": "ProbabilityTheory.indep_biSup_limsup", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [75, 9], "def_end_pos": [75, 27]}]], "state_before": "case refine'_1\n\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)\n\u22a2 \u2200 (i : \u03b1), Indep (\u2a06 n \u2208 ns i, s n) (limsup s f)", "state_after": "no goals"}, {"tactic": "exact fun a => iSup\u2082_le fun n _ => h_le n", "annotated_tactic": ["exact fun a => <a>iSup\u2082_le</a> fun n _ => h_le n", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}]], "state_before": "case refine'_2\n\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)\n\u22a2 \u2200 (i : \u03b1), \u2a06 n \u2208 ns i, s n \u2264 m0", "state_after": "no goals"}, {"tactic": "exact limsup_le_iSup.trans (iSup_le h_le)", "annotated_tactic": ["exact limsup_le_iSup.trans (<a>iSup_le</a> h_le)", [{"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\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)\n\u22a2 limsup s f \u2264 m0", "state_after": "no goals"}, {"tactic": "intro a b", "annotated_tactic": ["intro a b", []], "state_before": "case refine'_4\n\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)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => \u2a06 n \u2208 ns a, s n", "state_after": "case refine'_4\n\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)\na b : \u03b1\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) a) ((fun a => \u2a06 n \u2208 ns a, s n) z) \u2227\n      (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) b) ((fun a => \u2a06 n \u2208 ns a, s n) z)"}, {"tactic": "obtain \u27e8c, hc\u27e9 := hns a b", "annotated_tactic": ["obtain \u27e8c, hc\u27e9 := hns a b", []], "state_before": "case refine'_4\n\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)\na b : \u03b1\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) a) ((fun a => \u2a06 n \u2208 ns a, s n) z) \u2227\n      (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) b) ((fun a => \u2a06 n \u2208 ns a, s n) z)", "state_after": "case refine'_4.intro\n\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)\na b c : \u03b1\nhc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c)\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) a) ((fun a => \u2a06 n \u2208 ns a, s n) z) \u2227\n      (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) b) ((fun a => \u2a06 n \u2208 ns a, s n) z)"}, {"tactic": "refine' \u27e8c, _, _\u27e9 <;> refine' iSup_mono fun n => iSup_mono' fun hn => \u27e8_, le_rfl\u27e9", "annotated_tactic": ["refine' \u27e8c, _, _\u27e9 <;> refine' <a>iSup_mono</a> fun n => <a>iSup_mono'</a> fun hn => \u27e8_, <a>le_rfl</a>\u27e9", [{"full_name": "iSup_mono", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [900, 9], "def_end_pos": [900, 18]}, {"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": "case refine'_4.intro\n\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)\na b c : \u03b1\nhc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c)\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) a) ((fun a => \u2a06 n \u2208 ns a, s n) z) \u2227\n      (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) b) ((fun a => \u2a06 n \u2208 ns a, s n) z)", "state_after": "case refine'_4.intro.refine'_1\n\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)\na b c : \u03b1\nhc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c)\nn : \u03b9\nhn : n \u2208 ns a\n\u22a2 n \u2208 ns c\n\ncase refine'_4.intro.refine'_2\n\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)\na b c : \u03b1\nhc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c)\nn : \u03b9\nhn : n \u2208 ns b\n\u22a2 n \u2208 ns c"}, {"tactic": "exact hc.1 hn", "annotated_tactic": ["exact hc.1 hn", []], "state_before": "case refine'_4.intro.refine'_1\n\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)\na b c : \u03b1\nhc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c)\nn : \u03b9\nhn : n \u2208 ns a\n\u22a2 n \u2208 ns c", "state_after": "no goals"}, {"tactic": "exact hc.2 hn", "annotated_tactic": ["exact hc.2 hn", []], "state_before": "case refine'_4.intro.refine'_2\n\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)\na b c : \u03b1\nhc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c)\nn : \u03b9\nhn : n \u2208 ns b\n\u22a2 n \u2208 ns c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "UnitAddCircle.lintegral_preimage", "start": [213, 11], "end": [215, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "stronglyMeasurable_of_restrict_of_restrict_compl", "start": [816, 1], "end": [821, 7], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "OneOneEquiv.of_equiv", "start": [230, 1], "end": [232, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "full_name": "MeasureTheory.AEDisjoint.measure_diff_right", "start": [134, 1], "end": [135, 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.Ordered.zoom'", "start": [282, 1], "end": [287, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.Nontrivial.sdiff_singleton_nonempty", "start": [2328, 1], "end": [2332, 79], "traced_tactics": [{"tactic": "rw [Finset.sdiff_nonempty, Finset.subset_singleton_iff]", "annotated_tactic": ["rw [<a>Finset.sdiff_nonempty</a>, <a>Finset.subset_singleton_iff</a>]", [{"full_name": "Finset.sdiff_nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2201, 9], "def_end_pos": [2201, 23]}, {"full_name": "Finset.subset_singleton_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 29]}]], "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 b c : \u03b1\ns : Finset \u03b1\nhS : Finset.Nontrivial s\n\u22a2 Finset.Nonempty (s \\ {c})", "state_after": "\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 b c : \u03b1\ns : Finset \u03b1\nhS : Finset.Nontrivial s\n\u22a2 \u00ac(s = \u2205 \u2228 s = {c})"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "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 b c : \u03b1\ns : Finset \u03b1\nhS : Finset.Nontrivial s\n\u22a2 \u00ac(s = \u2205 \u2228 s = {c})", "state_after": "\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 b c : \u03b1\ns : Finset \u03b1\nhS : Finset.Nontrivial s\n\u22a2 s \u2260 \u2205 \u2227 s \u2260 {c}"}, {"tactic": "exact \u27e8by rintro rfl; exact Finset.not_nontrivial_empty hS, hS.ne_singleton\u27e9", "annotated_tactic": ["exact \u27e8by rintro rfl; exact <a>Finset.not_nontrivial_empty</a> hS, hS.ne_singleton\u27e9", [{"full_name": "Finset.not_nontrivial_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 29]}]], "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 b c : \u03b1\ns : Finset \u03b1\nhS : Finset.Nontrivial s\n\u22a2 s \u2260 \u2205 \u2227 s \u2260 {c}", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "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 b c : \u03b1\ns : Finset \u03b1\nhS : Finset.Nontrivial s\n\u22a2 s \u2260 \u2205", "state_after": "\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 c : \u03b1\nhS : Finset.Nontrivial \u2205\n\u22a2 False"}, {"tactic": "exact Finset.not_nontrivial_empty hS", "annotated_tactic": ["exact <a>Finset.not_nontrivial_empty</a> hS", [{"full_name": "Finset.not_nontrivial_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 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 c : \u03b1\nhS : Finset.Nontrivial \u2205\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurableSet.compl_toMeasurable_compl_ae_eq", "start": [241, 1], "end": [242, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "full_name": "integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt", "start": [54, 1], "end": [77, 31], "traced_tactics": [{"tactic": "obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) 1", "annotated_tactic": ["obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, <a>IsOpen</a> u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 <a>ball</a> (g x\u2080) 1", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) 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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\ncase 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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "exact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ zero_lt_one))", "annotated_tactic": ["exact <a>mem_nhdsWithin</a>.1 (hcg (<a>ball_mem_nhds</a> _ <a>zero_lt_one</a>))", [{"full_name": "mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [93, 9], "def_end_pos": [93, 23]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"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\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\ncase 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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) 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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "filter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) 1 zero_lt_one, hi\u03c6] with i hi h'i", "annotated_tactic": ["filter_upwards [<a>tendstoUniformlyOn_iff</a>.1 (hl\u03c6 u u_open x\u2080u) 1 <a>zero_lt_one</a>, hi\u03c6] with i hi h'i", [{"full_name": "Metric.tendstoUniformlyOn_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [922, 9], "def_end_pos": [922, 31]}, {"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\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "have A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u03bc := by\n  refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _\n  apply\n    mem\u2112p_top_of_bound\n      ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1\n  filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx\n  simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le", "annotated_tactic": ["have A : <a>IntegrableOn</a> (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u03bc := by\n    refine' <a>Integrable.smul_of_top_right</a> (hmg.mono (<a>diff_subset</a> _ _) <a>le_rfl</a>) _\n    apply\n      <a>mem\u2112p_top_of_bound</a>\n        ((<a>integrable_of_integral_eq_one</a> h'i).aestronglyMeasurable.mono_set (<a>diff_subset</a> _ _)) 1\n    filter_upwards [<a>self_mem_ae_restrict</a> (hs.diff u_open.measurableSet)] with x hx\n    simpa only [<a>Pi.zero_apply</a>, <a>dist_zero_left</a>] using (hi x hx).<a>le</a>", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "MeasureTheory.Integrable.smul_of_top_right", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1088, 9], "def_end_pos": [1088, 37]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasureTheory.mem\u2112p_top_of_bound", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [584, 9], "def_end_pos": [584, 27]}, {"full_name": "MeasureTheory.integrable_of_integral_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [864, 9], "def_end_pos": [864, 38]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"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": "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": "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\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "convert A.union B", "annotated_tactic": ["convert A.union B", []], "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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "state_after": "case h.e'_6\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 s = s \\ u \u222a s \u2229 u"}, {"tactic": "simp only [diff_union_inter]", "annotated_tactic": ["simp only [<a>diff_union_inter</a>]", [{"full_name": "Set.diff_union_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1894, 9], "def_end_pos": [1894, 25]}]], "state_before": "case h.e'_6\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 s = s \\ u \u222a s \u2229 u", "state_after": "no goals"}, {"tactic": "refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _", "annotated_tactic": ["refine' <a>Integrable.smul_of_top_right</a> (hmg.mono (<a>diff_subset</a> _ _) <a>le_rfl</a>) _", [{"full_name": "MeasureTheory.Integrable.smul_of_top_right", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1088, 9], "def_end_pos": [1088, 37]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}, {"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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 Mem\u2112p (fun x => \u03c6 i x) \u22a4"}, {"tactic": "apply\n  mem\u2112p_top_of_bound\n    ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1", "annotated_tactic": ["apply\n      <a>mem\u2112p_top_of_bound</a>\n        ((<a>integrable_of_integral_eq_one</a> h'i).aestronglyMeasurable.mono_set (<a>diff_subset</a> _ _)) 1", [{"full_name": "MeasureTheory.mem\u2112p_top_of_bound", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [584, 9], "def_end_pos": [584, 27]}, {"full_name": "MeasureTheory.integrable_of_integral_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [864, 9], "def_end_pos": [864, 38]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 Mem\u2112p (fun x => \u03c6 i x) \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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \\ u), \u2016\u03c6 i x\u2016 \u2264 1"}, {"tactic": "filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx", "annotated_tactic": ["filter_upwards [<a>self_mem_ae_restrict</a> (hs.diff u_open.measurableSet)] with x hx", [{"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \\ u), \u2016\u03c6 i x\u2016 \u2264 1", "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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 1"}, {"tactic": "simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le", "annotated_tactic": ["simpa only [<a>Pi.zero_apply</a>, <a>dist_zero_left</a>] using (hi x hx).<a>le</a>", [{"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": "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\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 1", "state_after": "no goals"}, {"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": "\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)", "state_after": "case h\u03c6\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Integrable fun x => \u03c6 i x\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Mem\u2112p (fun x => g x) \u22a4"}, {"tactic": "exact IntegrableOn.mono_set (integrable_of_integral_eq_one h'i) (inter_subset_left _ _)", "annotated_tactic": ["exact <a>IntegrableOn.mono_set</a> (<a>integrable_of_integral_eq_one</a> h'i) (<a>inter_subset_left</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": "MeasureTheory.integrable_of_integral_eq_one", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [864, 9], "def_end_pos": [864, 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": "case h\u03c6\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Integrable fun x => \u03c6 i x", "state_after": "no goals"}, {"tactic": "apply\n  mem\u2112p_top_of_bound (hmg.mono_set (inter_subset_left _ _)).aestronglyMeasurable (\u2016g x\u2080\u2016 + 1)", "annotated_tactic": ["apply\n        <a>mem\u2112p_top_of_bound</a> (hmg.mono_set (<a>inter_subset_left</a> _ _)).<a>aestronglyMeasurable</a> (\u2016g x\u2080\u2016 + 1)", [{"full_name": "MeasureTheory.mem\u2112p_top_of_bound", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [584, 9], "def_end_pos": [584, 27]}, {"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.Integrable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [457, 9], "def_end_pos": [457, 40]}]], "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Mem\u2112p (fun x => g x) \u22a4", "state_after": "case hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u2229 u), \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1"}, {"tactic": "filter_upwards [self_mem_ae_restrict (hs.inter u_open.measurableSet)] with x hx", "annotated_tactic": ["filter_upwards [<a>self_mem_ae_restrict</a> (hs.inter u_open.measurableSet)] with x hx", [{"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2681, 9], "def_end_pos": [2681, 29]}]], "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u2229 u), \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1", "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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1"}, {"tactic": "rw [inter_comm] at hx", "annotated_tactic": ["rw [<a>inter_comm</a>] at hx", [{"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\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1", "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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 u \u2229 s\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1"}, {"tactic": "exact (norm_lt_of_mem_ball (hu x hx)).le", "annotated_tactic": ["exact (<a>norm_lt_of_mem_ball</a> (hu x hx)).<a>le</a>", [{"full_name": "norm_lt_of_mem_ball", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [676, 15], "def_end_pos": [676, 34]}, {"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\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 u \u2229 s\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr_eq_map_of_unused_state", "start": [286, 1], "end": [289, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.upcrossingStrat_le_one", "start": [377, 1], "end": [395, 58], "traced_tactics": [{"tactic": "rw [upcrossingStrat, \u2190 Set.indicator_finset_biUnion_apply]", "annotated_tactic": ["rw [<a>upcrossingStrat</a>, \u2190 <a>Set.indicator_finset_biUnion_apply</a>]", [{"full_name": "MeasureTheory.upcrossingStrat", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [368, 19], "def_end_pos": [368, 34]}, {"full_name": "Set.indicator_finset_biUnion_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [691, 3], "def_end_pos": [691, 14]}]], "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\n\u22a2 upcrossingStrat a b f N n \u03c9 \u2264 1", "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\n\u22a2 Set.indicator (\u22c3 i \u2208 Finset.range N, Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9)) 1\n      n \u2264\n    1\n\ncase 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 : \u03a9\n\u2131 : Filtration \u2115 m0\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\n            Disjoint (Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9))\n              (Set.Ico (lowerCrossingTime a b f N j \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9))"}, {"tactic": "exact Set.indicator_le_self' (fun _ _ => zero_le_one) _", "annotated_tactic": ["exact <a>Set.indicator_le_self'</a> (fun _ _ => <a>zero_le_one</a>) _", [{"full_name": "Set.indicator_le_self'", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [858, 3], "def_end_pos": [858, 14]}, {"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\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\n\u22a2 Set.indicator (\u22c3 i \u2208 Finset.range N, Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9)) 1\n      n \u2264\n    1", "state_after": "no goals"}, {"tactic": "intro i _ j _ hij", "annotated_tactic": ["intro i _ j _ hij", []], "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 : \u03a9\n\u2131 : Filtration \u2115 m0\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\n            Disjoint (Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9))\n              (Set.Ico (lowerCrossingTime a b f N j \u03c9) (upperCrossingTime a b f N (j + 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 : \u03a9\n\u2131 : Filtration \u2115 m0\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\n\u22a2 Disjoint (Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9))\n    (Set.Ico (lowerCrossingTime a b f N j \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9))"}, {"tactic": "rw [Set.Ico_disjoint_Ico]", "annotated_tactic": ["rw [<a>Set.Ico_disjoint_Ico</a>]", [{"full_name": "Set.Ico_disjoint_Ico", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [137, 9], "def_end_pos": [137, 25]}]], "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 : \u03a9\n\u2131 : Filtration \u2115 m0\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\n\u22a2 Disjoint (Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9))\n    (Set.Ico (lowerCrossingTime a b f N j \u03c9) (upperCrossingTime a b f N (j + 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 : \u03a9\n\u2131 : Filtration \u2115 m0\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)"}, {"tactic": "obtain hij' | hij' := lt_or_gt_of_ne hij", "annotated_tactic": ["obtain hij' | hij' := <a>lt_or_gt_of_ne</a> hij", [{"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": "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 : \u03a9\n\u2131 : Filtration \u2115 m0\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)", "state_after": "case h.inl\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : i < j\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)\n\ncase h.inr\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : i > j\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)"}, {"tactic": "rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :\n  upperCrossingTime a b f N _ \u03c9 \u2264 upperCrossingTime a b f N _ \u03c9),\n  max_eq_right (lowerCrossingTime_mono hij'.le :\n    lowerCrossingTime a b f N _ _ \u2264 lowerCrossingTime _ _ _ _ _ _)]", "annotated_tactic": ["rw [<a>min_eq_left</a> (<a>upperCrossingTime_mono</a> (<a>Nat.succ_le_succ</a> hij'.le) :\n        <a>upperCrossingTime</a> a b f N _ \u03c9 \u2264 <a>upperCrossingTime</a> a b f N _ \u03c9),\n        <a>max_eq_right</a> (<a>lowerCrossingTime_mono</a> hij'.le :\n          <a>lowerCrossingTime</a> a b f N _ _ \u2264 <a>lowerCrossingTime</a> _ _ _ _ _ _)]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "MeasureTheory.upperCrossingTime_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [220, 9], "def_end_pos": [220, 31]}, {"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": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}, {"full_name": "MeasureTheory.lowerCrossingTime_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [213, 9], "def_end_pos": [213, 31]}, {"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}, {"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}]], "state_before": "case h.inl\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : i < j\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)", "state_after": "case h.inl\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : i < j\n\u22a2 upperCrossingTime a b f N (Nat.succ i) \u03c9 \u2264 lowerCrossingTime a b f N j \u03c9"}, {"tactic": "refine' le_trans upperCrossingTime_le_lowerCrossingTime\n  (lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))", "annotated_tactic": ["refine' <a>le_trans</a> <a>upperCrossingTime_le_lowerCrossingTime</a>\n        (<a>lowerCrossingTime_mono</a> (<a>Nat.succ_le_of_lt</a> hij'))", [{"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]}, {"full_name": "MeasureTheory.lowerCrossingTime_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [213, 9], "def_end_pos": [213, 31]}, {"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]}]], "state_before": "case h.inl\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : i < j\n\u22a2 upperCrossingTime a b f N (Nat.succ i) \u03c9 \u2264 lowerCrossingTime a b f N j \u03c9", "state_after": "no goals"}, {"tactic": "rw [gt_iff_lt] at hij'", "annotated_tactic": ["rw [<a>gt_iff_lt</a>] at hij'", [{"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}]], "state_before": "case h.inr\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : i > j\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)", "state_after": "case h.inr\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : j < i\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)"}, {"tactic": "rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :\n  upperCrossingTime a b f N _ \u03c9 \u2264 upperCrossingTime a b f N _ \u03c9),\n  max_eq_left (lowerCrossingTime_mono hij'.le :\n    lowerCrossingTime a b f N _ _ \u2264 lowerCrossingTime _ _ _ _ _ _)]", "annotated_tactic": ["rw [<a>min_eq_right</a> (<a>upperCrossingTime_mono</a> (<a>Nat.succ_le_succ</a> hij'.le) :\n        <a>upperCrossingTime</a> a b f N _ \u03c9 \u2264 <a>upperCrossingTime</a> a b f N _ \u03c9),\n        <a>max_eq_left</a> (<a>lowerCrossingTime_mono</a> hij'.le :\n          <a>lowerCrossingTime</a> a b f N _ _ \u2264 <a>lowerCrossingTime</a> _ _ _ _ _ _)]", [{"full_name": "min_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}, {"full_name": "MeasureTheory.upperCrossingTime_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [220, 9], "def_end_pos": [220, 31]}, {"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": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "MeasureTheory.upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [139, 19], "def_end_pos": [139, 36]}, {"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "MeasureTheory.lowerCrossingTime_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [213, 9], "def_end_pos": [213, 31]}, {"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}, {"full_name": "MeasureTheory.lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [148, 19], "def_end_pos": [148, 36]}]], "state_before": "case h.inr\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : j < i\n\u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264\n    max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9)", "state_after": "case h.inr\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : j < i\n\u22a2 upperCrossingTime a b f N (Nat.succ j) \u03c9 \u2264 lowerCrossingTime a b f N i \u03c9"}, {"tactic": "refine' le_trans upperCrossingTime_le_lowerCrossingTime\n  (lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))", "annotated_tactic": ["refine' <a>le_trans</a> <a>upperCrossingTime_le_lowerCrossingTime</a>\n        (<a>lowerCrossingTime_mono</a> (<a>Nat.succ_le_of_lt</a> hij'))", [{"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]}, {"full_name": "MeasureTheory.lowerCrossingTime_mono", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [213, 9], "def_end_pos": [213, 31]}, {"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]}]], "state_before": "case h.inr\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\ni : \u2115\na\u271d\u00b9 : i \u2208 Finset.range N\nj : \u2115\na\u271d : j \u2208 Finset.range N\nhij : i \u2260 j\nhij' : j < i\n\u22a2 upperCrossingTime a b f N (Nat.succ j) \u03c9 \u2264 lowerCrossingTime a b f N i \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snormEssSup_map_measure", "start": [915, 1], "end": [917, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Kernel.lean", "full_name": "ProbabilityTheory.kernel.indep_iSup_of_directed_le", "start": [464, 1], "end": [485, 51], "traced_tactics": [{"tactic": "let p : \u03b9 \u2192 Set (Set \u03a9) := fun n => { t | MeasurableSet[m n] t }", "annotated_tactic": ["let p : \u03b9 \u2192 <a>Set</a> (<a>Set</a> \u03a9) := fun n => { t | MeasurableSet[m n] t }", [{"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]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "have hp : \u2200 n, IsPiSystem (p n) := fun n => @isPiSystem_measurableSet \u03a9 (m n)", "annotated_tactic": ["have hp : \u2200 n, <a>IsPiSystem</a> (p n) := fun n => @<a>isPiSystem_measurableSet</a> \u03a9 (m n)", [{"full_name": "IsPiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [66, 5], "def_end_pos": [66, 15]}, {"full_name": "MeasurableSpace.isPiSystem_measurableSet", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [72, 9], "def_end_pos": [72, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "have h_gen_n : \u2200 n, m n = generateFrom (p n) := fun n =>\n  (@generateFrom_measurableSet \u03a9 (m n)).symm", "annotated_tactic": ["have h_gen_n : \u2200 n, m n = <a>generateFrom</a> (p n) := fun n =>\n    (@<a>generateFrom_measurableSet</a> \u03a9 (m n)).<a>symm</a>", [{"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}, {"full_name": "MeasurableSpace.generateFrom_measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [396, 9], "def_end_pos": [396, 35]}, {"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\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "have hp_supr_pi : IsPiSystem (\u22c3 n, p n) := isPiSystem_iUnion_of_directed_le p hp hm", "annotated_tactic": ["have hp_supr_pi : <a>IsPiSystem</a> (\u22c3 n, p n) := <a>isPiSystem_iUnion_of_directed_le</a> p hp hm", [{"full_name": "IsPiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [66, 5], "def_end_pos": [66, 15]}, {"full_name": "isPiSystem_iUnion_of_directed_le", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [115, 9], "def_end_pos": [115, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "let p' := { t : Set \u03a9 | MeasurableSet[m'] t }", "annotated_tactic": ["let p' := { t : <a>Set</a> \u03a9 | MeasurableSet[m'] t }", [{"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\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "have hp'_pi : IsPiSystem p' := @isPiSystem_measurableSet \u03a9 m'", "annotated_tactic": ["have hp'_pi : <a>IsPiSystem</a> p' := @<a>isPiSystem_measurableSet</a> \u03a9 m'", [{"full_name": "IsPiSystem", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [66, 5], "def_end_pos": [66, 15]}, {"full_name": "MeasurableSpace.isPiSystem_measurableSet", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [72, 9], "def_end_pos": [72, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "have h_gen' : m' = generateFrom p' := (@generateFrom_measurableSet \u03a9 m').symm", "annotated_tactic": ["have h_gen' : m' = <a>generateFrom</a> p' := (@<a>generateFrom_measurableSet</a> \u03a9 m').<a>symm</a>", [{"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}, {"full_name": "MeasurableSpace.generateFrom_measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [396, 9], "def_end_pos": [396, 35]}, {"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\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "have h_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba \u03bc := by\n  refine IndepSets.iUnion ?_\n  conv at h_indep =>\n    intro i\n    rw [h_gen_n i, h_gen']\n  exact fun n => (h_indep n).indepSets", "annotated_tactic": ["have h_pi_system_indep : <a>IndepSets</a> (\u22c3 n, p n) p' \u03ba \u03bc := by\n    refine <a>IndepSets.iUnion</a> ?_\n    conv at h_indep =>\n      intro i\n      rw [h_gen_n i, h_gen']\n    exact fun n => (h_indep n).<a>indepSets</a>", [{"full_name": "ProbabilityTheory.kernel.IndepSets", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [69, 5], "def_end_pos": [69, 14]}, {"full_name": "ProbabilityTheory.kernel.IndepSets.iUnion", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [198, 9], "def_end_pos": [198, 25]}, {"full_name": "ProbabilityTheory.kernel.Indep.indepSets", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [306, 9], "def_end_pos": [306, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\nh_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba"}, {"tactic": "refine' IndepSets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep", "annotated_tactic": ["refine' <a>IndepSets.indep</a> (<a>iSup_le</a> h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep", [{"full_name": "ProbabilityTheory.kernel.IndepSets.indep", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [353, 9], "def_end_pos": [353, 24]}, {"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\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\nh_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba\n\u22a2 Indep (\u2a06 i, m i) m' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\nh_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba\n\u22a2 \u2a06 i, m i = generateFrom (\u22c3 n, p n)"}, {"tactic": "exact (generateFrom_iUnion_measurableSet _).symm", "annotated_tactic": ["exact (<a>generateFrom_iUnion_measurableSet</a> _).<a>symm</a>", [{"full_name": "MeasurableSpace.generateFrom_iUnion_measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [524, 9], "def_end_pos": [524, 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": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\nh_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba\n\u22a2 \u2a06 i, m i = generateFrom (\u22c3 n, p n)", "state_after": "no goals"}, {"tactic": "refine IndepSets.iUnion ?_", "annotated_tactic": ["refine <a>IndepSets.iUnion</a> ?_", [{"full_name": "ProbabilityTheory.kernel.IndepSets.iUnion", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [198, 9], "def_end_pos": [198, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 IndepSets (\u22c3 n, p n) p' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 \u2200 (n : \u03b9), IndepSets (p n) p' \u03ba"}, {"tactic": "conv at h_indep =>\n  intro i\n  rw [h_gen_n i, h_gen']", "annotated_tactic": ["conv at h_indep =>\n      intro i\n      rw [h_gen_n i, h_gen']", []], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 \u2200 (n : \u03b9), IndepSets (p n) p' \u03ba", "state_after": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nh_indep : \u2200 (i : \u03b9), Indep (generateFrom (p i)) (generateFrom p') \u03ba\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 \u2200 (n : \u03b9), IndepSets (p n) p' \u03ba"}, {"tactic": "exact fun n => (h_indep n).indepSets", "annotated_tactic": ["exact fun n => (h_indep n).<a>indepSets</a>", [{"full_name": "ProbabilityTheory.kernel.Indep.indepSets", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [306, 9], "def_end_pos": [306, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03a9\u271d : Type u_2\n\u03b9 : Type u_3\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\u271d\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d }\n\u03bc\u271d : Measure \u03b1\n\u03a9 : Type u_4\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nm' m0 : MeasurableSpace \u03a9\n\u03ba : { x // x \u2208 kernel \u03b1 \u03a9 }\n\u03bc : Measure \u03b1\ninst\u271d : IsMarkovKernel \u03ba\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_le' : m' \u2264 m0\nhm : Directed (fun x x_1 => x \u2264 x_1) m\np : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t}\nhp : \u2200 (n : \u03b9), IsPiSystem (p n)\nh_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n)\nhp_supr_pi : IsPiSystem (\u22c3 n, p n)\np' : Set (Set \u03a9) := {t | MeasurableSet t}\nh_indep : \u2200 (i : \u03b9), Indep (generateFrom (p i)) (generateFrom p') \u03ba\nhp'_pi : IsPiSystem p'\nh_gen' : m' = generateFrom p'\n\u22a2 \u2200 (n : \u03b9), IndepSets (p n) p' \u03ba", "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_coe_smul", "start": [900, 1], "end": [915, 86], "traced_tactics": [{"tactic": "by_cases H : AEStronglyMeasurable (fun x : \u03b1 => (f x : \u211d) \u2022 g x) \u03bc", "annotated_tactic": ["by_cases H : <a>AEStronglyMeasurable</a> (fun x : \u03b1 => (f x : \u211d) \u2022 g x) \u03bc", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x", "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x\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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : \u00acAEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x"}, {"tactic": "simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, HasFiniteIntegral, H,\n  true_and_iff]", "annotated_tactic": ["simp only [<a>Integrable</a>, <a>aestronglyMeasurable_withDensity_iff</a> hf, <a>HasFiniteIntegral</a>, H,\n      <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": "aestronglyMeasurable_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1843, 9], "def_end_pos": [1843, 52]}, {"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x", "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "rw [lintegral_withDensity_eq_lintegral_mul\u2080' hf.coe_nnreal_ennreal.aemeasurable]", "annotated_tactic": ["rw [<a>lintegral_withDensity_eq_lintegral_mul\u2080'</a> hf.coe_nnreal_ennreal.aemeasurable]", [{"full_name": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul\u2080'", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [298, 9], "def_end_pos": [298, 49]}]], "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4", "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4\n\ncase 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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 AEMeasurable fun a => \u2191\u2016g a\u2016\u208a"}, {"tactic": "rw [iff_iff_eq]", "annotated_tactic": ["rw [<a>iff_iff_eq</a>]", [{"full_name": "iff_iff_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [51, 9], "def_end_pos": [51, 19]}]], "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4", "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a \u2202\u03bc < \u22a4) = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a \u2202\u03bc < \u22a4) = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4)", "state_after": "case pos.e_a.e_f\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (fun a => ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a) = fun a => \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case pos.e_a.e_f\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (fun a => ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a) = fun a => \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a", "state_after": "case pos.e_a.e_f.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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\nx : \u03b1\n\u22a2 ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) x = \u2191\u2016\u2191(f x) \u2022 g x\u2016\u208a"}, {"tactic": "simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul, Pi.mul_apply]", "annotated_tactic": ["simp only [<a>nnnorm_smul</a>, <a>NNReal.nnnorm_eq</a>, <a>coe_mul</a>, <a>Pi.mul_apply</a>]", [{"full_name": "nnnorm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}, {"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "Pi.mul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case pos.e_a.e_f.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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\nx : \u03b1\n\u22a2 ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) x = \u2191\u2016\u2191(f x) \u2022 g x\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [aemeasurable_withDensity_ennreal_iff hf]", "annotated_tactic": ["rw [<a>aemeasurable_withDensity_ennreal_iff</a> hf]", [{"full_name": "MeasureTheory.aemeasurable_withDensity_ennreal_iff", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [238, 9], "def_end_pos": [238, 45]}]], "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 AEMeasurable fun a => \u2191\u2016g a\u2016\u208a", "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 AEMeasurable fun x => \u2191(f x) * \u2191\u2016g x\u2016\u208a"}, {"tactic": "convert H.ennnorm using 1", "annotated_tactic": ["convert H.ennnorm using 1", []], "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 AEMeasurable fun x => \u2191(f x) * \u2191\u2016g x\u2016\u208a", "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (fun x => \u2191(f x) * \u2191\u2016g x\u2016\u208a) = fun a => \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 (fun x => \u2191(f x) * \u2191\u2016g x\u2016\u208a) = fun a => \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a", "state_after": "case h.e'_5.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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\nx : \u03b1\n\u22a2 \u2191(f x) * \u2191\u2016g x\u2016\u208a = \u2191\u2016\u2191(f x) \u2022 g x\u2016\u208a"}, {"tactic": "simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul]", "annotated_tactic": ["simp only [<a>nnnorm_smul</a>, <a>NNReal.nnnorm_eq</a>, <a>coe_mul</a>]", [{"full_name": "nnnorm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}, {"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "state_before": "case h.e'_5.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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\nx : \u03b1\n\u22a2 \u2191(f x) * \u2191\u2016g x\u2016\u208a = \u2191\u2016\u2191(f x) \u2022 g x\u2016\u208a", "state_after": "no goals"}, {"tactic": "simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and_iff]", "annotated_tactic": ["simp only [<a>Integrable</a>, <a>aestronglyMeasurable_withDensity_iff</a> hf, H, <a>false_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": "aestronglyMeasurable_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1843, 9], "def_end_pos": [1843, 52]}, {"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\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\nhf : Measurable f\ng : \u03b1 \u2192 E\nH : \u00acAEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.monotone_eapprox", "start": [892, 1], "end": [893, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.sort_singleton", "start": [73, 1], "end": [74, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Division.lean", "full_name": "MvPolynomial.zero_divMonomial", "start": [63, 1], "end": [64, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.pi_pi_aux", "start": [314, 1], "end": [326, 26], "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": "\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "case refine'_1\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\n\ncase refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u2264 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s)"}, {"tactic": "rw [Measure.pi, toMeasure_apply _ _ (MeasurableSet.pi countable_univ fun i _ => hs i)]", "annotated_tactic": ["rw [<a>Measure.pi</a>, <a>toMeasure_apply</a> _ _ (<a>MeasurableSet.pi</a> <a>countable_univ</a> fun i _ => hs i)]", [{"full_name": "MeasureTheory.Measure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [303, 27], "def_end_pos": [303, 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.pi", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [955, 19], "def_end_pos": [955, 35]}, {"full_name": "Set.countable_univ", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [112, 9], "def_end_pos": [112, 23]}]], "state_before": "case refine'_1\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "case refine'_1\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "apply OuterMeasure.pi_pi_le", "annotated_tactic": ["apply <a>OuterMeasure.pi_pi_le</a>", [{"full_name": "MeasureTheory.OuterMeasure.pi_pi_le", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [198, 9], "def_end_pos": [198, 17]}]], "state_before": "case refine'_1\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "no goals"}, {"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": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u2264 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s)", "state_after": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : Encodable \u03b9\n\u22a2 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u2264 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s)"}, {"tactic": "simp_rw [\u2190 pi'_pi \u03bc s, Measure.pi,\n  toMeasure_apply _ _ (MeasurableSet.pi countable_univ fun i _ => hs i)]", "annotated_tactic": ["simp_rw [\u2190 <a>pi'_pi</a> \u03bc s, <a>Measure.pi</a>,\n      <a>toMeasure_apply</a> _ _ (<a>MeasurableSet.pi</a> <a>countable_univ</a> fun i _ => hs i)]", [{"full_name": "MeasureTheory.Measure.pi'_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [274, 9], "def_end_pos": [274, 15]}, {"full_name": "MeasureTheory.Measure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [303, 27], "def_end_pos": [303, 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.pi", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [955, 19], "def_end_pos": [955, 35]}, {"full_name": "Set.countable_univ", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [112, 9], "def_end_pos": [112, 23]}]], "state_before": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : Encodable \u03b9\n\u22a2 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u2264 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s)", "state_after": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : Encodable \u03b9\n\u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i)"}, {"tactic": "suffices (pi' \u03bc).toOuterMeasure \u2264 OuterMeasure.pi fun i => (\u03bc i).toOuterMeasure by exact this _", "annotated_tactic": ["suffices (<a>pi'</a> \u03bc).toOuterMeasure \u2264 <a>OuterMeasure.pi</a> fun i => (\u03bc i).toOuterMeasure by exact this _", [{"full_name": "MeasureTheory.Measure.pi'", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [270, 5], "def_end_pos": [270, 8]}, {"full_name": "MeasureTheory.OuterMeasure.pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [194, 15], "def_end_pos": [194, 17]}]], "state_before": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : Encodable \u03b9\n\u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i)", "state_after": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : Encodable \u03b9\n\u22a2 \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i)"}, {"tactic": "clear hs s", "annotated_tactic": ["clear hs s", []], "state_before": "case refine'_2\n\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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : Encodable \u03b9\n\u22a2 \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i)", "state_after": "case refine'_2\n\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)\nthis : Encodable \u03b9\n\u22a2 \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i)"}, {"tactic": "rw [OuterMeasure.le_pi]", "annotated_tactic": ["rw [<a>OuterMeasure.le_pi</a>]", [{"full_name": "MeasureTheory.OuterMeasure.le_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [204, 9], "def_end_pos": [204, 14]}]], "state_before": "case refine'_2\n\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)\nthis : Encodable \u03b9\n\u22a2 \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i)", "state_after": "case refine'_2\n\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)\nthis : Encodable \u03b9\n\u22a2 \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), Set.Nonempty (Set.pi univ s) \u2192 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "intro s _", "annotated_tactic": ["intro s _", []], "state_before": "case refine'_2\n\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)\nthis : Encodable \u03b9\n\u22a2 \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), Set.Nonempty (Set.pi univ s) \u2192 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "case refine'_2\n\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)\nthis : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\na\u271d : Set.Nonempty (Set.pi univ s)\n\u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "exact (pi'_pi \u03bc s).le", "annotated_tactic": ["exact (<a>pi'_pi</a> \u03bc s).<a>le</a>", [{"full_name": "MeasureTheory.Measure.pi'_pi", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [274, 9], "def_end_pos": [274, 15]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "state_before": "case refine'_2\n\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)\nthis : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\na\u271d : Set.Nonempty (Set.pi univ s)\n\u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)", "state_after": "no goals"}, {"tactic": "exact this _", "annotated_tactic": ["exact this _", []], "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)\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis\u271d : Encodable \u03b9\nthis : \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i)\n\u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Enumerate.lean", "full_name": "Set.enumerate_inj", "start": [74, 1], "end": [100, 23], "traced_tactics": [{"tactic": "rcases le_total n\u2081 n\u2082 with (hn|hn)", "annotated_tactic": ["rcases <a>le_total</a> n\u2081 n\u2082 with (hn|hn)", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\n\u22a2 n\u2081 = n\u2082", "state_after": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082"}, {"tactic": "on_goal 2 => swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082", "annotated_tactic": ["on_goal 2 => swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082", []], "state_before": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082", "state_after": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081"}, {"tactic": "all_goals\n  rcases Nat.le.dest hn with \u27e8m, rfl\u27e9\n  clear hn\n  induction n\u2081 generalizing s\n  case zero =>\n    cases m\n    case zero => rfl\n    case succ m =>\n      have h' : enumerate sel (s \\ {a}) m = some a := by\n        simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n      have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n      simp_all [Set.mem_diff_singleton]\n  case succ k ih =>\n    cases h : sel s\n    \n    case none =>\n      simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n    case some _ =>\n      simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                     Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n      exact ih h\u2081 h\u2082", "annotated_tactic": ["all_goals\n    rcases <a>Nat.le.dest</a> hn with \u27e8m, rfl\u27e9\n    clear hn\n    induction n\u2081 generalizing s\n    case zero =>\n      cases m\n      case zero => rfl\n      case succ m =>\n        have h' : <a>enumerate</a> sel (s \\ {a}) m = <a>some</a> a := by\n          simp_all only [<a>enumerate</a>, <a>Nat.zero_eq</a>, <a>Nat.add_eq</a>, <a>zero_add</a>]; exact h\u2082\n        have : a \u2208 s \\ {a} := <a>enumerate_mem</a> sel h_sel h'\n        simp_all [<a>Set.mem_diff_singleton</a>]\n    case succ k ih =>\n      cases h : sel s\n      /- porting note : The original covered both goals with just `simp_all <;> tauto` -/\n      case none =>\n        simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>Nat.add_succ</a>, <a>enumerate_eq_none_of_sel</a> _ h]\n      case some _ =>\n        simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>enumerate</a>, Option.some.injEq,\n                       <a>Nat.add_succ</a>, enumerate._eq_2, Nat.succ.injEq]\n        exact ih h\u2081 h\u2082", [{"full_name": "Nat.le.dest", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 16]}, {"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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.add_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [84, 17], "def_end_pos": [84, 23]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.enumerate_mem", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "Set.mem_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2101, 9], "def_end_pos": [2101, 27]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"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": "Set.enumerate_eq_none_of_sel", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [41, 9], "def_end_pos": [41, 33]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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]}]], "state_before": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081", "state_after": "no goals"}, {"tactic": "swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082", "annotated_tactic": ["swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082", []], "state_before": "case inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082", "state_after": "case inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081"}, {"tactic": "rcases Nat.le.dest hn with \u27e8m, rfl\u27e9", "annotated_tactic": ["rcases <a>Nat.le.dest</a> hn with \u27e8m, rfl\u27e9", [{"full_name": "Nat.le.dest", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 16]}]], "state_before": "case inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081", "state_after": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\nhn : n\u2081 \u2264 n\u2081 + m\n\u22a2 n\u2081 + m = n\u2081"}, {"tactic": "clear hn", "annotated_tactic": ["clear hn", []], "state_before": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\nhn : n\u2081 \u2264 n\u2081 + m\n\u22a2 n\u2081 + m = n\u2081", "state_after": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\n\u22a2 n\u2081 + m = n\u2081"}, {"tactic": "induction n\u2081 generalizing s", "annotated_tactic": ["induction n\u2081 generalizing s", []], "state_before": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\n\u22a2 n\u2081 + m = n\u2081", "state_after": "case inr.intro.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero + m = Nat.zero\n\ncase inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d"}, {"tactic": "case zero =>\n  cases m\n  case zero => rfl\n  case succ m =>\n    have h' : enumerate sel (s \\ {a}) m = some a := by\n      simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n    have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n    simp_all [Set.mem_diff_singleton]", "annotated_tactic": ["case zero =>\n      cases m\n      case zero => rfl\n      case succ m =>\n        have h' : <a>enumerate</a> sel (s \\ {a}) m = <a>some</a> a := by\n          simp_all only [<a>enumerate</a>, <a>Nat.zero_eq</a>, <a>Nat.add_eq</a>, <a>zero_add</a>]; exact h\u2082\n        have : a \u2208 s \\ {a} := <a>enumerate_mem</a> sel h_sel h'\n        simp_all [<a>Set.mem_diff_singleton</a>]", [{"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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.add_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [84, 17], "def_end_pos": [84, 23]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.enumerate_mem", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "Set.mem_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2101, 9], "def_end_pos": [2101, 27]}]], "state_before": "case inr.intro.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero + m = Nat.zero\n\ncase inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d", "state_after": "case inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d"}, {"tactic": "case succ k ih =>\n  cases h : sel s\n  \n  case none =>\n    simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n  case some _ =>\n    simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                   Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n    exact ih h\u2081 h\u2082", "annotated_tactic": ["case succ k ih =>\n      cases h : sel s\n      /- porting note : The original covered both goals with just `simp_all <;> tauto` -/\n      case none =>\n        simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>Nat.add_succ</a>, <a>enumerate_eq_none_of_sel</a> _ h]\n      case some _ =>\n        simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>enumerate</a>, Option.some.injEq,\n                       <a>Nat.add_succ</a>, enumerate._eq_2, Nat.succ.injEq]\n        exact ih h\u2081 h\u2082", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"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": "Set.enumerate_eq_none_of_sel", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [41, 9], "def_end_pos": [41, 33]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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]}]], "state_before": "case inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d", "state_after": "no goals"}, {"tactic": "cases m", "annotated_tactic": ["cases m", []], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero + m = Nat.zero", "state_after": "case zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero + Nat.zero = Nat.zero\n\ncase succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero"}, {"tactic": "case zero => rfl", "annotated_tactic": ["case zero => rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero + Nat.zero = Nat.zero\n\ncase succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero", "state_after": "case succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero"}, {"tactic": "case succ m =>\n  have h' : enumerate sel (s \\ {a}) m = some a := by\n    simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n  have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n  simp_all [Set.mem_diff_singleton]", "annotated_tactic": ["case succ m =>\n        have h' : <a>enumerate</a> sel (s \\ {a}) m = <a>some</a> a := by\n          simp_all only [<a>enumerate</a>, <a>Nat.zero_eq</a>, <a>Nat.add_eq</a>, <a>zero_add</a>]; exact h\u2082\n        have : a \u2208 s \\ {a} := <a>enumerate_mem</a> sel h_sel h'\n        simp_all [<a>Set.mem_diff_singleton</a>]", [{"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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.add_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [84, 17], "def_end_pos": [84, 23]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.enumerate_mem", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "Set.mem_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2101, 9], "def_end_pos": [2101, 27]}]], "state_before": "case succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero + Nat.zero = Nat.zero", "state_after": "no goals"}, {"tactic": "have h' : enumerate sel (s \\ {a}) m = some a := by\n  simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082", "annotated_tactic": ["have h' : <a>enumerate</a> sel (s \\ {a}) m = <a>some</a> a := by\n          simp_all only [<a>enumerate</a>, <a>Nat.zero_eq</a>, <a>Nat.add_eq</a>, <a>zero_add</a>]; exact h\u2082", [{"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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.add_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [84, 17], "def_end_pos": [84, 23]}, {"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\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 Nat.zero + Nat.succ m = Nat.zero", "state_after": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\n\u22a2 Nat.zero + Nat.succ m = Nat.zero"}, {"tactic": "have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'", "annotated_tactic": ["have : a \u2208 s \\ {a} := <a>enumerate_mem</a> sel h_sel h'", [{"full_name": "Set.enumerate_mem", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}]], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\n\u22a2 Nat.zero + Nat.succ m = Nat.zero", "state_after": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\nthis : a \u2208 s \\ {a}\n\u22a2 Nat.zero + Nat.succ m = Nat.zero"}, {"tactic": "simp_all [Set.mem_diff_singleton]", "annotated_tactic": ["simp_all [<a>Set.mem_diff_singleton</a>]", [{"full_name": "Set.mem_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2101, 9], "def_end_pos": [2101, 27]}]], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\nthis : a \u2208 s \\ {a}\n\u22a2 Nat.zero + Nat.succ m = Nat.zero", "state_after": "no goals"}, {"tactic": "simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]", "annotated_tactic": ["simp_all only [<a>enumerate</a>, <a>Nat.zero_eq</a>, <a>Nat.add_eq</a>, <a>zero_add</a>]", [{"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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.add_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [84, 17], "def_end_pos": [84, 23]}, {"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\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 enumerate sel (s \\ {a}) m = some a", "state_after": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nm : \u2115\nh\u2081 : sel s = some a\nh\u2082 :\n  (do\n      let a \u2190 some a\n      enumerate sel (s \\ {a}) m) =\n    some a\n\u22a2 enumerate sel (s \\ {a}) m = some a"}, {"tactic": "exact h\u2082", "annotated_tactic": ["exact h\u2082", []], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nm : \u2115\nh\u2081 : sel s = some a\nh\u2082 :\n  (do\n      let a \u2190 some a\n      enumerate sel (s \\ {a}) m) =\n    some a\n\u22a2 enumerate sel (s \\ {a}) m = some a", "state_after": "no goals"}, {"tactic": "cases h : sel s", "annotated_tactic": ["cases h : sel s", []], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\n\u22a2 Nat.succ k + m = Nat.succ k", "state_after": "case none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k + m = Nat.succ k\n\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k"}, {"tactic": "case none =>\n  simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]", "annotated_tactic": ["case none =>\n        simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>Nat.add_succ</a>, <a>enumerate_eq_none_of_sel</a> _ h]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"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": "Set.enumerate_eq_none_of_sel", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [41, 9], "def_end_pos": [41, 33]}]], "state_before": "case none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k + m = Nat.succ k\n\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k", "state_after": "case some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k"}, {"tactic": "case some _ =>\n  simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                 Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n  exact ih h\u2081 h\u2082", "annotated_tactic": ["case some _ =>\n        simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>enumerate</a>, Option.some.injEq,\n                       <a>Nat.add_succ</a>, enumerate._eq_2, Nat.succ.injEq]\n        exact ih h\u2081 h\u2082", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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]}]], "state_before": "case some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k", "state_after": "no goals"}, {"tactic": "simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]", "annotated_tactic": ["simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>Nat.add_succ</a>, <a>enumerate_eq_none_of_sel</a> _ h]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"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": "Set.enumerate_eq_none_of_sel", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [41, 9], "def_end_pos": [41, 33]}]], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k + m = Nat.succ k", "state_after": "no goals"}, {"tactic": "simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n               Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]", "annotated_tactic": ["simp_all only [<a>add_comm</a>, <a>self_eq_add_left</a>, <a>enumerate</a>, Option.some.injEq,\n                       <a>Nat.add_succ</a>, enumerate._eq_2, Nat.succ.injEq]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "self_eq_add_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [216, 3], "def_end_pos": [216, 14]}, {"full_name": "Set.enumerate", "def_path": "Mathlib/Data/Set/Enumerate.lean", "def_pos": [34, 5], "def_end_pos": [34, 14]}, {"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]}]], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k", "state_after": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\ns : Set \u03b1\nval\u271d : \u03b1\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m + k = k\nh\u2081 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) k) =\n    some a\nh\u2082 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) (m + k)) =\n    some a\nh : sel s = some val\u271d\n\u22a2 m + k = k"}, {"tactic": "exact ih h\u2081 h\u2082", "annotated_tactic": ["exact ih h\u2081 h\u2082", []], "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\ns : Set \u03b1\nval\u271d : \u03b1\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m + k = k\nh\u2081 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) k) =\n    some a\nh\u2082 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) (m + k)) =\n    some a\nh : sel s = some val\u271d\n\u22a2 m + k = 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_comp_add_mul", "start": [763, 1], "end": [765, 63], "traced_tactics": [{"tactic": "rw [\u2190 integral_comp_add_left, \u2190 integral_comp_mul_left _ hc]", "annotated_tactic": ["rw [\u2190 <a>integral_comp_add_left</a>, \u2190 <a>integral_comp_mul_left</a> _ hc]", [{"full_name": "intervalIntegral.integral_comp_add_left", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [745, 16], "def_end_pos": [745, 38]}, {"full_name": "intervalIntegral.integral_comp_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [713, 9], "def_end_pos": [713, 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\na b c d\u271d : \u211d\nf : \u211d \u2192 E\nhc : c \u2260 0\nd : \u211d\n\u22a2 \u222b (x : \u211d) in a..b, f (d + c * x) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in d + c * a..d + c * b, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/ListOfFn.lean", "full_name": "Set.mem_list_prod", "start": [36, 1], "end": [47, 27], "traced_tactics": [{"tactic": "induction' l using List.ofFnRec with n f", "annotated_tactic": ["induction' l using <a>List.ofFnRec</a> with n f", [{"full_name": "List.ofFnRec", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [246, 5], "def_end_pos": [246, 12]}]], "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\u271d : \u03b1\nm n : \u2115\nl : List (Set \u03b1)\na : \u03b1\n\u22a2 a \u2208 List.prod l \u2194 \u2203 l', List.prod (List.map (fun x => \u2191x.snd) l') = a \u2227 List.map Sigma.fst l' = l", "state_after": "case h\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 a \u2208 List.prod (List.ofFn f) \u2194\n    \u2203 l', List.prod (List.map (fun x => \u2191x.snd) l') = a \u2227 List.map Sigma.fst l' = List.ofFn f"}, {"tactic": "simp only [mem_prod_list_ofFn, List.exists_iff_exists_tuple, List.map_ofFn, Function.comp,\n  List.ofFn_inj', Sigma.mk.inj_iff, and_left_comm, exists_and_left, exists_eq_left, heq_eq_eq]", "annotated_tactic": ["simp only [<a>mem_prod_list_ofFn</a>, <a>List.exists_iff_exists_tuple</a>, <a>List.map_ofFn</a>, <a>Function.comp</a>,\n    <a>List.ofFn_inj'</a>, <a>Sigma.mk.inj_iff</a>, <a>and_left_comm</a>, <a>exists_and_left</a>, <a>exists_eq_left</a>, <a>heq_eq_eq</a>]", [{"full_name": "Set.mem_prod_list_ofFn", "def_path": "Mathlib/Data/Set/Pointwise/ListOfFn.lean", "def_pos": [26, 9], "def_end_pos": [26, 27]}, {"full_name": "List.exists_iff_exists_tuple", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [263, 9], "def_end_pos": [263, 32]}, {"full_name": "List.map_ofFn", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [84, 9], "def_end_pos": [84, 17]}, {"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": "List.ofFn_inj'", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [274, 9], "def_end_pos": [274, 18]}, {"full_name": "Sigma.mk.inj_iff", "def_path": "Mathlib/Data/Sigma/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 19]}, {"full_name": "and_left_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}, {"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_eq_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [459, 17], "def_end_pos": [459, 31]}, {"full_name": "heq_eq_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [143, 17], "def_end_pos": [143, 26]}]], "state_before": "case h\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 a \u2208 List.prod (List.ofFn f) \u2194\n    \u2203 l', List.prod (List.map (fun x => \u2191x.snd) l') = a \u2227 List.map Sigma.fst l' = List.ofFn f", "state_after": "case h\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 (\u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a) \u2194\n    \u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 (\u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a) \u2194\n    \u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f", "state_after": "case h.mp\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 (\u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a) \u2192\n    \u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f\n\ncase h.mpr\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 (\u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f) \u2192\n    \u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a"}, {"tactic": "rintro \u27e8fi, rfl\u27e9", "annotated_tactic": ["rintro \u27e8fi, rfl\u27e9", []], "state_before": "case h.mp\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 (\u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a) \u2192\n    \u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f", "state_after": "case h.mp.intro\nF : 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\nf : Fin n \u2192 Set \u03b1\nfi : (i : Fin n) \u2192 \u2191(f i)\n\u22a2 \u2203 x,\n    List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = List.prod (List.ofFn fun i => \u2191(fi i)) \u2227\n      (fun x_1 => (x x_1).fst) = f"}, {"tactic": "exact \u27e8fun i \u21a6 \u27e8_, fi i\u27e9, rfl, rfl\u27e9", "annotated_tactic": ["exact \u27e8fun i \u21a6 \u27e8_, fi i\u27e9, <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": "case h.mp.intro\nF : 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\nf : Fin n \u2192 Set \u03b1\nfi : (i : Fin n) \u2192 \u2191(f i)\n\u22a2 \u2203 x,\n    List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = List.prod (List.ofFn fun i => \u2191(fi i)) \u2227\n      (fun x_1 => (x x_1).fst) = f", "state_after": "no goals"}, {"tactic": "rintro \u27e8fi, rfl, rfl\u27e9", "annotated_tactic": ["rintro \u27e8fi, rfl, rfl\u27e9", []], "state_before": "case h.mpr\nF : 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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\nf : Fin n \u2192 Set \u03b1\n\u22a2 (\u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f) \u2192\n    \u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a", "state_after": "case h.mpr.intro.intro\nF : 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\nfi : Fin n \u2192 (s : Set \u03b1) \u00d7 \u2191s\n\u22a2 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = List.prod (List.ofFn fun x => \u2191(fi x).snd)"}, {"tactic": "exact \u27e8fun i \u21a6 _, rfl\u27e9", "annotated_tactic": ["exact \u27e8fun i \u21a6 _, <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 h.mpr.intro.intro\nF : 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\nfi : Fin n \u2192 (s : Set \u03b1) \u00d7 \u2191s\n\u22a2 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = List.prod (List.ofFn fun x => \u2191(fi x).snd)", "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_smul_sub_smul", "start": [368, 1], "end": [374, 51], "traced_tactics": [{"tactic": "rcases eq_or_ne R 0 with (rfl | hR)", "annotated_tactic": ["rcases <a>eq_or_ne</a> R 0 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": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z", "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 w : \u2102\n\u22a2 (\u222e (z : \u2102) in C(c, 0), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, 0), f z\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 w : \u2102\nR : \u211d\nhR : R \u2260 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z"}, {"tactic": "have : (circleMap c R \u207b\u00b9' {w}).Countable := (countable_singleton _).preimage_circleMap c hR", "annotated_tactic": ["have : (<a>circleMap</a> c R \u207b\u00b9' {w}).<a>Countable</a> := (<a>countable_singleton</a> _).<a>preimage_circleMap</a> c hR", [{"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}, {"full_name": "Set.Countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [31, 15], "def_end_pos": [31, 24]}, {"full_name": "Set.countable_singleton", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [131, 17], "def_end_pos": [131, 36]}, {"full_name": "Set.Countable.preimage_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [97, 9], "def_end_pos": [97, 41]}]], "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 w : \u2102\nR : \u211d\nhR : R \u2260 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z", "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 w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z"}, {"tactic": "refine' intervalIntegral.integral_congr_ae ((this.ae_not_mem _).mono fun \u03b8 h\u03b8 _' => _)", "annotated_tactic": ["refine' <a>intervalIntegral.integral_congr_ae</a> ((this.ae_not_mem _).<a>mono</a> fun \u03b8 h\u03b8 _' => _)", [{"full_name": "intervalIntegral.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [990, 9], "def_end_pos": [990, 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 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 w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z", "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 w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\nh\u03b8 : \u00ac\u03b8 \u2208 circleMap c R \u207b\u00b9' {w}\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)"}, {"tactic": "change circleMap c R \u03b8 \u2260 w at h\u03b8", "annotated_tactic": ["change <a>circleMap</a> c R \u03b8 \u2260 w at h\u03b8", [{"full_name": "circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [89, 5], "def_end_pos": [89, 14]}]], "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 w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\nh\u03b8 : \u00ac\u03b8 \u2208 circleMap c R \u207b\u00b9' {w}\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)", "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 w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nh\u03b8 : circleMap c R \u03b8 \u2260 w\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)"}, {"tactic": "simp only [inv_smul_smul\u2080 (sub_ne_zero.2 <| h\u03b8)]", "annotated_tactic": ["simp only [<a>inv_smul_smul\u2080</a> (<a>sub_ne_zero</a>.2 <| h\u03b8)]", [{"full_name": "inv_smul_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [192, 9], "def_end_pos": [192, 23]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}]], "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 w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nh\u03b8 : circleMap c R \u03b8 \u2260 w\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)", "state_after": "no goals"}, {"tactic": "simp only [integral_radius_zero]", "annotated_tactic": ["simp only [<a>integral_radius_zero</a>]", [{"full_name": "circleIntegral.integral_radius_zero", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [359, 9], "def_end_pos": [359, 29]}]], "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 w : \u2102\n\u22a2 (\u222e (z : \u2102) in C(c, 0), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, 0), f z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essSup_map_measure", "start": [252, 1], "end": [258, 18], "traced_tactics": [{"tactic": "rw [essSup_congr_ae hg.ae_eq_mk, essSup_map_measure_of_measurable hg.measurable_mk hf]", "annotated_tactic": ["rw [<a>essSup_congr_ae</a> hg.ae_eq_mk, <a>essSup_map_measure_of_measurable</a> hg.measurable_mk hf]", [{"full_name": "essSup_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [54, 9], "def_end_pos": [54, 24]}, {"full_name": "essSup_map_measure_of_measurable", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [243, 9], "def_end_pos": [243, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 essSup g (Measure.map f \u03bc) = essSup (g \u2218 f) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 essSup (AEMeasurable.mk g hg \u2218 f) \u03bc = essSup (g \u2218 f) \u03bc"}, {"tactic": "refine' essSup_congr_ae _", "annotated_tactic": ["refine' <a>essSup_congr_ae</a> _", [{"full_name": "essSup_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [54, 9], "def_end_pos": [54, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 essSup (AEMeasurable.mk g hg \u2218 f) \u03bc = essSup (g \u2218 f) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f"}, {"tactic": "have h_eq := ae_of_ae_map hf hg.ae_eq_mk", "annotated_tactic": ["have h_eq := <a>ae_of_ae_map</a> hf hg.ae_eq_mk", [{"full_name": "MeasureTheory.ae_of_ae_map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2475, 9], "def_end_pos": [2475, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\nh_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) = AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f"}, {"tactic": "rw [\u2190 EventuallyEq] at h_eq", "annotated_tactic": ["rw [\u2190 <a>EventuallyEq</a>] at h_eq", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\nh_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) = AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\nh_eq : (fun x => g (f x)) =\u1d50[\u03bc] fun x => AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f"}, {"tactic": "exact h_eq.symm", "annotated_tactic": ["exact h_eq.symm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\nh_eq : (fun x => g (f x)) =\u1d50[\u03bc] fun x => AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_singleton", "start": [343, 1], "end": [344, 25], "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_subset", "start": [1952, 1], "end": [1958, 47], "traced_tactics": [{"tactic": "refine' ext_of_generateFrom_of_cover h_gen hc h_inter hU htop _ fun t ht => h_eq t (h_sub ht)", "annotated_tactic": ["refine' <a>ext_of_generateFrom_of_cover</a> h_gen hc h_inter hU htop _ fun t ht => h_eq t (h_sub ht)", [{"full_name": "MeasureTheory.Measure.ext_of_generateFrom_of_cover", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 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\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\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\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 : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \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)"}, {"tactic": "intro t ht s hs", "annotated_tactic": ["intro t ht s hs", []], "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\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \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)", "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\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)"}, {"tactic": "cases' (s \u2229 t).eq_empty_or_nonempty with H H", "annotated_tactic": ["cases' (s \u2229 t).<a>eq_empty_or_nonempty</a> 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\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 T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)", "state_after": "case 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\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\nH : s \u2229 t = \u2205\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\n\ncase 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\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\nH : Set.Nonempty (s \u2229 t)\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)"}, {"tactic": "simp only [H, measure_empty]", "annotated_tactic": ["simp only [H, <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 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\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\nH : s \u2229 t = \u2205\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)", "state_after": "no goals"}, {"tactic": "exact h_eq _ (h_inter _ hs _ (h_sub ht) H)", "annotated_tactic": ["exact h_eq _ (h_inter _ hs _ (h_sub ht) H)", []], "state_before": "case 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\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\nH : Set.Nonempty (s \u2229 t)\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 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": "Substring.ValidFor.contains", "start": [950, 1], "end": [951, 65], "traced_tactics": [{"tactic": "simp [Substring.contains, h.any, String.contains]", "annotated_tactic": ["simp [<a>Substring.contains</a>, h.any, <a>String.contains</a>]", [{"full_name": "Substring.contains", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [633, 5], "def_end_pos": [633, 13]}, {"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]}]], "state_before": "l m r : List Char\nc : Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 Substring.contains x\u271d c = true \u2194 c \u2208 m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.inclusionBool\u0393'_injective", "start": [91, 1], "end": [92, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm'_le_snorm'_mul_rpow_measure_univ", "start": [1042, 1], "end": [1063, 22], "traced_tactics": [{"tactic": "have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq", "annotated_tactic": ["have hq0_lt : 0 < q := <a>lt_of_lt_of_le</a> hp0_lt hpq", [{"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": "\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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 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\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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "by_cases hpq_eq : p = q", "annotated_tactic": ["by_cases hpq_eq : p = q", []], "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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : p = q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "have hpq : p < q := lt_of_le_of_ne hpq hpq_eq", "annotated_tactic": ["have hpq : p < q := <a>lt_of_le_of_ne</a> hpq hpq_eq", [{"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\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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "let g := fun _ : \u03b1 => (1 : \u211d\u22650\u221e)", "annotated_tactic": ["let g := fun _ : \u03b1 => (1 : \u211d\u22650\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "have h_rw : (\u222b\u207b a, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, ((\u2016f a\u2016\u208a : \u211d\u22650\u221e) * g a) ^ p \u2202\u03bc :=\n  lintegral_congr fun a => by simp", "annotated_tactic": ["have h_rw : (\u222b\u207b a, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, ((\u2016f a\u2016\u208a : \u211d\u22650\u221e) * g a) ^ p \u2202\u03bc :=\n    <a>lintegral_congr</a> fun a => by simp", [{"full_name": "MeasureTheory.lintegral_congr", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [308, 9], "def_end_pos": [308, 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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "repeat' rw [snorm']", "annotated_tactic": ["repeat' rw [snorm']", []], "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "rw [h_rw]", "annotated_tactic": ["rw [h_rw]", []], "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "let r := p * q / (q - p)", "annotated_tactic": ["let r := p * q / (q - p)", []], "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "have hpqr : 1 / p = 1 / q + 1 / r := by\n  field_simp [(ne_of_lt hp0_lt).symm, (ne_of_lt hq0_lt).symm]\n  ring", "annotated_tactic": ["have hpqr : 1 / p = 1 / q + 1 / r := by\n    field_simp [(<a>ne_of_lt</a> hp0_lt).<a>symm</a>, (<a>ne_of_lt</a> hq0_lt).<a>symm</a>]\n    ring", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"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": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 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": "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "calc\n  (\u222b\u207b a : \u03b1, (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n      (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b a : \u03b1, g a ^ r \u2202\u03bc) ^ (1 / r) :=\n    ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr \u03bc hf.ennnorm aemeasurable_const\n  _ = (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u03bc Set.univ ^ (1 / p - 1 / q) := by\n    rw [hpqr]; simp", "annotated_tactic": ["calc\n    (\u222b\u207b a : \u03b1, (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n        (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b a : \u03b1, g a ^ r \u2202\u03bc) ^ (1 / r) :=\n      <a>ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr</a> hp0_lt hpq hpqr \u03bc hf.ennnorm <a>aemeasurable_const</a>\n    _ = (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u03bc <a>Set.univ</a> ^ (1 / p - 1 / q) := by\n      rw [hpqr]; simp", [{"full_name": "ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [216, 9], "def_end_pos": [216, 38]}, {"full_name": "aemeasurable_const", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [746, 9], "def_end_pos": [746, 27]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)", "state_after": "no goals"}, {"tactic": "rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]", "annotated_tactic": ["rw [hpq_eq, <a>sub_self</a>, <a>ENNReal.rpow_zero</a>, <a>mul_one</a>]", [{"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": "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\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 : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : p = q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)", "state_after": "no goals"}, {"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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\na : \u03b1\n\u22a2 \u2191\u2016f a\u2016\u208a ^ p = (\u2191\u2016f a\u2016\u208a * g a) ^ p", "state_after": "no goals"}, {"tactic": "rw [snorm']", "annotated_tactic": ["rw [snorm']", []], "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)"}, {"tactic": "field_simp [(ne_of_lt hp0_lt).symm, (ne_of_lt hq0_lt).symm]", "annotated_tactic": ["field_simp [(<a>ne_of_lt</a> hp0_lt).<a>symm</a>, (<a>ne_of_lt</a> hq0_lt).<a>symm</a>]", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"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": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 1 / p = 1 / q + 1 / r", "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 p * q = q * p"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 p * q = q * p", "state_after": "no goals"}, {"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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / q + 1 / r - 1 / q)"}, {"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\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 : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / q + 1 / r - 1 / q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_of_pair_subset", "start": [2493, 1], "end": [2494, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.NullMeasurableSet.image", "start": [1372, 1], "end": [1381, 53], "traced_tactics": [{"tactic": "refine' \u27e8toMeasurable \u03bc (f '' toMeasurable (\u03bc.comap f) s), measurableSet_toMeasurable _ _, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>toMeasurable</a> \u03bc (f '' <a>toMeasurable</a> (\u03bc.comap f) s), <a>measurableSet_toMeasurable</a> _ _, _\u27e9", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}, {"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' s)", "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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] toMeasurable \u03bc (f '' toMeasurable (comap f \u03bc) s)"}, {"tactic": "refine' EventuallyEq.trans _ (NullMeasurableSet.toMeasurable_ae_eq _).symm", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>NullMeasurableSet.toMeasurable_ae_eq</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.NullMeasurableSet.toMeasurable_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [236, 9], "def_end_pos": [236, 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\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] toMeasurable \u03bc (f '' toMeasurable (comap f \u03bc) s)", "state_after": "case refine'_1\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s\n\ncase refine'_2\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine'_1\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s\n\ncase refine'_2\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s)", "state_after": "case refine'_2\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s)\n\ncase refine'_1\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s"}, {"tactic": "have h : toMeasurable (comap f \u03bc) s =\u1d50[comap f \u03bc] s :=\n  @NullMeasurableSet.toMeasurable_ae_eq _ _ (\u03bc.comap f : Measure \u03b1) s hs", "annotated_tactic": ["have h : <a>toMeasurable</a> (<a>comap</a> f \u03bc) s =\u1d50[<a>comap</a> f \u03bc] s :=\n    @<a>NullMeasurableSet.toMeasurable_ae_eq</a> _ _ (\u03bc.comap f : <a>Measure</a> \u03b1) s hs", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}, {"full_name": "MeasureTheory.Measure.comap", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1313, 5], "def_end_pos": [1313, 10]}, {"full_name": "MeasureTheory.Measure.comap", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1313, 5], "def_end_pos": [1313, 10]}, {"full_name": "MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [236, 9], "def_end_pos": [236, 27]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}]], "state_before": "case refine'_1\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s", "state_after": "case refine'_1\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\nh : toMeasurable (comap f \u03bc) s =\u1da0[ae (comap f \u03bc)] s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s"}, {"tactic": "exact ae_eq_image_of_ae_eq_comap f \u03bc hfi hf h.symm", "annotated_tactic": ["exact <a>ae_eq_image_of_ae_eq_comap</a> f \u03bc hfi hf h.symm", [{"full_name": "MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1354, 9], "def_end_pos": [1354, 35]}]], "state_before": "case refine'_1\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\nh : toMeasurable (comap f \u03bc) s =\u1da0[ae (comap f \u03bc)] s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s", "state_after": "no goals"}, {"tactic": "exact hf _ (measurableSet_toMeasurable _ _)", "annotated_tactic": ["exact hf _ (<a>measurableSet_toMeasurable</a> _ _)", [{"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case refine'_2\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 : 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 : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' toMeasurable (comap f \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.smul_finset_subset_iff\u2080", "start": [2080, 1], "end": [2081, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.join_pmap_eq_pmap_join", "start": [269, 1], "end": [271, 37], "traced_tactics": [{"tactic": "rcases x with (_ | _ | x) <;> simp", "annotated_tactic": ["rcases x with (_ | _ | x) <;> simp", []], "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 : (a : \u03b1) \u2192 p a \u2192 \u03b2\nx : Option (Option \u03b1)\nH : \u2200 (a : Option \u03b1), a \u2208 x \u2192 \u2200 (a_2 : \u03b1), a_2 \u2208 a \u2192 p a_2\n\u22a2 join (pmap (pmap f) x H) = pmap f (join x) (_ : \u2200 (a : \u03b1), a \u2208 join x \u2192 p a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2to1.addBottom_nth_snd", "start": [2386, 1], "end": [2388, 57], "traced_tactics": [{"tactic": "conv => rhs; rw [\u2190 addBottom_map L, ListBlank.nth_map]", "annotated_tactic": ["conv => rhs; rw [\u2190 <a>addBottom_map</a> L, <a>ListBlank.nth_map</a>]", [{"full_name": "Turing.TM2to1.addBottom_map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2370, 9], "def_end_pos": [2370, 22]}, {"full_name": "Turing.ListBlank.nth_map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [416, 9], "def_end_pos": [416, 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\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 (ListBlank.nth (addBottom L) n).2 = ListBlank.nth L n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.ae_eq_trim_iff_of_aeStronglyMeasurable'", "start": [144, 1], "end": [150, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff", "start": [3016, 1], "end": [3019, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.prod_apply", "start": [307, 1], "end": [310, 42], "traced_tactics": [{"tactic": "simp_rw [Measure.prod, bind_apply hs (Measurable.map_prod_mk_left (\u03bd := \u03bd)),\n  map_apply measurable_prod_mk_left hs]", "annotated_tactic": ["simp_rw [<a>Measure.prod</a>, <a>bind_apply</a> hs (<a>Measurable.map_prod_mk_left</a> (\u03bd := \u03bd)),\n    <a>map_apply</a> <a>measurable_prod_mk_left</a> hs]", [{"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [292, 27], "def_end_pos": [292, 31]}, {"full_name": "MeasureTheory.Measure.bind_apply", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [174, 9], "def_end_pos": [174, 19]}, {"full_name": "Measurable.map_prod_mk_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 36]}, {"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_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 \u2191\u2191(Measure.prod \u03bc \u03bd) s = \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.ofReal_condCdf_ae_eq", "start": [846, 1], "end": [850, 55], "traced_tactics": [{"tactic": "filter_upwards [condCdf_ae_eq \u03c1 r, preCdf_le_one \u03c1] with a ha ha_le_one", "annotated_tactic": ["filter_upwards [<a>condCdf_ae_eq</a> \u03c1 r, <a>preCdf_le_one</a> \u03c1] with a ha ha_le_one", [{"full_name": "ProbabilityTheory.condCdf_ae_eq", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [840, 9], "def_end_pos": [840, 22]}, {"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 (\u2191(condCdf \u03c1 a) \u2191r)) =\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 : \u2191(condCdf \u03c1 a) \u2191r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = 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 : \u2191(condCdf \u03c1 a) \u2191r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = 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 : \u2191(condCdf \u03c1 a) \u2191r = 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 : \u2191(condCdf \u03c1 a) \u2191r = 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/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.eval_neg", "start": [146, 1], "end": [147, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_aux5", "start": [576, 1], "end": [598, 23], "traced_tactics": [{"tactic": "have A : (\u2211' j : \u2115, \u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi (j : \u211d)}) < \u221e := by\n  convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2\n  ext1 j\n  exact (hident j).measure_mem_eq measurableSet_Ioi", "annotated_tactic": ["have A : (\u2211' j : \u2115, \u2119 {\u03c9 | X j \u03c9 \u2208 <a>Set.Ioi</a> (j : \u211d)}) < \u221e := by\n    convert <a>tsum_prob_mem_Ioi_lt_top</a> hint (hnonneg 0) using 2\n    ext1 j\n    exact (hident j).<a>measure_mem_eq</a> <a>measurableSet_Ioi</a>", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ProbabilityTheory.tsum_prob_mem_Ioi_lt_top", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}, {"full_name": "ProbabilityTheory.IdentDistrib.measure_mem_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [132, 9], "def_end_pos": [132, 23]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 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\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n", "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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n"}, {"tactic": "filter_upwards [B] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [B] with \u03c9 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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n", "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n"}, {"tactic": "convert isLittleO_sum_range_of_tendsto_zero h\u03c9 using 1", "annotated_tactic": ["convert <a>isLittleO_sum_range_of_tendsto_zero</a> h\u03c9 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]}]], "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n", "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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) = fun n =>\n    \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9)"}, {"tactic": "ext n", "annotated_tactic": ["ext 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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) = fun n =>\n    \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9)", "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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9 = \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9)"}, {"tactic": "rw [sum_sub_distrib]", "annotated_tactic": ["rw [<a>sum_sub_distrib</a>]", [{"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.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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9 = \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9)", "state_after": "no goals"}, {"tactic": "convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2", "annotated_tactic": ["convert <a>tsum_prob_mem_Ioi_lt_top</a> hint (hnonneg 0) using 2", [{"full_name": "ProbabilityTheory.tsum_prob_mem_Ioi_lt_top", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}]], "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 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4", "state_after": "case h.e'_3.h.e'_5\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 j => \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j}) = fun j => \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j}"}, {"tactic": "ext1 j", "annotated_tactic": ["ext1 j", []], "state_before": "case h.e'_3.h.e'_5\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 j => \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j}) = fun j => \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j}", "state_after": "case h.e'_3.h.e'_5.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\nj : \u2115\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} = \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j}"}, {"tactic": "exact (hident j).measure_mem_eq measurableSet_Ioi", "annotated_tactic": ["exact (hident j).<a>measure_mem_eq</a> <a>measurableSet_Ioi</a>", [{"full_name": "ProbabilityTheory.IdentDistrib.measure_mem_eq", "def_path": "Mathlib/Probability/IdentDistrib.lean", "def_pos": [132, 9], "def_end_pos": [132, 23]}, {"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.e'_3.h.e'_5.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\nj : \u2115\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} = \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j}", "state_after": "no goals"}, {"tactic": "filter_upwards [ae_eventually_not_mem A.ne] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [<a>ae_eventually_not_mem</a> A.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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)", "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\n\u22a2 Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)"}, {"tactic": "apply tendsto_const_nhds.congr' _", "annotated_tactic": ["apply tendsto_const_nhds.congr' _", []], "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\n\u22a2 Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)", "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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\n\u22a2 (fun x => 0) =\u1da0[atTop] fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9"}, {"tactic": "filter_upwards [h\u03c9, Ioi_mem_atTop 0] with n hn npos", "annotated_tactic": ["filter_upwards [h\u03c9, <a>Ioi_mem_atTop</a> 0] with n hn npos", [{"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}]], "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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\n\u22a2 (fun x => 0) =\u1da0[atTop] fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9", "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 0 = truncation (X n) (\u2191n) \u03c9 - X n \u03c9"}, {"tactic": "simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]", "annotated_tactic": ["simp only [<a>truncation</a>, <a>indicator</a>, <a>Set.mem_Ioc</a>, <a>id.def</a>, <a>Function.comp_apply</a>]", [{"full_name": "ProbabilityTheory.truncation", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [78, 5], "def_end_pos": [78, 15]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"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]}]], "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 0 = truncation (X n) (\u2191n) \u03c9 - X n \u03c9", "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 0 = (if -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n then X n \u03c9 else 0) - X n \u03c9"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 0 = (if -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n then X n \u03c9 else 0) - X n \u03c9", "state_after": "case pos\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n\n\u22a2 0 = X n \u03c9 - X n \u03c9\n\ncase neg\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 0 = 0 - X n \u03c9"}, {"tactic": "exact (sub_self _).symm", "annotated_tactic": ["exact (<a>sub_self</a> _).<a>symm</a>", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"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\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n\n\u22a2 0 = X n \u03c9 - X n \u03c9", "state_after": "no goals"}, {"tactic": "have : -(n : \u211d) < X n \u03c9 := by\n  apply lt_of_lt_of_le _ (hnonneg n \u03c9)\n  simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos", "annotated_tactic": ["have : -(n : \u211d) < X n \u03c9 := by\n        apply <a>lt_of_lt_of_le</a> _ (hnonneg n \u03c9)\n        simpa only [<a>Right.neg_neg_iff</a>, <a>Nat.cast_pos</a>] using npos", [{"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": "Right.neg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [279, 3], "def_end_pos": [279, 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 neg\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 0 = 0 - X n \u03c9", "state_after": "case neg\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\nthis : -\u2191n < X n \u03c9\n\u22a2 0 = 0 - X n \u03c9"}, {"tactic": "simp only [this, true_and_iff, not_le] at h", "annotated_tactic": ["simp only [this, <a>true_and_iff</a>, <a>not_le</a>] at h", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"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\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\nthis : -\u2191n < X n \u03c9\n\u22a2 0 = 0 - X n \u03c9", "state_after": "case neg\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nthis : -\u2191n < X n \u03c9\nh : \u2191n < X n \u03c9\n\u22a2 0 = 0 - X n \u03c9"}, {"tactic": "exact (hn h).elim", "annotated_tactic": ["exact (hn 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\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nthis : -\u2191n < X n \u03c9\nh : \u2191n < X n \u03c9\n\u22a2 0 = 0 - X n \u03c9", "state_after": "no goals"}, {"tactic": "apply lt_of_lt_of_le _ (hnonneg n \u03c9)", "annotated_tactic": ["apply <a>lt_of_lt_of_le</a> _ (hnonneg n \u03c9)", [{"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": "\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 -\u2191n < X n \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\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 -\u2191n < 0"}, {"tactic": "simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos", "annotated_tactic": ["simpa only [<a>Right.neg_neg_iff</a>, <a>Nat.cast_pos</a>] using npos", [{"full_name": "Right.neg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [279, 3], "def_end_pos": [279, 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": "\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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 -\u2191n < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.measure_iUnionNotConvergentSeq", "start": [147, 1], "end": [158, 33], "traced_tactics": [{"tactic": "refine' le_trans (measure_iUnion_le _) (le_trans\n  (ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos h\u03b5) hf hg hsm hs hfg) _)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>measure_iUnion_le</a> _) (<a>le_trans</a>\n    (<a>ENNReal.tsum_le_tsum</a> <| <a>notConvergentSeqLTIndex_spec</a> (<a>half_pos</a> h\u03b5) hf hg hsm hs hfg) _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [240, 9], "def_end_pos": [240, 26]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"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.Egorov.notConvergentSeqLTIndex_spec", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [122, 9], "def_end_pos": [122, 37]}, {"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\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \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))\n\u22a2 \u2191\u2191\u03bc (iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg) \u2264 ENNReal.ofReal \u03b5", "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 : \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))\n\u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2 * 2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "simp_rw [ENNReal.ofReal_mul (half_pos h\u03b5).le]", "annotated_tactic": ["simp_rw [<a>ENNReal.ofReal_mul</a> (<a>half_pos</a> h\u03b5).<a>le</a>]", [{"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 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]}]], "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 : \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))\n\u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2 * 2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5", "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 : \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))\n\u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2) * ENNReal.ofReal (2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "rw [ENNReal.tsum_mul_left, \u2190 ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,\n  \u2190 ENNReal.ofReal_mul (half_pos h\u03b5).le, div_mul_cancel \u03b5 two_ne_zero]", "annotated_tactic": ["rw [<a>ENNReal.tsum_mul_left</a>, \u2190 <a>ENNReal.ofReal_tsum_of_nonneg</a>, <a>inv_eq_one_div</a>, <a>tsum_geometric_two</a>,\n    \u2190 <a>ENNReal.ofReal_mul</a> (<a>half_pos</a> h\u03b5).<a>le</a>, <a>div_mul_cancel</a> \u03b5 <a>two_ne_zero</a>]", [{"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [897, 19], "def_end_pos": [897, 32]}, {"full_name": "ENNReal.ofReal_tsum_of_nonneg", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 38]}, {"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 23]}, {"full_name": "tsum_geometric_two", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [237, 9], "def_end_pos": [237, 27]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2225, 9], "def_end_pos": [2225, 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": "div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}]], "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 : \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))\n\u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2) * ENNReal.ofReal (2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5", "state_after": "case hf_nonneg\n\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 : \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))\n\u22a2 \u2200 (n : \u2115), 0 \u2264 2\u207b\u00b9 ^ n\n\ncase hf\n\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 : \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))\n\u22a2 Summable fun i => 2\u207b\u00b9 ^ i"}, {"tactic": "exact fun n => pow_nonneg (by norm_num) _", "annotated_tactic": ["exact fun n => <a>pow_nonneg</a> (by norm_num) _", [{"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "case hf_nonneg\n\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 : \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))\n\u22a2 \u2200 (n : \u2115), 0 \u2264 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 \u2264 2\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rw [inv_eq_one_div]", "annotated_tactic": ["rw [<a>inv_eq_one_div</a>]", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 23]}]], "state_before": "case hf\n\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 : \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))\n\u22a2 Summable fun i => 2\u207b\u00b9 ^ i", "state_after": "case hf\n\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 : \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))\n\u22a2 Summable fun i => (1 / 2) ^ i"}, {"tactic": "exact summable_geometric_two", "annotated_tactic": ["exact <a>summable_geometric_two</a>", [{"full_name": "summable_geometric_two", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [228, 9], "def_end_pos": [228, 31]}]], "state_before": "case hf\n\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 : \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))\n\u22a2 Summable fun i => (1 / 2) ^ i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.cond_cond_eq_cond_inter", "start": [135, 1], "end": [137, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.dvd_of_mul_dvd_mul_right", "start": [230, 1], "end": [231, 83], "traced_tactics": [{"tactic": "rw [mul_comm i k, mul_comm j k] at H", "annotated_tactic": ["rw [<a>mul_comm</a> i k, <a>mul_comm</a> j k] at H", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "i j k : \u2124\nk_non_zero : k \u2260 0\nH : i * k \u2223 j * k\n\u22a2 i \u2223 j", "state_after": "i j k : \u2124\nk_non_zero : k \u2260 0\nH : k * i \u2223 k * j\n\u22a2 i \u2223 j"}, {"tactic": "exact dvd_of_mul_dvd_mul_left k_non_zero H", "annotated_tactic": ["exact <a>dvd_of_mul_dvd_mul_left</a> k_non_zero H", [{"full_name": "Int.dvd_of_mul_dvd_mul_left", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [226, 9], "def_end_pos": [226, 32]}]], "state_before": "i j k : \u2124\nk_non_zero : k \u2260 0\nH : k * i \u2223 k * j\n\u22a2 i \u2223 j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepSet.iIndepFun_indicator", "start": [634, 1], "end": [637, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero", "start": [81, 1], "end": [94, 96], "traced_tactics": [{"tactic": "cases' isEmpty_or_nonempty \u03b9 with h h", "annotated_tactic": ["cases' <a>isEmpty_or_nonempty</a> \u03b9 with h h", [{"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\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : Nonempty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]", "annotated_tactic": ["rw [\u2190 <a>measure_inter_notConvergentSeq_eq_zero</a> hfg n, <a>Set.inter_iInter</a>]", [{"full_name": "MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [59, 9], "def_end_pos": [59, 47]}, {"full_name": "Set.inter_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [661, 9], "def_end_pos": [661, 21]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : Nonempty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : Nonempty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop\n    (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 i, s \u2229 notConvergentSeq (fun n x => f n x) (fun x => g x) n i)))"}, {"tactic": "refine' tendsto_measure_iInter (fun n => hsm.inter <| notConvergentSeq_measurableSet hf hg)\n  (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)\n  \u27e8h.some,\n    (lt_of_le_of_lt (measure_mono <| Set.inter_subset_left _ _) (lt_top_iff_ne_top.2 hs)).ne\u27e9", "annotated_tactic": ["refine' <a>tendsto_measure_iInter</a> (fun n => hsm.inter <| <a>notConvergentSeq_measurableSet</a> hf hg)\n    (fun k l hkl => <a>Set.inter_subset_inter_right</a> _ <| <a>notConvergentSeq_antitone</a> hkl)\n    \u27e8h.some,\n      (<a>lt_of_le_of_lt</a> (<a>measure_mono</a> <| <a>Set.inter_subset_left</a> _ _) (<a>lt_top_iff_ne_top</a>.2 hs)).<a>ne</a>\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": "MeasureTheory.Egorov.notConvergentSeq_measurableSet", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [73, 9], "def_end_pos": [73, 39]}, {"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.Egorov.notConvergentSeq_antitone", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [55, 9], "def_end_pos": [55, 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": "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": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : Nonempty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop\n    (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 i, s \u2229 notConvergentSeq (fun n x => f n x) (fun x => g x) n i)))", "state_after": "no goals"}, {"tactic": "have : (fun j => \u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0 := by\n  simp only [eq_iff_true_of_subsingleton]", "annotated_tactic": ["have : (fun j => \u03bc (s \u2229 <a>notConvergentSeq</a> f g n j)) = fun j => 0 := by\n      simp only [<a>eq_iff_true_of_subsingleton</a>]", [{"full_name": "MeasureTheory.Egorov.notConvergentSeq", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [43, 5], "def_end_pos": [43, 21]}, {"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 inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\nthis : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\nthis : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\nthis : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n\u22a2 Tendsto (fun j => 0) atTop (\ud835\udcdd 0)"}, {"tactic": "exact tendsto_const_nhds", "annotated_tactic": ["exact <a>tendsto_const_nhds</a>", [{"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\nthis : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n\u22a2 Tendsto (fun j => 0) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simp only [eq_iff_true_of_subsingleton]", "annotated_tactic": ["simp only [<a>eq_iff_true_of_subsingleton</a>]", [{"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\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\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\nh : IsEmpty \u03b9\n\u22a2 (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 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_smul_measure", "start": [587, 1], "end": [588, 100], "traced_tactics": [{"tactic": "simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul]", "annotated_tactic": ["simp only [<a>lintegral</a>, <a>iSup_subtype'</a>, <a>SimpleFunc.lintegral_smul</a>, <a>ENNReal.mul_iSup</a>, <a>smul_eq_mul</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_smul", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 23]}, {"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}, {"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\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202c \u2022 \u03bc = c * \u222b\u207b (a : \u03b1), f a \u2202\u03bc", "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": [1030, 1], "end": [1032, 56], "traced_tactics": [{"tactic": "refine' hs.cons_induction _ _ <;> intros <;> simp [*]", "annotated_tactic": ["refine' hs.cons_induction _ _ <;> intros <;> simp [*]", []], "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\ns\u271d : Finset \u03b2\nH : Finset.Nonempty s\u271d\nf\u271d : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : InfHomClass F \u03b1 \u03b2\nf : F\ns : Finset \u03b9\nhs : Finset.Nonempty s\ng : \u03b9 \u2192 \u03b1\n\u22a2 \u2191f (inf' s hs g) = inf' s hs (\u2191f \u2218 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.comp_assoc", "start": [620, 1], "end": [621, 59], "traced_tactics": [{"tactic": "simp only [comp_apply, Part.bind_comp]", "annotated_tactic": ["simp only [<a>comp_apply</a>, <a>Part.bind_comp</a>]", [{"full_name": "PFun.comp_apply", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [579, 9], "def_end_pos": [579, 19]}, {"full_name": "PFun.Part.bind_comp", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [611, 9], "def_end_pos": [611, 23]}]], "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 : \u03b1 \u2192. \u03b2\nf : \u03b3 \u2192. \u03b4\ng : \u03b2 \u2192. \u03b3\nh : \u03b1 \u2192. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 comp (comp f g) h x\u271d\u00b9 \u2194 x\u271d \u2208 comp f (comp g h) x\u271d\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.ListBlank.cons_bind", "start": [485, 1], "end": [490, 88], "traced_tactics": [{"tactic": "refine' l.inductionOn fun l \u21a6 _", "annotated_tactic": ["refine' l.inductionOn fun l \u21a6 _", []], "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\n\u22a2 bind (cons a l) f hf = append (f a) (bind l f hf)", "state_after": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\n\u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf)"}, {"tactic": "suffices ((mk l).cons a).bind f hf = ((mk l).bind f hf).append (f a) by exact this", "annotated_tactic": ["suffices ((<a>mk</a> l).<a>cons</a> a).<a>bind</a> f hf = ((<a>mk</a> l).<a>bind</a> f hf).<a>append</a> (f a) by exact this", [{"full_name": "Turing.ListBlank.mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [198, 5], "def_end_pos": [198, 17]}, {"full_name": "Turing.ListBlank.cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [240, 5], "def_end_pos": [240, 19]}, {"full_name": "Turing.ListBlank.bind", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [469, 5], "def_end_pos": [469, 19]}, {"full_name": "Turing.ListBlank.mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [198, 5], "def_end_pos": [198, 17]}, {"full_name": "Turing.ListBlank.bind", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [469, 5], "def_end_pos": [469, 19]}, {"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}]], "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\n\u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf)", "state_after": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\n\u22a2 bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf)"}, {"tactic": "simp only [ListBlank.append_mk, ListBlank.bind_mk, ListBlank.cons_mk, List.cons_bind]", "annotated_tactic": ["simp only [<a>ListBlank.append_mk</a>, <a>ListBlank.bind_mk</a>, <a>ListBlank.cons_mk</a>, <a>List.cons_bind</a>]", [{"full_name": "Turing.ListBlank.append_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [453, 9], "def_end_pos": [453, 28]}, {"full_name": "Turing.ListBlank.bind_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [479, 9], "def_end_pos": [479, 26]}, {"full_name": "Turing.ListBlank.cons_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [247, 9], "def_end_pos": [247, 26]}, {"full_name": "List.cons_bind", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [103, 17], "def_end_pos": [103, 26]}]], "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\n\u22a2 bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf)", "state_after": "no goals"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\nthis : bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf)\n\u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.fundamentalInterior", "start": [692, 11], "end": [707, 96], "traced_tactics": [{"tactic": "simp_rw [ae_iff, not_exists, \u2190 mem_inv_smul_set_iff, setOf_forall, \u2190 compl_setOf,\n  setOf_mem_eq, \u2190 compl_iUnion]", "annotated_tactic": ["simp_rw [<a>ae_iff</a>, <a>not_exists</a>, \u2190 <a>mem_inv_smul_set_iff</a>, <a>setOf_forall</a>, \u2190 <a>compl_setOf</a>,\n      <a>setOf_mem_eq</a>, \u2190 <a>compl_iUnion</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"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]}, {"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}, {"full_name": "Set.compl_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1649, 9], "def_end_pos": [1649, 20]}, {"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.compl_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [604, 9], "def_end_pos": [604, 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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 g, g \u2022 x \u2208 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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 0"}, {"tactic": "have :\n  ((\u22c3 g : G, g\u207b\u00b9 \u2022 s) \\ \u22c3 g : G, g\u207b\u00b9 \u2022 fundamentalFrontier G s) \u2286\n    \u22c3 g : G, g\u207b\u00b9 \u2022 fundamentalInterior G s := by\n  simp_rw [diff_subset_iff, \u2190 iUnion_union_distrib, \u2190 smul_set_union (\u03b1 := G) (\u03b2 := \u03b1),\n    fundamentalFrontier_union_fundamentalInterior]; rfl", "annotated_tactic": ["have :\n      ((\u22c3 g : G, g\u207b\u00b9 \u2022 s) \\ \u22c3 g : G, g\u207b\u00b9 \u2022 <a>fundamentalFrontier</a> G s) \u2286\n        \u22c3 g : G, g\u207b\u00b9 \u2022 <a>fundamentalInterior</a> G s := by\n      simp_rw [<a>diff_subset_iff</a>, \u2190 <a>iUnion_union_distrib</a>, \u2190 <a>smul_set_union</a> (\u03b1 := G) (\u03b2 := \u03b1),\n        <a>fundamentalFrontier_union_fundamentalInterior</a>]; rfl", [{"full_name": "MeasureTheory.fundamentalFrontier", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [544, 5], "def_end_pos": [544, 24]}, {"full_name": "MeasureTheory.fundamentalInterior", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [552, 5], "def_end_pos": [552, 24]}, {"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_union_distrib", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [643, 9], "def_end_pos": [643, 29]}, {"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": "MeasureTheory.fundamentalFrontier_union_fundamentalInterior", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [602, 9], "def_end_pos": [602, 54]}]], "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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 0"}, {"tactic": "refine' eq_bot_mono (\u03bc.mono <| compl_subset_compl.2 this) _", "annotated_tactic": ["refine' <a>eq_bot_mono</a> (\u03bc.mono <| <a>compl_subset_compl</a>.2 this) _", [{"full_name": "eq_bot_mono", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [367, 9], "def_end_pos": [367, 20]}, {"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1756, 9], "def_end_pos": [1756, 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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc ((\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s)\u1d9c = \u22a5"}, {"tactic": "simp only [iUnion_inv_smul, compl_sdiff, ENNReal.bot_eq_zero, himp_eq, sup_eq_union,\n  @iUnion_smul_eq_setOf_exists _ _ _ _ s]", "annotated_tactic": ["simp only [<a>iUnion_inv_smul</a>, <a>compl_sdiff</a>, <a>ENNReal.bot_eq_zero</a>, <a>himp_eq</a>, <a>sup_eq_union</a>,\n      @<a>iUnion_smul_eq_setOf_exists</a> _ _ _ _ s]", [{"full_name": "Set.iUnion_inv_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [997, 9], "def_end_pos": [997, 24]}, {"full_name": "compl_sdiff", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [723, 9], "def_end_pos": [723, 20]}, {"full_name": "ENNReal.bot_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [682, 9], "def_end_pos": [682, 20]}, {"full_name": "himp_eq", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [577, 9], "def_end_pos": [577, 16]}, {"full_name": "Set.sup_eq_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}, {"full_name": "Set.iUnion_smul_eq_setOf_exists", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [1003, 9], "def_end_pos": [1003, 36]}]], "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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc ((\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s)\u1d9c = \u22a5", "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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc ((\u22c3 g, g \u2022 fundamentalFrontier G s) \u222a {a | \u2203 g, g \u2022 a \u2208 s}\u1d9c) = 0"}, {"tactic": "exact measure_union_null\n  (measure_iUnion_null fun _ => measure_smul_null hs.measure_fundamentalFrontier _) hs.ae_covers", "annotated_tactic": ["exact <a>measure_union_null</a>\n      (<a>measure_iUnion_null</a> fun _ => <a>measure_smul_null</a> hs.measure_fundamentalFrontier _) hs.ae_covers", [{"full_name": "MeasureTheory.measure_union_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [302, 9], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.measure_iUnion_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [270, 9], "def_end_pos": [270, 28]}, {"full_name": "MeasureTheory.measure_smul_null", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [231, 9], "def_end_pos": [231, 26]}]], "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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc ((\u22c3 g, g \u2022 fundamentalFrontier G s) \u222a {a | \u2203 g, g \u2022 a \u2208 s}\u1d9c) = 0", "state_after": "no goals"}, {"tactic": "simp_rw [diff_subset_iff, \u2190 iUnion_union_distrib, \u2190 smul_set_union (\u03b1 := G) (\u03b2 := \u03b1),\n  fundamentalFrontier_union_fundamentalInterior]", "annotated_tactic": ["simp_rw [<a>diff_subset_iff</a>, \u2190 <a>iUnion_union_distrib</a>, \u2190 <a>smul_set_union</a> (\u03b1 := G) (\u03b2 := \u03b1),\n        <a>fundamentalFrontier_union_fundamentalInterior</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_union_distrib", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [643, 9], "def_end_pos": [643, 29]}, {"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": "MeasureTheory.fundamentalFrontier_union_fundamentalInterior", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [602, 9], "def_end_pos": [602, 54]}]], "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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u22c3 g, g\u207b\u00b9 \u2022 s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u22c3 g, g\u207b\u00b9 \u2022 s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "Basis.parallelepiped_map", "start": [206, 1], "end": [215, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurableSet.exists_measurable_subset_ae_eq", "start": [245, 1], "end": [248, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.swapAt!_def", "start": [130, 1], "end": [131, 67], "traced_tactics": [{"tactic": "simp [swapAt!, h]", "annotated_tactic": ["simp [<a>swapAt!</a>, h]", [{"full_name": "Array.swapAt!", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [94, 5], "def_end_pos": [94, 12]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni : Nat\nv : \u03b1\nh : i < size a\n\u22a2 swapAt! a i v = (a[i], set a { val := i, isLt := h } v)", "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.insertNth_eq_insertNthTR", "start": [589, 10], "end": [590, 56], "traced_tactics": [{"tactic": "funext \u03b1 f n l", "annotated_tactic": ["funext \u03b1 f n l", []], "state_before": "\u22a2 @insertNth = @insertNthTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nf : Nat\nn : \u03b1\nl : List \u03b1\n\u22a2 insertNth f n l = insertNthTR f n l"}, {"tactic": "simp [insertNthTR, insertNthTR_go_eq]", "annotated_tactic": ["simp [<a>insertNthTR</a>, <a>insertNthTR_go_eq</a>]", [{"full_name": "List.insertNthTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [578, 15], "def_end_pos": [578, 26]}, {"full_name": "List.insertNthTR_go_eq", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [585, 9], "def_end_pos": [585, 26]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nf : Nat\nn : \u03b1\nl : List \u03b1\n\u22a2 insertNth f n l = insertNthTR f n l", "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_of_mem_eraseP", "start": [1122, 1], "end": [1122, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Basic.lean", "full_name": "Int.succ_neg_succ", "start": [170, 1], "end": [170, 83], "traced_tactics": [{"tactic": "rw [neg_succ, succ_pred]", "annotated_tactic": ["rw [<a>neg_succ</a>, <a>succ_pred</a>]", [{"full_name": "Int.neg_succ", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 17]}, {"full_name": "Int.succ_pred", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [164, 9], "def_end_pos": [164, 18]}]], "state_before": "a : \u2124\n\u22a2 succ (-succ a) = -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.restrict_union_add_inter'", "start": [1728, 1], "end": [1730, 84], "traced_tactics": [{"tactic": "simpa only [union_comm, inter_comm, add_comm] using restrict_union_add_inter t hs", "annotated_tactic": ["simpa only [<a>union_comm</a>, <a>inter_comm</a>, <a>add_comm</a>] using <a>restrict_union_add_inter</a> t hs", [{"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": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.Measure.restrict_union_add_inter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 restrict \u03bc (s \u222a t) + restrict \u03bc (s \u2229 t) = restrict \u03bc s + restrict \u03bc t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.exists_integral_le", "start": [583, 1], "end": [584, 95], "traced_tactics": [{"tactic": "simpa only [average_eq_integral] using exists_average_le (IsProbabilityMeasure.ne_zero \u03bc) hf", "annotated_tactic": ["simpa only [<a>average_eq_integral</a>] using <a>exists_average_le</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hf", [{"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_average_le", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [535, 9], "def_end_pos": [535, 26]}, {"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\n\u22a2 \u2203 x, \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/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.deriv_integral_left", "start": [805, 1], "end": [808, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.ret_supports", "start": [1953, 1], "end": [1965, 80], "traced_tactics": [{"tactic": "have W := fun {q} => trStmts\u2081_self q", "annotated_tactic": ["have W := fun {q} => <a>trStmts\u2081_self</a> q", [{"full_name": "Turing.PartrecToTM2.trStmts\u2081_self", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 22]}]], "state_before": "S : Finset \u039b'\nk : Cont'\nH\u2081 : contSupp k \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret k))", "state_after": "S : Finset \u039b'\nk : Cont'\nH\u2081 : contSupp k \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret k))"}, {"tactic": "cases k", "annotated_tactic": ["cases k", []], "state_before": "S : Finset \u039b'\nk : Cont'\nH\u2081 : contSupp k \u2286 S\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret k))", "state_after": "case halt\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nH\u2081 : contSupp Cont'.halt \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret Cont'.halt))\n\ncase cons\u2081\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))\n\ncase cons\u2082\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))\n\ncase comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "case halt => trivial", "annotated_tactic": ["case halt => trivial", []], "state_before": "case halt\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nH\u2081 : contSupp Cont'.halt \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret Cont'.halt))\n\ncase cons\u2081\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))\n\ncase cons\u2082\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))\n\ncase comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "case cons\u2081\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))\n\ncase cons\u2082\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))\n\ncase comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "case cons\u2081 => rw [contSupp_cons\u2081, Finset.union_subset_iff] at H\u2081; exact fun _ => H\u2081.1 W", "annotated_tactic": ["case cons\u2081 => rw [<a>contSupp_cons\u2081</a>, <a>Finset.union_subset_iff</a>] at H\u2081; exact fun _ => H\u2081.1 W", [{"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_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 25]}]], "state_before": "case cons\u2081\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))\n\ncase cons\u2082\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))\n\ncase comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "case cons\u2082\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))\n\ncase comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "case cons\u2082 => rw [contSupp_cons\u2082, Finset.union_subset_iff] at H\u2081; exact fun _ => H\u2081.1 W", "annotated_tactic": ["case cons\u2082 => rw [<a>contSupp_cons\u2082</a>, <a>Finset.union_subset_iff</a>] at H\u2081; exact fun _ => H\u2081.1 W", [{"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": "case cons\u2082\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))\n\ncase comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "case comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "case comp => rw [contSupp_comp] at H\u2081; exact fun _ => H\u2081 (codeSupp_self _ _ W)", "annotated_tactic": ["case comp => rw [<a>contSupp_comp</a>] at H\u2081; exact fun _ => H\u2081 (<a>codeSupp_self</a> _ _ W)", [{"full_name": "Turing.PartrecToTM2.contSupp_comp", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 22]}, {"full_name": "Turing.PartrecToTM2.codeSupp_self", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1828, 9], "def_end_pos": [1828, 22]}]], "state_before": "case comp\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))\n\ncase fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "case fix\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "trivial", "annotated_tactic": ["trivial", []], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\nH\u2081 : contSupp Cont'.halt \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret Cont'.halt))", "state_after": "no goals"}, {"tactic": "rw [contSupp_cons\u2081, Finset.union_subset_iff] at H\u2081", "annotated_tactic": ["rw [<a>contSupp_cons\u2081</a>, <a>Finset.union_subset_iff</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]}, {"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'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\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 a\u271d\u00b9 (Cont'.cons\u2082 a\u271d))))) \u2286\n      S \u2227\n    codeSupp a\u271d\u00b9 (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))"}, {"tactic": "exact fun _ => H\u2081.1 W", "annotated_tactic": ["exact fun _ => H\u2081.1 W", []], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\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 a\u271d\u00b9 (Cont'.cons\u2082 a\u271d))))) \u2286\n      S \u2227\n    codeSupp a\u271d\u00b9 (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))", "state_after": "no goals"}, {"tactic": "rw [contSupp_cons\u2082, Finset.union_subset_iff] at H\u2081", "annotated_tactic": ["rw [<a>contSupp_cons\u2082</a>, <a>Finset.union_subset_iff</a>] at H\u2081", [{"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'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : trStmts\u2081 (head stack (\u039b'.ret a\u271d)) \u2286 S \u2227 contSupp a\u271d \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))"}, {"tactic": "exact fun _ => H\u2081.1 W", "annotated_tactic": ["exact fun _ => H\u2081.1 W", []], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d : Cont'\nH\u2081 : trStmts\u2081 (head stack (\u039b'.ret a\u271d)) \u2286 S \u2227 contSupp a\u271d \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))", "state_after": "no goals"}, {"tactic": "rw [contSupp_comp] at H\u2081", "annotated_tactic": ["rw [<a>contSupp_comp</a>] at H\u2081", [{"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'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 a\u271d \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))"}, {"tactic": "exact fun _ => H\u2081 (codeSupp_self _ _ W)", "annotated_tactic": ["exact fun _ => H\u2081 (<a>codeSupp_self</a> _ _ W)", [{"full_name": "Turing.PartrecToTM2.codeSupp_self", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1828, 9], "def_end_pos": [1828, 22]}]], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 a\u271d \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))", "state_after": "no goals"}, {"tactic": "rw [contSupp_fix] at H\u2081", "annotated_tactic": ["rw [<a>contSupp_fix</a>] at H\u2081", [{"full_name": "Turing.PartrecToTM2.contSupp_fix", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 21]}]], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "have L := @Finset.mem_union_left", "annotated_tactic": ["have L := @<a>Finset.mem_union_left</a>", [{"full_name": "Finset.mem_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1382, 9], "def_end_pos": [1382, 23]}]], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "have R := @Finset.mem_union_right", "annotated_tactic": ["have R := @<a>Finset.mem_union_right</a>", [{"full_name": "Finset.mem_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}]], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))"}, {"tactic": "intro s", "annotated_tactic": ["intro s", []], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\n\u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d)))", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (fun s => bif natEnd (Option.iget s) then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) s \u2208 S"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (fun s => bif natEnd (Option.iget s) then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) s \u2208 S", "state_after": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (bif natEnd (Option.iget s) then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S"}, {"tactic": "cases natEnd s.iget", "annotated_tactic": ["cases <a>natEnd</a> s.iget", [{"full_name": "Turing.PartrecToTM2.natEnd", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [992, 5], "def_end_pos": [992, 11]}]], "state_before": "S : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (bif natEnd (Option.iget s) then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S", "state_after": "case false\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (bif false then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S\n\ncase true\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (bif true then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S"}, {"tactic": "refine' H\u2081 (R _ <| L _ <| R _ <| R _ <| L _ W)", "annotated_tactic": ["refine' H\u2081 (R _ <| L _ <| R _ <| R _ <| L _ W)", []], "state_before": "case false\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (bif false then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S", "state_after": "no goals"}, {"tactic": "exact H\u2081 (R _ <| L _ <| R _ <| R _ <| R _ <| Finset.mem_singleton_self _)", "annotated_tactic": ["exact H\u2081 (R _ <| L _ <| R _ <| R _ <| R _ <| <a>Finset.mem_singleton_self</a> _)", [{"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [690, 9], "def_end_pos": [690, 27]}]], "state_before": "case true\nS : Finset \u039b'\nW : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q\na\u271d\u00b9 : Code\na\u271d : Cont'\nH\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S\nL : \u2200 {\u03b1 : Type} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t\nR : \u2200 {\u03b1 : Type} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t\ns : Option \u0393'\n\u22a2 (bif true then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S", "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.getLast?_concat", "start": [726, 9], "end": [727, 26], "traced_tactics": [{"tactic": "simp [getLast?_eq_get?]", "annotated_tactic": ["simp [<a>getLast?_eq_get?</a>]", [{"full_name": "List.getLast?_eq_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [718, 9], "def_end_pos": [718, 25]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\n\u22a2 getLast? (l ++ [a]) = some a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_trim", "start": [1869, 1], "end": [1898, 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 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm\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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm\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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm\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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"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\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm\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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm\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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm\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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "haveI : SeparableSpace (range f \u222a {0} : Set G) := hf.separableSpace_range_union_singleton", "annotated_tactic": ["haveI : <a>SeparableSpace</a> (<a>range</a> f \u222a {0} : <a>Set</a> G) := 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": "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "let f_seq := @SimpleFunc.approxOn G \u03b2 _ _ _ m _ hf.measurable (range f \u222a {0}) 0 (by simp) _", "annotated_tactic": ["let f_seq := @<a>SimpleFunc.approxOn</a> G \u03b2 _ _ _ m _ hf.measurable (<a>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": "case pos\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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "have hf_seq_meas : \u2200 n, StronglyMeasurable[m] (f_seq n) := fun n =>\n  @SimpleFunc.stronglyMeasurable \u03b2 G m _ (f_seq n)", "annotated_tactic": ["have hf_seq_meas : \u2200 n, StronglyMeasurable[m] (f_seq n) := fun n =>\n    @<a>SimpleFunc.stronglyMeasurable</a> \u03b2 G m _ (f_seq n)", [{"full_name": "MeasureTheory.SimpleFunc.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 38]}]], "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "have hf_seq_int : \u2200 n, Integrable (f_seq n) \u03bc :=\n  SimpleFunc.integrable_approxOn_range (hf.mono hm).measurable hf_int", "annotated_tactic": ["have hf_seq_int : \u2200 n, <a>Integrable</a> (f_seq n) \u03bc :=\n    <a>SimpleFunc.integrable_approxOn_range</a> (hf.mono hm).<a>measurable</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.SimpleFunc.integrable_approxOn_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [261, 9], "def_end_pos": [261, 34]}, {"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 pos\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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "have hf_seq_int_m : \u2200 n, Integrable (f_seq n) (\u03bc.trim hm) := fun n =>\n  (hf_seq_int n).trim hm (hf_seq_meas n)", "annotated_tactic": ["have hf_seq_int_m : \u2200 n, <a>Integrable</a> (f_seq n) (\u03bc.trim hm) := fun n =>\n    (hf_seq_int n).<a>trim</a> hm (hf_seq_meas 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.trim", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1212, 9], "def_end_pos": [1212, 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\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "have hf_seq_eq : \u2200 n, \u222b x, f_seq n x \u2202\u03bc = \u222b x, f_seq n x \u2202\u03bc.trim hm := fun n =>\n  integral_trim_simpleFunc hm (f_seq n) (hf_seq_int n)", "annotated_tactic": ["have hf_seq_eq : \u2200 n, \u222b x, f_seq n x \u2202\u03bc = \u222b x, f_seq n x \u2202\u03bc.trim hm := fun n =>\n    <a>integral_trim_simpleFunc</a> hm (f_seq n) (hf_seq_int n)", [{"full_name": "MeasureTheory.integral_trim_simpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1858, 9], "def_end_pos": [1858, 33]}]], "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "have h_lim_1 : atTop.Tendsto (fun n => \u222b x, f_seq n x \u2202\u03bc) (\ud835\udcdd (\u222b x, f x \u2202\u03bc)) := by\n  refine' tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _\n  exact SimpleFunc.tendsto_approxOn_range_L1_nnnorm (hf.mono hm).measurable hf_int", "annotated_tactic": ["have h_lim_1 : atTop.Tendsto (fun n => \u222b x, f_seq n x \u2202\u03bc) (\ud835\udcdd (\u222b x, f x \u2202\u03bc)) := by\n    refine' <a>tendsto_integral_of_L1</a> f hf_int (<a>eventually_of_forall</a> hf_seq_int) _\n    exact <a>SimpleFunc.tendsto_approxOn_range_L1_nnnorm</a> (hf.mono hm).<a>measurable</a> hf_int", [{"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": "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_range_L1_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [249, 9], "def_end_pos": [249, 41]}, {"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 pos\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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "have h_lim_2 : atTop.Tendsto (fun n => \u222b x, f_seq n x \u2202\u03bc) (\ud835\udcdd (\u222b x, f x \u2202\u03bc.trim hm)) := by\n  simp_rw [hf_seq_eq]\n  refine' @tendsto_integral_of_L1 \u03b2 G _ _ m (\u03bc.trim hm) _ f (hf_int.trim hm hf) _ _\n    (eventually_of_forall hf_seq_int_m) _\n  exact @SimpleFunc.tendsto_approxOn_range_L1_nnnorm \u03b2 G m _ _ _ f _ _ hf.measurable\n    (hf_int.trim hm hf)", "annotated_tactic": ["have h_lim_2 : atTop.Tendsto (fun n => \u222b x, f_seq n x \u2202\u03bc) (\ud835\udcdd (\u222b x, f x \u2202\u03bc.trim hm)) := by\n    simp_rw [hf_seq_eq]\n    refine' @<a>tendsto_integral_of_L1</a> \u03b2 G _ _ m (\u03bc.trim hm) _ f (hf_int.trim hm hf) _ _\n      (<a>eventually_of_forall</a> hf_seq_int_m) _\n    exact @<a>SimpleFunc.tendsto_approxOn_range_L1_nnnorm</a> \u03b2 G m _ _ _ f _ _ hf.measurable\n      (hf_int.trim hm hf)", [{"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": "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_range_L1_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [249, 9], "def_end_pos": [249, 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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\nh_lim_2 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm))\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "exact tendsto_nhds_unique h_lim_1 h_lim_2", "annotated_tactic": ["exact <a>tendsto_nhds_unique</a> h_lim_1 h_lim_2", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [994, 9], "def_end_pos": [994, 28]}]], "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\nh_lim_2 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm))\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "state_after": "no goals"}, {"tactic": "have hf_int_m : \u00acIntegrable f (\u03bc.trim hm) := fun hf_int_m =>\n  hf_int (integrable_of_integrable_trim hm hf_int_m)", "annotated_tactic": ["have hf_int_m : \u00ac<a>Integrable</a> f (\u03bc.trim hm) := fun hf_int_m =>\n      hf_int (<a>integrable_of_integrable_trim</a> hm hf_int_m)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integrable_of_integrable_trim", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1220, 9], "def_end_pos": [1220, 38]}]], "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : \u00acIntegrable f\nhf_int_m : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm"}, {"tactic": "rw [integral_undef hf_int, integral_undef hf_int_m]", "annotated_tactic": ["rw [<a>integral_undef</a> hf_int, <a>integral_undef</a> hf_int_m]", [{"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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : \u00acIntegrable f\nhf_int_m : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\n\u22a2 0 \u2208 range f \u222a {0}", "state_after": "no goals"}, {"tactic": "refine' tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _", "annotated_tactic": ["refine' <a>tendsto_integral_of_L1</a> f hf_int (<a>eventually_of_forall</a> hf_seq_int) _", [{"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": "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\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\n\u22a2 Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f 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\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(f_seq i) x - f x\u2016\u208a \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "exact SimpleFunc.tendsto_approxOn_range_L1_nnnorm (hf.mono hm).measurable hf_int", "annotated_tactic": ["exact <a>SimpleFunc.tendsto_approxOn_range_L1_nnnorm</a> (hf.mono hm).<a>measurable</a> hf_int", [{"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [249, 9], "def_end_pos": [249, 41]}, {"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\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(f_seq i) x - f x\u2016\u208a \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simp_rw [hf_seq_eq]", "annotated_tactic": ["simp_rw [hf_seq_eq]", []], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm))", "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 Tendsto\n    (fun n =>\n      \u222b (x : \u03b2),\n        \u2191(SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n) x \u2202Measure.trim \u03bc hm)\n    atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm))"}, {"tactic": "refine' @tendsto_integral_of_L1 \u03b2 G _ _ m (\u03bc.trim hm) _ f (hf_int.trim hm hf) _ _\n  (eventually_of_forall hf_seq_int_m) _", "annotated_tactic": ["refine' @<a>tendsto_integral_of_L1</a> \u03b2 G _ _ m (\u03bc.trim hm) _ f (hf_int.trim hm hf) _ _\n      (<a>eventually_of_forall</a> hf_seq_int_m) _", [{"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": "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\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 Tendsto\n    (fun n =>\n      \u222b (x : \u03b2),\n        \u2191(SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n) x \u2202Measure.trim \u03bc hm)\n    atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm))", "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 Tendsto\n    (fun i =>\n      \u222b\u207b (x : \u03b2),\n        \u2191\u2016\u2191(SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) x -\n              f x\u2016\u208a \u2202Measure.trim \u03bc hm)\n    atTop (\ud835\udcdd 0)"}, {"tactic": "exact @SimpleFunc.tendsto_approxOn_range_L1_nnnorm \u03b2 G m _ _ _ f _ _ hf.measurable\n  (hf_int.trim hm hf)", "annotated_tactic": ["exact @<a>SimpleFunc.tendsto_approxOn_range_L1_nnnorm</a> \u03b2 G m _ _ _ f _ _ hf.measurable\n      (hf_int.trim hm hf)", [{"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [249, 9], "def_end_pos": [249, 41]}]], "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 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\nH : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : NormedAddCommGroup H\nm m0 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b2\nhm : m \u2264 m0\nf : \u03b2 \u2192 G\nhf : StronglyMeasurable f\nhG : CompleteSpace G\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhf_int : Integrable f\nthis : SeparableSpace \u2191(range f \u222a {0})\nf_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0})\nhf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n)\nhf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n)\nhf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm\nh_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc))\n\u22a2 Tendsto\n    (fun i =>\n      \u222b\u207b (x : \u03b2),\n        \u2191\u2016\u2191(SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) x -\n              f x\u2016\u208a \u2202Measure.trim \u03bc hm)\n    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.mem_insert_of_eq", "start": [688, 1], "end": [689, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd", "start": [111, 1], "end": [123, 17], "traced_tactics": [{"tactic": "have heq : \u03bc.map X = (\u03bc.map (fun x => (X x, Y x))).fst", "annotated_tactic": ["have heq : \u03bc.map X = (\u03bc.map (fun x => (X x, Y x))).<a>fst</a>", [{"full_name": "MeasureTheory.Measure.fst", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [910, 19], "def_end_pos": [910, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map X \u03bc, \u2191\u03ba x = \u2191(condDistrib Y X \u03bc) x", "state_after": "case heq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\n\u22a2 Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map X \u03bc, \u2191\u03ba x = \u2191(condDistrib Y X \u03bc) x"}, {"tactic": "rw [heq, condDistrib]", "annotated_tactic": ["rw [heq, <a>condDistrib</a>]", [{"full_name": "ProbabilityTheory.condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [62, 31], "def_end_pos": [62, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map X \u03bc, \u2191\u03ba x = \u2191(condDistrib Y X \u03bc) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc),\n    \u2191\u03ba x = \u2191(Measure.condKernel (Measure.map (fun a => (X a, Y a)) \u03bc)) x"}, {"tactic": "refine' eq_condKernel_of_measure_eq_compProd _ _ _", "annotated_tactic": ["refine' <a>eq_condKernel_of_measure_eq_compProd</a> _ _ _", [{"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [530, 9], "def_end_pos": [530, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc),\n    \u2191\u03ba x = \u2191(Measure.condKernel (Measure.map (fun a => (X a, Y a)) \u03bc)) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 Measure.map (fun a => (X a, Y a)) \u03bc =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (fun a => (X a, Y a)) \u03bc)) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()"}, {"tactic": "convert h\u03ba", "annotated_tactic": ["convert h\u03ba", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 Measure.map (fun a => (X a, Y a)) \u03bc =\n    \u2191(kernel.const Unit (Measure.fst (Measure.map (fun a => (X a, Y a)) \u03bc)) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()", "state_after": "case h.e'_3.h.e'_5.h.e'_7.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 Measure.fst (Measure.map (fun a => (X a, Y a)) \u03bc) = Measure.map X \u03bc"}, {"tactic": "exact heq.symm", "annotated_tactic": ["exact heq.symm", []], "state_before": "case h.e'_3.h.e'_5.h.e'_7.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\nheq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)\n\u22a2 Measure.fst (Measure.map (fun a => (X a, Y a)) \u03bc) = Measure.map X \u03bc", "state_after": "no goals"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case heq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\n\u22a2 Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)", "state_after": "case heq.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.map X \u03bc) s = \u2191\u2191(Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)) s"}, {"tactic": "rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> hX hs, <a>Measure.fst_apply</a> hs, <a>Measure.map_apply</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.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 heq.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.map X \u03bc) s = \u2191\u2191(Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)) s", "state_after": "case heq.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (X \u207b\u00b9' s) = \u2191\u2191\u03bc ((fun x => (X x, Y x)) \u207b\u00b9' (Prod.fst \u207b\u00b9' s))\n\ncase heq.h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun x => (X x, Y x)\n\ncase heq.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)"}, {"tactic": "exacts [rfl, Measurable.prod hX hY, measurable_fst hs]", "annotated_tactic": ["exacts [<a>rfl</a>, <a>Measurable.prod</a> hX hY, <a>measurable_fst</a> 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]}]], "state_before": "case heq.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (X \u207b\u00b9' s) = \u2191\u2191\u03bc ((fun x => (X x, Y x)) \u207b\u00b9' (Prod.fst \u207b\u00b9' s))\n\ncase heq.h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun x => (X x, Y x)\n\ncase heq.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nF : Type u_4\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup F\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : 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\nhX : Measurable X\nhY : Measurable Y\n\u03ba : { x // x \u2208 kernel \u03b2 \u03a9 }\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' 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_piecewise_const", "start": [1140, 1], "end": [1143, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/NatPrime.lean", "full_name": "Int.not_prime_of_int_mul", "start": [19, 1], "end": [21, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.core_restrict", "start": [500, 1], "end": [501, 25], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "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 : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 core (\u2191f) s = f \u207b\u00b9' 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\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 core (\u2191f) s \u2194 x \u2208 f \u207b\u00b9' s"}, {"tactic": "simp [core_def]", "annotated_tactic": ["simp [<a>core_def</a>]", [{"full_name": "PFun.core_def", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [463, 9], "def_end_pos": [463, 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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 core (\u2191f) s \u2194 x \u2208 f \u207b\u00b9' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "AddCircle.volume_closedBall", "start": [96, 1], "end": [114, 53], "traced_tactics": [{"tactic": "have hT' : |T| = T := abs_eq_self.mpr hT.out.le", "annotated_tactic": ["have hT' : |T| = T := abs_eq_self.mpr hT.out.le", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "let I := Ioc (-(T / 2)) (T / 2)", "annotated_tactic": ["let I := <a>Ioc</a> (-(T / 2)) (T / 2)", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "have h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall (0 : \u211d) \u03b5 \u2229 I = Metric.closedBall (0 : \u211d) \u03b5 := by\n  intro h\u03b5\n  rw [inter_eq_left, Real.closedBall_eq_Icc, zero_sub, zero_add]\n  rintro y \u27e8hy\u2081, hy\u2082\u27e9; constructor <;> linarith", "annotated_tactic": ["have h\u2081 : \u03b5 < T / 2 \u2192 <a>Metric.closedBall</a> (0 : \u211d) \u03b5 \u2229 I = <a>Metric.closedBall</a> (0 : \u211d) \u03b5 := by\n    intro h\u03b5\n    rw [<a>inter_eq_left</a>, <a>Real.closedBall_eq_Icc</a>, <a>zero_sub</a>, <a>zero_add</a>]\n    rintro y \u27e8hy\u2081, hy\u2082\u27e9; constructor <;> 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": "Set.inter_eq_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 15], "def_end_pos": [981, 28]}, {"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "have h\u2082 : (\u2191) \u207b\u00b9' Metric.closedBall (0 : AddCircle T) \u03b5 \u2229 I =\n    if \u03b5 < T / 2 then Metric.closedBall (0 : \u211d) \u03b5 else I := by\n  conv_rhs => rw [\u2190 if_ctx_congr (Iff.rfl : \u03b5 < T / 2 \u2194 \u03b5 < T / 2) h\u2081 fun _ => rfl, \u2190 hT']\n  apply coe_real_preimage_closedBall_inter_eq\n  simpa only [hT', Real.closedBall_eq_Icc, zero_add, zero_sub] using Ioc_subset_Icc_self", "annotated_tactic": ["have h\u2082 : (\u2191) \u207b\u00b9' <a>Metric.closedBall</a> (0 : <a>AddCircle</a> T) \u03b5 \u2229 I =\n      if \u03b5 < T / 2 then <a>Metric.closedBall</a> (0 : \u211d) \u03b5 else I := by\n    conv_rhs => rw [\u2190 <a>if_ctx_congr</a> (<a>Iff.rfl</a> : \u03b5 < T / 2 \u2194 \u03b5 < T / 2) h\u2081 fun _ => <a>rfl</a>, \u2190 hT']\n    apply <a>coe_real_preimage_closedBall_inter_eq</a>\n    simpa only [hT', <a>Real.closedBall_eq_Icc</a>, <a>zero_add</a>, <a>zero_sub</a>] using <a>Ioc_subset_Icc_self</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [124, 8], "def_end_pos": [124, 17]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "if_ctx_congr", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [367, 9], "def_end_pos": [367, 21]}, {"full_name": "Iff.rfl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [663, 19], "def_end_pos": [663, 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": "AddCircle.coe_real_preimage_closedBall_inter_eq", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [199, 9], "def_end_pos": [199, 46]}, {"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"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": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "rw [addHaar_closedBall_center]", "annotated_tactic": ["rw [<a>addHaar_closedBall_center</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]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 \u2191\u2191volume (Metric.closedBall 0 \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "simp only [restrict_apply' measurableSet_Ioc, (by linarith : -(T / 2) + T = T / 2), h\u2082, \u2190\n  (AddCircle.measurePreserving_mk T (-(T / 2))).measure_preimage measurableSet_closedBall]", "annotated_tactic": ["simp only [<a>restrict_apply'</a> <a>measurableSet_Ioc</a>, (by linarith : -(T / 2) + T = T / 2), h\u2082, \u2190\n    (<a>AddCircle.measurePreserving_mk</a> T (-(T / 2))).<a>measure_preimage</a> <a>measurableSet_closedBall</a>]", [{"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}, {"full_name": "AddCircle.measurePreserving_mk", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [86, 19], "def_end_pos": [86, 39]}, {"full_name": "MeasureTheory.MeasurePreserving.measure_preimage", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [126, 9], "def_end_pos": [126, 25]}, {"full_name": "measurableSet_closedBall", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1681, 9], "def_end_pos": [1681, 33]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 \u2191\u2191volume (Metric.closedBall 0 \u03b5) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "by_cases h\u03b5 : \u03b5 < T / 2", "annotated_tactic": ["by_cases h\u03b5 : \u03b5 < T / 2", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "case pos\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\nh\u03b5 : \u03b5 < T / 2\n\u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5))\n\ncase neg\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\nh\u03b5 : \u00ac\u03b5 < T / 2\n\u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5))"}, {"tactic": "intro h\u03b5", "annotated_tactic": ["intro h\u03b5", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\n\u22a2 \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u03b5 : \u03b5 < T / 2\n\u22a2 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5"}, {"tactic": "rw [inter_eq_left, Real.closedBall_eq_Icc, zero_sub, zero_add]", "annotated_tactic": ["rw [<a>inter_eq_left</a>, <a>Real.closedBall_eq_Icc</a>, <a>zero_sub</a>, <a>zero_add</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": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u03b5 : \u03b5 < T / 2\n\u22a2 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u03b5 : \u03b5 < T / 2\n\u22a2 Icc (-\u03b5) \u03b5 \u2286 I"}, {"tactic": "rintro y \u27e8hy\u2081, hy\u2082\u27e9", "annotated_tactic": ["rintro y \u27e8hy\u2081, hy\u2082\u27e9", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u03b5 : \u03b5 < T / 2\n\u22a2 Icc (-\u03b5) \u03b5 \u2286 I", "state_after": "case intro\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u03b5 : \u03b5 < T / 2\ny : \u211d\nhy\u2081 : -\u03b5 \u2264 y\nhy\u2082 : y \u2264 \u03b5\n\u22a2 y \u2208 I"}, {"tactic": "constructor <;> linarith", "annotated_tactic": ["constructor <;> linarith", []], "state_before": "case intro\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u03b5 : \u03b5 < T / 2\ny : \u211d\nhy\u2081 : -\u03b5 \u2264 y\nhy\u2082 : y \u2264 \u03b5\n\u22a2 y \u2208 I", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [\u2190 if_ctx_congr (Iff.rfl : \u03b5 < T / 2 \u2194 \u03b5 < T / 2) h\u2081 fun _ => rfl, \u2190 hT']", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>if_ctx_congr</a> (<a>Iff.rfl</a> : \u03b5 < T / 2 \u2194 \u03b5 < T / 2) h\u2081 fun _ => <a>rfl</a>, \u2190 hT']", [{"full_name": "if_ctx_congr", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [367, 9], "def_end_pos": [367, 21]}, {"full_name": "Iff.rfl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [663, 19], "def_end_pos": [663, 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": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I", "state_after": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < |T| / 2 then Metric.closedBall 0 \u03b5 \u2229 I else I"}, {"tactic": "apply coe_real_preimage_closedBall_inter_eq", "annotated_tactic": ["apply <a>coe_real_preimage_closedBall_inter_eq</a>", [{"full_name": "AddCircle.coe_real_preimage_closedBall_inter_eq", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [199, 9], "def_end_pos": [199, 46]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < |T| / 2 then Metric.closedBall 0 \u03b5 \u2229 I else I", "state_after": "case hs\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 I \u2286 Metric.closedBall 0 (|T| / 2)"}, {"tactic": "simpa only [hT', Real.closedBall_eq_Icc, zero_add, zero_sub] using Ioc_subset_Icc_self", "annotated_tactic": ["simpa only [hT', <a>Real.closedBall_eq_Icc</a>, <a>zero_add</a>, <a>zero_sub</a>] using <a>Ioc_subset_Icc_self</a>", [{"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"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": "case hs\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\n\u22a2 I \u2286 Metric.closedBall 0 (|T| / 2)", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\n\u22a2 -(T / 2) + T = T / 2", "state_after": "no goals"}, {"tactic": "simp [h\u03b5, min_eq_right (by linarith : 2 * \u03b5 \u2264 T)]", "annotated_tactic": ["simp [h\u03b5, <a>min_eq_right</a> (by linarith : 2 * \u03b5 \u2264 T)]", [{"full_name": "min_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}]], "state_before": "case pos\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\nh\u03b5 : \u03b5 < T / 2\n\u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\nh\u03b5 : \u03b5 < T / 2\n\u22a2 2 * \u03b5 \u2264 T", "state_after": "no goals"}, {"tactic": "simp [h\u03b5, min_eq_left (by linarith : T \u2264 2 * \u03b5)]", "annotated_tactic": ["simp [h\u03b5, <a>min_eq_left</a> (by linarith : T \u2264 2 * \u03b5)]", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "state_before": "case neg\nT : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\nh\u03b5 : \u00ac\u03b5 < T / 2\n\u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5))", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nx : AddCircle T\n\u03b5 : \u211d\nhT' : |T| = T\nI : Set \u211d := Ioc (-(T / 2)) (T / 2)\nh\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5\nh\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I\nh\u03b5 : \u00ac\u03b5 < T / 2\n\u22a2 T \u2264 2 * \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.preimage_comp", "start": [202, 1], "end": [203, 48], "traced_tactics": [{"tactic": "simp only [preimage, inv_comp, image_comp]", "annotated_tactic": ["simp only [<a>preimage</a>, <a>inv_comp</a>, <a>image_comp</a>]", [{"full_name": "Rel.preimage", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [174, 5], "def_end_pos": [174, 13]}, {"full_name": "Rel.inv_comp", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [124, 9], "def_end_pos": [124, 17]}, {"full_name": "Rel.image_comp", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [162, 9], "def_end_pos": [162, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\n\u22a2 preimage (r \u2022 s) t = preimage r (preimage s t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "full_name": "Std.RBNode.any_iff", "start": [185, 1], "end": [186, 47], "traced_tactics": [{"tactic": "induction t <;> simp [*, any, Any, or_assoc]", "annotated_tactic": ["induction t <;> simp [*, <a>any</a>, <a>Any</a>, <a>or_assoc</a>]", [{"full_name": "Std.RBNode.any", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [161, 19], "def_end_pos": [161, 22]}, {"full_name": "Std.RBNode.Any", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [181, 5], "def_end_pos": [181, 8]}, {"full_name": "or_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [265, 9], "def_end_pos": [265, 17]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nt : RBNode \u03b1\n\u22a2 any p t = true \u2194 Any (fun x => p x = true) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Agrees.set", "start": [104, 1], "end": [113, 61], "traced_tactics": [{"tactic": "cases H", "annotated_tactic": ["cases H", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nn : \u2115\nm : Fin n \u2192 \u03b2\nH : Agrees arr f m\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin n \u2192 \u03b2\nhm\u2081 : \u2200 (j : Fin n), \u2191j \u2260 \u2191i \u2192 m' j = m j\nhm\u2082 : \u2200 (h : \u2191i < n), f x = m' { val := \u2191i, isLt := h }\n\u22a2 Agrees (Array.set arr i x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\n\u22a2 Agrees (Array.set arr i x) f m'"}, {"tactic": "refine mk' (by simp) fun j hj\u2081 hj\u2082 \u21a6 ?_", "annotated_tactic": ["refine <a>mk'</a> (by simp) fun j hj\u2081 hj\u2082 \u21a6 ?_", [{"full_name": "UFModel.Agrees.mk'", "def_path": "Mathlib/Data/UnionFind.lean", "def_pos": [74, 9], "def_end_pos": [74, 12]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\n\u22a2 Agrees (Array.set arr i x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\n\u22a2 f (Array.get (Array.set arr i x) { val := j, isLt := hj\u2081 }) = m' { val := j, isLt := hj\u2082 }"}, {"tactic": "suffices f (Array.set arr i x)[j] = m' \u27e8j, hj\u2082\u27e9 by simp_all [Array.get_set]", "annotated_tactic": ["suffices f (<a>Array.set</a> arr i x)[j] = m' \u27e8j, hj\u2082\u27e9 by simp_all [<a>Array.get_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": "Array.get_set", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [106, 9], "def_end_pos": [106, 16]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\n\u22a2 f (Array.get (Array.set arr i x) { val := j, isLt := hj\u2081 }) = m' { val := j, isLt := hj\u2082 }", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\n\u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }"}, {"tactic": "by_cases h : i = j", "annotated_tactic": ["by_cases h : i = j", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\n\u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u2191i = j\n\u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\n\u22a2 Array.size arr = Array.size (Array.set arr i x)", "state_after": "no goals"}, {"tactic": "simp_all [Array.get_set]", "annotated_tactic": ["simp_all [<a>Array.get_set</a>]", [{"full_name": "Array.get_set", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [106, 9], "def_end_pos": [106, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nthis : f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }\n\u22a2 f (Array.get (Array.set arr i x) { val := j, isLt := hj\u2081 }) = m' { val := j, isLt := hj\u2082 }", "state_after": "no goals"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u2191i = j\n\u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nhj\u2081 : \u2191i < Array.size (Array.set arr i x)\nhj\u2082 : \u2191i < Array.size arr\n\u22a2 f (Array.set arr i x)[\u2191i] = m' { val := \u2191i, isLt := hj\u2082 }"}, {"tactic": "rw [Array.get_set_eq, \u2190 hm\u2082]", "annotated_tactic": ["rw [<a>Array.get_set_eq</a>, \u2190 hm\u2082]", [{"full_name": "Array.get_set_eq", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [87, 17], "def_end_pos": [87, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nhj\u2081 : \u2191i < Array.size (Array.set arr i x)\nhj\u2082 : \u2191i < Array.size arr\n\u22a2 f (Array.set arr i x)[\u2191i] = m' { val := \u2191i, isLt := hj\u2082 }", "state_after": "no goals"}, {"tactic": "rw [arr.get_set_ne _ _ _ h, hm\u2081 \u27e8j, _\u27e9 (Ne.symm h)]", "annotated_tactic": ["rw [arr.get_set_ne _ _ _ h, hm\u2081 \u27e8j, _\u27e9 (<a>Ne.symm</a> h)]", [{"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\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 }", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 f arr[j] = (fun i => f (Array.get arr i)) { val := j, isLt := hj\u2082 }\n\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 j < Array.size arr\n\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 j < Array.size arr"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 f arr[j] = (fun i => f (Array.get arr i)) { val := j, isLt := hj\u2082 }\n\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 j < Array.size arr\n\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\ni : Fin (Array.size arr)\nx : \u03b1\nm' : Fin (Array.size arr) \u2192 \u03b2\nhm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h }\nhm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j\nj : \u2115\nhj\u2081 : j < Array.size (Array.set arr i x)\nhj\u2082 : j < Array.size arr\nh : \u00ac\u2191i = j\n\u22a2 j < Array.size arr", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_spanningSets_lt_top", "start": [3329, 1], "end": [3331, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.aefinStronglyMeasurable_zero", "start": [1147, 1], "end": [1149, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.mk'_right", "start": [560, 1], "end": [561, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.mem_lookup_union", "start": [600, 1], "end": [602, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_iff_exists_subset_erase", "start": [1996, 1], "end": [1999, 42], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_subset_of_ssubset h <| erase_ssubset ha\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => <a>ssubset_of_subset_of_ssubset</a> h <| <a>erase_ssubset</a> ha\u27e9", [{"full_name": "Finset.ssubset_of_subset_of_ssubset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 37]}, {"full_name": "Finset.erase_ssubset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1990, 9], "def_end_pos": [1990, 22]}]], "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\n\u22a2 s \u2282 t \u2194 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a", "state_after": "\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\nh : s \u2282 t\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a"}, {"tactic": "obtain \u27e8a, ht, hs\u27e9 := not_subset.1 h.2", "annotated_tactic": ["obtain \u27e8a, ht, hs\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\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a", "state_after": "case intro.intro\n\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\nh : s \u2282 t\na : \u03b1\nht : a \u2208 t\nhs : \u00aca \u2208 s\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a"}, {"tactic": "exact \u27e8a, ht, subset_erase.2 \u27e8h.1, hs\u27e9\u27e9", "annotated_tactic": ["exact \u27e8a, ht, <a>subset_erase</a>.2 \u27e8h.1, hs\u27e9\u27e9", [{"full_name": "Finset.subset_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1980, 9], "def_end_pos": [1980, 21]}]], "state_before": "case intro.intro\n\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\nh : s \u2282 t\na : \u03b1\nht : a \u2208 t\nhs : \u00aca \u2208 s\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.card_Ioi_eq_card_Ici_sub_one", "start": [717, 1], "end": [718, 57], "traced_tactics": [{"tactic": "rw [Ici_eq_cons_Ioi, card_cons, add_tsub_cancel_right]", "annotated_tactic": ["rw [<a>Ici_eq_cons_Ioi</a>, <a>card_cons</a>, <a>add_tsub_cancel_right</a>]", [{"full_name": "Finset.Ici_eq_cons_Ioi", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [713, 9], "def_end_pos": [713, 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 : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 card (Ioi a) = card (Ici a) - 1", "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_sup_eq", "start": [484, 1], "end": [513, 68], "traced_tactics": [{"tactic": "rcases isCompact_isCompact_separated K\u2081.2 K\u2082.2 h with \u27e8U\u2081, U\u2082, h1U\u2081, h1U\u2082, h2U\u2081, h2U\u2082, hU\u27e9", "annotated_tactic": ["rcases <a>isCompact_isCompact_separated</a> K\u2081.2 K\u2082.2 h with \u27e8U\u2081, U\u2082, h1U\u2081, h1U\u2082, h2U\u2081, h2U\u2082, hU\u27e9", [{"full_name": "isCompact_isCompact_separated", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1280, 9], "def_end_pos": [1280, 38]}]], "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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "rcases compact_open_separated_mul_right K\u2081.2 h1U\u2081 h2U\u2081 with \u27e8L\u2081, h1L\u2081, h2L\u2081\u27e9", "annotated_tactic": ["rcases <a>compact_open_separated_mul_right</a> K\u2081.2 h1U\u2081 h2U\u2081 with \u27e8L\u2081, h1L\u2081, h2L\u2081\u27e9", [{"full_name": "compact_open_separated_mul_right", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1617, 9], "def_end_pos": [1617, 41]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nh2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "rcases mem_nhds_iff.mp h1L\u2081 with \u27e8V\u2081, h1V\u2081, h2V\u2081, h3V\u2081\u27e9", "annotated_tactic": ["rcases mem_nhds_iff.mp h1L\u2081 with \u27e8V\u2081, h1V\u2081, h2V\u2081, h3V\u2081\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nh2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nh2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "replace h2L\u2081 := Subset.trans (mul_subset_mul_left h1V\u2081) h2L\u2081", "annotated_tactic": ["replace h2L\u2081 := <a>Subset.trans</a> (<a>mul_subset_mul_left</a> h1V\u2081) h2L\u2081", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.mul_subset_mul_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [422, 9], "def_end_pos": [422, 28]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nh2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "rcases compact_open_separated_mul_right K\u2082.2 h1U\u2082 h2U\u2082 with \u27e8L\u2082, h1L\u2082, h2L\u2082\u27e9", "annotated_tactic": ["rcases <a>compact_open_separated_mul_right</a> K\u2082.2 h1U\u2082 h2U\u2082 with \u27e8L\u2082, h1L\u2082, h2L\u2082\u27e9", [{"full_name": "compact_open_separated_mul_right", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1617, 9], "def_end_pos": [1617, 41]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nh2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "rcases mem_nhds_iff.mp h1L\u2082 with \u27e8V\u2082, h1V\u2082, h2V\u2082, h3V\u2082\u27e9", "annotated_tactic": ["rcases mem_nhds_iff.mp h1L\u2082 with \u27e8V\u2082, h1V\u2082, h2V\u2082, h3V\u2082\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nh2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nh2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "replace h2L\u2082 := Subset.trans (mul_subset_mul_left h1V\u2082) h2L\u2082", "annotated_tactic": ["replace h2L\u2082 := <a>Subset.trans</a> (<a>mul_subset_mul_left</a> h1V\u2082) h2L\u2082", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.mul_subset_mul_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [422, 9], "def_end_pos": [422, 28]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nh2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"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": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "have : Continuous eval :=\n  ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082))", "annotated_tactic": ["have : <a>Continuous</a> eval :=\n    ((<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": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082"}, {"tactic": "rw [eq_comm, \u2190 sub_eq_zero]", "annotated_tactic": ["rw [<a>eq_comm</a>, \u2190 <a>sub_eq_zero</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"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.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082) = 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": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082) = 0", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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' {0}"}, {"tactic": "let V := V\u2081 \u2229 V\u2082", "annotated_tactic": ["let V := V\u2081 \u2229 V\u2082", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\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' {0}", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}"}, {"tactic": "apply\n  mem_of_subset_of_mem _\n    (chaar_mem_clPrehaar K\u2080\n      \u27e8\u27e8V\u207b\u00b9, (h2V\u2081.inter h2V\u2082).preimage continuous_inv\u27e9, by\n        simp only [mem_inv, inv_one, h3V\u2081, h3V\u2082, mem_inter_iff, true_and_iff]\u27e9)", "annotated_tactic": ["apply\n    <a>mem_of_subset_of_mem</a> _\n      (<a>chaar_mem_clPrehaar</a> K\u2080\n        \u27e8\u27e8V\u207b\u00b9, (h2V\u2081.inter h2V\u2082).<a>preimage</a> <a>continuous_inv</a>\u27e9, by\n          simp only [<a>mem_inv</a>, <a>inv_one</a>, h3V\u2081, h3V\u2082, <a>mem_inter_iff</a>, <a>true_and_iff</a>]\u27e9)", [{"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]}, {"full_name": "IsOpen.preimage", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1607, 9], "def_end_pos": [1607, 24]}, {"full_name": "ContinuousInv.continuous_inv", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [186, 3], "def_end_pos": [186, 17]}, {"full_name": "Set.mem_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}", "state_after": "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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 clPrehaar \u2191K\u2080\n      { toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n        mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) } \u2286\n    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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 clPrehaar \u2191K\u2080\n      { toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n        mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) } \u2286\n    eval \u207b\u00b9' {0}", "state_after": "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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 closure\n      (prehaar \u2191K\u2080 ''\n        {U |\n          U \u2286\n              \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n                    mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227\n            IsOpen U \u2227 1 \u2208 U}) \u2286\n    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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 closure\n      (prehaar \u2191K\u2080 ''\n        {U |\n          U \u2286\n              \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n                    mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227\n            IsOpen U \u2227 1 \u2208 U}) \u2286\n    eval \u207b\u00b9' {0}", "state_after": "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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 prehaar \u2191K\u2080 ''\n      {U |\n        U \u2286\n            \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n                  mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227\n          IsOpen U \u2227 1 \u2208 U} \u2286\n    eval \u207b\u00b9' {0}\n\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 IsClosed (eval \u207b\u00b9' {0})"}, {"tactic": "simp only [mem_inv, inv_one, h3V\u2081, h3V\u2082, mem_inter_iff, true_and_iff]", "annotated_tactic": ["simp only [<a>mem_inv</a>, <a>inv_one</a>, h3V\u2081, h3V\u2082, <a>mem_inter_iff</a>, <a>true_and_iff</a>]", [{"full_name": "Set.mem_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}]], "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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 1 \u2208 { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) }.carrier", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8U, \u27e8h1U, h2U, h3U\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, \u27e8h1U, h2U, h3U\u27e9, rfl\u27e9", []], "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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 prehaar \u2191K\u2080 ''\n      {U |\n        U \u2286\n            \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n                  mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227\n          IsOpen U \u2227 1 \u2208 U} \u2286\n    eval \u207b\u00b9' {0}", "state_after": "case intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}"}, {"tactic": "simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>sub_eq_zero</a>, <a>mem_singleton_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": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"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 intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082 = prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082)"}, {"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": "case intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082 = prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082)", "state_after": "case intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082"}, {"tactic": "apply prehaar_sup_eq", "annotated_tactic": ["apply <a>prehaar_sup_eq</a>", [{"full_name": "MeasureTheory.Measure.haar.prehaar_sup_eq", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [341, 9], "def_end_pos": [341, 23]}]], "state_before": "case intro.intro.intro.intro\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = 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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)\n\ncase intro.intro.intro.intro.h\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)"}, {"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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U", "state_after": "no goals"}, {"tactic": "refine' disjoint_of_subset _ _ hU", "annotated_tactic": ["refine' <a>disjoint_of_subset</a> _ _ hU", [{"full_name": "Set.disjoint_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1579, 7], "def_end_pos": [1579, 25]}]], "state_before": "case intro.intro.intro.intro.h\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)", "state_after": "case intro.intro.intro.intro.h.refine'_1\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 K\u2081.carrier * U\u207b\u00b9 \u2286 U\u2081\n\ncase intro.intro.intro.intro.h.refine'_2\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 K\u2082.carrier * U\u207b\u00b9 \u2286 U\u2082"}, {"tactic": "refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L\u2081", "annotated_tactic": ["refine' <a>Subset.trans</a> (<a>mul_subset_mul</a> <a>Subset.rfl</a> _) h2L\u2081", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"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 intro.intro.intro.intro.h.refine'_1\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 K\u2081.carrier * U\u207b\u00b9 \u2286 U\u2081", "state_after": "case intro.intro.intro.intro.h.refine'_1\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 U\u207b\u00b9 \u2286 V\u2081"}, {"tactic": "exact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _)", "annotated_tactic": ["exact <a>Subset.trans</a> (inv_subset.mpr h1U) (<a>inter_subset_left</a> _ _)", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 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 intro.intro.intro.intro.h.refine'_1\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 U\u207b\u00b9 \u2286 V\u2081", "state_after": "no goals"}, {"tactic": "refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L\u2082", "annotated_tactic": ["refine' <a>Subset.trans</a> (<a>mul_subset_mul</a> <a>Subset.rfl</a> _) h2L\u2082", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"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 intro.intro.intro.intro.h.refine'_2\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 K\u2082.carrier * U\u207b\u00b9 \u2286 U\u2082", "state_after": "case intro.intro.intro.intro.h.refine'_2\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 U\u207b\u00b9 \u2286 V\u2082"}, {"tactic": "exact Subset.trans (inv_subset.mpr h1U) (inter_subset_right _ _)", "annotated_tactic": ["exact <a>Subset.trans</a> (inv_subset.mpr h1U) (<a>inter_subset_right</a> _ _)", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 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 intro.intro.intro.intro.h.refine'_2\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n  U \u2286\n    \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n          mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 U\u207b\u00b9 \u2286 V\u2082", "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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 IsClosed (eval \u207b\u00b9' {0})", "state_after": "case a\nG : 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\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\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\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 IsClosed {0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.map_mul_right_eq_self", "start": [91, 1], "end": [92, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "Filter.EventuallyEq.restrict", "start": [2578, 1], "end": [2583, 68], "traced_tactics": [{"tactic": "refine' hfg.filter_mono _", "annotated_tactic": ["refine' hfg.filter_mono _", []], "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\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 f =\u1da0[ae (Measure.restrict \u03bc s)] g", "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\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 ae (Measure.restrict \u03bc s) \u2264 ae \u03bc"}, {"tactic": "rw [Measure.ae_le_iff_absolutelyContinuous]", "annotated_tactic": ["rw [<a>Measure.ae_le_iff_absolutelyContinuous</a>]", [{"full_name": "MeasureTheory.Measure.ae_le_iff_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2174, 9], "def_end_pos": [2174, 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\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\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 ae (Measure.restrict \u03bc s) \u2264 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\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\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 Measure.restrict \u03bc s \u226a \u03bc"}, {"tactic": "exact Measure.absolutelyContinuous_of_le Measure.restrict_le_self", "annotated_tactic": ["exact <a>Measure.absolutelyContinuous_of_le</a> <a>Measure.restrict_le_self</a>", [{"full_name": "MeasureTheory.Measure.absolutelyContinuous_of_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2119, 9], "def_end_pos": [2119, 35]}, {"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\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\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 Measure.restrict \u03bc s \u226a \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_iff_exists_ne", "start": [2520, 1], "end": [2521, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_add_left'", "start": [1091, 1], "end": [1102, 45], "traced_tactics": [{"tactic": "suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply]", "annotated_tactic": ["suffices <a>setToL1</a> hT'' = <a>setToL1</a> hT + <a>setToL1</a> hT' by 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'\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 : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 hT'') 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'\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 : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT'' = 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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\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 : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT'' = 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'\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 : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT + 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'\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 : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT + 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'\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\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 = \u2191(setToL1SCLM \u03b1 E \u03bc hT'') f"}, {"tactic": "suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT'' f by rw [\u2190 this]; congr", "annotated_tactic": ["suffices <a>setToL1</a> hT f + <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.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'\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\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 = \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'\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\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 hT'') f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM,\n  setToL1SCLM_add_left' hT hT' hT'' h_add]", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>, <a>setToL1_eq_setToL1SCLM</a>,\n    <a>setToL1SCLM_add_left'</a> hT hT' hT'' h_add]", [{"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": [925, 9], "def_end_pos": [925, 30]}]], "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'\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\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 hT'') 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'\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 : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT'' = setToL1 hT + setToL1 hT'\n\u22a2 \u2191(setToL1 hT'') 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'\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\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 hT'') f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + 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'\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\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 hT'') f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT) \u2191f + \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'\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\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 hT'') 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/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.sub_ediv_of_dvd_sub", "start": [811, 1], "end": [813, 93], "traced_tactics": [{"tactic": "rw [\u2190 Int.add_sub_cancel ((a-b) / c), \u2190 Int.add_ediv_of_dvd_left hcab, Int.sub_add_cancel]", "annotated_tactic": ["rw [\u2190 <a>Int.add_sub_cancel</a> ((a-b) / c), \u2190 <a>Int.add_ediv_of_dvd_left</a> hcab, <a>Int.sub_add_cancel</a>]", [{"full_name": "Int.add_sub_cancel", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [489, 27], "def_end_pos": [489, 41]}, {"full_name": "Int.add_ediv_of_dvd_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [182, 9], "def_end_pos": [182, 29]}, {"full_name": "Int.sub_add_cancel", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [486, 27], "def_end_pos": [486, 41]}]], "state_before": "a b c : Int\nhcab : c \u2223 a - b\n\u22a2 (a - b) / c = a / c - b / c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "Real.measure_ext_Ioo_rat", "start": [1941, 1], "end": [1946, 12], "traced_tactics": [{"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": "\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\u03bc \u03bd : Measure \u211d\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)\n\u22a2 \u2200 (s : Set \u211d), s \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s", "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\u03bc \u03bd : Measure \u211d\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)\n\u22a2 \u2200 (s : Set \u211d), (\u2203 i i_1 h, s = Ioo \u2191i \u2191i_1) \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s"}, {"tactic": "rintro _ \u27e8a, b, -, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8a, b, -, 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\u03bc \u03bd : Measure \u211d\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)\n\u22a2 \u2200 (s : Set \u211d), (\u2203 i i_1 h, s = Ioo \u2191i \u2191i_1) \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s", "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 t u : Set \u03b1\n\u03bc \u03bd : Measure \u211d\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)\na b : \u211a\n\u22a2 \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "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 t u : Set \u03b1\n\u03bc \u03bd : Measure \u211d\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)\na b : \u211a\n\u22a2 \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_nonneg_of_forall", "start": [1364, 1], "end": [1365, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_emetric_ball", "start": [118, 1], "end": [123, 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.ball a r) = 2 * r", "state_after": "case inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (EMetric.ball 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.ball a r) = 2 * r"}, {"tactic": "rw [Metric.emetric_ball_top, volume_univ, two_mul, _root_.top_add]", "annotated_tactic": ["rw [<a>Metric.emetric_ball_top</a>, <a>volume_univ</a>, <a>two_mul</a>, <a>_root_.top_add</a>]", [{"full_name": "Metric.emetric_ball_top", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1233, 9], "def_end_pos": [1233, 32]}, {"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.ball 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.ball 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.ball a \u2191r) = 2 * \u2191r"}, {"tactic": "rw [Metric.emetric_ball_nnreal, volume_ball, two_mul, \u2190 NNReal.coe_add,\n  ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]", "annotated_tactic": ["rw [<a>Metric.emetric_ball_nnreal</a>, <a>volume_ball</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_ball_nnreal", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1214, 9], "def_end_pos": [1214, 35]}, {"full_name": "Real.volume_ball", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [108, 9], "def_end_pos": [108, 20]}, {"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.ball a \u2191r) = 2 * \u2191r", "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_add_mkRat", "start": [200, 1], "end": [202, 101], "traced_tactics": [{"tactic": "rw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_add_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_add_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_add_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [191, 9], "def_end_pos": [191, 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 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\n\u22a2 mkRat n\u2081 d\u2081 + mkRat n\u2082 d\u2082 = mkRat (n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) (d\u2081 * d\u2082)", "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_condexpL1", "start": [558, 1], "end": [562, 78], "traced_tactics": [{"tactic": "simp_rw [condexpL1_eq hf]", "annotated_tactic": ["simp_rw [<a>condexpL1_eq</a> hf]", [{"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\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": "\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\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hf)) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "rw [set_integral_condexpL1Clm (hf.toL1 f) hs]", "annotated_tactic": ["rw [<a>set_integral_condexpL1Clm</a> (hf.toL1 f) hs]", [{"full_name": "MeasureTheory.set_integral_condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [430, 9], "def_end_pos": [430, 34]}]], "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\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hf)) x \u2202\u03bc = \u222b (x : \u03b1) in s, 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 g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(Integrable.toL1 f hf) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "exact set_integral_congr_ae (hm s hs) (hf.coeFn_toL1.mono fun x hx _ => hx)", "annotated_tactic": ["exact <a>set_integral_congr_ae</a> (hm s hs) (hf.coeFn_toL1.mono 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]}]], "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\nhf : Integrable f\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(Integrable.toL1 f hf) 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/Int/DivMod.lean", "full_name": "Int.div_def'", "start": [81, 9], "end": [82, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_compl", "start": [164, 1], "end": [167, 58], "traced_tactics": [{"tactic": "rw [\u2190 condCount_union hs disjoint_compl_right, Set.union_compl_self,\n  (condCount_isProbabilityMeasure hs hs').measure_univ]", "annotated_tactic": ["rw [\u2190 <a>condCount_union</a> hs <a>disjoint_compl_right</a>, <a>Set.union_compl_self</a>,\n    (<a>condCount_isProbabilityMeasure</a> hs hs').<a>measure_univ</a>]", [{"full_name": "ProbabilityTheory.condCount_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [157, 9], "def_end_pos": [157, 24]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 29]}, {"full_name": "Set.union_compl_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1739, 9], "def_end_pos": [1739, 25]}, {"full_name": "ProbabilityTheory.condCount_isProbabilityMeasure", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [81, 9], "def_end_pos": [81, 39]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u t : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191(condCount s) t + \u2191\u2191(condCount s) t\u1d9c = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2.stmts_supportsStmt", "start": [2242, 1], "end": [2246, 60], "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.TM2.stmts", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2223, 19], "def_end_pos": [2223, 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": "K : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq : Stmt\u2082\nss : Supports M S\n\u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q", "state_after": "K : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq : Stmt\u2082\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q"}, {"tactic": "exact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls)", "annotated_tactic": ["exact fun l ls h \u21a6 <a>stmts\u2081_supportsStmt_mono</a> h (ss.2 _ ls)", [{"full_name": "Turing.TM2.stmts\u2081_supportsStmt_mono", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2211, 9], "def_end_pos": [2211, 33]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq : Stmt\u2082\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q", "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.eraseP_toList", "start": [174, 9], "end": [176, 63], "traced_tactics": [{"tactic": "induction l <;> simp [List.eraseP, cond]", "annotated_tactic": ["induction l <;> simp [<a>List.eraseP</a>, <a>cond</a>]", [{"full_name": "List.eraseP", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [980, 5], "def_end_pos": [980, 11]}, {"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\np : \u03b1 \u2192 \u03b2 \u2192 Bool\nl : AssocList \u03b1 \u03b2\n\u22a2 toList (eraseP p l) =\n    List.eraseP\n      (fun x =>\n        match x with\n        | (a, b) => p a b)\n      (toList l)", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np : \u03b1 \u2192 \u03b2 \u2192 Bool\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  toList (eraseP p tail\u271d) =\n    List.eraseP\n      (fun x =>\n        match x with\n        | (a, b) => p a b)\n      (toList tail\u271d)\n\u22a2 toList\n      (match p key\u271d value\u271d with\n      | true => tail\u271d\n      | false => cons key\u271d value\u271d (eraseP p tail\u271d)) =\n    match p key\u271d value\u271d with\n    | true => toList tail\u271d\n    | false => (key\u271d, value\u271d) :: List.eraseP (fun x => p x.fst x.snd) (toList tail\u271d)"}, {"tactic": "split <;> simp [*]", "annotated_tactic": ["split <;> simp [*]", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np : \u03b1 \u2192 \u03b2 \u2192 Bool\nkey\u271d : \u03b1\nvalue\u271d : \u03b2\ntail\u271d : AssocList \u03b1 \u03b2\ntail_ih\u271d :\n  toList (eraseP p tail\u271d) =\n    List.eraseP\n      (fun x =>\n        match x with\n        | (a, b) => p a b)\n      (toList tail\u271d)\n\u22a2 toList\n      (match p key\u271d value\u271d with\n      | true => tail\u271d\n      | false => cons key\u271d value\u271d (eraseP p tail\u271d)) =\n    match p key\u271d value\u271d with\n    | true => toList tail\u271d\n    | false => (key\u271d, value\u271d) :: List.eraseP (fun x => p x.fst x.snd) (toList tail\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.Measure.map_div_left_ae", "start": [534, 1], "end": [536, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector.lean", "full_name": "Vector.toList_take", "start": [286, 1], "end": [288, 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 (take n v) = List.take 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 (take n { val := val\u271d, property := property\u271d }) = List.take 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 (take n { val := val\u271d, property := property\u271d }) = List.take n (toList { val := val\u271d, property := property\u271d })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.prod_set'", "start": [649, 1], "end": [652, 36], "traced_tactics": [{"tactic": "refine' (List.prod_set' v.toList i a).trans _", "annotated_tactic": ["refine' (<a>List.prod_set'</a> v.toList i a).<a>trans</a> _", [{"full_name": "List.prod_set'", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 18]}, {"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": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.prod (toList (set v i a)) = List.prod (toList v) * (get v i)\u207b\u00b9 * a", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 (List.prod (toList v) * if hn : \u2191i < List.length (toList v) then (List.nthLe (toList v) (\u2191i) hn)\u207b\u00b9 * a else 1) =\n    List.prod (toList v) * (get v i)\u207b\u00b9 * a"}, {"tactic": "simp [get_eq_get, mul_assoc]", "annotated_tactic": ["simp [<a>get_eq_get</a>, <a>mul_assoc</a>]", [{"full_name": "Vector.get_eq_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 19]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 (List.prod (toList v) * if hn : \u2191i < List.length (toList v) then (List.nthLe (toList v) (\u2191i) hn)\u207b\u00b9 * a else 1) =\n    List.prod (toList v) * (get v i)\u207b\u00b9 * a", "state_after": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.nthLe (toList v) \u2191i (_ : \u2191i < List.length (toList v)) =\n    List.get (toList v) (Fin.cast (_ : n = List.length (toList v)) i)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.nthLe (toList v) \u2191i (_ : \u2191i < List.length (toList v)) =\n    List.get (toList v) (Fin.cast (_ : n = List.length (toList v)) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Infinite.lean", "full_name": "Set.Ico.infinite", "start": [52, 1], "end": [53, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict", "start": [787, 1], "end": [792, 52], "traced_tactics": [{"tactic": "simpa [mem_restrict_range hs, h0, -mem_range] using hr", "annotated_tactic": ["simpa [<a>mem_restrict_range</a> hs, h0, -<a>mem_range</a>] using hr", [{"full_name": "MeasureTheory.SimpleFunc.mem_restrict_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [782, 9], "def_end_pos": [782, 27]}, {"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\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d : Zero \u03b2\nr : \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192\u209b \u03b2\nhr : r \u2208 SimpleFunc.range (restrict f s)\nh0 : r \u2260 0\nhs : MeasurableSet s\n\u22a2 r \u2208 \u2191f '' s", "state_after": "no goals"}, {"tactic": "rw [restrict_of_not_measurable hs] at hr", "annotated_tactic": ["rw [<a>restrict_of_not_measurable</a> hs] at hr", [{"full_name": "MeasureTheory.SimpleFunc.restrict_of_not_measurable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [732, 9], "def_end_pos": [732, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d : Zero \u03b2\nr : \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192\u209b \u03b2\nhr : r \u2208 SimpleFunc.range (restrict f s)\nh0 : r \u2260 0\nhs : \u00acMeasurableSet s\n\u22a2 r \u2208 \u2191f '' s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d : Zero \u03b2\nr : \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192\u209b \u03b2\nhr : r \u2208 SimpleFunc.range 0\nh0 : r \u2260 0\nhs : \u00acMeasurableSet s\n\u22a2 r \u2208 \u2191f '' s"}, {"tactic": "exact (h0 <| eq_zero_of_mem_range_zero hr).elim", "annotated_tactic": ["exact (h0 <| <a>eq_zero_of_mem_range_zero</a> hr).<a>elim</a>", [{"full_name": "MeasureTheory.SimpleFunc.eq_zero_of_mem_range_zero", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [545, 9], "def_end_pos": [545, 34]}, {"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nK : Type u_5\ninst\u271d : Zero \u03b2\nr : \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192\u209b \u03b2\nhr : r \u2208 SimpleFunc.range 0\nh0 : r \u2260 0\nhs : \u00acMeasurableSet s\n\u22a2 r \u2208 \u2191f '' s", "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_cons_addChar", "start": [446, 1], "end": [449, 49], "traced_tactics": [{"tactic": "simp [extract, Nat.add_le_add_iff_right]", "annotated_tactic": ["simp [<a>extract</a>, <a>Nat.add_le_add_iff_right</a>]", [{"full_name": "String.extract", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [220, 5], "def_end_pos": [220, 12]}, {"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]}]], "state_before": "c : Char\ncs : List Char\nb e : Pos\n\u22a2 extract { data := c :: cs } (b + c) (e + c) = extract { data := cs } b e", "state_after": "c : Char\ncs : List Char\nb e : Pos\n\u22a2 (if e.byteIdx \u2264 b.byteIdx then \"\" else { data := extract.go\u2081 (c :: cs) 0 (b + c) (e + c) }) =\n    if b.byteIdx \u2265 e.byteIdx then \"\" else { data := extract.go\u2081 cs 0 b e }"}, {"tactic": "split <;> [rfl; rw [extract.go\u2081_cons_addChar]]", "annotated_tactic": ["split <;> [rfl; rw [<a>extract.go\u2081_cons_addChar</a>]]", [{"full_name": "String.extract.go\u2081_cons_addChar", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [430, 9], "def_end_pos": [430, 33]}]], "state_before": "c : Char\ncs : List Char\nb e : Pos\n\u22a2 (if e.byteIdx \u2264 b.byteIdx then \"\" else { data := extract.go\u2081 (c :: cs) 0 (b + c) (e + c) }) =\n    if b.byteIdx \u2265 e.byteIdx then \"\" else { data := extract.go\u2081 cs 0 b e }", "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_cast", "start": [1557, 1], "end": [1558, 53], "traced_tactics": [{"tactic": "rw [\u2190 Int.cast_ofNat, of_int_cast, Int.cast_ofNat]", "annotated_tactic": ["rw [\u2190 <a>Int.cast_ofNat</a>, <a>of_int_cast</a>, <a>Int.cast_ofNat</a>]", [{"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}, {"full_name": "ZNum.of_int_cast", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1552, 9], "def_end_pos": [1552, 20]}, {"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : \u2115\n\u22a2 \u2191\u2191n = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.measurable_of_pdf_ne_zero", "start": [102, 1], "end": [105, 45], "traced_tactics": [{"tactic": "by_contra hX", "annotated_tactic": ["by_contra hX", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nh : pdf X \u2119 \u2260 0\n\u22a2 Measurable X", "state_after": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nh : pdf X \u2119 \u2260 0\nhX : \u00acMeasurable X\n\u22a2 False"}, {"tactic": "exact h (pdf_eq_zero_of_not_measurable hX)", "annotated_tactic": ["exact h (<a>pdf_eq_zero_of_not_measurable</a> hX)", [{"full_name": "MeasureTheory.pdf_eq_zero_of_not_measurable", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [97, 9], "def_end_pos": [97, 38]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nh : pdf X \u2119 \u2260 0\nhX : \u00acMeasurable X\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux", "start": [723, 1], "end": [731, 6], "traced_tactics": [{"tactic": "refine' le_trans (le_of_eq _)\n  (integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_const _ _ _)).pos\n    (fun \u03c9 => LatticeOrderedGroup.pos_nonneg _)\n    (fun \u03c9 => LatticeOrderedGroup.pos_nonneg _) (sub_pos.2 hab))", "annotated_tactic": ["refine' <a>le_trans</a> (<a>le_of_eq</a> _)\n    (<a>integral_mul_upcrossingsBefore_le_integral</a> (hf.sub_martingale (<a>martingale_const</a> _ _ _)).<a>pos</a>\n      (fun \u03c9 => <a>LatticeOrderedGroup.pos_nonneg</a> _)\n      (fun \u03c9 => <a>LatticeOrderedGroup.pos_nonneg</a> _) (<a>sub_pos</a>.2 hab))", [{"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.integral_mul_upcrossingsBefore_le_integral", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [655, 9], "def_end_pos": [655, 51]}, {"full_name": "MeasureTheory.martingale_const", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [70, 9], "def_end_pos": [70, 25]}, {"full_name": "MeasureTheory.Submartingale.pos", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [280, 19], "def_end_pos": [280, 22]}, {"full_name": "LatticeOrderedGroup.pos_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [200, 15], "def_end_pos": [200, 25]}, {"full_name": "LatticeOrderedGroup.pos_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [200, 15], "def_end_pos": [200, 25]}, {"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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\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": "\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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc =\n    (b - a - 0) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (f - fun x x => a)\u207a N x) \u2202\u03bc"}, {"tactic": "simp_rw [sub_zero, \u2190 upcrossingsBefore_pos_eq hab]", "annotated_tactic": ["simp_rw [<a>sub_zero</a>, \u2190 <a>upcrossingsBefore_pos_eq</a> hab]", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "MeasureTheory.upcrossingsBefore_pos_eq", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [718, 9], "def_end_pos": [718, 33]}]], "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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc =\n    (b - a - 0) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (f - fun x x => a)\u207a N x) \u2202\u03bc", "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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N x) \u2202\u03bc =\n    (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (f - fun x x => a)\u207a N x) \u2202\u03bc"}, {"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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N x) \u2202\u03bc =\n    (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (f - fun x x => a)\u207a N x) \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.monotone_powerset", "start": [2178, 1], "end": [2178, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min'_lt_max'", "start": [1485, 1], "end": [1487, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id", "start": [44, 1], "end": [49, 11], "traced_tactics": [{"tactic": "rw [\u2190 aestronglyMeasurable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf", "annotated_tactic": ["rw [\u2190 <a>aestronglyMeasurable_comp_snd_map_prod_mk_iff</a> (<a>measurable_id''</a> hm)] at hf", [{"full_name": "ProbabilityTheory.aestronglyMeasurable_comp_snd_map_prod_mk_iff", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [325, 9], "def_end_pos": [325, 54]}, {"full_name": "measurable_id''", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [239, 9], "def_end_pos": [239, 24]}]], "state_before": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : TopologicalSpace F\nhm : m \u2264 m\u03a9\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, id \u03c9)) \u03bc)", "state_after": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : TopologicalSpace F\nhm : m \u2264 m\u03a9\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, \u03c9)) \u03bc)\n\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, id \u03c9)) \u03bc)"}, {"tactic": "simp_rw [id.def] at hf \u22a2", "annotated_tactic": ["simp_rw [<a>id.def</a>] at hf \u22a2", [{"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": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : TopologicalSpace F\nhm : m \u2264 m\u03a9\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, \u03c9)) \u03bc)\n\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, id \u03c9)) \u03bc)", "state_after": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : TopologicalSpace F\nhm : m \u2264 m\u03a9\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)\n\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)"}, {"tactic": "exact hf", "annotated_tactic": ["exact hf", []], "state_before": "\u03a9 : Type u_1\nF : Type u_2\nm m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nf : \u03a9 \u2192 F\ninst\u271d : TopologicalSpace F\nhm : m \u2264 m\u03a9\nhf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)\n\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "full_name": "MeasureTheory.L2.integrable_inner", "start": [191, 1], "end": [195, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.mk'_left", "start": [550, 1], "end": [551, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "Function.Periodic.tendsto_atTop_intervalIntegral_of_pos", "start": [340, 1], "end": [345, 79], "traced_tactics": [{"tactic": "apply tendsto_atTop_mono (hg.sInf_add_zsmul_le_integral_of_pos h_int hT)", "annotated_tactic": ["apply <a>tendsto_atTop_mono</a> (hg.sInf_add_zsmul_le_integral_of_pos h_int hT)", [{"full_name": "Filter.tendsto_atTop_mono", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [409, 9], "def_end_pos": [409, 27]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun t => \u222b (x : \u211d) in 0 ..t, g x) atTop atTop", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun n => sInf ((fun t => \u222b (x : \u211d) in 0 ..t, g x) '' Icc 0 T) + \u230an / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atTop\n    atTop"}, {"tactic": "apply atTop.tendsto_atTop_add_const_left (sInf <| (fun t => \u222b x in (0)..t, g x) '' Icc 0 T)", "annotated_tactic": ["apply atTop.tendsto_atTop_add_const_left (<a>sInf</a> <| (fun t => \u222b x in (0)..t, g x) '' <a>Icc</a> 0 T)", [{"full_name": "InfSet.sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [62, 3], "def_end_pos": [62, 7]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun n => sInf ((fun t => \u222b (x : \u211d) in 0 ..t, g x) '' Icc 0 T) + \u230an / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atTop\n    atTop", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atTop atTop"}, {"tactic": "apply Tendsto.atTop_zsmul_const h\u2080", "annotated_tactic": ["apply <a>Tendsto.atTop_zsmul_const</a> h\u2080", [{"full_name": "Filter.Tendsto.atTop_zsmul_const", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [248, 9], "def_end_pos": [248, 34]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atTop atTop", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b) atTop atTop"}, {"tactic": "exact tendsto_floor_atTop.comp (tendsto_id.atTop_mul_const (inv_pos.mpr hT))", "annotated_tactic": ["exact tendsto_floor_atTop.comp (tendsto_id.atTop_mul_const (inv_pos.mpr hT))", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\ng : \u211d \u2192 \u211d\nhg : Periodic g T\nh_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082\nh\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x\nhT : 0 < T\n\u22a2 Tendsto (fun x => \u230ax / T\u230b) atTop atTop", "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_linearMap_volume_pi_eq_smul_volume_pi", "start": [446, 1], "end": [456, 18], "traced_tactics": [{"tactic": "classical\n  let M := LinearMap.toMatrix' f\n  have A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']\n  have B : f = toLin' M := by simp only [toLin'_toMatrix']\n  rw [A, B]\n  apply map_matrix_volume_pi_eq_smul_volume_pi\n  rwa [A] at hf", "annotated_tactic": ["classical\n    -- this is deduced from the matrix case\n    let M := <a>LinearMap.toMatrix'</a> f\n    have A : <a>LinearMap.det</a> f = <a>det</a> M := by simp only [<a>LinearMap.det_toMatrix'</a>]\n    have B : f = <a>toLin'</a> M := by simp only [<a>toLin'_toMatrix'</a>]\n    rw [A, B]\n    apply <a>map_matrix_volume_pi_eq_smul_volume_pi</a>\n    rwa [A] at hf", [{"full_name": "LinearMap.toMatrix'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [284, 5], "def_end_pos": [284, 24]}, {"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [178, 27], "def_end_pos": [178, 30]}, {"full_name": "Matrix.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [65, 8], "def_end_pos": [65, 11]}, {"full_name": "LinearMap.det_toMatrix'", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [217, 9], "def_end_pos": [217, 22]}, {"full_name": "Matrix.toLin'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [305, 5], "def_end_pos": [305, 18]}, {"full_name": "Matrix.toLin'_toMatrix'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [331, 9], "def_end_pos": [331, 32]}, {"full_name": "Real.map_matrix_volume_pi_eq_smul_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [423, 9], "def_end_pos": [423, 47]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume", "state_after": "no goals"}, {"tactic": "let M := LinearMap.toMatrix' f", "annotated_tactic": ["let M := <a>LinearMap.toMatrix'</a> f", [{"full_name": "LinearMap.toMatrix'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [284, 5], "def_end_pos": [284, 24]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume"}, {"tactic": "have A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']", "annotated_tactic": ["have A : <a>LinearMap.det</a> f = <a>det</a> M := by simp only [<a>LinearMap.det_toMatrix'</a>]", [{"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [178, 27], "def_end_pos": [178, 30]}, {"full_name": "Matrix.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [65, 8], "def_end_pos": [65, 11]}, {"full_name": "LinearMap.det_toMatrix'", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [217, 9], "def_end_pos": [217, 22]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume"}, {"tactic": "have B : f = toLin' M := by simp only [toLin'_toMatrix']", "annotated_tactic": ["have B : f = <a>toLin'</a> M := by simp only [<a>toLin'_toMatrix'</a>]", [{"full_name": "Matrix.toLin'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [305, 5], "def_end_pos": [305, 18]}, {"full_name": "Matrix.toLin'_toMatrix'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [331, 9], "def_end_pos": [331, 32]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume"}, {"tactic": "rw [A, B]", "annotated_tactic": ["rw [A, B]", []], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume"}, {"tactic": "apply map_matrix_volume_pi_eq_smul_volume_pi", "annotated_tactic": ["apply <a>map_matrix_volume_pi_eq_smul_volume_pi</a>", [{"full_name": "Real.map_matrix_volume_pi_eq_smul_volume_pi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [423, 9], "def_end_pos": [423, 47]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume", "state_after": "case hM\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 det M \u2260 0"}, {"tactic": "rwa [A] at hf", "annotated_tactic": ["rwa [A] at hf", []], "state_before": "case hM\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 det M \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [LinearMap.det_toMatrix']", "annotated_tactic": ["simp only [<a>LinearMap.det_toMatrix'</a>]", [{"full_name": "LinearMap.det_toMatrix'", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [217, 9], "def_end_pos": [217, 22]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 \u2191LinearMap.det f = det M", "state_after": "no goals"}, {"tactic": "simp only [toLin'_toMatrix']", "annotated_tactic": ["simp only [<a>toLin'_toMatrix'</a>]", [{"full_name": 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[]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.finStronglyMeasurable_iff_measurable", "start": [2008, 1], "end": [2010, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image_image\u2082_right_comm", "start": [440, 1], "end": [443, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.trim_smul", "start": [1749, 1], "end": [1751, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.char_rmatch_iff", "start": [224, 1], "end": [235, 65], "traced_tactics": [{"tactic": "cases' x with _ x", "annotated_tactic": ["cases' x with _ x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a) x = true \u2194 x = [a]", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a : \u03b1\n\u22a2 rmatch (char a) [] = true \u2194 [] = [a]\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a) (head\u271d :: x) = true \u2194 head\u271d :: x = [a]"}, {"tactic": "cases' x with head tail", "annotated_tactic": ["cases' x with head tail", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nx : List \u03b1\n\u22a2 rmatch (char a) (head\u271d :: x) = true \u2194 head\u271d :: x = [a]", "state_after": "case cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\n\u22a2 rmatch (char a) [head\u271d] = true \u2194 [head\u271d] = [a]\n\ncase cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\n\u22a2 rmatch (char a) (head\u271d :: head :: tail) = true \u2194 head\u271d :: head :: tail = [a]"}, {"tactic": "exact of_decide_eq_true rfl", "annotated_tactic": ["exact <a>of_decide_eq_true</a> <a>rfl</a>", [{"full_name": "of_decide_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [855, 9], "def_end_pos": [855, 26]}, {"full_name": "rfl", "def_path": 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: Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\n\u22a2 rmatch (if a = head\u271d then 1 else 0) [] = true \u2194 [head\u271d] = [a]"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\n\u22a2 rmatch (if a = head\u271d then 1 else 0) [] = true \u2194 [head\u271d] = [a]", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nh\u271d : a = head\u271d\n\u22a2 rmatch 1 [] = true \u2194 [head\u271d] = [a]\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nh\u271d : \u00aca = head\u271d\n\u22a2 rmatch 0 [] = true \u2194 [head\u271d] = [a]"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nh\u271d : a = head\u271d\n\u22a2 rmatch 1 [] = true \u2194 [head\u271d] = [a]", "state_after": "no goals"}, {"tactic": "simp [List.singleton_inj]", "annotated_tactic": ["simp [<a>List.singleton_inj</a>]", [{"full_name": "List.singleton_inj", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [70, 9], "def_end_pos": [70, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nh\u271d : \u00aca = head\u271d\n\u22a2 rmatch 0 [] = true \u2194 [head\u271d] = [a]", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nh\u271d : \u00aca = head\u271d\n\u22a2 \u00achead\u271d = a"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d : \u03b1\nh\u271d : \u00aca = head\u271d\n\u22a2 \u00achead\u271d = a", "state_after": "no goals"}, {"tactic": "rw [rmatch, rmatch, deriv]", "annotated_tactic": ["rw [<a>rmatch</a>, <a>rmatch</a>, <a>deriv</a>]", [{"full_name": "RegularExpression.rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [210, 5], "def_end_pos": [210, 11]}, {"full_name": "RegularExpression.rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [210, 5], "def_end_pos": [210, 11]}, {"full_name": "RegularExpression.deriv", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [169, 5], "def_end_pos": [169, 10]}]], "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\n\u22a2 rmatch (char a) (head\u271d :: head :: tail) = true \u2194 head\u271d :: head :: tail = [a]", "state_after": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\n\u22a2 rmatch (deriv (if a = head\u271d then 1 else 0) head) tail = true \u2194 head\u271d :: head :: tail = [a]"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\n\u22a2 rmatch (deriv (if a = head\u271d then 1 else 0) head) tail = true \u2194 head\u271d :: head :: tail = [a]", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\nh : a = head\u271d\n\u22a2 rmatch (deriv 1 head) tail = true \u2194 head\u271d :: head :: tail = [a]\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\nh : \u00aca = head\u271d\n\u22a2 rmatch (deriv 0 head) tail = true \u2194 head\u271d :: head :: tail = [a]"}, {"tactic": "simp only [deriv_one, zero_rmatch, cons.injEq, and_false]", "annotated_tactic": ["simp only [<a>deriv_one</a>, <a>zero_rmatch</a>, cons.injEq, <a>and_false</a>]", [{"full_name": "RegularExpression.deriv_one", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [184, 9], "def_end_pos": [184, 18]}, {"full_name": "RegularExpression.zero_rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [216, 9], "def_end_pos": [216, 20]}, {"full_name": "and_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [85, 17], "def_end_pos": [85, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\nh : a = head\u271d\n\u22a2 rmatch (deriv 1 head) tail = true \u2194 head\u271d :: head :: tail = [a]", "state_after": "no goals"}, {"tactic": "simp only [deriv_zero, zero_rmatch, cons.injEq, and_false]", "annotated_tactic": ["simp only [<a>deriv_zero</a>, <a>zero_rmatch</a>, cons.injEq, <a>and_false</a>]", [{"full_name": "RegularExpression.deriv_zero", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [179, 9], "def_end_pos": [179, 19]}, {"full_name": "RegularExpression.zero_rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [216, 9], "def_end_pos": [216, 20]}, {"full_name": "and_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [85, 17], "def_end_pos": [85, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na\u271d b a head\u271d head : \u03b1\ntail : List \u03b1\nh : \u00aca = head\u271d\n\u22a2 rmatch (deriv 0 head) tail = true \u2194 head\u271d :: head :: tail = [a]", "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.size_uset", "start": [229, 1], "end": [229, 82], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nv : \u03b1\ni : USize\nh : USize.toNat i < size a\n\u22a2 size (uset a i v h) = size a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr", "start": [380, 1], "end": [389, 17], "traced_tactics": [{"tactic": "refine' setToSimpleFunc_congr' T h_add hf ((integrable_congr h).mp hf) _", "annotated_tactic": ["refine' <a>setToSimpleFunc_congr'</a> T h_add hf ((<a>integrable_congr</a> h).<a>mp</a> hf) _", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [361, 9], "def_end_pos": [361, 31]}, {"full_name": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}, {"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\nE : Type u_2\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191f =\u1d50[\u03bc] \u2191g\n\u22a2 setToSimpleFunc T f = setToSimpleFunc 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\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191f =\u1d50[\u03bc] \u2191g\n\u22a2 \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0"}, {"tactic": "refine' fun x y hxy => h_zero _ ((measurableSet_fiber f x).inter (measurableSet_fiber g y)) _", "annotated_tactic": ["refine' fun x y hxy => h_zero _ ((<a>measurableSet_fiber</a> f x).<a>inter</a> (<a>measurableSet_fiber</a> g y)) _", [{"full_name": "MeasureTheory.SimpleFunc.measurableSet_fiber", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"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\nE : Type u_2\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191f =\u1d50[\u03bc] \u2191g\n\u22a2 \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 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\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191f =\u1d50[\u03bc] \u2191g\nx y : E\nhxy : x \u2260 y\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0"}, {"tactic": "rw [EventuallyEq, ae_iff] at h", "annotated_tactic": ["rw [<a>EventuallyEq</a>, <a>ae_iff</a>] at h", [{"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\nE : Type u_2\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191f =\u1d50[\u03bc] \u2191g\nx y : E\nhxy : x \u2260 y\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 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\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0"}, {"tactic": "refine' measure_mono_null (fun z => _) h", "annotated_tactic": ["refine' <a>measure_mono_null</a> (fun z => _) h", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 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\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 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\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\nz : \u03b1\n\u22a2 z \u2208 \u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y} \u2192 z \u2208 {a | \u00ac\u2191f a = \u2191g a}"}, {"tactic": "simp_rw [Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_preimage, Set.mem_singleton_iff]", "annotated_tactic": ["simp_rw [<a>Set.mem_inter_iff</a>, <a>Set.mem_setOf_eq</a>, <a>Set.mem_preimage</a>, <a>Set.mem_singleton_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_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": "\u03b1 : Type u_1\nE : Type u_2\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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\nz : \u03b1\n\u22a2 z \u2208 \u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y} \u2192 z \u2208 {a | \u00ac\u2191f a = \u2191g a}", "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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\nz : \u03b1\n\u22a2 \u2191f z = x \u2227 \u2191g z = y \u2192 \u00ac\u2191f z = \u2191g z"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\nz : \u03b1\n\u22a2 \u2191f z = x \u2227 \u2191g z = y \u2192 \u00ac\u2191f z = \u2191g z", "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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh\u271d : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\nz : \u03b1\nh : \u2191f z = x \u2227 \u2191g z = y\n\u22a2 \u00ac\u2191f z = \u2191g z"}, {"tactic": "rwa [h.1, h.2]", "annotated_tactic": ["rwa [h.1, h.2]", []], "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\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\nf g : \u03b1 \u2192\u209b E\nhf : Integrable \u2191f\nh\u271d : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0\nx y : E\nhxy : x \u2260 y\nz : \u03b1\nh : \u2191f z = x \u2227 \u2191g z = y\n\u22a2 \u00ac\u2191f z = \u2191g z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_strictMono", "start": [732, 1], "end": [733, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to1.stepAux_read", "start": [1818, 1], "end": [1846, 6], "traced_tactics": [{"tactic": "suffices \u2200 f, stepAux (readAux n f) v (trTape' enc0 L R) =\n    stepAux (f (enc R.head)) v (trTape' enc0 (L.cons R.head) R.tail) by\n  rw [read, this, stepAux_move, encdec, trTape'_move_left enc0]\n  simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons]", "annotated_tactic": ["suffices \u2200 f, <a>stepAux</a> (<a>readAux</a> n f) v (<a>trTape'</a> enc0 L R) =\n      <a>stepAux</a> (f (enc R.head)) v (<a>trTape'</a> enc0 (L.cons R.head) R.tail) by\n    rw [<a>read</a>, this, <a>stepAux_move</a>, encdec, <a>trTape'_move_left</a> enc0]\n    simp only [<a>ListBlank.head_cons</a>, <a>ListBlank.cons_head_tail</a>, <a>ListBlank.tail_cons</a>]", [{"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1657, 5], "def_end_pos": [1657, 12]}, {"full_name": "Turing.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1734, 5], "def_end_pos": [1734, 12]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1734, 5], "def_end_pos": [1734, 12]}, {"full_name": "Turing.TM1to1.read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1675, 5], "def_end_pos": [1675, 9]}, {"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.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]}]], "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L R) =\n      stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail 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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L R) =\n      stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail 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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L (ListBlank.cons a R)) =\n      stepAux (f (enc (ListBlank.head (ListBlank.cons a R)))) v\n        (trTape' enc0 (ListBlank.cons (ListBlank.head (ListBlank.cons a R)) L) (ListBlank.tail (ListBlank.cons a R)))"}, {"tactic": "simp only [ListBlank.head_cons, ListBlank.tail_cons, trTape', ListBlank.cons_bind,\n  ListBlank.append_assoc]", "annotated_tactic": ["simp only [<a>ListBlank.head_cons</a>, <a>ListBlank.tail_cons</a>, <a>trTape'</a>, <a>ListBlank.cons_bind</a>,\n    <a>ListBlank.append_assoc</a>]", [{"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.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1734, 5], "def_end_pos": [1734, 12]}, {"full_name": "Turing.ListBlank.cons_bind", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [485, 9], "def_end_pos": [485, 28]}, {"full_name": "Turing.ListBlank.append_assoc", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [459, 9], "def_end_pos": [459, 31]}]], "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L (ListBlank.cons a R)) =\n      stepAux (f (enc (ListBlank.head (ListBlank.cons a R)))) v\n        (trTape' enc0 (ListBlank.cons (ListBlank.head (ListBlank.cons a R)) L) (ListBlank.tail (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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v\n        (Tape.mk'\n          (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n            (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n          (ListBlank.append (Vector.toList (enc a))\n            (ListBlank.bind R (fun x => Vector.toList (enc x))\n              (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n      stepAux (f (enc a)) v\n        (Tape.mk'\n          (ListBlank.append (List.reverse (Vector.toList (enc a)))\n            (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n              (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n          (ListBlank.bind R (fun x => Vector.toList (enc x))\n            (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))"}, {"tactic": "suffices \u2200 i f L' R' l\u2081 l\u2082 h,\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f \u27e8l\u2082, h\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R') by\n  intro f\n  exact this n f (L.bind (fun x => (enc x).1.reverse) _)\n    (R.bind (fun x => (enc x).1) _) [] _ (enc a).2", "annotated_tactic": ["suffices \u2200 i f L' R' l\u2081 l\u2082 h,\n      <a>stepAux</a> (<a>readAux</a> i f) v (<a>Tape.mk'</a> (<a>ListBlank.append</a> l\u2081 L') (<a>ListBlank.append</a> l\u2082 R')) =\n      <a>stepAux</a> (f \u27e8l\u2082, h\u27e9) v (<a>Tape.mk'</a> (<a>ListBlank.append</a> (l\u2082.reverseAux l\u2081) L') R') by\n    intro f\n    -- Porting note: Here was `change`.\n    exact this n f (L.bind (fun x => (enc x).1.<a>reverse</a>) _)\n      (R.bind (fun x => (enc x).1) _) [] _ (enc a).2", [{"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1657, 5], "def_end_pos": [1657, 12]}, {"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [545, 5], "def_end_pos": [545, 13]}, {"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}, {"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}, {"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": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}, {"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": "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v\n        (Tape.mk'\n          (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n            (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n          (ListBlank.append (Vector.toList (enc a))\n            (ListBlank.bind R (fun x => Vector.toList (enc x))\n              (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n      stepAux (f (enc a)) v\n        (Tape.mk'\n          (ListBlank.append (List.reverse (Vector.toList (enc a)))\n            (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n              (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n          (ListBlank.bind R (fun x => Vector.toList (enc x))\n            (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))", "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')"}, {"tactic": "clear f L a R", "annotated_tactic": ["clear f L a R", []], "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) 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\nv : \u03c3\n\u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')"}, {"tactic": "intro i f L' R' l\u2081 l\u2082 _", "annotated_tactic": ["intro i f L' R' l\u2081 l\u2082 _", []], "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\nv : \u03c3\n\u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) 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\nv : \u03c3\ni : \u2115\nf : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 : List Bool\nh\u271d : List.length l\u2082 = i\n\u22a2 stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f { val := l\u2082, property := h\u271d }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')"}, {"tactic": "subst i", "annotated_tactic": ["subst i", []], "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\nv : \u03c3\ni : \u2115\nf : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 : List Bool\nh\u271d : List.length l\u2082 = i\n\u22a2 stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f { val := l\u2082, property := h\u271d }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 : List Bool\nf : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')"}, {"tactic": "induction' l\u2082 with a l\u2082 IH generalizing l\u2081", "annotated_tactic": ["induction' l\u2082 with a l\u2082 IH generalizing 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 : List Bool\nf : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')", "state_after": "case intro.intro.nil\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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082 : List Bool\nf\u271d : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3\nl\u2081 : List Bool\nf : Vector Bool (List.length []) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length []) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n    stepAux (f { val := [], property := (_ : List.length [] = List.length []) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux [] l\u2081) L') R')\n\ncase intro.intro.cons\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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')"}, {"tactic": "trans\n  stepAux (readAux l\u2082.length fun v \u21a6 f (a ::\u1d65 v)) v\n    (Tape.mk' ((L'.append l\u2081).cons a) (R'.append l\u2082))", "annotated_tactic": ["trans\n    <a>stepAux</a> (<a>readAux</a> l\u2082.length fun v \u21a6 f (a ::\u1d65 v)) v\n      (<a>Tape.mk'</a> ((L'.append l\u2081).<a>cons</a> a) (R'.append l\u2082))", [{"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1281, 5], "def_end_pos": [1281, 12]}, {"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1657, 5], "def_end_pos": [1657, 12]}, {"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [545, 5], "def_end_pos": [545, 13]}, {"full_name": "Turing.ListBlank.cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [240, 5], "def_end_pos": [240, 19]}]], "state_before": "case intro.intro.cons\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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n\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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) =\n    stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')"}, {"tactic": "rw [\u2190 ListBlank.append, IH]", "annotated_tactic": ["rw [\u2190 <a>ListBlank.append</a>, IH]", [{"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [447, 5], "def_end_pos": [447, 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) =\n    stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (f (a ::\u1d65 { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) })) v\n      (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 (a :: l\u2081)) L') R') =\n    stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (f (a ::\u1d65 { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) })) v\n      (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 (a :: l\u2081)) L') R') =\n    stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')", "state_after": "no goals"}, {"tactic": "rw [read, this, stepAux_move, encdec, trTape'_move_left enc0]", "annotated_tactic": ["rw [<a>read</a>, this, <a>stepAux_move</a>, encdec, <a>trTape'_move_left</a> enc0]", [{"full_name": "Turing.TM1to1.read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1675, 5], "def_end_pos": [1675, 9]}, {"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]}]], "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\nthis :\n  \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L R) =\n      stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n\u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\nthis :\n  \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L R) =\n      stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n\u22a2 stepAux (f (ListBlank.head R)) v\n      (trTape' enc0 (ListBlank.tail (ListBlank.cons (ListBlank.head R) L))\n        (ListBlank.cons (ListBlank.head (ListBlank.cons (ListBlank.head R) L)) (ListBlank.tail R))) =\n    stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)"}, {"tactic": "simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons]", "annotated_tactic": ["simp only [<a>ListBlank.head_cons</a>, <a>ListBlank.cons_head_tail</a>, <a>ListBlank.tail_cons</a>]", [{"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.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]}]], "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\nthis :\n  \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v (trTape' enc0 L R) =\n      stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n\u22a2 stepAux (f (ListBlank.head R)) v\n      (trTape' enc0 (ListBlank.tail (ListBlank.cons (ListBlank.head R) L))\n        (ListBlank.cons (ListBlank.head (ListBlank.cons (ListBlank.head R) L)) (ListBlank.tail R))) =\n    stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)", "state_after": "no goals"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "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\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\nthis :\n  \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v\n        (Tape.mk'\n          (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n            (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n          (ListBlank.append (Vector.toList (enc a))\n            (ListBlank.bind R (fun x => Vector.toList (enc x))\n              (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n      stepAux (f (enc a)) v\n        (Tape.mk'\n          (ListBlank.append (List.reverse (Vector.toList (enc a)))\n            (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n              (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n          (ListBlank.bind R (fun x => Vector.toList (enc x))\n            (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))", "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\nf\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\nthis :\n  \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nf : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux n f) v\n      (Tape.mk'\n        (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n          (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n        (ListBlank.append (Vector.toList (enc a))\n          (ListBlank.bind R (fun x => Vector.toList (enc x))\n            (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n    stepAux (f (enc a)) v\n      (Tape.mk'\n        (ListBlank.append (List.reverse (Vector.toList (enc a)))\n          (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n            (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n        (ListBlank.bind R (fun x => Vector.toList (enc x))\n          (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))"}, {"tactic": "exact this n f (L.bind (fun x => (enc x).1.reverse) _)\n  (R.bind (fun x => (enc x).1) _) [] _ (enc a).2", "annotated_tactic": ["exact this n f (L.bind (fun x => (enc x).1.<a>reverse</a>) _)\n      (R.bind (fun x => (enc x).1) _) [] _ (enc a).2", [{"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": "\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\nf\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\nthis :\n  \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nf : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux n f) v\n      (Tape.mk'\n        (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n          (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n        (ListBlank.append (Vector.toList (enc a))\n          (ListBlank.bind R (fun x => Vector.toList (enc x))\n            (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n    stepAux (f (enc a)) v\n      (Tape.mk'\n        (ListBlank.append (List.reverse (Vector.toList (enc a)))\n          (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n            (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n        (ListBlank.bind R (fun x => Vector.toList (enc x))\n          (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.nil\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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082 : List Bool\nf\u271d : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3\nl\u2081 : List Bool\nf : Vector Bool (List.length []) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length []) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n    stepAux (f { val := [], property := (_ : List.length [] = List.length []) }) v\n      (Tape.mk' (ListBlank.append (List.reverseAux [] l\u2081) L') R')", "state_after": "no goals"}, {"tactic": "dsimp [readAux, stepAux]", "annotated_tactic": ["dsimp [<a>readAux</a>, <a>stepAux</a>]", [{"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1657, 5], "def_end_pos": [1657, 12]}, {"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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif ListBlank.head (ListBlank.cons a (ListBlank.append l\u2082 R')) then\n      stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n        (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))\n    else\n      stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n        (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))) =\n    stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))"}, {"tactic": "simp only [ListBlank.head_cons, Tape.move_right_mk', ListBlank.tail_cons]", "annotated_tactic": ["simp only [<a>ListBlank.head_cons</a>, <a>Tape.move_right_mk'</a>, <a>ListBlank.tail_cons</a>]", [{"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_right_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [588, 9], "def_end_pos": [588, 28]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}]], "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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif ListBlank.head (ListBlank.cons a (ListBlank.append l\u2082 R')) then\n      stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n        (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))\n    else\n      stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n        (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))) =\n    stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif a then\n      stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n        (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n    else\n      stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n        (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))) =\n    stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))"}, {"tactic": "cases a <;> rfl", "annotated_tactic": ["cases a <;> rfl", []], "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\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n        (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif a then\n      stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n        (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n    else\n      stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n        (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))) =\n    stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n      (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))", "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_subset_singleton", "start": [1041, 1], "end": [1045, 50], "traced_tactics": [{"tactic": "simp_rw [ncard_eq_toFinset_card _ hs, Finset.card_le_one_iff_subset_singleton,\n  Finite.toFinset_subset, Finset.coe_singleton]", "annotated_tactic": ["simp_rw [<a>ncard_eq_toFinset_card</a> _ hs, <a>Finset.card_le_one_iff_subset_singleton</a>,\n    <a>Finite.toFinset_subset</a>, <a>Finset.coe_singleton</a>]", [{"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": "Finset.card_le_one_iff_subset_singleton", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [559, 9], "def_end_pos": [559, 41]}, {"full_name": "Set.Finite.toFinset_subset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [198, 9], "def_end_pos": [198, 24]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [725, 9], "def_end_pos": [725, 22]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : Nonempty \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard s \u2264 1 \u2194 \u2203 x, s \u2286 {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.encodePosNum_nonempty", "start": [128, 1], "end": [130, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/PairingHeap.lean", "full_name": "Std.PairingHeapImp.Heap.WF.deleteMin", "start": [253, 1], "end": [255, 60], "traced_tactics": [{"tactic": "cases h with cases eq | node h => exact Heap.WF.combine h", "annotated_tactic": ["cases h with cases eq | <a>node</a> h => exact <a>Heap.WF.combine</a> h", [{"full_name": "Std.PairingHeapImp.Heap.WF.node", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [228, 5], "def_end_pos": [228, 9]}, {"full_name": "Std.PairingHeapImp.Heap.WF.combine", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [247, 9], "def_end_pos": [247, 24]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na : \u03b1\ns' s : Heap \u03b1\nh : WF le s\neq : Heap.deleteMin le s = some (a, s')\n\u22a2 WF le s'", "state_after": "no goals"}, {"tactic": "exact Heap.WF.combine h", "annotated_tactic": ["exact <a>Heap.WF.combine</a> h", [{"full_name": "Std.PairingHeapImp.Heap.WF.combine", "def_path": "lake-packages/std/Std/Data/PairingHeap.lean", "def_pos": [247, 9], "def_end_pos": [247, 24]}]], "state_before": "case node.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na : \u03b1\nc\u271d : Heap \u03b1\nh : NodeWF le a c\u271d\n\u22a2 WF le (Heap.combine le c\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_iterate_eq", "start": [171, 1], "end": [173, 61], "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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b1\nn : \u2115\n\u22a2 preimage f^[n] = (preimage f)^[n]", "state_after": "case zero\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\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b1\n\u22a2 preimage f^[Nat.zero] = (preimage f)^[Nat.zero]\n\ncase succ\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\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nih : preimage f^[n] = (preimage f)^[n]\n\u22a2 preimage f^[Nat.succ n] = (preimage f)^[Nat.succ n]"}, {"tactic": "rw [iterate_succ, iterate_succ', Set.preimage_comp_eq, ih]", "annotated_tactic": ["rw [<a>iterate_succ</a>, <a>iterate_succ'</a>, <a>Set.preimage_comp_eq</a>, ih]", [{"full_name": "Function.iterate_succ", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [62, 9], "def_end_pos": [62, 21]}, {"full_name": "Function.iterate_succ'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "Set.preimage_comp_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [167, 9], "def_end_pos": [167, 25]}]], "state_before": "case succ\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\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nih : preimage f^[n] = (preimage f)^[n]\n\u22a2 preimage f^[Nat.succ n] = (preimage f)^[Nat.succ n]", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\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\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b1\n\u22a2 preimage f^[Nat.zero] = (preimage f)^[Nat.zero]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_diff", "start": [864, 1], "end": [866, 68], "traced_tactics": [{"tactic": "rw [\u2190 ncard_diff_add_ncard_of_subset h ht, add_tsub_cancel_right]", "annotated_tactic": ["rw [\u2190 <a>ncard_diff_add_ncard_of_subset</a> h ht, <a>add_tsub_cancel_right</a>]", [{"full_name": "Set.ncard_diff_add_ncard_of_subset", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [857, 9], "def_end_pos": [857, 39]}, {"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": "\u03b1 : Type u_1\ns t : Set \u03b1\nh : s \u2286 t\nht : autoParam (Set.Finite t) _auto\u271d\n\u22a2 ncard (t \\ s) = ncard t - ncard 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.isSFiniteKernel_withDensity_of_isFiniteKernel", "start": [156, 1], "end": [210, 33], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba f)"}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba f)\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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba f)"}, {"tactic": "let fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (n + 1) - min (f a b) n", "annotated_tactic": ["let fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => <a>min</a> (f a b) (n + 1) - <a>min</a> (f a b) n", [{"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": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\n\u22a2 IsSFiniteKernel (withDensity \u03ba f)"}, {"tactic": "have h_zero : \u2200 a b n, \u2308(f a b).toReal\u2309\u208a \u2264 n \u2192 fs n a b = 0 := by\n  intro a b n hn\n  suffices min (f a b) (n + 1) = f a b \u2227 min (f a b) n = f a b by\n    simp_rw [this.1, this.2, tsub_self (f a b)]\n  exact \u27e8min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),\n    min_eq_left (h_le a b n hn)\u27e9", "annotated_tactic": ["have h_zero : \u2200 a b n, \u2308(f a b).<a>toReal</a>\u2309\u208a \u2264 n \u2192 fs n a b = 0 := by\n    intro a b n hn\n    suffices <a>min</a> (f a b) (n + 1) = f a b \u2227 <a>min</a> (f a b) n = f a b by\n      simp_rw [this.1, this.2, <a>tsub_self</a> (f a b)]\n    exact \u27e8<a>min_eq_left</a> ((h_le a b n hn).<a>trans</a> (<a>le_add_of_nonneg_right</a> <a>zero_le_one</a>)),\n      <a>min_eq_left</a> (h_le a b n hn)\u27e9", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 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": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\n\u22a2 IsSFiniteKernel (withDensity \u03ba f)"}, {"tactic": "rw [hf_eq_tsum, withDensity_tsum _ fun n : \u2115 => _]", "annotated_tactic": ["rw [hf_eq_tsum, <a>withDensity_tsum</a> _ fun n : \u2115 => _]", [{"full_name": "ProbabilityTheory.kernel.withDensity_tsum", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [114, 9], "def_end_pos": [114, 25]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 IsSFiniteKernel (kernel.sum fun n => withDensity \u03ba (fs n))\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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 \u2200 (n : \u2115), Measurable (Function.uncurry (fs n))"}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 IsSFiniteKernel (kernel.sum fun n => withDensity \u03ba (fs n))\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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 \u2200 (n : \u2115), Measurable (Function.uncurry (fs 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 \u2200 (n : \u2115), Measurable (Function.uncurry (fs n))\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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 IsSFiniteKernel (kernel.sum fun n => withDensity \u03ba (fs n))"}, {"tactic": "refine' isSFiniteKernel_sum fun n => _", "annotated_tactic": ["refine' <a>isSFiniteKernel_sum</a> fun n => _", [{"full_name": "ProbabilityTheory.kernel.isSFiniteKernel_sum", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [343, 9], "def_end_pos": [343, 28]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 IsSFiniteKernel (kernel.sum fun n => withDensity \u03ba (fs n))", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\n\u22a2 IsSFiniteKernel (withDensity \u03ba (fs n))"}, {"tactic": "suffices IsFiniteKernel (withDensity \u03ba (fs n)) by haveI := this; infer_instance", "annotated_tactic": ["suffices <a>IsFiniteKernel</a> (<a>withDensity</a> \u03ba (fs n)) by haveI := this; infer_instance", [{"full_name": "ProbabilityTheory.IsFiniteKernel", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "ProbabilityTheory.kernel.withDensity", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [47, 19], "def_end_pos": [47, 30]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\n\u22a2 IsSFiniteKernel (withDensity \u03ba (fs n))", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\n\u22a2 IsFiniteKernel (withDensity \u03ba (fs n))"}, {"tactic": "refine' isFiniteKernel_withDensity_of_bounded _ (ENNReal.coe_ne_top : \u2191n + 1 \u2260 \u221e) fun a b => _", "annotated_tactic": ["refine' <a>isFiniteKernel_withDensity_of_bounded</a> _ (<a>ENNReal.coe_ne_top</a> : \u2191n + 1 \u2260 \u221e) fun a b => _", [{"full_name": "ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [139, 9], "def_end_pos": [139, 46]}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\n\u22a2 IsFiniteKernel (withDensity \u03ba (fs n))", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\na : \u03b1\nb : \u03b2\n\u22a2 fs n a b \u2264 \u2191((fun x x_1 => x + x_1) (\u2191n) 1)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\na : \u03b1\nb : \u03b2\n\u22a2 fs n a b \u2264 \u2191((fun x x_1 => x + x_1) (\u2191n) 1)", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\na : \u03b1\nb : \u03b2\n\u22a2 fs n a b \u2264 \u2191(n + 1)"}, {"tactic": "calc\n  fs n a b \u2264 min (f a b) (n + 1) := tsub_le_self\n  _ \u2264 n + 1 := (min_le_right _ _)\n  _ = \u2191(n + 1) := by norm_cast", "annotated_tactic": ["calc\n    fs n a b \u2264 <a>min</a> (f a b) (n + 1) := <a>tsub_le_self</a>\n    _ \u2264 n + 1 := (<a>min_le_right</a> _ _)\n    _ = \u2191(n + 1) := by norm_cast", [{"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": "tsub_le_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [337, 9], "def_end_pos": [337, 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 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\na : \u03b1\nb : \u03b2\n\u22a2 fs n a b \u2264 \u2191(n + 1)", "state_after": "no goals"}, {"tactic": "rw [withDensity_of_not_measurable _ hf]", "annotated_tactic": ["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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel 0"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : \u00acMeasurable (Function.uncurry f)\n\u22a2 IsSFiniteKernel 0", "state_after": "no goals"}, {"tactic": "intro a b n hn", "annotated_tactic": ["intro a b n hn", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\n\u22a2 \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 f a b \u2264 \u2191n"}, {"tactic": "have : (f a b).toReal \u2264 n := Nat.le_of_ceil_le hn", "annotated_tactic": ["have : (f a b).<a>toReal</a> \u2264 n := <a>Nat.le_of_ceil_le</a> hn", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Nat.le_of_ceil_le", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [340, 9], "def_end_pos": [340, 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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 f a b \u2264 \u2191n", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : ENNReal.toReal (f a b) \u2264 \u2191n\n\u22a2 f a b \u2264 \u2191n"}, {"tactic": "rw [\u2190 ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this", "annotated_tactic": ["rw [\u2190 <a>ENNReal.le_ofReal_iff_toReal_le</a> (hf_ne_top a b) _] at this", [{"full_name": "ENNReal.le_ofReal_iff_toReal_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2204, 9], "def_end_pos": [2204, 32]}]], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : ENNReal.toReal (f a b) \u2264 \u2191n\n\u22a2 f a b \u2264 \u2191n", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : f a b \u2264 ENNReal.ofReal \u2191n\n\u22a2 f a b \u2264 \u2191n\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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : ENNReal.toReal (f a b) \u2264 \u2191n\n\u22a2 0 \u2264 \u2191n"}, {"tactic": "refine' this.trans (le_of_eq _)", "annotated_tactic": ["refine' this.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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : f a b \u2264 ENNReal.ofReal \u2191n\n\u22a2 f a b \u2264 \u2191n", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : f a b \u2264 ENNReal.ofReal \u2191n\n\u22a2 ENNReal.ofReal \u2191n = \u2191n"}, {"tactic": "rw [ENNReal.ofReal_coe_nat]", "annotated_tactic": ["rw [<a>ENNReal.ofReal_coe_nat</a>]", [{"full_name": "ENNReal.ofReal_coe_nat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [710, 17], "def_end_pos": [710, 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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : f a b \u2264 ENNReal.ofReal \u2191n\n\u22a2 ENNReal.ofReal \u2191n = \u2191n", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : ENNReal.toReal (f a b) \u2264 \u2191n\n\u22a2 0 \u2264 \u2191n", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : ENNReal.toReal (f a b) \u2264 \u2191n\n\u22a2 0 \u2264 n"}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : ENNReal.toReal (f a b) \u2264 \u2191n\n\u22a2 0 \u2264 n", "state_after": "no goals"}, {"tactic": "intro a b n hn", "annotated_tactic": ["intro a b n hn", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\n\u22a2 \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 fs n a b = 0"}, {"tactic": "suffices min (f a b) (n + 1) = f a b \u2227 min (f a b) n = f a b by\n  simp_rw [this.1, this.2, tsub_self (f a b)]", "annotated_tactic": ["suffices <a>min</a> (f a b) (n + 1) = f a b \u2227 <a>min</a> (f a b) n = f a b by\n      simp_rw [this.1, this.2, <a>tsub_self</a> (f a b)]", [{"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": "Min.min", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1103, 3], "def_end_pos": [1103, 6]}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 fs n a b = 0", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 min (f a b) (\u2191n + 1) = f a b \u2227 min (f a b) \u2191n = f a b"}, {"tactic": "exact \u27e8min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),\n  min_eq_left (h_le a b n hn)\u27e9", "annotated_tactic": ["exact \u27e8<a>min_eq_left</a> ((h_le a b n hn).<a>trans</a> (<a>le_add_of_nonneg_right</a> <a>zero_le_one</a>)),\n      <a>min_eq_left</a> (h_le a b n hn)\u27e9", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 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 }\nf : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 min (f a b) (\u2191n + 1) = f a b \u2227 min (f a b) \u2191n = f a b", "state_after": "no goals"}, {"tactic": "simp_rw [this.1, this.2, tsub_self (f a b)]", "annotated_tactic": ["simp_rw [this.1, this.2, <a>tsub_self</a> (f a b)]", [{"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\nthis : min (f a b) (\u2191n + 1) = f a b \u2227 min (f a b) \u2191n = f a b\n\u22a2 fs n a b = 0", "state_after": "no goals"}, {"tactic": "have h_sum_a : \u2200 a, Summable fun n => fs n a := by\n  refine' fun a => Pi.summable.mpr fun b => _\n  suffices : \u2200 n, n \u2209 Finset.range \u2308(f a b).toReal\u2309\u208a \u2192 fs n a b = 0\n  exact summable_of_ne_finset_zero this\n  intro n hn_not_mem\n  rw [Finset.mem_range, not_lt] at hn_not_mem\n  exact h_zero a b n hn_not_mem", "annotated_tactic": ["have h_sum_a : \u2200 a, <a>Summable</a> fun n => fs n a := by\n      refine' fun a => Pi.summable.mpr fun b => _\n      suffices : \u2200 n, n \u2209 <a>Finset.range</a> \u2308(f a b).<a>toReal</a>\u2309\u208a \u2192 fs n a b = 0\n      exact <a>summable_of_ne_finset_zero</a> this\n      intro n hn_not_mem\n      rw [<a>Finset.mem_range</a>, <a>not_lt</a>] at hn_not_mem\n      exact h_zero a b n hn_not_mem", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "summable_of_ne_finset_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 35]}, {"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": "\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\n\u22a2 f = \u2211' (n : \u2115), fs 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\n\u22a2 f = \u2211' (n : \u2115), fs n"}, {"tactic": "ext a b : 2", "annotated_tactic": ["ext a b : 2", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\n\u22a2 f = \u2211' (n : \u2115), fs 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\n\u22a2 f a b = tsum (fun n => fs n) a b"}, {"tactic": "rw [tsum_apply (Pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),\n  ENNReal.tsum_eq_liminf_sum_nat]", "annotated_tactic": ["rw [<a>tsum_apply</a> (Pi.summable.mpr h_sum_a), <a>tsum_apply</a> (h_sum_a a),\n      <a>ENNReal.tsum_eq_liminf_sum_nat</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": "ENNReal.tsum_eq_liminf_sum_nat", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [848, 19], "def_end_pos": [848, 41]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\n\u22a2 f a b = tsum (fun n => fs n) a b", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\n\u22a2 f a b = Filter.liminf (fun n => \u2211 i in Finset.range n, fs i a b) Filter.atTop"}, {"tactic": "simp_rw [h_finset_sum]", "annotated_tactic": ["simp_rw [h_finset_sum]", []], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 f a b = Filter.liminf (fun n => \u2211 i in Finset.range n, fs i a b) Filter.atTop", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 f a b = Filter.liminf (fun n => min (f a b) \u2191n) Filter.atTop"}, {"tactic": "refine' (Filter.Tendsto.liminf_eq _).symm", "annotated_tactic": ["refine' (<a>Filter.Tendsto.liminf_eq</a> _).<a>symm</a>", [{"full_name": "Filter.Tendsto.liminf_eq", "def_path": "Mathlib/Topology/Algebra/Order/LiminfLimsup.lean", "def_pos": [158, 9], "def_end_pos": [158, 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 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 f a b = Filter.liminf (fun n => min (f a b) \u2191n) Filter.atTop", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 Filter.Tendsto (fun n => min (f a b) \u2191n) Filter.atTop (nhds (f a b))"}, {"tactic": "refine' Filter.Tendsto.congr' _ tendsto_const_nhds", "annotated_tactic": ["refine' <a>Filter.Tendsto.congr'</a> _ <a>tendsto_const_nhds</a>", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3009, 9], "def_end_pos": [3009, 23]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 Filter.Tendsto (fun n => min (f a b) \u2191n) Filter.atTop (nhds (f a b))", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 (fun x => f a b) =\u1da0[Filter.atTop] fun n => min (f a b) \u2191n"}, {"tactic": "rw [Filter.EventuallyEq, Filter.eventually_atTop]", "annotated_tactic": ["rw [<a>Filter.EventuallyEq</a>, <a>Filter.eventually_atTop</a>]", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 (fun x => f a b) =\u1da0[Filter.atTop] fun n => min (f a b) \u2191n", "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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 \u2203 a_1, \u2200 (b_1 : \u2115), b_1 \u2265 a_1 \u2192 f a b = min (f a b) \u2191b_1"}, {"tactic": "exact \u27e8\u2308(f a b).toReal\u2309\u208a, fun n hn => (min_eq_left (h_le a b n hn)).symm\u27e9", "annotated_tactic": ["exact \u27e8\u2308(f a b).<a>toReal</a>\u2309\u208a, fun n hn => (<a>min_eq_left</a> (h_le a b n hn)).<a>symm</a>\u27e9", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 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.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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nh_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 \u2203 a_1, \u2200 (b_1 : \u2115), b_1 \u2265 a_1 \u2192 f a b = min (f a b) \u2191b_1", "state_after": "no goals"}, {"tactic": "refine' fun a => Pi.summable.mpr fun b => _", "annotated_tactic": ["refine' fun a => Pi.summable.mpr fun 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\n\u22a2 \u2200 (a : \u03b1), Summable fun n => fs n 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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\n\u22a2 Summable fun i => fs i a b"}, {"tactic": "suffices : \u2200 n, n \u2209 Finset.range \u2308(f a b).toReal\u2309\u208a \u2192 fs n a b = 0", "annotated_tactic": ["suffices : \u2200 n, n \u2209 <a>Finset.range</a> \u2308(f a b).<a>toReal</a>\u2309\u208a \u2192 fs n a b = 0", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\n\u22a2 Summable fun i => fs i a 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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\nthis : \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0\n\u22a2 Summable fun i => fs i a b\n\ncase this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\n\u22a2 \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0"}, {"tactic": "exact summable_of_ne_finset_zero this", "annotated_tactic": ["exact <a>summable_of_ne_finset_zero</a> this", [{"full_name": "summable_of_ne_finset_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 35]}]], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\nthis : \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0\n\u22a2 Summable fun i => fs i a b\n\ncase this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\n\u22a2 \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0", "state_after": "case this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\n\u22a2 \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0"}, {"tactic": "intro n hn_not_mem", "annotated_tactic": ["intro n hn_not_mem", []], "state_before": "case this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\n\u22a2 \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0", "state_after": "case this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\nn : \u2115\nhn_not_mem : \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a\n\u22a2 fs n a b = 0"}, {"tactic": "rw [Finset.mem_range, not_lt] at hn_not_mem", "annotated_tactic": ["rw [<a>Finset.mem_range</a>, <a>not_lt</a>] at hn_not_mem", [{"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 this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\nn : \u2115\nhn_not_mem : \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a\n\u22a2 fs n a b = 0", "state_after": "case this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\nn : \u2115\nhn_not_mem : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 fs n a b = 0"}, {"tactic": "exact h_zero a b n hn_not_mem", "annotated_tactic": ["exact h_zero a b n hn_not_mem", []], "state_before": "case this\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\na : \u03b1\nb : \u03b2\nn : \u2115\nhn_not_mem : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n\n\u22a2 fs n a b = 0", "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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\n\u22a2 \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n", "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n"}, {"tactic": "induction' n with n hn", "annotated_tactic": ["induction' n with n hn", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n", "state_after": "case zero\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\n\u22a2 \u2211 i in Finset.range Nat.zero, fs i a b = min (f a b) \u2191Nat.zero\n\ncase succ\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 \u2211 i in Finset.range (Nat.succ n), fs i a b = min (f a b) \u2191(Nat.succ n)"}, {"tactic": "rw [Finset.sum_range_succ, hn]", "annotated_tactic": ["rw [<a>Finset.sum_range_succ</a>, hn]", [{"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}]], "state_before": "case succ\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 \u2211 i in Finset.range (Nat.succ n), fs i a b = min (f a b) \u2191(Nat.succ n)", "state_after": "case succ\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 min (f a b) \u2191n + fs n a b = min (f a b) \u2191(Nat.succ n)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case succ\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\nn : \u2115\nhn : \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n\n\u22a2 min (f a b) \u2191n + fs n a b = min (f a b) \u2191(Nat.succ n)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nh_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a\na : \u03b1\nb : \u03b2\n\u22a2 \u2211 i in Finset.range Nat.zero, fs i a b = min (f a b) \u2191Nat.zero", "state_after": "no goals"}, {"tactic": "exact fun _ => (hf.min measurable_const).sub (hf.min measurable_const)", "annotated_tactic": ["exact fun _ => (hf.min <a>measurable_const</a>).<a>sub</a> (hf.min <a>measurable_const</a>)", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}, {"full_name": "Measurable.sub", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [333, 3], "def_end_pos": [333, 14]}, {"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\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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\n\u22a2 \u2200 (n : \u2115), Measurable (Function.uncurry (fs n))", "state_after": "no goals"}, {"tactic": "haveI := this", "annotated_tactic": ["haveI := 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\nthis : IsFiniteKernel (withDensity \u03ba (fs n))\n\u22a2 IsSFiniteKernel (withDensity \u03ba (fs 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\nthis\u271d this : IsFiniteKernel (withDensity \u03ba (fs n))\n\u22a2 IsSFiniteKernel (withDensity \u03ba (fs n))"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\nthis\u271d this : IsFiniteKernel (withDensity \u03ba (fs n))\n\u22a2 IsSFiniteKernel (withDensity \u03ba (fs n))", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "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 : IsFiniteKernel \u03ba\nhf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4\nhf : Measurable (Function.uncurry f)\nfs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n\nh_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n\nh_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0\nhf_eq_tsum : f = \u2211' (n : \u2115), fs n\nn : \u2115\na : \u03b1\nb : \u03b2\n\u22a2 \u2191n + 1 = \u2191(n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_ofReal", "start": [1191, 1], "end": [1192, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.ae_mem_limsup_atTop_iff", "start": [376, 1], "end": [380, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "full_name": "MvQPF.Fix.rec_eq", "start": [225, 1], "end": [239, 86], "traced_tactics": [{"tactic": "have : recF g \u2218 fixToW = Fix.rec g := by\n  apply funext\n  apply Quotient.ind\n  intro x\n  apply recF_eq_of_wEquiv\n  apply wrepr_equiv", "annotated_tactic": ["have : <a>recF</a> g \u2218 <a>fixToW</a> = <a>Fix.rec</a> g := by\n    apply <a>funext</a>\n    apply <a>Quotient.ind</a>\n    intro x\n    apply <a>recF_eq_of_wEquiv</a>\n    apply <a>wrepr_equiv</a>", [{"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.fixToW", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [210, 5], "def_end_pos": [210, 11]}, {"full_name": "MvQPF.Fix.rec", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [205, 5], "def_end_pos": [205, 12]}, {"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "Quotient.ind", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1355, 19], "def_end_pos": [1355, 22]}, {"full_name": "MvQPF.recF_eq_of_wEquiv", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [92, 9], "def_end_pos": [92, 26]}, {"full_name": "MvQPF.wrepr_equiv", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [147, 9], "def_end_pos": [147, 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\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> 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\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)"}, {"tactic": "conv =>\n  lhs\n  rw [Fix.rec, Fix.mk]\n  dsimp", "annotated_tactic": ["conv =>\n    lhs\n    rw [<a>Fix.rec</a>, <a>Fix.mk</a>]\n    dsimp", [{"full_name": "MvQPF.Fix.rec", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [205, 5], "def_end_pos": [205, 12]}, {"full_name": "MvQPF.Fix.mk", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [216, 5], "def_end_pos": [216, 11]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> 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\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x)"}, {"tactic": "cases' h : repr x with a f", "annotated_tactic": ["cases' h : <a>repr</a> x with a f", [{"full_name": "MvQPF.repr", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [88, 3], "def_end_pos": [88, 7]}]], "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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x)", "state_after": "case mk\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1 ::: Fix F \u03b1\nh : repr x = { fst := a, snd := f }\n\u22a2 recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n    g ((TypeVec.id ::: rec g) <$$> x)"}, {"tactic": "rw [MvPFunctor.map_eq, recF_eq', \u2190 MvPFunctor.map_eq, MvPFunctor.wDest'_wMk']", "annotated_tactic": ["rw [<a>MvPFunctor.map_eq</a>, <a>recF_eq'</a>, \u2190 <a>MvPFunctor.map_eq</a>, <a>MvPFunctor.wDest'_wMk'</a>]", [{"full_name": "MvPFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Multivariate/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 15]}, {"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": "MvPFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Multivariate/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 15]}, {"full_name": "MvPFunctor.wDest'_wMk'", "def_path": "Mathlib/Data/PFunctor/Multivariate/W.lean", "def_pos": [311, 9], "def_end_pos": [311, 20]}]], "state_before": "case mk\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1 ::: Fix F \u03b1\nh : repr x = { fst := a, snd := f }\n\u22a2 recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n    g ((TypeVec.id ::: rec g) <$$> x)", "state_after": "case mk\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1 ::: Fix F \u03b1\nh : repr x = { fst := a, snd := f }\n\u22a2 g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n    g ((TypeVec.id ::: rec g) <$$> x)"}, {"tactic": "rw [\u2190 MvPFunctor.comp_map, abs_map, \u2190 h, abs_repr, \u2190 appendFun_comp, id_comp, this]", "annotated_tactic": ["rw [\u2190 <a>MvPFunctor.comp_map</a>, <a>abs_map</a>, \u2190 h, <a>abs_repr</a>, \u2190 <a>appendFun_comp</a>, <a>id_comp</a>, this]", [{"full_name": "MvPFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Multivariate/Basic.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}, {"full_name": "MvQPF.abs_repr", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [89, 3], "def_end_pos": [89, 11]}, {"full_name": "TypeVec.appendFun_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [252, 9], "def_end_pos": [252, 23]}, {"full_name": "TypeVec.id_comp", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}]], "state_before": "case mk\nn : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1 ::: Fix F \u03b1\nh : repr x = { fst := a, snd := f }\n\u22a2 g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n    g ((TypeVec.id ::: rec g) <$$> x)", "state_after": "no goals"}, {"tactic": "apply funext", "annotated_tactic": ["apply <a>funext</a>", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 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\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 recF g \u2218 fixToW = rec g", "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\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 \u2200 (x : Fix F \u03b1), (recF g \u2218 fixToW) x = rec g x"}, {"tactic": "apply Quotient.ind", "annotated_tactic": ["apply <a>Quotient.ind</a>", [{"full_name": "Quotient.ind", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1355, 19], "def_end_pos": [1355, 22]}]], "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\n\u03b2 : Type u\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 \u2200 (x : Fix F \u03b1), (recF g \u2218 fixToW) x = rec g x", "state_after": "case h.a\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 \u2200 (a : MvPFunctor.W (P F) \u03b1), (recF g \u2218 fixToW) (Quotient.mk (wSetoid \u03b1) a) = rec g (Quotient.mk (wSetoid \u03b1) a)"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case h.a\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx : F (\u03b1 ::: Fix F \u03b1)\n\u22a2 \u2200 (a : MvPFunctor.W (P F) \u03b1), (recF g \u2218 fixToW) (Quotient.mk (wSetoid \u03b1) a) = rec g (Quotient.mk (wSetoid \u03b1) a)", "state_after": "case h.a\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx\u271d : F (\u03b1 ::: Fix F \u03b1)\nx : MvPFunctor.W (P F) \u03b1\n\u22a2 (recF g \u2218 fixToW) (Quotient.mk (wSetoid \u03b1) x) = rec g (Quotient.mk (wSetoid \u03b1) x)"}, {"tactic": "apply recF_eq_of_wEquiv", "annotated_tactic": ["apply <a>recF_eq_of_wEquiv</a>", [{"full_name": "MvQPF.recF_eq_of_wEquiv", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [92, 9], "def_end_pos": [92, 26]}]], "state_before": "case h.a\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx\u271d : F (\u03b1 ::: Fix F \u03b1)\nx : MvPFunctor.W (P F) \u03b1\n\u22a2 (recF g \u2218 fixToW) (Quotient.mk (wSetoid \u03b1) x) = rec g (Quotient.mk (wSetoid \u03b1) x)", "state_after": "case h.a.a\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx\u271d : F (\u03b1 ::: Fix F \u03b1)\nx : MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv (fixToW (Quotient.mk (wSetoid \u03b1) x)) x"}, {"tactic": "apply wrepr_equiv", "annotated_tactic": ["apply <a>wrepr_equiv</a>", [{"full_name": "MvQPF.wrepr_equiv", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [147, 9], "def_end_pos": [147, 20]}]], "state_before": "case h.a.a\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\ng : F (\u03b1 ::: \u03b2) \u2192 \u03b2\nx\u271d : F (\u03b1 ::: Fix F \u03b1)\nx : MvPFunctor.W (P F) \u03b1\n\u22a2 WEquiv (fixToW (Quotient.mk (wSetoid \u03b1) x)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.trans_assoc", "start": [142, 1], "end": [144, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasureTheory.Measure.ext_of_Ioc'", "start": [791, 1], "end": [796, 25], "traced_tactics": [{"tactic": "refine' @ext_of_Ico' \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ \u03bc \u03bd _ _ <;> intro a b hab <;> erw [dual_Ico (\u03b1 := \u03b1)]", "annotated_tactic": ["refine' @<a>ext_of_Ico'</a> \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ \u03bc \u03bd _ _ <;> intro a b hab <;> erw [<a>dual_Ico</a> (\u03b1 := \u03b1)]", [{"full_name": "MeasureTheory.Measure.ext_of_Ico'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [765, 9], "def_end_pos": [765, 20]}, {"full_name": "Set.dual_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [256, 9], "def_end_pos": [256, 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\ninst\u271d : NoMinOrder \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\n\u22a2 \u03bc = \u03bd", "state_after": "case refine'_1\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\ninst\u271d : NoMinOrder \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) \u2260 \u22a4\n\ncase refine'_2\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\ninst\u271d : NoMinOrder \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) = \u2191\u2191\u03bd (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a)"}, {"tactic": "exacts [h\u03bc hab, h hab]", "annotated_tactic": ["exacts [h\u03bc hab, h hab]", []], "state_before": "case refine'_1\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\ninst\u271d : NoMinOrder \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) \u2260 \u22a4\n\ncase refine'_2\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\ninst\u271d : NoMinOrder \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4\nh : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b)\na b : \u03b1\u1d52\u1d48\nhab : a < b\n\u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) = \u2191\u2191\u03bd (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Mem\u2112p.induction_dense", "start": [980, 1], "end": [1026, 9], "traced_tactics": [{"tactic": "rcases eq_or_ne p 0 with (rfl | hp_pos)", "annotated_tactic": ["rcases <a>eq_or_ne</a> p 0 with (rfl | hp_pos)", [{"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\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\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\n\u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g\n\ncase 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "suffices H :\n  \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e) (h\u03b4 : \u03b4 \u2260 0), Mem\u2112p f' p \u03bc \u2192 \u2203 g, snorm (\u21d1f' - g) p \u03bc \u2264 \u03b4 \u2227 P g", "annotated_tactic": ["suffices H :\n    \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e) (h\u03b4 : \u03b4 \u2260 0), <a>Mem\u2112p</a> f' p \u03bc \u2192 \u2203 g, <a>snorm</a> (\u21d1f' - g) p \u03bc \u2264 \u03b4 \u2227 P 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.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}]], "state_before": "case 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n\ncase 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g"}, {"tactic": "apply SimpleFunc.induction", "annotated_tactic": ["apply <a>SimpleFunc.induction</a>", [{"full_name": "MeasureTheory.SimpleFunc.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1266, 19], "def_end_pos": [1266, 28]}]], "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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_ind\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (\u03b4 : \u211d\u22650\u221e),\n    \u03b4 \u2260 0 \u2192\n      Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p \u2192\n        \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\ncase H.h_add\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g) \u2192\n        (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191g) p \u2192 \u2203 g_1, snorm (\u2191g - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1) \u2192\n          \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191(f + g)) p \u2192 \u2203 g_1, snorm (\u2191(f + g) - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1"}, {"tactic": "rcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top])\n    h\u03b5 with \u27e8g, _, Pg\u27e9", "annotated_tactic": ["rcases h0P (0 : E) <a>MeasurableSet.empty</a> (by simp only [<a>measure_empty</a>, <a>WithTop.zero_lt_top</a>])\n        h\u03b5 with \u27e8g, _, Pg\u27e9", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "case 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\n\u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\ng : \u03b1 \u2192 E\nleft\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "exact \u27e8g, by simp only [snorm_exponent_zero, zero_le'], Pg\u27e9", "annotated_tactic": ["exact \u27e8g, by simp only [<a>snorm_exponent_zero</a>, <a>zero_le'</a>], Pg\u27e9", [{"full_name": "MeasureTheory.snorm_exponent_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [176, 9], "def_end_pos": [176, 28]}, {"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\ng : \u03b1 \u2192 E\nleft\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "no goals"}, {"tactic": "simp only [measure_empty, WithTop.zero_lt_top]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>WithTop.zero_lt_top</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "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\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4", "state_after": "no goals"}, {"tactic": "simp only [snorm_exponent_zero, zero_le']", "annotated_tactic": ["simp only [<a>snorm_exponent_zero</a>, <a>zero_le'</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": "zero_le'", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [93, 9], "def_end_pos": [93, 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\ng : \u03b1 \u2192 E\nleft\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 snorm (f - g) 0 \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := exists_Lp_half E \u03bc p h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := <a>exists_Lp_half</a> E \u03bc p h\u03b5", [{"full_name": "MeasureTheory.exists_Lp_half", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [862, 9], "def_end_pos": [862, 23]}]], "state_before": "case 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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case inr.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rcases hf.exists_simpleFunc_snorm_sub_lt hp_ne_top \u03b7pos.ne' with \u27e8f', hf', f'_mem\u27e9", "annotated_tactic": ["rcases hf.exists_simpleFunc_snorm_sub_lt hp_ne_top \u03b7pos.ne' with \u27e8f', hf', f'_mem\u27e9", []], "state_before": "case inr.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case inr.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rcases H f' \u03b7 \u03b7pos.ne' f'_mem with \u27e8g, hg, Pg\u27e9", "annotated_tactic": ["rcases H f' \u03b7 \u03b7pos.ne' f'_mem with \u27e8g, hg, Pg\u27e9", []], "state_before": "case inr.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case inr.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "refine' \u27e8g, _, Pg\u27e9", "annotated_tactic": ["refine' \u27e8g, _, Pg\u27e9", []], "state_before": "case inr.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case inr.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 snorm (f - g) p \u03bc \u2264 \u03b5"}, {"tactic": "convert (h\u03b7 _ _ (hf.aestronglyMeasurable.sub f'.aestronglyMeasurable)\n      (f'.aestronglyMeasurable.sub (h2P g Pg)) hf'.le hg).le using 2", "annotated_tactic": ["convert (h\u03b7 _ _ (hf.aestronglyMeasurable.sub f'.aestronglyMeasurable)\n          (f'.aestronglyMeasurable.sub (h2P g Pg)) hf'.le hg).<a>le</a> using 2", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case inr.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 snorm (f - g) p \u03bc \u2264 \u03b5", "state_after": "case h.e'_3.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 f - g = f - \u2191f' + (\u2191f' - g)"}, {"tactic": "simp only [sub_add_sub_cancel]", "annotated_tactic": ["simp only [<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": "case h.e'_3.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 f - g = f - \u2191f' + (\u2191f' - g)", "state_after": "no goals"}, {"tactic": "intro c s hs \u03b5 \u03b5pos Hs", "annotated_tactic": ["intro c s hs \u03b5 \u03b5pos Hs", []], "state_before": "case H.h_ind\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (\u03b4 : \u211d\u22650\u221e),\n    \u03b4 \u2260 0 \u2192\n      Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p \u2192\n        \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_ind\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rcases eq_or_ne c 0 with (rfl | hc)", "annotated_tactic": ["rcases <a>eq_or_ne</a> c 0 with (rfl | hc)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case H.h_ind\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n\ncase H.h_ind.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top])\n    \u03b5pos with \u27e8g, hg, Pg\u27e9", "annotated_tactic": ["rcases h0P (0 : E) <a>MeasurableSet.empty</a> (by simp only [<a>measure_empty</a>, <a>WithTop.zero_lt_top</a>])\n          \u03b5pos with \u27e8g, hg, Pg\u27e9", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "state_before": "case H.h_ind.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm (g - Set.indicator \u2205 fun x => 0) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rw [\u2190 snorm_neg, neg_sub] at hg", "annotated_tactic": ["rw [\u2190 <a>snorm_neg</a>, <a>neg_sub</a>] at hg", [{"full_name": "MeasureTheory.snorm_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [261, 9], "def_end_pos": [261, 18]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "case H.h_ind.inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm (g - Set.indicator \u2205 fun x => 0) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "refine' \u27e8g, _, Pg\u27e9", "annotated_tactic": ["refine' \u27e8g, _, Pg\u27e9", []], "state_before": "case H.h_ind.inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5"}, {"tactic": "convert hg", "annotated_tactic": ["convert hg", []], "state_before": "case H.h_ind.inl.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5", "state_after": "case h.e'_3.h.e'_5.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) = Set.indicator \u2205 fun x => 0"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_3.h.e'_5.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) = Set.indicator \u2205 fun x => 0", "state_after": "case h.e'_3.h.e'_5.h.e'_5.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) x = Set.indicator \u2205 (fun x => 0) x"}, {"tactic": "simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_zero,\n  piecewise_eq_indicator, indicator_zero', Pi.zero_apply, indicator_zero]", "annotated_tactic": ["simp only [<a>SimpleFunc.const_zero</a>, <a>SimpleFunc.coe_piecewise</a>, <a>SimpleFunc.coe_zero</a>,\n        <a>piecewise_eq_indicator</a>, <a>indicator_zero'</a>, <a>Pi.zero_apply</a>, <a>indicator_zero</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_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": "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": "Set.indicator_zero", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [207, 3], "def_end_pos": [207, 14]}]], "state_before": "case h.e'_3.h.e'_5.h.e'_5.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) x = Set.indicator \u2205 (fun x => 0) x", "state_after": "no goals"}, {"tactic": "simp only [measure_empty, WithTop.zero_lt_top]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>WithTop.zero_lt_top</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "WithTop.zero_lt_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [431, 9], "def_end_pos": [431, 20]}]], "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\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4", "state_after": "no goals"}, {"tactic": "have : \u03bc s < \u221e := SimpleFunc.measure_lt_top_of_mem\u2112p_indicator hp_pos hp_ne_top hc hs Hs", "annotated_tactic": ["have : \u03bc s < \u221e := <a>SimpleFunc.measure_lt_top_of_mem\u2112p_indicator</a> hp_pos hp_ne_top hc hs Hs", [{"full_name": "MeasureTheory.SimpleFunc.measure_lt_top_of_mem\u2112p_indicator", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [417, 9], "def_end_pos": [417, 42]}]], "state_before": "case H.h_ind.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rcases h0P c hs this \u03b5pos with \u27e8g, hg, Pg\u27e9", "annotated_tactic": ["rcases h0P c hs this \u03b5pos with \u27e8g, hg, Pg\u27e9", []], "state_before": "case H.h_ind.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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.inr.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\ng : \u03b1 \u2192 E\nhg : snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "rw [\u2190 snorm_neg, neg_sub] at hg", "annotated_tactic": ["rw [\u2190 <a>snorm_neg</a>, <a>neg_sub</a>] at hg", [{"full_name": "MeasureTheory.snorm_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [261, 9], "def_end_pos": [261, 18]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "case H.h_ind.inr.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\ng : \u03b1 \u2192 E\nhg : snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "case H.h_ind.inr.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator s fun x => c) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g"}, {"tactic": "exact \u27e8g, hg, Pg\u27e9", "annotated_tactic": ["exact \u27e8g, hg, Pg\u27e9", []], "state_before": "case H.h_ind.inr.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator s fun x => c) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g", "state_after": "no goals"}, {"tactic": "intro f f' hff' hf hf' \u03b4 \u03b4pos int_ff'", "annotated_tactic": ["intro f f' hff' hf hf' \u03b4 \u03b4pos int_ff'", []], "state_before": "case H.h_add\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g) \u2192\n        (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191g) p \u2192 \u2203 g_1, snorm (\u2191g - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1) \u2192\n          \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191(f + g)) p \u2192 \u2203 g_1, snorm (\u2191(f + g) - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1", "state_after": "case H.h_add\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191(f + f')) p\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g"}, {"tactic": "obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := exists_Lp_half E \u03bc p \u03b4pos", "annotated_tactic": ["obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := <a>exists_Lp_half</a> E \u03bc p \u03b4pos", [{"full_name": "MeasureTheory.exists_Lp_half", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [862, 9], "def_end_pos": [862, 23]}]], "state_before": "case H.h_add\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191(f + f')) p\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_add.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191(f + f')) p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g"}, {"tactic": "rw [SimpleFunc.coe_add,\n  mem\u2112p_add_of_disjoint hff' f.stronglyMeasurable f'.stronglyMeasurable] at int_ff'", "annotated_tactic": ["rw [<a>SimpleFunc.coe_add</a>,\n      <a>mem\u2112p_add_of_disjoint</a> hff' f.stronglyMeasurable f'.stronglyMeasurable] at int_ff'", [{"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.mem\u2112p_add_of_disjoint", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [810, 9], "def_end_pos": [810, 30]}]], "state_before": "case H.h_add.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191(f + f')) p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_add.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g"}, {"tactic": "rcases hf \u03b7 \u03b7pos.ne' int_ff'.1 with \u27e8g, hg, Pg\u27e9", "annotated_tactic": ["rcases hf \u03b7 \u03b7pos.ne' int_ff'.1 with \u27e8g, hg, Pg\u27e9", []], "state_before": "case H.h_add.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_add.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g"}, {"tactic": "rcases hf' \u03b7 \u03b7pos.ne' int_ff'.2 with \u27e8g', hg', Pg'\u27e9", "annotated_tactic": ["rcases hf' \u03b7 \u03b7pos.ne' int_ff'.2 with \u27e8g', hg', Pg'\u27e9", []], "state_before": "case H.h_add.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_add.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g"}, {"tactic": "refine' \u27e8g + g', _, h1P g g' Pg Pg'\u27e9", "annotated_tactic": ["refine' \u27e8g + g', _, h1P g g' Pg Pg'\u27e9", []], "state_before": "case H.h_add.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g", "state_after": "case H.h_add.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 snorm (\u2191(f + f') - (g + g')) p \u03bc \u2264 \u03b4"}, {"tactic": "convert (h\u03b7 _ _ (f.aestronglyMeasurable.sub (h2P g Pg))\n      (f'.aestronglyMeasurable.sub (h2P g' Pg')) hg hg').le using 2", "annotated_tactic": ["convert (h\u03b7 _ _ (f.aestronglyMeasurable.sub (h2P g Pg))\n          (f'.aestronglyMeasurable.sub (h2P g' Pg')) hg hg').<a>le</a> using 2", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case H.h_add.intro.intro.intro.intro.intro.intro\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 snorm (\u2191(f + f') - (g + g')) p \u03bc \u2264 \u03b4", "state_after": "case h.e'_3.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191(f + f') - (g + g') = \u2191f - g + (\u2191f' - g')"}, {"tactic": "rw [SimpleFunc.coe_add]", "annotated_tactic": ["rw [<a>SimpleFunc.coe_add</a>]", [{"full_name": "MeasureTheory.SimpleFunc.coe_add", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case h.e'_3.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191(f + f') - (g + g') = \u2191f - g + (\u2191f' - g')", "state_after": "case h.e'_3.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191f + \u2191f' - (g + g') = \u2191f - g + (\u2191f' - g')"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h.e'_3.h.e'_5\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\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191f + \u2191f' - (g + g') = \u2191f - g + (\u2191f' - g')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.mod_mul_left_mod", "start": [325, 1], "end": [326, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.mul_self_kstar_comm", "start": [264, 1], "end": [265, 77], "traced_tactics": [{"tactic": "simp only [kstar_eq_iSup_pow, mul_iSup, iSup_mul, \u2190 pow_succ, \u2190 pow_succ']", "annotated_tactic": ["simp only [<a>kstar_eq_iSup_pow</a>, <a>mul_iSup</a>, <a>iSup_mul</a>, \u2190 <a>pow_succ</a>, \u2190 <a>pow_succ'</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.mul_iSup", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [214, 9], "def_end_pos": [214, 17]}, {"full_name": "Language.iSup_mul", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x : List \u03b1\nl : Language \u03b1\n\u22a2 l\u2217 * l = l * l\u2217", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Constructions.lean", "full_name": "WType.leftInverse_list", "start": [163, 1], "end": [170, 8], "traced_tactics": [{"tactic": "simp only [toList, ofList, mk.injEq, heq_eq_eq, true_and]", "annotated_tactic": ["simp only [<a>toList</a>, <a>ofList</a>, mk.injEq, <a>heq_eq_eq</a>, <a>true_and</a>]", [{"full_name": "WType.toList", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [158, 5], "def_end_pos": [158, 11]}, {"full_name": "WType.ofList", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [151, 5], "def_end_pos": [151, 11]}, {"full_name": "heq_eq_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [143, 17], "def_end_pos": [143, 26]}, {"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": "\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\n\u22a2 ofList \u03b3 (toList \u03b3 (mk List\u03b1.nil f)) = mk List\u03b1.nil f", "state_after": "\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\n\u22a2 PEmpty.elim = f"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\n\u22a2 PEmpty.elim = f", "state_after": "case h\n\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\nx : PEmpty.{u + 1}\n\u22a2 PEmpty.elim x = f x"}, {"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "case h\n\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\nx : PEmpty.{u + 1}\n\u22a2 PEmpty.elim x = f x", "state_after": "no goals"}, {"tactic": "simp only [ofList, leftInverse_list (f PUnit.unit), mk.injEq, heq_eq_eq, true_and]", "annotated_tactic": ["simp only [<a>ofList</a>, leftInverse_list (f <a>PUnit.unit</a>), mk.injEq, <a>heq_eq_eq</a>, <a>true_and</a>]", [{"full_name": "WType.ofList", "def_path": "Mathlib/Data/W/Constructions.lean", "def_pos": [151, 5], "def_end_pos": [151, 11]}, {"full_name": "PUnit.unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [111, 5], "def_end_pos": [111, 9]}, {"full_name": "heq_eq_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [143, 17], "def_end_pos": [143, 26]}, {"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": "\u03b3 : Type u\nx : \u03b3\nf : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3)\n\u22a2 ofList \u03b3 (toList \u03b3 (mk (List\u03b1.cons x) f)) = mk (List\u03b1.cons x) f", "state_after": "\u03b3 : Type u\nx : \u03b3\nf : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3)\n\u22a2 (fun x => f PUnit.unit) = f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b3 : Type u\nx : \u03b3\nf : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3)\n\u22a2 (fun x => f PUnit.unit) = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.of_graph", "start": [806, 1], "end": [810, 72], "traced_tactics": [{"tactic": "rcases h\u2081 with \u27e8g, pg, hg : \u2200 x, f x \u2264 g x\u27e9", "annotated_tactic": ["rcases h\u2081 with \u27e8g, pg, hg : \u2200 x, f x \u2264 g x\u27e9", []], "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 \u2115\nh\u2081 : PrimrecBounded f\nh\u2082 : PrimrecRel fun a b => f a = b\n\u22a2 Primrec f", "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\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 \u2115\nh\u2082 : PrimrecRel fun a b => f a = b\ng : \u03b1 \u2192 \u2115\npg : Primrec g\nhg : \u2200 (x : \u03b1), f x \u2264 g x\n\u22a2 Primrec f"}, {"tactic": "refine (nat_findGreatest pg h\u2082).of_eq fun n => ?_", "annotated_tactic": ["refine (<a>nat_findGreatest</a> pg h\u2082).<a>of_eq</a> fun n => ?_", [{"full_name": "Primrec.nat_findGreatest", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [797, 9], "def_end_pos": [797, 25]}, {"full_name": "Primrec.of_eq", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [246, 9], "def_end_pos": [246, 14]}]], "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\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 \u2115\nh\u2082 : PrimrecRel fun a b => f a = b\ng : \u03b1 \u2192 \u2115\npg : Primrec g\nhg : \u2200 (x : \u03b1), f x \u2264 g x\n\u22a2 Primrec f", "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\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 \u2115\nh\u2082 : PrimrecRel fun a b => f a = b\ng : \u03b1 \u2192 \u2115\npg : Primrec g\nhg : \u2200 (x : \u03b1), f x \u2264 g x\nn : \u03b1\n\u22a2 Nat.findGreatest (fun b => f n = b) (g n) = f n"}, {"tactic": "exact (Nat.findGreatest_spec (P := fun b => f n = b) (hg n) rfl).symm", "annotated_tactic": ["exact (<a>Nat.findGreatest_spec</a> (P := fun b => f n = b) (hg n) <a>rfl</a>).<a>symm</a>", [{"full_name": "Nat.findGreatest_spec", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [628, 9], "def_end_pos": [628, 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": "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\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 \u2115\nh\u2082 : PrimrecRel fun a b => f a = b\ng : \u03b1 \u2192 \u2115\npg : Primrec g\nhg : \u2200 (x : \u03b1), f x \u2264 g x\nn : \u03b1\n\u22a2 Nat.findGreatest (fun b => f n = b) (g n) = f n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.indep_limsup_self", "start": [110, 1], "end": [113, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.ofNat'_bit", "start": [239, 1], "end": [240, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.isometryEquiv_map_mkMetric", "start": [409, 1], "end": [411, 80], "traced_tactics": [{"tactic": "rw [\u2190 isometryEquiv_comap_mkMetric _ f, map_comap_of_surjective f.surjective]", "annotated_tactic": ["rw [\u2190 <a>isometryEquiv_comap_mkMetric</a> _ f, <a>map_comap_of_surjective</a> f.surjective]", [{"full_name": "MeasureTheory.OuterMeasure.isometryEquiv_comap_mkMetric", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [404, 9], "def_end_pos": [404, 37]}, {"full_name": "MeasureTheory.OuterMeasure.map_comap_of_surjective", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [605, 9], "def_end_pos": [605, 32]}]], "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 \u2243\u1d62 Y\n\u22a2 \u2191(map \u2191f) (mkMetric m) = mkMetric m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.ulower_up", "start": [1263, 1], "end": [1265, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.support_finSuccEquiv_nonempty", "start": [480, 1], "end": [489, 21], "traced_tactics": [{"tactic": "simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]", "annotated_tactic": ["simp only [<a>Finset.nonempty_iff_ne_empty</a>, <a>Ne</a>, <a>Polynomial.support_eq_empty</a>]", [{"full_name": "Finset.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [604, 9], "def_end_pos": [604, 30]}, {"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "Polynomial.support_eq_empty", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [420, 9], "def_end_pos": [420, 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 Finset.Nonempty (Polynomial.support (\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 : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\n\u22a2 \u00ac\u2191(finSuccEquiv R n) f = 0"}, {"tactic": "refine fun c => h ?_", "annotated_tactic": ["refine fun c => h ?_", []], "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 \u00ac\u2191(finSuccEquiv R n) f = 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\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nc : \u2191(finSuccEquiv R n) f = 0\n\u22a2 f = 0"}, {"tactic": "let ii := (finSuccEquiv R n).symm", "annotated_tactic": ["let ii := (<a>finSuccEquiv</a> R n).<a>symm</a>", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "AlgEquiv.symm", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [317, 5], "def_end_pos": [317, 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 : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nc : \u2191(finSuccEquiv R n) f = 0\n\u22a2 f = 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\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nc : \u2191(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n\u22a2 f = 0"}, {"tactic": "calc\n  f = ii (finSuccEquiv R n f) := by\n    simpa only [\u2190 AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm\n  _ = ii 0 := by rw [c]\n  _ = 0 := by simp", "annotated_tactic": ["calc\n    f = ii (<a>finSuccEquiv</a> R n f) := by\n      simpa only [\u2190 <a>AlgEquiv.invFun_eq_symm</a>] using ((<a>finSuccEquiv</a> R n).<a>left_inv</a> f).<a>symm</a>\n    _ = ii 0 := by rw [c]\n    _ = 0 := by simp", [{"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "AlgEquiv.invFun_eq_symm", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [350, 9], "def_end_pos": [350, 23]}, {"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "Equiv.left_inv", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [69, 13], "def_end_pos": [69, 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": "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\nc : \u2191(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n\u22a2 f = 0", "state_after": "no goals"}, {"tactic": "simpa only [\u2190 AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm", "annotated_tactic": ["simpa only [\u2190 <a>AlgEquiv.invFun_eq_symm</a>] using ((<a>finSuccEquiv</a> R n).<a>left_inv</a> f).<a>symm</a>", [{"full_name": "AlgEquiv.invFun_eq_symm", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [350, 9], "def_end_pos": [350, 23]}, {"full_name": "MvPolynomial.finSuccEquiv", "def_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "def_pos": [316, 5], "def_end_pos": [316, 17]}, {"full_name": "Equiv.left_inv", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [69, 13], "def_end_pos": [69, 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": "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\nc : \u2191(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n\u22a2 f = \u2191ii (\u2191(finSuccEquiv R n) f)", "state_after": "no goals"}, {"tactic": "rw [c]", "annotated_tactic": ["rw [c]", []], "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\nc : \u2191(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n\u22a2 \u2191ii (\u2191(finSuccEquiv R n) f) = \u2191ii 0", "state_after": "no goals"}, {"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\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nc : \u2191(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n\u22a2 \u2191ii 0 = 0", "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_right", "start": [85, 1], "end": [87, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.pred_add_one", "start": [526, 1], "end": [529, 33], "traced_tactics": [{"tactic": "rw [ext_iff, coe_pred, coe_castLT, val_add, val_one, Nat.mod_eq_of_lt, Nat.add_sub_cancel]", "annotated_tactic": ["rw [<a>ext_iff</a>, <a>coe_pred</a>, <a>coe_castLT</a>, <a>val_add</a>, <a>val_one</a>, <a>Nat.mod_eq_of_lt</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": "Fin.coe_castLT", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [273, 17], "def_end_pos": [273, 27]}, {"full_name": "Fin.val_add", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [178, 9], "def_end_pos": [178, 16]}, {"full_name": "Fin.val_one", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [160, 17], "def_end_pos": [160, 24]}, {"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": "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 : Nat\ni : Fin (n + 2)\nh : \u2191i < n + 1\n\u22a2 pred (i + 1) (_ : i + 1 \u2260 0) = castLT i h", "state_after": "n : Nat\ni : Fin (n + 2)\nh : \u2191i < n + 1\n\u22a2 \u2191i + 1 < n + 1 + 1"}, {"tactic": "exact Nat.add_lt_add_right h 1", "annotated_tactic": ["exact <a>Nat.add_lt_add_right</a> h 1", [{"full_name": "Nat.add_lt_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [398, 19], "def_end_pos": [398, 35]}]], "state_before": "n : Nat\ni : Fin (n + 2)\nh : \u2191i < n + 1\n\u22a2 \u2191i + 1 < n + 1 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indepFun_iff_indepSet_preimage", "start": [554, 1], "end": [559, 48], "traced_tactics": [{"tactic": "simp only [IndepFun, IndepSet, kernel.indepFun_iff_indepSet_preimage hf hg, ae_dirac_eq,\n  Filter.eventually_pure, kernel.const_apply]", "annotated_tactic": ["simp only [<a>IndepFun</a>, <a>IndepSet</a>, <a>kernel.indepFun_iff_indepSet_preimage</a> hf hg, <a>ae_dirac_eq</a>,\n    <a>Filter.eventually_pure</a>, <a>kernel.const_apply</a>]", [{"full_name": "ProbabilityTheory.IndepFun", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 13]}, {"full_name": "ProbabilityTheory.IndepSet", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [123, 5], "def_end_pos": [123, 13]}, {"full_name": "ProbabilityTheory.kernel.indepFun_iff_indepSet_preimage", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [702, 9], "def_end_pos": [702, 39]}, {"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 : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Measurable f\nhg : Measurable g\n\u22a2 IndepFun f g \u2194 \u2200 (s : Set \u03b2) (t : Set \u03b2'), MeasurableSet s \u2192 MeasurableSet t \u2192 IndepSet (f \u207b\u00b9' s) (g \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.fiber_ncard_ne_zero_iff_mem_image", "start": [676, 1], "end": [681, 33], "traced_tactics": [{"tactic": "refine' \u27e8nonempty_of_ncard_ne_zero, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>nonempty_of_ncard_ne_zero</a>, _\u27e9", [{"full_name": "Set.nonempty_of_ncard_ne_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [554, 9], "def_end_pos": [554, 34]}]], "state_before": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\ny : \u03b2\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard {x | x \u2208 s \u2227 f x = y} \u2260 0 \u2194 y \u2208 f '' s", "state_after": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\ny : \u03b2\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 y \u2208 f '' s \u2192 ncard {x | x \u2208 s \u2227 f x = y} \u2260 0"}, {"tactic": "rintro \u27e8z, hz, rfl\u27e9", "annotated_tactic": ["rintro \u27e8z, hz, rfl\u27e9", []], "state_before": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\ny : \u03b2\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 y \u2208 f '' s \u2192 ncard {x | x \u2208 s \u2227 f x = y} \u2260 0", "state_after": "case intro.intro\n\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nhs : autoParam (Set.Finite s) _auto\u271d\nz : \u03b1\nhz : z \u2208 s\n\u22a2 ncard {x | x \u2208 s \u2227 f x = f z} \u2260 0"}, {"tactic": "exact @ncard_ne_zero_of_mem _ ({ x \u2208 s | f x = f z }) z (mem_sep hz rfl)\n  (hs.subset (sep_subset _ _))", "annotated_tactic": ["exact @<a>ncard_ne_zero_of_mem</a> _ ({ x \u2208 s | f x = f z }) z (<a>mem_sep</a> hz <a>rfl</a>)\n    (hs.subset (<a>sep_subset</a> _ _))", [{"full_name": "Set.ncard_ne_zero_of_mem", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [542, 9], "def_end_pos": [542, 29]}, {"full_name": "Set.mem_sep", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1423, 9], "def_end_pos": [1423, 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": "Set.sep_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1446, 9], "def_end_pos": [1446, 19]}]], "state_before": "case intro.intro\n\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nhs : autoParam (Set.Finite s) _auto\u271d\nz : \u03b1\nhz : z \u2208 s\n\u22a2 ncard {x | x \u2208 s \u2227 f x = f z} \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_homothety_preimage", "start": [1129, 1], "end": [1134, 31], "traced_tactics": [{"tactic": "change \u03bcH[d] (AffineEquiv.homothetyUnitsMulHom x (Units.mk0 c hc) \u207b\u00b9' s) = _", "annotated_tactic": ["change \u03bcH[d] (<a>AffineEquiv.homothetyUnitsMulHom</a> x (<a>Units.mk0</a> c hc) \u207b\u00b9' s) = _", [{"full_name": "AffineEquiv.homothetyUnitsMulHom", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "def_pos": [523, 5], "def_end_pos": [523, 25]}, {"full_name": "Units.mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [184, 5], "def_end_pos": [184, 8]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(AffineMap.homothety x c) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(\u2191(AffineEquiv.homothetyUnitsMulHom x) (Units.mk0 c hc)) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s"}, {"tactic": "rw [\u2190 AffineEquiv.image_symm, AffineEquiv.coe_homothetyUnitsMulHom_apply_symm,\n  hausdorffMeasure_homothety_image hd x (_ : \ud835\udd5c\u02e3).isUnit.ne_zero, Units.val_inv_eq_inv_val,\n  Units.val_mk0, nnnorm_inv]", "annotated_tactic": ["rw [\u2190 <a>AffineEquiv.image_symm</a>, <a>AffineEquiv.coe_homothetyUnitsMulHom_apply_symm</a>,\n    <a>hausdorffMeasure_homothety_image</a> hd x (_ : \ud835\udd5c\u02e3).isUnit.ne_zero, <a>Units.val_inv_eq_inv_val</a>,\n    <a>Units.val_mk0</a>, <a>nnnorm_inv</a>]", [{"full_name": "AffineEquiv.image_symm", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "def_pos": [273, 9], "def_end_pos": [273, 19]}, {"full_name": "AffineEquiv.coe_homothetyUnitsMulHom_apply_symm", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "def_pos": [534, 9], "def_end_pos": [534, 44]}, {"full_name": "MeasureTheory.hausdorffMeasure_homothety_image", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1118, 9], "def_end_pos": [1118, 41]}, {"full_name": "Units.val_inv_eq_inv_val", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [406, 9], "def_end_pos": [406, 27]}, {"full_name": "Units.val_mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 16]}, {"full_name": "nnnorm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 19]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(\u2191(AffineEquiv.homothetyUnitsMulHom x) (Units.mk0 c hc)) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.outerMeasure_preimage", "start": [300, 1], "end": [305, 43], "traced_tactics": [{"tactic": "refine' inducedOuterMeasure_preimage _ \u03bc.innerContent_iUnion_nat \u03bc.innerContent_mono _\n  (fun _ => f.isOpen_preimage) _", "annotated_tactic": ["refine' <a>inducedOuterMeasure_preimage</a> _ \u03bc.innerContent_iUnion_nat \u03bc.innerContent_mono _\n    (fun _ => f.isOpen_preimage) _", [{"full_name": "MeasureTheory.inducedOuterMeasure_preimage", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1485, 9], "def_end_pos": [1485, 37]}]], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nf : G \u2243\u209c G\nh : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K\nA : Set G\n\u22a2 \u2191(Content.outerMeasure \u03bc) (\u2191f \u207b\u00b9' A) = \u2191(Content.outerMeasure \u03bc) A", "state_after": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nf : G \u2243\u209c G\nh : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K\nA : Set G\n\u22a2 \u2200 (s : Set G) (hs : IsOpen s),\n    innerContent \u03bc { carrier := \u2191f.toEquiv \u207b\u00b9' s, is_open' := (_ : IsOpen (\u2191f.toEquiv \u207b\u00b9' s)) } =\n      innerContent \u03bc { carrier := s, is_open' := hs }"}, {"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nf : G \u2243\u209c G\nh : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K\nA : Set G\n\u22a2 \u2200 (s : Set G) (hs : IsOpen s),\n    innerContent \u03bc { carrier := \u2191f.toEquiv \u207b\u00b9' s, is_open' := (_ : IsOpen (\u2191f.toEquiv \u207b\u00b9' s)) } =\n      innerContent \u03bc { carrier := s, is_open' := hs }", "state_after": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nf : G \u2243\u209c G\nh : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K\nA s : Set G\nhs : IsOpen s\n\u22a2 innerContent \u03bc { carrier := \u2191f.toEquiv \u207b\u00b9' s, is_open' := (_ : IsOpen (\u2191f.toEquiv \u207b\u00b9' s)) } =\n    innerContent \u03bc { carrier := s, is_open' := hs }"}, {"tactic": "convert \u03bc.innerContent_comap f h \u27e8s, hs\u27e9", "annotated_tactic": ["convert \u03bc.innerContent_comap f h \u27e8s, hs\u27e9", []], "state_before": "G : Type w\ninst\u271d\u00b9 : TopologicalSpace G\n\u03bc : Content G\ninst\u271d : T2Space G\nf : G \u2243\u209c G\nh : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K\nA s : Set G\nhs : IsOpen s\n\u22a2 innerContent \u03bc { carrier := \u2191f.toEquiv \u207b\u00b9' s, is_open' := (_ : IsOpen (\u2191f.toEquiv \u207b\u00b9' s)) } =\n    innerContent \u03bc { carrier := s, is_open' := hs }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "full_name": "MeasureTheory.Submartingale.stoppedProcess", "start": [95, 11], "end": [105, 93], "traced_tactics": [{"tactic": "rw [submartingale_iff_expected_stoppedValue_mono]", "annotated_tactic": ["rw [<a>submartingale_iff_expected_stoppedValue_mono</a>]", [{"full_name": "MeasureTheory.submartingale_iff_expected_stoppedValue_mono", "def_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "def_pos": [86, 9], "def_end_pos": [86, 53]}]], "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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 Submartingale (stoppedProcess f \u03c4) \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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 \u2200 (\u03c4_1 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \ud835\udca2 \u03c4_1 \u2192\n      IsStoppingTime \ud835\udca2 \u03c0 \u2192\n        \u03c4_1 \u2264 \u03c0 \u2192\n          (\u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N) \u2192\n            \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c4_1 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c0 x \u2202\u03bc\n\ncase hadp\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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 Adapted \ud835\udca2 (stoppedProcess f \u03c4)\n\ncase hint\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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 \u2200 (i : \u2115), Integrable (stoppedProcess f \u03c4 i)"}, {"tactic": "intro \u03c3 \u03c0 h\u03c3 h\u03c0 h\u03c3_le_\u03c0 h\u03c0_bdd", "annotated_tactic": ["intro \u03c3 \u03c0 h\u03c3 h\u03c0 h\u03c3_le_\u03c0 h\u03c0_bdd", []], "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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 \u2200 (\u03c4_1 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \ud835\udca2 \u03c4_1 \u2192\n      IsStoppingTime \ud835\udca2 \u03c0 \u2192\n        \u03c4_1 \u2264 \u03c0 \u2192\n          (\u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N) \u2192\n            \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c4_1 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c0 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\u271d : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \ud835\udca2 \u03c3\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nh\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N\n\u22a2 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c3 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c0 x \u2202\u03bc"}, {"tactic": "simp_rw [stoppedValue_stoppedProcess]", "annotated_tactic": ["simp_rw [<a>stoppedValue_stoppedProcess</a>]", [{"full_name": "MeasureTheory.stoppedValue_stoppedProcess", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [789, 9], "def_end_pos": [789, 36]}]], "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\u271d : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \ud835\udca2 \u03c3\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nh\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N\n\u22a2 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c3 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c0 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\u271d : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \ud835\udca2 \u03c3\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nh\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N\n\u22a2 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c0 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc"}, {"tactic": "obtain \u27e8n, h\u03c0_le_n\u27e9 := h\u03c0_bdd", "annotated_tactic": ["obtain \u27e8n, h\u03c0_le_n\u27e9 := h\u03c0_bdd", []], "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\u271d : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \ud835\udca2 \u03c3\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nh\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N\n\u22a2 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c0 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc", "state_after": "case 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\u271d : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \ud835\udca2 \u03c3\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c0 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc"}, {"tactic": "exact h.expected_stoppedValue_mono (h\u03c3.min h\u03c4) (h\u03c0.min h\u03c4)\n  (fun \u03c9 => min_le_min (h\u03c3_le_\u03c0 \u03c9) le_rfl) fun \u03c9 => (min_le_left _ _).trans (h\u03c0_le_n \u03c9)", "annotated_tactic": ["exact h.expected_stoppedValue_mono (h\u03c3.min h\u03c4) (h\u03c0.min h\u03c4)\n      (fun \u03c9 => <a>min_le_min</a> (h\u03c3_le_\u03c0 \u03c9) <a>le_rfl</a>) fun \u03c9 => (<a>min_le_left</a> _ _).<a>trans</a> (h\u03c0_le_n \u03c9)", [{"full_name": "min_le_min", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}, {"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]}, {"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\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\u271d : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \ud835\udca2 \u03c3\nh\u03c0 : IsStoppingTime \ud835\udca2 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c0 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact Adapted.stoppedProcess_of_discrete h.adapted h\u03c4", "annotated_tactic": ["exact <a>Adapted.stoppedProcess_of_discrete</a> h.adapted h\u03c4", [{"full_name": "MeasureTheory.Adapted.stoppedProcess_of_discrete", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [1035, 9], "def_end_pos": [1035, 43]}]], "state_before": "case hadp\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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 Adapted \ud835\udca2 (stoppedProcess f \u03c4)", "state_after": "no goals"}, {"tactic": "exact fun i =>\n  h.integrable_stoppedValue ((isStoppingTime_const _ i).min h\u03c4) fun \u03c9 => min_le_left _ _", "annotated_tactic": ["exact fun i =>\n      h.integrable_stoppedValue ((<a>isStoppingTime_const</a> _ i).<a>min</a> h\u03c4) fun \u03c9 => <a>min_le_left</a> _ _", [{"full_name": "MeasureTheory.isStoppingTime_const", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [57, 9], "def_end_pos": [57, 29]}, {"full_name": "MeasureTheory.IsStoppingTime.min", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [265, 19], "def_end_pos": [265, 22]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case hint\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\nh : Submartingale f \ud835\udca2 \u03bc\nh\u03c4 : IsStoppingTime \ud835\udca2 \u03c4\n\u22a2 \u2200 (i : \u2115), Integrable (stoppedProcess f \u03c4 i)", "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_Ioi", "start": [89, 1], "end": [90, 83], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ioi</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>, <a>Ioi_inter_Iio</a>]", [{"full_name": "WithTop.preimage_coe_Ioi", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [39, 9], "def_end_pos": [39, 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.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 '' Ioi a = Ioo \u2191a \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.RightInvOn.comp", "start": [1152, 1], "end": [1154, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Nat.Primrec'.prec'", "start": [1443, 1], "end": [1445, 47], "traced_tactics": [{"tactic": "simpa using comp' (prec hg hh) (hf.cons idv)", "annotated_tactic": ["simpa using <a>comp'</a> (<a>prec</a> hg hh) (hf.cons <a>idv</a>)", [{"full_name": "Nat.Primrec'.comp'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1429, 9], "def_end_pos": [1429, 14]}, {"full_name": "Nat.Primrec'.prec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1367, 5], "def_end_pos": [1367, 9]}, {"full_name": "Nat.Primrec'.idv", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 12]}]], "state_before": "n : \u2115\nf g : Vector \u2115 n \u2192 \u2115\nh : Vector \u2115 (n + 2) \u2192 \u2115\nhf : Primrec' f\nhg : Primrec' g\nhh : Primrec' h\n\u22a2 Primrec' fun v => Nat.rec (g v) (fun y IH => h (y ::\u1d65 IH ::\u1d65 v)) (f v)", "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_eq_sum_compProd_right", "start": [501, 1], "end": [509, 23], "traced_tactics": [{"tactic": "by_cases h\u03ba : IsSFiniteKernel \u03ba", "annotated_tactic": ["by_cases h\u03ba : <a>IsSFiniteKernel</a> \u03ba", [{"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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n", "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n\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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : \u00acIsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n"}, {"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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n\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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : \u00acIsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n", "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : \u00acIsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n\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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n"}, {"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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n", "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n"}, {"tactic": "simp_rw [compProd_eq_sum_compProd_left \u03ba _]", "annotated_tactic": ["simp_rw [<a>compProd_eq_sum_compProd_left</a> \u03ba _]", [{"full_name": "ProbabilityTheory.kernel.compProd_eq_sum_compProd_left", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [489, 9], "def_end_pos": [489, 38]}]], "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n", "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) =\n    kernel.sum fun n => kernel.sum fun n_1 => seq \u03ba n_1 \u2297\u2096 seq \u03b7 n"}, {"tactic": "rw [kernel.sum_comm]", "annotated_tactic": ["rw [<a>kernel.sum_comm</a>]", [{"full_name": "ProbabilityTheory.kernel.sum_comm", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 17]}]], "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : IsSFiniteKernel \u03ba\n\u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) =\n    kernel.sum fun n => kernel.sum fun n_1 => seq \u03ba n_1 \u2297\u2096 seq \u03b7 n", "state_after": "no goals"}, {"tactic": "simp_rw [compProd_of_not_isSFiniteKernel_left _ _ h\u03ba]", "annotated_tactic": ["simp_rw [<a>compProd_of_not_isSFiniteKernel_left</a> _ _ h\u03ba]", [{"full_name": "ProbabilityTheory.kernel.compProd_of_not_isSFiniteKernel_left", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [230, 9], "def_end_pos": [230, 45]}]], "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : \u00acIsSFiniteKernel \u03ba\n\u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n", "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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : \u00acIsSFiniteKernel \u03ba\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 }\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\nh\u03ba : \u00acIsSFiniteKernel \u03ba\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/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero", "start": [360, 1], "end": [377, 30], "traced_tactics": [{"tactic": "by_cases h : IntegrableOn f t \u03bc", "annotated_tactic": ["by_cases h : <a>IntegrableOn</a> f t \u03bc", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : \u00acIntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : \u00acIntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : \u00acIntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "let f' := h.1.mk f", "annotated_tactic": ["let f' := h.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": "case pos\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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "calc\n  \u222b x in t, f x \u2202\u03bc = \u222b x in t, f' x \u2202\u03bc := integral_congr_ae h.1.ae_eq_mk\n  _ = \u222b x in s, f' x \u2202\u03bc := by\n    apply\n      set_integral_eq_of_subset_of_ae_diff_eq_zero_aux hts _ h.1.stronglyMeasurable_mk\n        (h.congr h.1.ae_eq_mk)\n    filter_upwards [h't, ae_imp_of_ae_restrict h.1.ae_eq_mk] with x hx h'x h''x\n    rw [\u2190 h'x h''x.1, hx h''x]\n  _ = \u222b x in s, f x \u2202\u03bc := by\n    apply integral_congr_ae\n    apply ae_restrict_of_ae_restrict_of_subset hts\n    exact h.1.ae_eq_mk.symm", "annotated_tactic": ["calc\n    \u222b x in t, f x \u2202\u03bc = \u222b x in t, f' x \u2202\u03bc := <a>integral_congr_ae</a> h.1.<a>ae_eq_mk</a>\n    _ = \u222b x in s, f' x \u2202\u03bc := by\n      apply\n        <a>set_integral_eq_of_subset_of_ae_diff_eq_zero_aux</a> hts _ h.1.<a>stronglyMeasurable_mk</a>\n          (h.congr h.1.<a>ae_eq_mk</a>)\n      filter_upwards [h't, <a>ae_imp_of_ae_restrict</a> h.1.<a>ae_eq_mk</a>] with x hx h'x h''x\n      rw [\u2190 h'x h''x.1, hx h''x]\n    _ = \u222b x in s, f x \u2202\u03bc := by\n      apply <a>integral_congr_ae</a>\n      apply <a>ae_restrict_of_ae_restrict_of_subset</a> hts\n      exact h.1.ae_eq_mk.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.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero_aux", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [331, 9], "def_end_pos": [331, 57]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 17]}, {"full_name": "MeasureTheory.ae_imp_of_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2566, 9], "def_end_pos": [2566, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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": "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 pos\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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "have : \u00acIntegrableOn f s \u03bc := fun H => h (H.of_ae_diff_eq_zero ht h't)", "annotated_tactic": ["have : \u00ac<a>IntegrableOn</a> f s \u03bc := fun H => h (H.of_ae_diff_eq_zero ht h't)", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "state_before": "case neg\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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : \u00acIntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : \u00acIntegrableOn f t\nthis : \u00acIntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc"}, {"tactic": "rw [integral_undef h, integral_undef this]", "annotated_tactic": ["rw [<a>integral_undef</a> h, <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]}, {"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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : \u00acIntegrableOn f t\nthis : \u00acIntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply\n  set_integral_eq_of_subset_of_ae_diff_eq_zero_aux hts _ h.1.stronglyMeasurable_mk\n    (h.congr h.1.ae_eq_mk)", "annotated_tactic": ["apply\n        <a>set_integral_eq_of_subset_of_ae_diff_eq_zero_aux</a> hts _ h.1.<a>stronglyMeasurable_mk</a>\n          (h.congr h.1.<a>ae_eq_mk</a>)", [{"full_name": "MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero_aux", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [331, 9], "def_end_pos": [331, 57]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u222b (x : \u03b1) in t, f' x \u2202\u03bc = \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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x = 0"}, {"tactic": "filter_upwards [h't, ae_imp_of_ae_restrict h.1.ae_eq_mk] with x hx h'x h''x", "annotated_tactic": ["filter_upwards [h't, <a>ae_imp_of_ae_restrict</a> h.1.<a>ae_eq_mk</a>] with x hx h'x h''x", [{"full_name": "MeasureTheory.ae_imp_of_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2566, 9], "def_end_pos": [2566, 30]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1215, 9], "def_end_pos": [1215, 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 l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x = 0", "state_after": "case 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\nx : \u03b1\nhx : x \u2208 t \\ s \u2192 f x = 0\nh'x : x \u2208 t \u2192 f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x\nh''x : x \u2208 t \\ s\n\u22a2 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x = 0"}, {"tactic": "rw [\u2190 h'x h''x.1, hx h''x]", "annotated_tactic": ["rw [\u2190 h'x h''x.1, hx h''x]", []], "state_before": "case 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\nx : \u03b1\nhx : x \u2208 t \\ s \u2192 f x = 0\nh'x : x \u2208 t \u2192 f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x\nh''x : x \u2208 t \\ s\n\u22a2 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x = 0", "state_after": "no goals"}, {"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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u222b (x : \u03b1) in s, f' x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc", "state_after": "case 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 (fun a => f' a) =\u1d50[Measure.restrict \u03bc s] fun a => f a"}, {"tactic": "apply ae_restrict_of_ae_restrict_of_subset hts", "annotated_tactic": ["apply <a>ae_restrict_of_ae_restrict_of_subset</a> hts", [{"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 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 (fun a => f' a) =\u1d50[Measure.restrict \u03bc s] fun a => f 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\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\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, (fun a => f' a) x = (fun a => f a) x"}, {"tactic": "exact h.1.ae_eq_mk.symm", "annotated_tactic": ["exact h.1.ae_eq_mk.symm", []], "state_before": "case 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 g : \u03b1 \u2192 E\ns t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \u211d E\nht : NullMeasurableSet t\nhts : s \u2286 t\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0\nh : IntegrableOn f t\nf' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, (fun a => f' a) x = (fun a => f a) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_finset_subset_smul_finset_iff\u2080", "start": [2076, 1], "end": [2077, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.forall_sigma_iff", "start": [79, 1], "end": [81, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage", "start": [1368, 1], "end": [1371, 58], "traced_tactics": [{"tactic": "rw [\u2190 hg.map_eq, set_lintegral_map hs hf hg.measurable]", "annotated_tactic": ["rw [\u2190 hg.map_eq, <a>set_lintegral_map</a> hs hf hg.measurable]", [{"full_name": "MeasureTheory.set_lintegral_map", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1316, 9], "def_end_pos": [1316, 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\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\ns : Set \u03b2\nhs : MeasurableSet s\nf : \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1) in g \u207b\u00b9' s, f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2) in s, f b \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurable.measurable_of_complete", "start": [471, 1], "end": [472, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_image", "start": [1036, 1], "end": [1039, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant'", "start": [641, 1], "end": [644, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "EMetric.measure_closedBall_pos", "start": [249, 1], "end": [250, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpIndL1_disjoint_union", "start": [248, 1], "end": [256, 59], "traced_tactics": [{"tactic": "have h\u03bcst : \u03bc (s \u222a t) \u2260 \u221e :=\n  ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr \u27e8h\u03bcs, h\u03bct\u27e9))).ne", "annotated_tactic": ["have h\u03bcst : \u03bc (s \u222a t) \u2260 \u221e :=\n    ((<a>measure_union_le</a> s t).<a>trans_lt</a> (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr \u27e8h\u03bcs, h\u03bct\u27e9))).<a>ne</a>", [{"full_name": "MeasureTheory.measure_union_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}, {"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\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 condexpIndL1 hm \u03bc (s \u222a t) x = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc t 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\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 condexpIndL1 hm \u03bc (s \u222a t) x = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc t x"}, {"tactic": "rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs x,\n  condexpIndL1_of_measurableSet_of_measure_ne_top ht h\u03bct x,\n  condexpIndL1_of_measurableSet_of_measure_ne_top (hs.union ht) h\u03bcst x]", "annotated_tactic": ["rw [<a>condexpIndL1_of_measurableSet_of_measure_ne_top</a> hs h\u03bcs x,\n    <a>condexpIndL1_of_measurableSet_of_measure_ne_top</a> ht h\u03bct x,\n    <a>condexpIndL1_of_measurableSet_of_measure_ne_top</a> (hs.union ht) h\u03bcst 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]}, {"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]}, {"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": "\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 condexpIndL1 hm \u03bc (s \u222a t) x = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc t 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\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x"}, {"tactic": "exact condexpIndL1Fin_disjoint_union hs ht h\u03bcs h\u03bct hst x", "annotated_tactic": ["exact <a>condexpIndL1Fin_disjoint_union</a> hs ht h\u03bcs h\u03bct hst x", [{"full_name": "MeasureTheory.condexpIndL1Fin_disjoint_union", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [147, 9], "def_end_pos": [147, 39]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct 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.compl_mem_cofinite", "start": [2402, 1], "end": [2402, 92], "traced_tactics": [{"tactic": "rw [mem_cofinite, compl_compl]", "annotated_tactic": ["rw [<a>mem_cofinite</a>, <a>compl_compl</a>]", [{"full_name": "MeasureTheory.Measure.mem_cofinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2398, 9], "def_end_pos": [2398, 21]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 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\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\n\u22a2 s\u1d9c \u2208 cofinite \u03bc \u2194 \u2191\u2191\u03bc s < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Prime.lean", "full_name": "PosNum.minFac_to_nat", "start": [65, 1], "end": [82, 9], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "n : PosNum\n\u22a2 \u2191(minFac n) = Nat.minFac \u2191n", "state_after": "case one\n\n\u22a2 \u2191(minFac one) = Nat.minFac \u2191one\n\ncase bit1\nn : PosNum\n\u22a2 \u2191(minFac (bit1 n)) = Nat.minFac \u2191(bit1 n)\n\ncase bit0\na\u271d : PosNum\n\u22a2 \u2191(minFac (bit0 a\u271d)) = Nat.minFac \u2191(bit0 a\u271d)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case one\n\n\u22a2 \u2191(minFac one) = Nat.minFac \u2191one", "state_after": "no goals"}, {"tactic": "rw [minFac, Nat.minFac_eq, if_neg]", "annotated_tactic": ["rw [<a>minFac</a>, <a>Nat.minFac_eq</a>, <a>if_neg</a>]", [{"full_name": "PosNum.minFac", "def_path": "Mathlib/Data/Num/Prime.lean", "def_pos": [58, 5], "def_end_pos": [58, 11]}, {"full_name": "Nat.minFac_eq", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [276, 9], "def_end_pos": [276, 18]}, {"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 bit1\nn : PosNum\n\u22a2 \u2191(minFac (bit1 n)) = Nat.minFac \u2191(bit1 n)", "state_after": "case bit1\nn : PosNum\n\u22a2 \u2191(match bit1 n with\n      | one => 1\n      | bit0 a => 2\n      | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =\n    Nat.minFacAux (\u2191(bit1 n)) 3\n\ncase bit1.hnc\nn : PosNum\n\u22a2 \u00ac2 \u2223 \u2191(bit1 n)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case bit1\nn : PosNum\n\u22a2 \u2191(match bit1 n with\n      | one => 1\n      | bit0 a => 2\n      | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =\n    Nat.minFacAux (\u2191(bit1 n)) 3\n\ncase bit1.hnc\nn : PosNum\n\u22a2 \u00ac2 \u2223 \u2191(bit1 n)", "state_after": "case bit1.hnc\nn : PosNum\n\u22a2 \u00ac2 \u2223 \u2191(bit1 n)\n\ncase bit1\nn : PosNum\n\u22a2 \u2191(match bit1 n with\n      | one => 1\n      | bit0 a => 2\n      | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =\n    Nat.minFacAux (\u2191(bit1 n)) 3"}, {"tactic": "rw [minFacAux_to_nat]", "annotated_tactic": ["rw [<a>minFacAux_to_nat</a>]", [{"full_name": "PosNum.minFacAux_to_nat", "def_path": "Mathlib/Data/Num/Prime.lean", "def_pos": [44, 9], "def_end_pos": [44, 25]}]], "state_before": "case bit1\nn : PosNum\n\u22a2 \u2191(match bit1 n with\n      | one => 1\n      | bit0 a => 2\n      | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =\n    Nat.minFacAux (\u2191(bit1 n)) 3", "state_after": "case bit1\nn : PosNum\n\u22a2 Nat.minFacAux \u2191(bit1 n) \u2191(bit1 1) = Nat.minFacAux (\u2191(bit1 n)) 3\n\ncase bit1\nn : PosNum\n\u22a2 Nat.sqrt \u2191(bit1 n) < \u2191n + \u2191(bit1 1)"}, {"tactic": "simp only [cast_one, cast_bit1]", "annotated_tactic": ["simp only [<a>cast_one</a>, <a>cast_bit1</a>]", [{"full_name": "PosNum.cast_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}, {"full_name": "PosNum.cast_bit1", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [52, 9], "def_end_pos": [52, 18]}]], "state_before": "case bit1\nn : PosNum\n\u22a2 Nat.sqrt \u2191(bit1 n) < \u2191n + \u2191(bit1 1)", "state_after": "case bit1\nn : PosNum\n\u22a2 Nat.sqrt (_root_.bit1 \u2191n) < \u2191n + _root_.bit1 1"}, {"tactic": "unfold _root_.bit1 _root_.bit0", "annotated_tactic": ["unfold <a>_root_.bit1</a> <a>_root_.bit0</a>", [{"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": "case bit1\nn : PosNum\n\u22a2 Nat.sqrt (_root_.bit1 \u2191n) < \u2191n + _root_.bit1 1", "state_after": "case bit1\nn : PosNum\n\u22a2 Nat.sqrt (\u2191n + \u2191n + 1) < \u2191n + (1 + 1 + 1)"}, {"tactic": "rw [Nat.sqrt_lt]", "annotated_tactic": ["rw [<a>Nat.sqrt_lt</a>]", [{"full_name": "Nat.sqrt_lt", "def_path": "Mathlib/Data/Nat/Sqrt.lean", "def_pos": [92, 9], "def_end_pos": [92, 16]}]], "state_before": "case bit1\nn : PosNum\n\u22a2 Nat.sqrt (\u2191n + \u2191n + 1) < \u2191n + (1 + 1 + 1)", "state_after": "case bit1\nn : PosNum\n\u22a2 \u2191n + \u2191n + 1 < (\u2191n + (1 + 1 + 1)) * (\u2191n + (1 + 1 + 1))"}, {"tactic": "calc\n  (n : \u2115) + (n : \u2115) + 1 \u2264 (n : \u2115) + (n : \u2115) + (n : \u2115) := by simp\n  _ = (n : \u2115) * (1 + 1 + 1) := by simp only [mul_add, mul_one]\n  _ < _ := by simp [mul_lt_mul]", "annotated_tactic": ["calc\n      (n : \u2115) + (n : \u2115) + 1 \u2264 (n : \u2115) + (n : \u2115) + (n : \u2115) := by simp\n      _ = (n : \u2115) * (1 + 1 + 1) := by simp only [<a>mul_add</a>, <a>mul_one</a>]\n      _ < _ := by simp [<a>mul_lt_mul</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"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", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}]], "state_before": "case bit1\nn : PosNum\n\u22a2 \u2191n + \u2191n + 1 < (\u2191n + (1 + 1 + 1)) * (\u2191n + (1 + 1 + 1))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case bit1.hnc\nn : PosNum\n\u22a2 \u00ac2 \u2223 \u2191(bit1 n)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case bit1\nn : PosNum\n\u22a2 Nat.minFacAux \u2191(bit1 n) \u2191(bit1 1) = Nat.minFacAux (\u2191(bit1 n)) 3", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : PosNum\n\u22a2 \u2191n + \u2191n + 1 \u2264 \u2191n + \u2191n + \u2191n", "state_after": "no goals"}, {"tactic": "simp only [mul_add, mul_one]", "annotated_tactic": ["simp only [<a>mul_add</a>, <a>mul_one</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "n : PosNum\n\u22a2 \u2191n + \u2191n + \u2191n = \u2191n * (1 + 1 + 1)", "state_after": "no goals"}, {"tactic": "simp [mul_lt_mul]", "annotated_tactic": ["simp [<a>mul_lt_mul</a>]", [{"full_name": "mul_lt_mul", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}]], "state_before": "n : PosNum\n\u22a2 \u2191n * (1 + 1 + 1) < (\u2191n + (1 + 1 + 1)) * (\u2191n + (1 + 1 + 1))", "state_after": "no goals"}, {"tactic": "rw [minFac, Nat.minFac_eq, if_pos]", "annotated_tactic": ["rw [<a>minFac</a>, <a>Nat.minFac_eq</a>, <a>if_pos</a>]", [{"full_name": "PosNum.minFac", "def_path": "Mathlib/Data/Num/Prime.lean", "def_pos": [58, 5], "def_end_pos": [58, 11]}, {"full_name": "Nat.minFac_eq", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [276, 9], "def_end_pos": [276, 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 bit0\na\u271d : PosNum\n\u22a2 \u2191(minFac (bit0 a\u271d)) = Nat.minFac \u2191(bit0 a\u271d)", "state_after": "case bit0\na\u271d : PosNum\n\u22a2 \u2191(match bit0 a\u271d with\n      | one => 1\n      | bit0 a => 2\n      | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =\n    2\n\ncase bit0.hc\na\u271d : PosNum\n\u22a2 2 \u2223 \u2191(bit0 a\u271d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case bit0.hc\na\u271d : PosNum\n\u22a2 2 \u2223 \u2191(bit0 a\u271d)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case bit0\na\u271d : PosNum\n\u22a2 \u2191(match bit0 a\u271d with\n      | one => 1\n      | bit0 a => 2\n      | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =\n    2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.Subsingleton.aestronglyMeasurable", "start": [1168, 1], "end": [1170, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.getLastD_concat", "start": [729, 9], "end": [730, 50], "traced_tactics": [{"tactic": "rw [getLastD_eq_getLast?, getLast?_concat]", "annotated_tactic": ["rw [<a>getLastD_eq_getLast?</a>, <a>getLast?_concat</a>]", [{"full_name": "List.getLastD_eq_getLast?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [544, 9], "def_end_pos": [544, 29]}, {"full_name": "List.getLast?_concat", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [726, 17], "def_end_pos": [726, 32]}]], "state_before": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\n\u22a2 getLastD (l ++ [b]) a = b", "state_after": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\n\u22a2 Option.getD (some b) a = b"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\na b : \u03b1\nl : List \u03b1\n\u22a2 Option.getD (some b) a = b", "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_one_of_density_one_aux", "start": [775, 1], "end": [824, 19], "traced_tactics": [{"tactic": "have L' : Tendsto (fun r : \u211d => \u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n  tendsto_addHaar_inter_smul_zero_of_density_zero \u03bc s\u1d9c x L t ht h''t", "annotated_tactic": ["have L' : <a>Tendsto</a> (fun r : \u211d => \u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n    <a>tendsto_addHaar_inter_smul_zero_of_density_zero</a> \u03bc s\u1d9c x L t ht h''t", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [715, 9], "def_end_pos": [715, 56]}]], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \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 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\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 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)"}, {"tactic": "have L'' : Tendsto (fun r : \u211d => \u03bc ({x} + r \u2022 t) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) := by\n  apply tendsto_const_nhds.congr' _\n  filter_upwards [self_mem_nhdsWithin]\n  rintro r (rpos : 0 < r)\n  rw [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_self h't h''t]", "annotated_tactic": ["have L'' : <a>Tendsto</a> (fun r : \u211d => \u03bc ({x} + r \u2022 t) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) := by\n    apply tendsto_const_nhds.congr' _\n    filter_upwards [<a>self_mem_nhdsWithin</a>]\n    rintro r (rpos : 0 < r)\n    rw [<a>addHaar_singleton_add_smul_div_singleton_add_smul</a> \u03bc rpos.ne', <a>ENNReal.div_self</a> h't h''t]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"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_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1746, 19], "def_end_pos": [1746, 27]}]], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\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 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\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 1)"}, {"tactic": "have := ENNReal.Tendsto.sub L'' L' (Or.inl ENNReal.one_ne_top)", "annotated_tactic": ["have := <a>ENNReal.Tendsto.sub</a> L'' L' (<a>Or.inl</a> <a>ENNReal.one_ne_top</a>)", [{"full_name": "ENNReal.Tendsto.sub", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [324, 19], "def_end_pos": [324, 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": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}]], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\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 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (1 - 0))\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 1)"}, {"tactic": "simp only [tsub_zero] at this", "annotated_tactic": ["simp only [<a>tsub_zero</a>] at this", [{"full_name": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd (1 - 0))\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 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\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 1)"}, {"tactic": "apply this.congr' _", "annotated_tactic": ["apply this.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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\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 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\n\u22a2 (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n    fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)"}, {"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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\n\u22a2 (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n    fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) =\n        \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)"}, {"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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) =\n        \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)"}, {"tactic": "refine' I ({x} + r \u2022 t) s _ _ hs", "annotated_tactic": ["refine' I ({x} + r \u2022 t) s _ _ hs", []], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) =\n    \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)", "state_after": "case h.refine'_1\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 0\n\ncase h.refine'_2\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 \u22a4"}, {"tactic": "intro u v uzero utop vmeas", "annotated_tactic": ["intro u v uzero utop vmeas", []], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u"}, {"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": "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u * (\u2191\u2191\u03bc u)\u207b\u00b9 - \u2191\u2191\u03bc (v\u1d9c \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9"}, {"tactic": "rw [\u2190 ENNReal.sub_mul]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.sub_mul</a>]", [{"full_name": "ENNReal.sub_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1235, 9], "def_end_pos": [1235, 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u * (\u2191\u2191\u03bc u)\u207b\u00b9 - \u2191\u2191\u03bc (v\u1d9c \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9\n\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9\n\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4\n\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9", "state_after": "case e_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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc (v \u2229 u)"}, {"tactic": "apply\n  ENNReal.sub_eq_of_add_eq (ne_top_of_le_ne_top utop (measure_mono (inter_subset_right _ _)))", "annotated_tactic": ["apply\n      <a>ENNReal.sub_eq_of_add_eq</a> (<a>ne_top_of_le_ne_top</a> utop (<a>measure_mono</a> (<a>inter_subset_right</a> _ _)))", [{"full_name": "ENNReal.sub_eq_of_add_eq", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1172, 9], "def_end_pos": [1172, 25]}, {"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": "case e_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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc (v \u2229 u)", "state_after": "case e_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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc (v \u2229 u) + \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc u"}, {"tactic": "rw [inter_comm _ u, inter_comm _ u]", "annotated_tactic": ["rw [<a>inter_comm</a> _ u, <a>inter_comm</a> _ u]", [{"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 e_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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc (v \u2229 u) + \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc u", "state_after": "case e_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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc (u \u2229 v) + \u2191\u2191\u03bc (u \u2229 v\u1d9c) = \u2191\u2191\u03bc u"}, {"tactic": "exact measure_inter_add_diff u vmeas", "annotated_tactic": ["exact <a>measure_inter_add_diff</a> u vmeas", [{"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": "case e_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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc (u \u2229 v) + \u2191\u2191\u03bc (u \u2229 v\u1d9c) = \u2191\u2191\u03bc u", "state_after": "no goals"}, {"tactic": "simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [uzero, <a>ENNReal.inv_eq_top</a>, <a>imp_true_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.inv_eq_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1449, 17], "def_end_pos": [1449, 27]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 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": "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "have B := ENNReal.Tendsto.sub A h (Or.inl ENNReal.one_ne_top)", "annotated_tactic": ["have B := <a>ENNReal.Tendsto.sub</a> A h (<a>Or.inl</a> <a>ENNReal.one_ne_top</a>)", [{"full_name": "ENNReal.Tendsto.sub", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [324, 19], "def_end_pos": [324, 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": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}]], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (1 - 1))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "simp only [tsub_self] at B", "annotated_tactic": ["simp only [<a>tsub_self</a>] at B", [{"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (1 - 1))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)"}, {"tactic": "apply B.congr' _", "annotated_tactic": ["apply B.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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun a =>\n      \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n    fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x 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\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun a =>\n      \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n    fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) =\n        \u2191\u2191\u03bc (s\u1d9c \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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n    a \u2208 Ioi 0 \u2192\n      \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) =\n        \u2191\u2191\u03bc (s\u1d9c \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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\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) - \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) =\n    \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "convert I (closedBall x r) s\u1d9c (measure_closedBall_pos \u03bc _ rpos).ne'\n  measure_closedBall_lt_top.ne hs.compl", "annotated_tactic": ["convert I (<a>closedBall</a> x r) s\u1d9c (<a>measure_closedBall_pos</a> \u03bc _ rpos).<a>ne'</a>\n      measure_closedBall_lt_top.ne hs.compl", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"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]}]], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\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) - \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) =\n    \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)", "state_after": "case h.e'_2.h.e'_6.h.e'_5.h.e'_3.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\ns\u271d s : Set E\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 s = s\u1d9c\u1d9c"}, {"tactic": "rw [compl_compl]", "annotated_tactic": ["rw [<a>compl_compl</a>]", [{"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}]], "state_before": "case h.e'_2.h.e'_6.h.e'_5.h.e'_3.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\ns\u271d s : Set E\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n  Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n    (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 s = s\u1d9c\u1d9c", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x 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\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a)"}, {"tactic": "intro r hr", "annotated_tactic": ["intro r hr", []], "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc (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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 1 = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)"}, {"tactic": "rw [div_eq_mul_inv, ENNReal.mul_inv_cancel]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>ENNReal.mul_inv_cancel</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": "ENNReal.mul_inv_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1418, 19], "def_end_pos": [1418, 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 1 = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)", "state_after": "case h.h0\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 0\n\ncase h.ht\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 \u22a4"}, {"tactic": "exact (measure_closedBall_pos \u03bc _ hr).ne'", "annotated_tactic": ["exact (<a>measure_closedBall_pos</a> \u03bc _ hr).<a>ne'</a>", [{"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]}]], "state_before": "case h.h0\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 0", "state_after": "no goals"}, {"tactic": "exact measure_closedBall_lt_top.ne", "annotated_tactic": ["exact measure_closedBall_lt_top.ne", []], "state_before": "case h.ht\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 \u22a4", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)"}, {"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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t)"}, {"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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t)", "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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 1 = \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)"}, {"tactic": "rw [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_self h't h''t]", "annotated_tactic": ["rw [<a>addHaar_singleton_add_smul_div_singleton_add_smul</a> \u03bc rpos.ne', <a>ENNReal.div_self</a> h't 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_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1746, 19], "def_end_pos": [1746, 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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 1 = \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)", "state_after": "no goals"}, {"tactic": "simp only [h't, abs_of_nonneg rpos.le, pow_pos rpos, addHaar_smul, image_add_left,\n  ENNReal.ofReal_eq_zero, not_le, or_false_iff, Ne.def, measure_preimage_add, abs_pow,\n  singleton_add, mul_eq_zero]", "annotated_tactic": ["simp only [h't, <a>abs_of_nonneg</a> rpos.le, <a>pow_pos</a> rpos, <a>addHaar_smul</a>, <a>image_add_left</a>,\n      <a>ENNReal.ofReal_eq_zero</a>, <a>not_le</a>, <a>or_false_iff</a>, <a>Ne.def</a>, <a>measure_preimage_add</a>, <a>abs_pow</a>,\n      <a>singleton_add</a>, <a>mul_eq_zero</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": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}, {"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": "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]}, {"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": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 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": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}]], "state_before": "case h.refine'_1\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 0", "state_after": "no goals"}, {"tactic": "simp [h''t, ENNReal.ofReal_ne_top, addHaar_smul, image_add_left, ENNReal.mul_eq_top,\n  Ne.def, not_false_iff, measure_preimage_add, singleton_add, and_false_iff, false_and_iff,\n  or_self_iff]", "annotated_tactic": ["simp [h''t, <a>ENNReal.ofReal_ne_top</a>, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>ENNReal.mul_eq_top</a>,\n      <a>Ne.def</a>, <a>not_false_iff</a>, <a>measure_preimage_add</a>, <a>singleton_add</a>, <a>and_false_iff</a>, <a>false_and_iff</a>,\n      <a>or_self_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": "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": "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": "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": "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 h.refine'_2\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\nhs : MeasurableSet s\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 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n  \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n  Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n    (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \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.Lp.ae_eq_of_forall_set_integral_eq", "start": [480, 1], "end": [487, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_finset_subset_smul_finset_iff", "start": [1966, 1], "end": [1967, 50], "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_mem_haarProduct", "start": [364, 1], "end": [366, 90], "traced_tactics": [{"tactic": "rintro \u27e8K, hK\u27e9 _", "annotated_tactic": ["rintro \u27e8K, hK\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\nU : Set G\nhU : Set.Nonempty (interior U)\n\u22a2 prehaar (\u2191K\u2080) U \u2208 haarProduct \u2191K\u2080", "state_after": "case mk\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 : Set G\nhK : IsCompact K\na\u271d : { carrier := K, isCompact' := hK } \u2208 univ\n\u22a2 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2208\n    (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) { carrier := K, isCompact' := hK }"}, {"tactic": "rw [mem_Icc]", "annotated_tactic": ["rw [<a>mem_Icc</a>]", [{"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 mk\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 : Set G\nhK : IsCompact K\na\u271d : { carrier := K, isCompact' := hK } \u2208 univ\n\u22a2 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2208\n    (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) { carrier := K, isCompact' := hK }", "state_after": "case mk\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 : Set G\nhK : IsCompact K\na\u271d : { carrier := K, isCompact' := hK } \u2208 univ\n\u22a2 0 \u2264 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2227\n    prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2264 \u2191(index \u2191{ carrier := K, isCompact' := hK } \u2191K\u2080)"}, {"tactic": "exact \u27e8prehaar_nonneg K\u2080 _, prehaar_le_index K\u2080 _ hU\u27e9", "annotated_tactic": ["exact \u27e8<a>prehaar_nonneg</a> K\u2080 _, <a>prehaar_le_index</a> K\u2080 _ hU\u27e9", [{"full_name": "MeasureTheory.Measure.haar.prehaar_nonneg", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 23]}, {"full_name": "MeasureTheory.Measure.haar.prehaar_le_index", "def_path": 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"annotated_tactic": ["simp_rw [<a>smul_inter_ne_empty_iff</a>, <a>div_eq_mul_inv</a>]", [{"full_name": "Set.smul_inter_ne_empty_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [963, 9], "def_end_pos": [963, 32]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 x \u2022 s \u2229 t \u2260 \u2205 \u2194 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a / b = x", "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_eq_foldr_data", "start": [71, 1], "end": [73, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.toLp_injective", "start": [1813, 1], "end": [1815, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.comp_def", "start": [102, 1], "end": [103, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithTop.image_coe_Icc", "start": [107, 1], "end": [110, 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_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]", "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_Iic_self</a> <| <a>Iic_subset_Iio</a>.2 <| <a>coe_lt_top</a> b)]", [{"full_name": "WithTop.preimage_coe_Icc", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [59, 9], "def_end_pos": [59, 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.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.Icc_subset_Iic_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [474, 9], "def_end_pos": [474, 28]}, {"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 '' Icc a b = Icc \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.Countable.biUnion_iff", "start": [198, 1], "end": [201, 63], "traced_tactics": [{"tactic": "haveI := hs.to_subtype", "annotated_tactic": ["haveI := hs.to_subtype", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nt : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b2\nhs : Set.Countable s\n\u22a2 Set.Countable (\u22c3 a, \u22c3 (h : a \u2208 s), t a h) \u2194 \u2200 (a : \u03b1) (ha : a \u2208 s), Set.Countable (t a ha)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nt : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b2\nhs : Set.Countable s\nthis : Countable \u2191s\n\u22a2 Set.Countable (\u22c3 a, \u22c3 (h : a \u2208 s), t a h) \u2194 \u2200 (a : \u03b1) (ha : a \u2208 s), Set.Countable (t a ha)"}, {"tactic": "rw [biUnion_eq_iUnion, countable_iUnion_iff, SetCoe.forall']", "annotated_tactic": ["rw [<a>biUnion_eq_iUnion</a>, <a>countable_iUnion_iff</a>, <a>SetCoe.forall'</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": "Set.countable_iUnion_iff", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [193, 9], "def_end_pos": [193, 29]}, {"full_name": "SetCoe.forall'", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nt : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b2\nhs : Set.Countable s\nthis : Countable \u2191s\n\u22a2 Set.Countable (\u22c3 a, \u22c3 (h : a \u2208 s), t a h) \u2194 \u2200 (a : \u03b1) (ha : a \u2208 s), Set.Countable (t a ha)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sigma.lean", "full_name": "Finset.inf_sigma", "start": [107, 1], "end": [109, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.piPremeasure_pi_eval", "start": [183, 1], "end": [185, 42], "traced_tactics": [{"tactic": "simp only [eval, piPremeasure_pi']", "annotated_tactic": ["simp only [<a>eval</a>, <a>piPremeasure_pi'</a>]", [{"full_name": "Function.eval", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [29, 24], "def_end_pos": [29, 28]}, {"full_name": "MeasureTheory.piPremeasure_pi'", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [167, 9], "def_end_pos": [167, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 piPremeasure m (Set.pi univ fun i => eval i '' s) = piPremeasure m s", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 \u220f i : \u03b9, \u2191(m i) ((fun a => a i) '' s) = piPremeasure m s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 \u220f i : \u03b9, \u2191(m i) ((fun a => a i) '' s) = piPremeasure m 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_add", "start": [280, 1], "end": [290, 82], "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\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \u03bc[g|m]"}, {"tactic": "refine' (condexp_ae_eq_condexpL1 hm _).trans _", "annotated_tactic": ["refine' (<a>condexp_ae_eq_condexpL1</a> hm _).<a>trans</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]}]], "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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (f + g)) =\u1d50[\u03bc] \u03bc[f|m] + \u03bc[g|m]"}, {"tactic": "rw [condexpL1_add hf hg]", "annotated_tactic": ["rw [<a>condexpL1_add</a> hf hg]", [{"full_name": "MeasureTheory.condexpL1_add", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [565, 9], "def_end_pos": [565, 22]}]], "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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (f + g)) =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc f + condexpL1 hm \u03bc g) =\u1d50[\u03bc] \u03bc[f|m] + \u03bc[g|m]"}, {"tactic": "exact (coeFn_add _ _).trans\n  ((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)", "annotated_tactic": ["exact (<a>coeFn_add</a> _ _).<a>trans</a>\n    ((<a>condexp_ae_eq_condexpL1</a> hm _).symm.add (<a>condexp_ae_eq_condexpL1</a> hm _).<a>symm</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": "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]}, {"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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc f + condexpL1 hm \u03bc g) =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] 0 + 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nhf : Integrable f\nhg : Integrable g\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1d50[\u03bc] 0 + 0", "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\nhf : Integrable f\nhg : Integrable 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\nhf : Integrable f\nhg : Integrable 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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0 + 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0 + 0", "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\nhf : Integrable f\nhg : Integrable 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\nhf : Integrable f\nhg : Integrable g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 =\u1d50[\u03bc] 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.tr_eval_rev", "start": [931, 1], "end": [943, 24], "traced_tactics": [{"tactic": "cases' mem_eval.1 ab with ab b0", "annotated_tactic": ["cases' <a>mem_eval</a>.1 ab with ab b0", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab : b\u2082 \u2208 eval f\u2082 a\u2082\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081", "state_after": "case intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081"}, {"tactic": "rcases tr_reaches_rev H aa ab with \u27e8c\u2081, c\u2082, bc, cc, ac\u27e9", "annotated_tactic": ["rcases <a>tr_reaches_rev</a> H aa ab with \u27e8c\u2081, c\u2082, bc, cc, ac\u27e9", [{"full_name": "Turing.tr_reaches_rev", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [903, 9], "def_end_pos": [903, 23]}]], "state_before": "case intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081", "state_after": "case intro.intro.intro.intro.intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nc\u2082 : \u03c3\u2082\nbc : Reaches f\u2082 b\u2082 c\u2082\ncc : tr c\u2081 c\u2082\nac : Reaches f\u2081 a\u2081 c\u2081\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081"}, {"tactic": "cases (reflTransGen_iff_eq (Option.eq_none_iff_forall_not_mem.1 b0)).1 bc", "annotated_tactic": ["cases (<a>reflTransGen_iff_eq</a> (<a>Option.eq_none_iff_forall_not_mem</a>.1 b0)).1 bc", [{"full_name": "Relation.reflTransGen_iff_eq", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [531, 9], "def_end_pos": [531, 28]}, {"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]}]], "state_before": "case intro.intro.intro.intro.intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nc\u2082 : \u03c3\u2082\nbc : Reaches f\u2082 b\u2082 c\u2082\ncc : tr c\u2081 c\u2082\nac : Reaches f\u2081 a\u2081 c\u2081\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081", "state_after": "case intro.intro.intro.intro.intro.refl\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081"}, {"tactic": "refine' \u27e8_, cc, mem_eval.2 \u27e8ac, _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8_, cc, <a>mem_eval</a>.2 \u27e8ac, _\u27e9\u27e9", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "state_before": "case intro.intro.intro.intro.intro.refl\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081", "state_after": "case intro.intro.intro.intro.intro.refl\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\n\u22a2 f\u2081 c\u2081 = none"}, {"tactic": "have := H cc", "annotated_tactic": ["have := H cc", []], "state_before": "case intro.intro.intro.intro.intro.refl\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\n\u22a2 f\u2081 c\u2081 = none", "state_after": "case intro.intro.intro.intro.intro.refl\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n  match f\u2081 c\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\n\u22a2 f\u2081 c\u2081 = none"}, {"tactic": "cases' hfc : f\u2081 c\u2081 with d\u2081", "annotated_tactic": ["cases' hfc : f\u2081 c\u2081 with d\u2081", []], "state_before": "case intro.intro.intro.intro.intro.refl\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n  match f\u2081 c\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\n\u22a2 f\u2081 c\u2081 = none", "state_after": "case intro.intro.intro.intro.intro.refl.none\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n  match f\u2081 c\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\nhfc : f\u2081 c\u2081 = none\n\u22a2 none = none\n\ncase intro.intro.intro.intro.intro.refl.some\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n  match f\u2081 c\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\n\u22a2 some d\u2081 = none"}, {"tactic": "rw [hfc] at this", "annotated_tactic": ["rw [hfc] at this", []], "state_before": "case intro.intro.intro.intro.intro.refl.some\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n  match f\u2081 c\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\n\u22a2 some d\u2081 = none", "state_after": "case intro.intro.intro.intro.intro.refl.some\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nthis :\n  match some d\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\nhfc : f\u2081 c\u2081 = some d\u2081\n\u22a2 some d\u2081 = none"}, {"tactic": "rcases this with \u27e8d\u2082, _, bd\u27e9", "annotated_tactic": ["rcases this with \u27e8d\u2082, _, bd\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.refl.some\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nthis :\n  match some d\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\nhfc : f\u2081 c\u2081 = some d\u2081\n\u22a2 some d\u2081 = none", "state_after": "case intro.intro.intro.intro.intro.refl.some.intro.intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\nd\u2082 : \u03c3\u2082\nleft\u271d : tr d\u2081 d\u2082\nbd : Reaches\u2081 f\u2082 b\u2082 d\u2082\n\u22a2 some d\u2081 = none"}, {"tactic": "rcases TransGen.head'_iff.1 bd with \u27e8e, h, _\u27e9", "annotated_tactic": ["rcases <a>TransGen.head'_iff</a>.1 bd with \u27e8e, h, _\u27e9", [{"full_name": "Relation.TransGen.head'_iff", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [428, 9], "def_end_pos": [428, 18]}]], "state_before": "case intro.intro.intro.intro.intro.refl.some.intro.intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\nd\u2082 : \u03c3\u2082\nleft\u271d : tr d\u2081 d\u2082\nbd : Reaches\u2081 f\u2082 b\u2082 d\u2082\n\u22a2 some d\u2081 = none", "state_after": "case intro.intro.intro.intro.intro.refl.some.intro.intro.intro.intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\nd\u2082 : \u03c3\u2082\nleft\u271d : tr d\u2081 d\u2082\nbd : Reaches\u2081 f\u2082 b\u2082 d\u2082\ne : \u03c3\u2082\nh : e \u2208 f\u2082 b\u2082\nright\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) e d\u2082\n\u22a2 some d\u2081 = none"}, {"tactic": "cases b0.symm.trans h", "annotated_tactic": ["cases b0.symm.trans h", []], "state_before": "case intro.intro.intro.intro.intro.refl.some.intro.intro.intro.intro\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\nd\u2082 : \u03c3\u2082\nleft\u271d : tr d\u2081 d\u2082\nbd : Reaches\u2081 f\u2082 b\u2082 d\u2082\ne : \u03c3\u2082\nh : e \u2208 f\u2082 b\u2082\nright\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) e d\u2082\n\u22a2 some d\u2081 = none", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.intro.intro.intro.refl.none\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\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n  match f\u2081 c\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\nhfc : f\u2081 c\u2081 = none\n\u22a2 none = none", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.Martingale.ae_not_tendsto_atTop_atTop", "start": [257, 1], "end": [261, 89], "traced_tactics": [{"tactic": "filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n  unbounded_of_tendsto_atTop htop (h\u03c9.2 <| bddBelow_range_of_tendsto_atTop_atTop htop)", "annotated_tactic": ["filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n    <a>unbounded_of_tendsto_atTop</a> htop (h\u03c9.2 <| <a>bddBelow_range_of_tendsto_atTop_atTop</a> htop)", [{"full_name": "Filter.unbounded_of_tendsto_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1749, 9], "def_end_pos": [1749, 35]}, {"full_name": "Filter.bddBelow_range_of_tendsto_atTop_atTop", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [291, 9], "def_end_pos": [291, 46]}]], "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 : Martingale 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, \u00acTendsto (fun n => f n \u03c9) atTop atTop", "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_aux\u2082", "start": [141, 1], "end": [244, 96], "traced_tactics": [{"tactic": "rcases I.exists_seq_mono_tendsto with \u27e8J, hJ_sub, hJl, hJu\u27e9", "annotated_tactic": ["rcases I.exists_seq_mono_tendsto with \u27e8J, hJ_sub, hJl, hJu\u27e9", []], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "have hJ_sub' : \u2200 k, Box.Icc (J k) \u2286 Box.Icc I := fun k => (hJ_sub k).trans I.Ioo_subset_Icc", "annotated_tactic": ["have hJ_sub' : \u2200 k, <a>Box.Icc</a> (J k) \u2286 <a>Box.Icc</a> I := fun k => (hJ_sub k).<a>trans</a> I.Ioo_subset_Icc", [{"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"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\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "have hJ_le : \u2200 k, J k \u2264 I := fun k => Box.le_iff_Icc.2 (hJ_sub' k)", "annotated_tactic": ["have hJ_le : \u2200 k, J k \u2264 I := fun k => <a>Box.le_iff_Icc</a>.2 (hJ_sub' k)", [{"full_name": "BoxIntegral.Box.le_iff_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 19]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "have HcJ : \u2200 k, ContinuousOn f (Box.Icc (J k)) := fun k => Hc.mono (hJ_sub' k)", "annotated_tactic": ["have HcJ : \u2200 k, <a>ContinuousOn</a> f (<a>Box.Icc</a> (J k)) := fun k => Hc.mono (hJ_sub' k)", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [532, 5], "def_end_pos": [532, 17]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "have HdJ : \u2200 (k), \u2200 x \u2208 (Box.Icc (J k)) \\ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x :=\n  fun k x hx => (Hd x \u27e8hJ_sub k hx.1, hx.2\u27e9).hasFDerivWithinAt", "annotated_tactic": ["have HdJ : \u2200 (k), \u2200 x \u2208 (<a>Box.Icc</a> (J k)) \\ s, <a>HasFDerivWithinAt</a> f (f' x) (<a>Box.Icc</a> (J k)) x :=\n    fun k x hx => (Hd x \u27e8hJ_sub k hx.1, hx.2\u27e9).<a>hasFDerivWithinAt</a>", [{"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "HasFDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [150, 5], "def_end_pos": [150, 22]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "HasFDerivAt.hasFDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [390, 9], "def_end_pos": [390, 38]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "have HJ_eq := fun k =>\n  integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081 (J k) f f' s hs (HcJ k) (HdJ k)\n    (HiJ k)", "annotated_tactic": ["have HJ_eq := fun k =>\n    <a>integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081</a> (J k) f f' s hs (HcJ k) (HdJ k)\n      (HiJ k)", [{"full_name": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081", "def_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "def_pos": [109, 9], "def_end_pos": [109, 68]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "have hI_tendsto :\n  Tendsto (fun k => \u222b x in Box.Icc (J k), \u2211 i, f' x (e i) i) atTop\n    (\ud835\udcdd (\u222b x in Box.Icc I, \u2211 i, f' x (e i) i)) := by\n  simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi \u22a2\n  rw [\u2190 Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi \u22a2\n  exact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo)\n    (Box.Ioo.comp J).monotone Hi", "annotated_tactic": ["have hI_tendsto :\n    <a>Tendsto</a> (fun k => \u222b x in <a>Box.Icc</a> (J k), \u2211 i, f' x (e i) i) <a>atTop</a>\n      (\ud835\udcdd (\u222b x in <a>Box.Icc</a> I, \u2211 i, f' x (e i) i)) := by\n    simp only [<a>IntegrableOn</a>, \u2190 <a>Measure.restrict_congr_set</a> (<a>Box.Ioo_ae_eq_Icc</a> _)] at Hi \u22a2\n    rw [\u2190 <a>Box.iUnion_Ioo_of_tendsto</a> J.monotone hJl hJu] at Hi \u22a2\n    exact <a>tendsto_set_integral_of_monotone</a> (fun k => (J k).<a>measurableSet_Ioo</a>)\n      (Box.Ioo.comp J).<a>monotone</a> Hi", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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": "BoxIntegral.Box.Ioo_ae_eq_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [79, 9], "def_end_pos": [79, 22]}, {"full_name": "BoxIntegral.Box.iUnion_Ioo_of_tendsto", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 30]}, {"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": "BoxIntegral.Box.measurableSet_Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [66, 9], "def_end_pos": [66, 26]}, {"full_name": "OrderHom.monotone", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [231, 19], "def_end_pos": [231, 27]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nhI_tendsto :\n  Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)"}, {"tactic": "refine' tendsto_nhds_unique_of_eventuallyEq hI_tendsto _ (eventually_of_forall HJ_eq)", "annotated_tactic": ["refine' <a>tendsto_nhds_unique_of_eventuallyEq</a> hI_tendsto _ (<a>eventually_of_forall</a> HJ_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 intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nhI_tendsto :\n  Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nhI_tendsto :\n  Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))"}, {"tactic": "clear hI_tendsto", "annotated_tactic": ["clear hI_tendsto", []], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nhI_tendsto :\n  Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))"}, {"tactic": "rw [tendsto_pi_nhds] at hJl hJu", "annotated_tactic": ["rw [<a>tendsto_pi_nhds</a>] at hJl hJu", [{"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))"}, {"tactic": "suffices \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d), (\u2200 k, c k \u2208 Icc (I.lower i) (I.upper i)) \u2192\n  Tendsto c atTop (\ud835\udcdd d) \u2192\n    Tendsto (fun k => \u222b x in Box.Icc ((J k).face i), f (i.insertNth (c k) x) i) atTop\n      (\ud835\udcdd <| \u222b x in Box.Icc (I.face i), f (i.insertNth d x) i) by\n  rw [Box.Icc_eq_pi] at hJ_sub'\n  refine' tendsto_finset_sum _ fun i _ => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))\n  exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>\n    hJ_sub' k (J k).lower_mem_Icc _ trivial]", "annotated_tactic": ["suffices \u2200 (i : <a>Fin</a> (n + 1)) (c : \u2115 \u2192 \u211d) (d), (\u2200 k, c k \u2208 <a>Icc</a> (I.lower i) (I.upper i)) \u2192\n    <a>Tendsto</a> c <a>atTop</a> (\ud835\udcdd d) \u2192\n      <a>Tendsto</a> (fun k => \u222b x in <a>Box.Icc</a> ((J k).<a>face</a> i), f (i.insertNth (c k) x) i) <a>atTop</a>\n        (\ud835\udcdd <| \u222b x in <a>Box.Icc</a> (I.face i), f (i.insertNth d x) i) by\n    rw [<a>Box.Icc_eq_pi</a>] at hJ_sub'\n    refine' <a>tendsto_finset_sum</a> _ fun i _ => (this _ _ _ _ (hJu _)).<a>sub</a> (this _ _ _ _ (hJl _))\n    exacts [fun k => hJ_sub' k (J k).<a>upper_mem_Icc</a> _ <a>trivial</a>, fun k =>\n      hJ_sub' k (J k).<a>lower_mem_Icc</a> _ <a>trivial</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": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"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", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.Icc_eq_pi", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 18]}, {"full_name": "tendsto_finset_sum", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [736, 3], "def_end_pos": [736, 14]}, {"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}, {"full_name": "BoxIntegral.Box.upper_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 22]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"full_name": "BoxIntegral.Box.lower_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}, {"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.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "intro i c d hc hcd", "annotated_tactic": ["intro i c d hc hcd", []], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "have hd : d \u2208 Icc (I.lower i) (I.upper i) :=\n  isClosed_Icc.mem_of_tendsto hcd (eventually_of_forall hc)", "annotated_tactic": ["have hd : d \u2208 <a>Icc</a> (I.lower i) (I.upper i) :=\n    isClosed_Icc.mem_of_tendsto hcd (<a>eventually_of_forall</a> hc)", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"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\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "have Hic : \u2200 k, IntegrableOn (fun x => f (i.insertNth (c k) x) i) (Box.Icc (I.face i)) := fun k =>\n  (Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) (hc k)).integrableOn_Icc", "annotated_tactic": ["have Hic : \u2200 k, <a>IntegrableOn</a> (fun x => f (i.insertNth (c k) x) i) (<a>Box.Icc</a> (I.face i)) := fun k =>\n    (<a>Box.continuousOn_face_Icc</a> ((<a>continuous_apply</a> i).<a>comp_continuousOn</a> Hc) (hc k)).<a>integrableOn_Icc</a>", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.continuousOn_face_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 30]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.comp_continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [957, 9], "def_end_pos": [957, 37]}, {"full_name": "ContinuousOn.integrableOn_Icc", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [401, 9], "def_end_pos": [401, 38]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "have Hid : IntegrableOn (fun x => f (i.insertNth d x) i) (Box.Icc (I.face i)) :=\n  (Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) hd).integrableOn_Icc", "annotated_tactic": ["have Hid : <a>IntegrableOn</a> (fun x => f (i.insertNth d x) i) (<a>Box.Icc</a> (I.face i)) :=\n    (<a>Box.continuousOn_face_Icc</a> ((<a>continuous_apply</a> i).<a>comp_continuousOn</a> Hc) hd).<a>integrableOn_Icc</a>", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.continuousOn_face_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 30]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}, {"full_name": "Continuous.comp_continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [957, 9], "def_end_pos": [957, 37]}, {"full_name": "ContinuousOn.integrableOn_Icc", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [401, 9], "def_end_pos": [401, 38]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "have H :\n  Tendsto (fun k => \u222b x in Box.Icc ((J k).face i), f (i.insertNth d x) i) atTop\n    (\ud835\udcdd <| \u222b x in Box.Icc (I.face i), f (i.insertNth d x) i) := by\n  have hIoo : (\u22c3 k, Box.Ioo ((J k).face i)) = Box.Ioo (I.face i) :=\n    Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone)\n      (tendsto_pi_nhds.2 fun _ => hJl _) (tendsto_pi_nhds.2 fun _ => hJu _)\n  simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), \u2190 hIoo] at Hid \u22a2\n  exact tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)\n    (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid", "annotated_tactic": ["have H :\n    <a>Tendsto</a> (fun k => \u222b x in <a>Box.Icc</a> ((J k).<a>face</a> i), f (i.insertNth d x) i) <a>atTop</a>\n      (\ud835\udcdd <| \u222b x in <a>Box.Icc</a> (I.face i), f (i.insertNth d x) i) := by\n    have hIoo : (\u22c3 k, <a>Box.Ioo</a> ((J k).<a>face</a> i)) = <a>Box.Ioo</a> (I.face i) :=\n      <a>Box.iUnion_Ioo_of_tendsto</a> ((<a>Box.monotone_face</a> i).<a>comp</a> J.monotone)\n        (<a>tendsto_pi_nhds</a>.2 fun _ => hJl _) (<a>tendsto_pi_nhds</a>.2 fun _ => hJu _)\n    simp only [<a>IntegrableOn</a>, \u2190 <a>Measure.restrict_congr_set</a> (<a>Box.Ioo_ae_eq_Icc</a> _), \u2190 hIoo] at Hid \u22a2\n    exact <a>tendsto_set_integral_of_monotone</a> (fun k => ((J k).<a>face</a> i).<a>measurableSet_Ioo</a>)\n      (Box.Ioo.monotone.comp ((<a>Box.monotone_face</a> i).<a>comp</a> J.monotone)) Hid", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [441, 15], "def_end_pos": [441, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [441, 15], "def_end_pos": [441, 18]}, {"full_name": "BoxIntegral.Box.iUnion_Ioo_of_tendsto", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 30]}, {"full_name": "BoxIntegral.Box.monotone_face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [409, 9], "def_end_pos": [409, 22]}, {"full_name": "Monotone.comp", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [671, 19], "def_end_pos": [671, 32]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}, {"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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": "BoxIntegral.Box.Ioo_ae_eq_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [79, 9], "def_end_pos": [79, 22]}, {"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": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.measurableSet_Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [66, 9], "def_end_pos": [66, 26]}, {"full_name": "BoxIntegral.Box.monotone_face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [409, 9], "def_end_pos": [409, 22]}, {"full_name": "Monotone.comp", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [671, 19], "def_end_pos": [671, 32]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "refine' H.congr_dist (Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b5pos => _)", "annotated_tactic": ["refine' H.congr_dist (Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b5pos => _)", []], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n        (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n      Metric.closedBall 0 \u03b5"}, {"tactic": "have hvol_pos : \u2200 J : Box (Fin n), 0 < \u220f j, (J.upper j - J.lower j) := fun J =>\n  prod_pos fun j hj => sub_pos.2 <| J.lower_lt_upper _", "annotated_tactic": ["have hvol_pos : \u2200 J : <a>Box</a> (<a>Fin</a> n), 0 < \u220f j, (J.upper j - J.lower j) := fun J =>\n    <a>prod_pos</a> fun j hj => <a>sub_pos</a>.2 <| J.lower_lt_upper _", [{"full_name": "BoxIntegral.Box", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [73, 11], "def_end_pos": [73, 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": "Finset.prod_pos", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [648, 9], "def_end_pos": [648, 17]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n        (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n      Metric.closedBall 0 \u03b5", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n        (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n      Metric.closedBall 0 \u03b5"}, {"tactic": "rcases Metric.uniformContinuousOn_iff_le.1 (I.isCompact_Icc.uniformContinuousOn_of_continuous Hc)\n    (\u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j)) (div_pos \u03b5pos (hvol_pos (I.face i)))\n  with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9", "annotated_tactic": ["rcases <a>Metric.uniformContinuousOn_iff_le</a>.1 (I.isCompact_Icc.uniformContinuousOn_of_continuous Hc)\n      (\u03b5 / \u220f j, ((I.face i).<a>upper</a> j - (I.face i).<a>lower</a> j)) (<a>div_pos</a> \u03b5pos (hvol_pos (I.face i)))\n    with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9", [{"full_name": "Metric.uniformContinuousOn_iff_le", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [841, 9], "def_end_pos": [841, 35]}, {"full_name": "BoxIntegral.Box.upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 10], "def_end_pos": [74, 15]}, {"full_name": "BoxIntegral.Box.lower", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 4], "def_end_pos": [74, 9]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}]], "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n        (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n      Metric.closedBall 0 \u03b5", "state_after": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n        (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n      Metric.closedBall 0 \u03b5"}, {"tactic": "refine' (hcd.eventually (Metric.ball_mem_nhds _ \u03b4pos)).mono fun k hk => _", "annotated_tactic": ["refine' (hcd.eventually (<a>Metric.ball_mem_nhds</a> _ \u03b4pos)).<a>mono</a> fun k hk => _", [{"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"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\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n        (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n      Metric.closedBall 0 \u03b5", "state_after": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\n\u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)\n      (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208\n    Metric.closedBall 0 \u03b5"}, {"tactic": "have Hsub : Box.Icc ((J k).face i) \u2286 Box.Icc (I.face i) :=\n  Box.le_iff_Icc.1 (Box.face_mono (hJ_le _) i)", "annotated_tactic": ["have Hsub : <a>Box.Icc</a> ((J k).<a>face</a> i) \u2286 <a>Box.Icc</a> (I.face i) :=\n    <a>Box.le_iff_Icc</a>.1 (<a>Box.face_mono</a> (hJ_le _) i)", [{"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.le_iff_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 19]}, {"full_name": "BoxIntegral.Box.face_mono", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [404, 9], "def_end_pos": [404, 18]}]], "state_before": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\n\u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)\n      (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208\n    Metric.closedBall 0 \u03b5", "state_after": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)\n      (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208\n    Metric.closedBall 0 \u03b5"}, {"tactic": "rw [mem_closedBall_zero_iff, Real.norm_eq_abs, abs_of_nonneg dist_nonneg, dist_eq_norm,\n  \u2190 integral_sub (Hid.mono_set Hsub) ((Hic _).mono_set Hsub)]", "annotated_tactic": ["rw [<a>mem_closedBall_zero_iff</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> <a>dist_nonneg</a>, <a>dist_eq_norm</a>,\n    \u2190 <a>integral_sub</a> (Hid.mono_set Hsub) ((Hic _).<a>mono_set</a> Hsub)]", [{"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": "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": "dist_nonneg", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [270, 9], "def_end_pos": [270, 20]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"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.mono_set", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}]], "state_before": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)\n      (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208\n    Metric.closedBall 0 \u03b5", "state_after": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u2016\u222b (a : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d a) i - f (Fin.insertNth i (c k) a) i\u2016 \u2264 \u03b5"}, {"tactic": "calc\n  \u2016\u222b x in Box.Icc ((J k).face i), f (i.insertNth d x) i - f (i.insertNth (c k) x) i\u2016 \u2264\n      (\u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j)) *\n        (volume (Box.Icc ((J k).face i))).toReal := by\n    refine norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)\n      ((J k).face i).measurableSet_Icc fun x hx => ?_\n    rw [\u2190 dist_eq_norm]\n    calc\n      dist (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264\n          dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n        dist_le_pi_dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n      _ \u2264 \u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j) :=\n        h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _\n          (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_\n    rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]\n    exact max_le hk.le \u03b4pos.lt.le\n  _ \u2264 \u03b5 := by\n    rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,\n      \u2190 le_div_iff (hvol_pos _)]\n    refine' div_le_div_of_le_left \u03b5pos.le (hvol_pos _)\n      (prod_le_prod (fun j _ => _) fun j _ => _)\n    exacts [sub_nonneg.2 (Box.lower_le_upper _ _),\n      sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]", "annotated_tactic": ["calc\n    \u2016\u222b x in <a>Box.Icc</a> ((J k).<a>face</a> i), f (i.insertNth d x) i - f (i.insertNth (c k) x) i\u2016 \u2264\n        (\u03b5 / \u220f j, ((I.face i).<a>upper</a> j - (I.face i).<a>lower</a> j)) *\n          (<a>volume</a> (<a>Box.Icc</a> ((J k).<a>face</a> i))).<a>toReal</a> := by\n      refine <a>norm_set_integral_le_of_norm_le_const'</a> (((J k).<a>face</a> i).<a>measure_Icc_lt_top</a> _)\n        ((J k).<a>face</a> i).<a>measurableSet_Icc</a> fun x hx => ?_\n      rw [\u2190 <a>dist_eq_norm</a>]\n      calc\n        <a>dist</a> (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264\n            <a>dist</a> (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n          <a>dist_le_pi_dist</a> (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n        _ \u2264 \u03b5 / \u220f j, ((I.face i).<a>upper</a> j - (I.face i).<a>lower</a> j) :=\n          h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _\n            (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_\n      rw [<a>Fin.dist_insertNth_insertNth</a>, <a>dist_self</a>, <a>dist_comm</a>]\n      exact <a>max_le</a> hk.le \u03b4pos.lt.le\n    _ \u2264 \u03b5 := by\n      rw [<a>Box.Icc_def</a>, <a>Real.volume_Icc_pi_toReal</a> ((J k).<a>face</a> i).<a>lower_le_upper</a>,\n        \u2190 <a>le_div_iff</a> (hvol_pos _)]\n      refine' <a>div_le_div_of_le_left</a> \u03b5pos.le (hvol_pos _)\n        (<a>prod_le_prod</a> (fun j _ => _) fun j _ => _)\n      exacts [<a>sub_nonneg</a>.2 (<a>Box.lower_le_upper</a> _ _),\n        <a>sub_le_sub</a> ((hJ_sub' _ (J _).<a>upper_mem_Icc</a>).2 _) ((hJ_sub' _ (J _).<a>lower_mem_Icc</a>).1 _)]", [{"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 10], "def_end_pos": [74, 15]}, {"full_name": "BoxIntegral.Box.lower", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 4], "def_end_pos": [74, 9]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.norm_set_integral_le_of_norm_le_const'", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [580, 9], "def_end_pos": [580, 47]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.measure_Icc_lt_top", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [45, 9], "def_end_pos": [45, 27]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.measurableSet_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [62, 9], "def_end_pos": [62, 26]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"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_le_pi_dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2102, 9], "def_end_pos": [2102, 24]}, {"full_name": "BoxIntegral.Box.upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 10], "def_end_pos": [74, 15]}, {"full_name": "BoxIntegral.Box.lower", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 4], "def_end_pos": [74, 9]}, {"full_name": "Fin.dist_insertNth_insertNth", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2162, 9], "def_end_pos": [2162, 37]}, {"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 18]}, {"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}, {"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "BoxIntegral.Box.Icc_def", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [203, 9], "def_end_pos": [203, 16]}, {"full_name": "Real.volume_Icc_pi_toReal", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 29]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.lower_le_upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 23]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}, {"full_name": "div_le_div_of_le_left", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [349, 9], "def_end_pos": [349, 30]}, {"full_name": "Finset.prod_le_prod", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [592, 9], "def_end_pos": [592, 21]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "BoxIntegral.Box.lower_le_upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 23]}, {"full_name": "sub_le_sub", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [850, 15], "def_end_pos": [850, 25]}, {"full_name": "BoxIntegral.Box.upper_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 22]}, {"full_name": "BoxIntegral.Box.lower_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "case intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u2016\u222b (a : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d a) i - f (Fin.insertNth i (c k) a) i\u2016 \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi \u22a2", "annotated_tactic": ["simp only [<a>IntegrableOn</a>, \u2190 <a>Measure.restrict_congr_set</a> (<a>Box.Ioo_ae_eq_Icc</a> _)] at Hi \u22a2", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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": "BoxIntegral.Box.Ioo_ae_eq_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [79, 9], "def_end_pos": [79, 22]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nHi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1))"}, {"tactic": "rw [\u2190 Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi \u22a2", "annotated_tactic": ["rw [\u2190 <a>Box.iUnion_Ioo_of_tendsto</a> J.monotone hJl hJu] at Hi \u22a2", [{"full_name": "BoxIntegral.Box.iUnion_Ioo_of_tendsto", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 30]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nHi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nHi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u22c3 n_1, \u2191Box.Ioo (\u2191J n_1), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1))"}, {"tactic": "exact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo)\n  (Box.Ioo.comp J).monotone Hi", "annotated_tactic": ["exact <a>tendsto_set_integral_of_monotone</a> (fun k => (J k).<a>measurableSet_Ioo</a>)\n      (Box.Ioo.comp J).<a>monotone</a> Hi", [{"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": "BoxIntegral.Box.measurableSet_Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [66, 9], "def_end_pos": [66, 26]}, {"full_name": "OrderHom.monotone", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [231, 19], "def_end_pos": [231, 27]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nHi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop\n    (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u22c3 n_1, \u2191Box.Ioo (\u2191J n_1), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1))", "state_after": "no goals"}, {"tactic": "rw [Box.Icc_eq_pi] at hJ_sub'", "annotated_tactic": ["rw [<a>Box.Icc_eq_pi</a>] at hJ_sub'", [{"full_name": "BoxIntegral.Box.Icc_eq_pi", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 18]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))"}, {"tactic": "refine' tendsto_finset_sum _ fun i _ => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))", "annotated_tactic": ["refine' <a>tendsto_finset_sum</a> _ fun i _ => (this _ _ _ _ (hJu _)).<a>sub</a> (this _ _ _ _ (hJl _))", [{"full_name": "tendsto_finset_sum", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [736, 3], "def_end_pos": [736, 14]}, {"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto\n    (fun x =>\n      \u2211 i : Fin (n + 1),\n        ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n          \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n    atTop\n    (\ud835\udcdd\n      (\u2211 i : Fin (n + 1),\n        ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))", "state_after": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (k : \u2115), (Box.upper \u2218 \u2191J) k i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\n\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (k : \u2115), Box.lower (\u2191J k) i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)"}, {"tactic": "exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>\n  hJ_sub' k (J k).lower_mem_Icc _ trivial]", "annotated_tactic": ["exacts [fun k => hJ_sub' k (J k).<a>upper_mem_Icc</a> _ <a>trivial</a>, fun k =>\n      hJ_sub' k (J k).<a>lower_mem_Icc</a> _ <a>trivial</a>]", [{"full_name": "BoxIntegral.Box.upper_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 22]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"full_name": "BoxIntegral.Box.lower_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}, {"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 refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (k : \u2115), (Box.upper \u2218 \u2191J) k i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\n\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n  \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n    (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n      Tendsto c atTop (\ud835\udcdd d) \u2192\n        Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n          (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (k : \u2115), Box.lower (\u2191J k) i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)", "state_after": "no goals"}, {"tactic": "have hIoo : (\u22c3 k, Box.Ioo ((J k).face i)) = Box.Ioo (I.face i) :=\n  Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone)\n    (tendsto_pi_nhds.2 fun _ => hJl _) (tendsto_pi_nhds.2 fun _ => hJu _)", "annotated_tactic": ["have hIoo : (\u22c3 k, <a>Box.Ioo</a> ((J k).<a>face</a> i)) = <a>Box.Ioo</a> (I.face i) :=\n      <a>Box.iUnion_Ioo_of_tendsto</a> ((<a>Box.monotone_face</a> i).<a>comp</a> J.monotone)\n        (<a>tendsto_pi_nhds</a>.2 fun _ => hJl _) (<a>tendsto_pi_nhds</a>.2 fun _ => hJu _)", [{"full_name": "BoxIntegral.Box.Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [441, 15], "def_end_pos": [441, 18]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [441, 15], "def_end_pos": [441, 18]}, {"full_name": "BoxIntegral.Box.iUnion_Ioo_of_tendsto", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 30]}, {"full_name": "BoxIntegral.Box.monotone_face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [409, 9], "def_end_pos": [409, 22]}, {"full_name": "Monotone.comp", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [671, 19], "def_end_pos": [671, 32]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1231, 9], "def_end_pos": [1231, 24]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nhIoo : \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))"}, {"tactic": "simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), \u2190 hIoo] at Hid \u22a2", "annotated_tactic": ["simp only [<a>IntegrableOn</a>, \u2190 <a>Measure.restrict_congr_set</a> (<a>Box.Ioo_ae_eq_Icc</a> _), \u2190 hIoo] at Hid \u22a2", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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": "BoxIntegral.Box.Ioo_ae_eq_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [79, 9], "def_end_pos": [79, 22]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nhIoo : \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nhIoo : \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i)\nHid : Integrable fun x => f (Fin.insertNth i d x) i\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i))"}, {"tactic": "exact tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)\n  (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid", "annotated_tactic": ["exact <a>tendsto_set_integral_of_monotone</a> (fun k => ((J k).<a>face</a> i).<a>measurableSet_Ioo</a>)\n      (Box.Ioo.monotone.comp ((<a>Box.monotone_face</a> i).<a>comp</a> J.monotone)) Hid", [{"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": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.measurableSet_Ioo", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [66, 9], "def_end_pos": [66, 26]}, {"full_name": "BoxIntegral.Box.monotone_face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [409, 9], "def_end_pos": [409, 22]}, {"full_name": "Monotone.comp", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [671, 19], "def_end_pos": [671, 32]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nhIoo : \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i)\nHid : Integrable fun x => f (Fin.insertNth i d x) i\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i))", "state_after": "no goals"}, {"tactic": "refine norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)\n  ((J k).face i).measurableSet_Icc fun x hx => ?_", "annotated_tactic": ["refine <a>norm_set_integral_le_of_norm_le_const'</a> (((J k).<a>face</a> i).<a>measure_Icc_lt_top</a> _)\n        ((J k).<a>face</a> i).<a>measurableSet_Icc</a> fun x hx => ?_", [{"full_name": "MeasureTheory.norm_set_integral_le_of_norm_le_const'", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [580, 9], "def_end_pos": [580, 47]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.measure_Icc_lt_top", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [45, 9], "def_end_pos": [45, 27]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.measurableSet_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [62, 9], "def_end_pos": [62, 26]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u2016\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264\n    (\u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)) *\n      ENNReal.toReal (\u2191\u2191volume (\u2191Box.Icc (Box.face (\u2191J k) i)))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 \u2016f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264\n    \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)"}, {"tactic": "rw [\u2190 dist_eq_norm]", "annotated_tactic": ["rw [\u2190 <a>dist_eq_norm</a>]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 \u2016f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264\n    \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 dist (f (Fin.insertNth i d x) i) (f (Fin.insertNth i (c k) x) i) \u2264\n    \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)"}, {"tactic": "calc\n  dist (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264\n      dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n    dist_le_pi_dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n  _ \u2264 \u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j) :=\n    h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _\n      (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_", "annotated_tactic": ["calc\n        <a>dist</a> (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264\n            <a>dist</a> (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n          <a>dist_le_pi_dist</a> (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n        _ \u2264 \u03b5 / \u220f j, ((I.face i).<a>upper</a> j - (I.face i).<a>lower</a> j) :=\n          h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _\n            (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_", [{"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_le_pi_dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2102, 9], "def_end_pos": [2102, 24]}, {"full_name": "BoxIntegral.Box.upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 10], "def_end_pos": [74, 15]}, {"full_name": "BoxIntegral.Box.lower", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [74, 4], "def_end_pos": [74, 9]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 dist (f (Fin.insertNth i d x) i) (f (Fin.insertNth i (c k) x) i) \u2264\n    \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 dist (Fin.insertNth i d x) (Fin.insertNth i (c k) x) \u2264 \u03b4"}, {"tactic": "rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]", "annotated_tactic": ["rw [<a>Fin.dist_insertNth_insertNth</a>, <a>dist_self</a>, <a>dist_comm</a>]", [{"full_name": "Fin.dist_insertNth_insertNth", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2162, 9], "def_end_pos": [2162, 37]}, {"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 18]}, {"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 dist (Fin.insertNth i d x) (Fin.insertNth i (c k) x) \u2264 \u03b4", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 max (dist (c k) d) 0 \u2264 \u03b4"}, {"tactic": "exact max_le hk.le \u03b4pos.lt.le", "annotated_tactic": ["exact <a>max_le</a> hk.le \u03b4pos.lt.le", [{"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 max (dist (c k) d) 0 \u2264 \u03b4", "state_after": "no goals"}, {"tactic": "rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,\n  \u2190 le_div_iff (hvol_pos _)]", "annotated_tactic": ["rw [<a>Box.Icc_def</a>, <a>Real.volume_Icc_pi_toReal</a> ((J k).<a>face</a> i).<a>lower_le_upper</a>,\n        \u2190 <a>le_div_iff</a> (hvol_pos _)]", [{"full_name": "BoxIntegral.Box.Icc_def", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [203, 9], "def_end_pos": [203, 16]}, {"full_name": "Real.volume_Icc_pi_toReal", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 29]}, {"full_name": "BoxIntegral.Box.face", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [394, 5], "def_end_pos": [394, 9]}, {"full_name": "BoxIntegral.Box.lower_le_upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 23]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 (\u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)) *\n      ENNReal.toReal (\u2191\u2191volume (\u2191Box.Icc (Box.face (\u2191J k) i))) \u2264\n    \u03b5", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) \u2264\n    \u03b5 / \u220f j : Fin n, (Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j)"}, {"tactic": "refine' div_le_div_of_le_left \u03b5pos.le (hvol_pos _)\n  (prod_le_prod (fun j _ => _) fun j _ => _)", "annotated_tactic": ["refine' <a>div_le_div_of_le_left</a> \u03b5pos.le (hvol_pos _)\n        (<a>prod_le_prod</a> (fun j _ => _) fun j _ => _)", [{"full_name": "div_le_div_of_le_left", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [349, 9], "def_end_pos": [349, 30]}, {"full_name": "Finset.prod_le_prod", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [592, 9], "def_end_pos": [592, 21]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) \u2264\n    \u03b5 / \u220f j : Fin n, (Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j)", "state_after": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nj : Fin n\nx\u271d : j \u2208 Finset.univ\n\u22a2 0 \u2264 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j\n\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nj : Fin n\nx\u271d : j \u2208 Finset.univ\n\u22a2 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j \u2264\n    Box.upper (Box.face I i) j - Box.lower (Box.face I i) j"}, {"tactic": "exacts [sub_nonneg.2 (Box.lower_le_upper _ _),\n  sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]", "annotated_tactic": ["exacts [<a>sub_nonneg</a>.2 (<a>Box.lower_le_upper</a> _ _),\n        <a>sub_le_sub</a> ((hJ_sub' _ (J _).<a>upper_mem_Icc</a>).2 _) ((hJ_sub' _ (J _).<a>lower_mem_Icc</a>).1 _)]", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "BoxIntegral.Box.lower_le_upper", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 23]}, {"full_name": "sub_le_sub", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [850, 15], "def_end_pos": [850, 25]}, {"full_name": "BoxIntegral.Box.upper_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 22]}, {"full_name": "BoxIntegral.Box.lower_mem_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nj : Fin n\nx\u271d : j \u2208 Finset.univ\n\u22a2 0 \u2264 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j\n\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n  \u2200 (k : \u2115),\n    \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \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 \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n          \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n  Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n    (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n  \u2200 (x : Fin (n + 1) \u2192 \u211d),\n    x \u2208 \u2191Box.Icc I \u2192\n      \u2200 (y : Fin (n + 1) \u2192 \u211d),\n        y \u2208 \u2191Box.Icc I \u2192\n          dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nj : Fin n\nx\u271d : j \u2208 Finset.univ\n\u22a2 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j \u2264\n    Box.upper (Box.face I i) j - Box.lower (Box.face I i) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.zneg_pred", "start": [1079, 1], "end": [1080, 53], "traced_tactics": [{"tactic": "rw [\u2190 zneg_zneg (succ (-n)), zneg_succ, zneg_zneg]", "annotated_tactic": ["rw [\u2190 <a>zneg_zneg</a> (<a>succ</a> (-n)), <a>zneg_succ</a>, <a>zneg_zneg</a>]", [{"full_name": "ZNum.zneg_zneg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 18]}, {"full_name": "ZNum.succ", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [361, 5], "def_end_pos": [361, 9]}, {"full_name": "ZNum.zneg_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1075, 9], "def_end_pos": [1075, 18]}, {"full_name": "ZNum.zneg_zneg", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 18]}]], "state_before": "\u03b1 : Type u_1\nn : ZNum\n\u22a2 -pred n = succ (-n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "full_name": "MvPolynomial.totalDegree_sub", "start": [209, 1], "end": [214, 67], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg]", "annotated_tactic": ["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": "R : Type u\nS : Type v\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q a b : MvPolynomial \u03c3 R\n\u22a2 totalDegree (a - b) = totalDegree (a + -b)", "state_after": "no goals"}, {"tactic": "rw [totalDegree_neg]", "annotated_tactic": ["rw [<a>totalDegree_neg</a>]", [{"full_name": "MvPolynomial.totalDegree_neg", "def_path": "Mathlib/Data/MvPolynomial/CommRing.lean", "def_pos": [205, 9], "def_end_pos": [205, 24]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q a b : MvPolynomial \u03c3 R\n\u22a2 max (totalDegree a) (totalDegree (-b)) = max (totalDegree a) (totalDegree b)", "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_biInter", "start": [386, 1], "end": [388, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/OneDim.lean", "full_name": "Real.tendsto_Icc_vitaliFamily_left", "start": [51, 1], "end": [59, 44], "traced_tactics": [{"tactic": "refine' (VitaliFamily.tendsto_filterAt_iff _).2 \u27e8_, _\u27e9", "annotated_tactic": ["refine' (<a>VitaliFamily.tendsto_filterAt_iff</a> _).2 \u27e8_, _\u27e9", [{"full_name": "VitaliFamily.tendsto_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [261, 9], "def_end_pos": [261, 29]}]], "state_before": "x : \u211d\n\u22a2 Tendsto (fun y => Icc y x) (\ud835\udcdd[Iio x] x) (VitaliFamily.filterAt (vitaliFamily volume 1) x)", "state_after": "case refine'_1\nx : \u211d\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) x\n\ncase refine'_2\nx : \u211d\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with y hy using <a>Icc_mem_vitaliFamily_at_left</a> hy", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "Real.Icc_mem_vitaliFamily_at_left", "def_path": "Mathlib/MeasureTheory/Covering/OneDim.lean", "def_pos": [44, 9], "def_end_pos": [44, 37]}]], "state_before": "case refine'_1\nx : \u211d\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) x", "state_after": "no goals"}, {"tactic": "intro \u03b5 \u03b5pos", "annotated_tactic": ["intro \u03b5 \u03b5pos", []], "state_before": "case refine'_2\nx : \u211d\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5", "state_after": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5"}, {"tactic": "have : x \u2208 Ioc (x - \u03b5) x := \u27e8by linarith, le_refl _\u27e9", "annotated_tactic": ["have : x \u2208 <a>Ioc</a> (x - \u03b5) x := \u27e8by linarith, <a>le_refl</a> _\u27e9", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5", "state_after": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5"}, {"tactic": "filter_upwards [Icc_mem_nhdsWithin_Iio this] with y hy", "annotated_tactic": ["filter_upwards [<a>Icc_mem_nhdsWithin_Iio</a> this] with y hy", [{"full_name": "Icc_mem_nhdsWithin_Iio", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [517, 9], "def_end_pos": [517, 31]}]], "state_before": "case refine'_2\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\n\u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5", "state_after": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\ny : \u211d\nhy : y \u2208 Icc (x - \u03b5) x\n\u22a2 Icc y x \u2286 Metric.closedBall x \u03b5"}, {"tactic": "rw [closedBall_eq_Icc]", "annotated_tactic": ["rw [<a>closedBall_eq_Icc</a>]", [{"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}]], "state_before": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\ny : \u211d\nhy : y \u2208 Icc (x - \u03b5) x\n\u22a2 Icc y x \u2286 Metric.closedBall x \u03b5", "state_after": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\ny : \u211d\nhy : y \u2208 Icc (x - \u03b5) x\n\u22a2 Icc y x \u2286 Icc (x - \u03b5) (x + \u03b5)"}, {"tactic": "exact Icc_subset_Icc hy.1 (by linarith)", "annotated_tactic": ["exact <a>Icc_subset_Icc</a> hy.1 (by linarith)", [{"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}]], "state_before": "case h\nx \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\ny : \u211d\nhy : y \u2208 Icc (x - \u03b5) x\n\u22a2 Icc y x \u2286 Icc (x - \u03b5) (x + \u03b5)", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "x \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 x - \u03b5 < x", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "x \u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nthis : x \u2208 Ioc (x - \u03b5) x\ny : \u211d\nhy : y \u2208 Icc (x - \u03b5) x\n\u22a2 x \u2264 x + \u03b5", "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.sub_add_min_cancel", "start": [490, 19], "end": [491, 64], "traced_tactics": [{"tactic": "rw [sub_eq_sub_min, Nat.sub_add_cancel (Nat.min_le_left n m)]", "annotated_tactic": ["rw [<a>sub_eq_sub_min</a>, <a>Nat.sub_add_cancel</a> (<a>Nat.min_le_left</a> n m)]", [{"full_name": "Nat.sub_eq_sub_min", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [485, 9], "def_end_pos": [485, 23]}, {"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.min_le_left", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [38, 19], "def_end_pos": [38, 30]}]], "state_before": "n m : Nat\n\u22a2 n - m + min n m = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coe_LpSubmodule", "start": [490, 1], "end": [491, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.decode_encodeNum", "start": [142, 1], "end": [148, 41], "traced_tactics": [{"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u22a2 \u2200 (n : Num), decodeNum (encodeNum n) = n", "state_after": "n : Num\n\u22a2 decodeNum (encodeNum n) = n"}, {"tactic": "cases' n with n <;> unfold encodeNum decodeNum", "annotated_tactic": ["cases' n with n <;> unfold <a>encodeNum</a> <a>decodeNum</a>", [{"full_name": "Computability.encodeNum", "def_path": "Mathlib/Computability/Encoding.lean", "def_pos": [103, 5], "def_end_pos": [103, 14]}, {"full_name": "Computability.decodeNum", "def_path": "Mathlib/Computability/Encoding.lean", "def_pos": [121, 5], "def_end_pos": [121, 14]}]], "state_before": "n : Num\n\u22a2 decodeNum (encodeNum n) = n", "state_after": "case zero\n\n\u22a2 (if\n        (match Num.zero with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else\n      \u2191(decodePosNum\n          (match Num.zero with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n))) =\n    Num.zero\n\ncase pos\nn : PosNum\n\u22a2 (if\n        (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else\n      \u2191(decodePosNum\n          (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n))) =\n    Num.pos n"}, {"tactic": "rw [decode_encodePosNum n]", "annotated_tactic": ["rw [<a>decode_encodePosNum</a> n]", [{"full_name": "Computability.decode_encodePosNum", "def_path": "Mathlib/Computability/Encoding.lean", "def_pos": [133, 9], "def_end_pos": [133, 28]}]], "state_before": "case pos\nn : PosNum\n\u22a2 (if\n        (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else\n      \u2191(decodePosNum\n          (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n))) =\n    Num.pos n", "state_after": "case pos\nn : PosNum\n\u22a2 (if\n        (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else \u2191n) =\n    Num.pos n"}, {"tactic": "rw [PosNum.cast_to_num]", "annotated_tactic": ["rw [<a>PosNum.cast_to_num</a>]", [{"full_name": "PosNum.cast_to_num", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [653, 9], "def_end_pos": [653, 20]}]], "state_before": "case pos\nn : PosNum\n\u22a2 (if\n        (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else \u2191n) =\n    Num.pos n", "state_after": "case pos\nn : PosNum\n\u22a2 (if\n        (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else Num.pos n) =\n    Num.pos n"}, {"tactic": "exact if_neg (encodePosNum_nonempty n)", "annotated_tactic": ["exact <a>if_neg</a> (<a>encodePosNum_nonempty</a> n)", [{"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": "Computability.encodePosNum_nonempty", "def_path": "Mathlib/Computability/Encoding.lean", "def_pos": [128, 9], "def_end_pos": [128, 30]}]], "state_before": "case pos\nn : PosNum\n\u22a2 (if\n        (match Num.pos n with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else Num.pos n) =\n    Num.pos n", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\n\u22a2 (if\n        (match Num.zero with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n) =\n          [] then\n      Num.zero\n    else\n      \u2191(decodePosNum\n          (match Num.zero with\n          | Num.zero => []\n          | Num.pos n => encodePosNum n))) =\n    Num.zero", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "full_name": "Complex.continuous_circleTransformDeriv", "start": [88, 1], "end": [92, 81], "traced_tactics": [{"tactic": "rw [circleTransformDeriv_eq]", "annotated_tactic": ["rw [<a>circleTransformDeriv_eq</a>]", [{"full_name": "Complex.circleTransformDeriv_eq", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [60, 9], "def_end_pos": [60, 32]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous (circleTransformDeriv R z w f)", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun \u03b8 => (circleMap z R \u03b8 - w)\u207b\u00b9 \u2022 circleTransform R z w f \u03b8"}, {"tactic": "exact (continuous_circleMap_inv hw).smul (continuous_circleTransform hR hf hw)", "annotated_tactic": ["exact (<a>continuous_circleMap_inv</a> hw).<a>smul</a> (<a>continuous_circleTransform</a> hR hf hw)", [{"full_name": "continuous_circleMap_inv", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [213, 9], "def_end_pos": [213, 33]}, {"full_name": "Continuous.smul", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "Complex.continuous_circleTransform", "def_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "def_pos": [77, 9], "def_end_pos": [77, 35]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun \u03b8 => (circleMap z R \u03b8 - w)\u207b\u00b9 \u2022 circleTransform R z w f \u03b8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_piCongrLeft", "start": [946, 1], "end": [950, 85], "traced_tactics": [{"tactic": "rw [measurable_pi_iff]", "annotated_tactic": ["rw [<a>measurable_pi_iff</a>]", [{"full_name": "measurable_pi_iff", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [890, 9], "def_end_pos": [890, 26]}]], "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\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\n\u22a2 Measurable \u2191(piCongrLeft \u03c0 f)", "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 t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\n\u22a2 \u2200 (a : \u03b4), Measurable fun x => \u2191(piCongrLeft \u03c0 f) x a"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "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\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\n\u22a2 \u2200 (a : \u03b4), Measurable fun x => \u2191(piCongrLeft \u03c0 f) x a", "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 t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\ni : \u03b4\n\u22a2 Measurable fun x => \u2191(piCongrLeft \u03c0 f) x i"}, {"tactic": "simp_rw [piCongrLeft_apply_eq_cast]", "annotated_tactic": ["simp_rw [<a>piCongrLeft_apply_eq_cast</a>]", [{"full_name": "Equiv.piCongrLeft_apply_eq_cast", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1888, 7], "def_end_pos": [1888, 32]}]], "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\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\ni : \u03b4\n\u22a2 Measurable fun x => \u2191(piCongrLeft \u03c0 f) x i", "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 t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\ni : \u03b4\n\u22a2 Measurable fun x => cast (_ : \u03c0 (\u2191f (\u2191f.symm i)) = \u03c0 i) (x (\u2191f.symm i))"}, {"tactic": "exact Measurable.eq_mp \u03c0 (f.apply_symm_apply i) <| measurable_pi_apply <| f.symm i", "annotated_tactic": ["exact <a>Measurable.eq_mp</a> \u03c0 (f.apply_symm_apply i) <| <a>measurable_pi_apply</a> <| f.symm i", [{"full_name": "Measurable.eq_mp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [942, 9], "def_end_pos": [942, 25]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [896, 9], "def_end_pos": [896, 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 t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b4' \u2243 \u03b4\ni : \u03b4\n\u22a2 Measurable fun x => cast (_ : \u03c0 (\u2191f (\u2191f.symm i)) = \u03c0 i) (x (\u2191f.symm 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_of_mem\u2112p_trim", "start": [1035, 1], "end": [1037, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "integrableOn_Ici_iff_integrableOn_Ioi", "start": [723, 1], "end": [725, 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/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.Indep.indepSet_of_measurableSet", "start": [511, 1], "end": [515, 55], "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.Buckets.WF.update", "start": [46, 1], "end": [60, 38], "traced_tactics": [{"tactic": "refine \u27e8fun l hl => ?_, fun i hi p hp => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun l hl => ?_, fun i hi p hp => ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i)\n      buckets.val[i] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i) d\n\u22a2 WF (Buckets.update buckets i d h)", "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\nbuckets : Buckets \u03b1 \u03b2\ni : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i)\n      buckets.val[i] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i) d\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nhl : l \u2208 (Buckets.update buckets i d h).val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList l)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nhp : p \u2208 AssocList.toList (Buckets.update buckets i\u271d d h).val[i]\n\u22a2 (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.update buckets i\u271d d h).val) = i) p.fst p.snd"}, {"tactic": "exact match List.mem_or_eq_of_mem_set hl with\n| .inl hl => H.1 _ hl\n| .inr rfl => h\u2081 (H.1 _ (Array.getElem_mem_data ..))", "annotated_tactic": ["exact match <a>List.mem_or_eq_of_mem_set</a> hl with\n    | .inl hl => H.1 _ hl\n    | .inr <a>rfl</a> => h\u2081 (H.1 _ (<a>Array.getElem_mem_data</a> ..))", [{"full_name": "List.mem_or_eq_of_mem_set", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [965, 9], "def_end_pos": [965, 29]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Array.getElem_mem_data", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "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\nbuckets : Buckets \u03b1 \u03b2\ni : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i)\n      buckets.val[i] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i) d\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nhl : l \u2208 (Buckets.update buckets i d h).val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList l)", "state_after": "no goals"}, {"tactic": "revert hp", "annotated_tactic": ["revert hp", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nhp : p \u2208 AssocList.toList (Buckets.update buckets i\u271d d h).val[i]\n\u22a2 (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.update buckets i\u271d d h).val) = i) p.fst p.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\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208 AssocList.toList (Buckets.update buckets i\u271d d h).val[i] \u2192\n    (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.update buckets i\u271d d h).val) = i) p.fst p.snd"}, {"tactic": "simp [update_data, Array.getElem_eq_data_get, List.get_set]", "annotated_tactic": ["simp [<a>update_data</a>, <a>Array.getElem_eq_data_get</a>, <a>List.get_set</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.update_data", "def_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "def_pos": [21, 9], "def_end_pos": [21, 20]}, {"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", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [961, 9], "def_end_pos": [961, 16]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208 AssocList.toList (Buckets.update buckets i\u271d d h).val[i] \u2192\n    (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.update buckets i\u271d d h).val) = i) p.fst p.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\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208\n      AssocList.toList\n        (if USize.toNat i\u271d = i then d\n        else List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) }) \u2192\n    USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i"}, {"tactic": "split <;> intro hp", "annotated_tactic": ["split <;> intro hp", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208\n      AssocList.toList\n        (if USize.toNat i\u271d = i then d\n        else List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) }) \u2192\n    USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i", "state_after": "case refine_2.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nh\u271d : USize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList d\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i\n\ncase refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nh\u271d : \u00acUSize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList (List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) })\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i"}, {"tactic": "next eq => exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", "annotated_tactic": ["next eq => exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", []], "state_before": "case refine_2.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nh\u271d : USize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList d\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i", "state_after": "no goals"}, {"tactic": "exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", "annotated_tactic": ["exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\neq : USize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList d\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i", "state_after": "no goals"}, {"tactic": "simp at hi", "annotated_tactic": ["simp at hi", []], "state_before": "case refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nh\u271d : \u00acUSize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList (List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) })\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i", "state_after": "case refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi\u271d : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nh\u271d : \u00acUSize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList (List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) })\nhi : i < Array.size buckets.val\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i"}, {"tactic": "exact H.2 i hi _ hp", "annotated_tactic": ["exact H.2 i hi _ hp", []], "state_before": "case refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : USize.toNat i\u271d < Array.size buckets.val\nH : WF buckets\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d)\nh\u2082 :\n  AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)\n      buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d\ni : Nat\nhi\u271d : i < Array.size (Buckets.update buckets i\u271d d h).val\np : \u03b1 \u00d7 \u03b2\nh\u271d : \u00acUSize.toNat i\u271d = i\nhp : p \u2208 AssocList.toList (List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) })\nhi : i < Array.size buckets.val\n\u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "Measurable.exists_continuous", "start": [324, 1], "end": [342, 41], "traced_tactics": [{"tactic": "obtain \u27e8b, b_count, -, hb\u27e9 :\n  \u2203 b : Set (Set (range f)), b.Countable \u2227 \u2205 \u2209 b \u2227 IsTopologicalBasis b :=\n  exists_countable_basis (range f)", "annotated_tactic": ["obtain \u27e8b, b_count, -, hb\u27e9 :\n    \u2203 b : <a>Set</a> (<a>Set</a> (<a>range</a> f)), b.Countable \u2227 \u2205 \u2209 b \u2227 <a>IsTopologicalBasis</a> b :=\n    <a>exists_countable_basis</a> (<a>range</a> f)", [{"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.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"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]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1"}, {"tactic": "haveI : Countable b := b_count.to_subtype", "annotated_tactic": ["haveI : <a>Countable</a> b := b_count.to_subtype", [{"full_name": "Countable", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}]], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1"}, {"tactic": "have : \u2200 s : b, IsClopenable (rangeFactorization f \u207b\u00b9' s) := fun s \u21a6 by\n  apply MeasurableSet.isClopenable\n  exact hf.subtype_mk (hb.isOpen s.2).measurableSet", "annotated_tactic": ["have : \u2200 s : b, <a>IsClopenable</a> (<a>rangeFactorization</a> f \u207b\u00b9' s) := fun s \u21a6 by\n    apply <a>MeasurableSet.isClopenable</a>\n    exact hf.subtype_mk (hb.isOpen s.2).<a>measurableSet</a>", [{"full_name": "PolishSpace.IsClopenable", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [386, 5], "def_end_pos": [386, 17]}, {"full_name": "Set.rangeFactorization", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1061, 5], "def_end_pos": [1061, 23]}, {"full_name": "MeasurableSet.isClopenable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [299, 9], "def_end_pos": [299, 42]}, {"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}]], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis\u271d : Countable \u2191b\nthis : \u2200 (s : \u2191b), IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s)\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1"}, {"tactic": "choose T Tt Tpolish _ Topen using this", "annotated_tactic": ["choose T Tt Tpolish _ Topen using this", []], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis\u271d : Countable \u2191b\nthis : \u2200 (s : \u2191b), IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s)\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1"}, {"tactic": "obtain \u27e8t', t'T, t't, t'_polish\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b1, (\u2200 i, t' \u2264 T i) \u2227 t' \u2264 t \u2227 @PolishSpace \u03b1 t' :=\n  exists_polishSpace_forall_le T Tt Tpolish", "annotated_tactic": ["obtain \u27e8t', t'T, t't, t'_polish\u27e9 :\n    \u2203 t' : <a>TopologicalSpace</a> \u03b1, (\u2200 i, t' \u2264 T i) \u2227 t' \u2264 t \u2227 @<a>PolishSpace</a> \u03b1 t' :=\n    <a>exists_polishSpace_forall_le</a> T Tt Tpolish", [{"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]}, {"full_name": "PolishSpace.exists_polishSpace_forall_le", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [201, 9], "def_end_pos": [201, 37]}]], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1"}, {"tactic": "refine' \u27e8t', t't, _, t'_polish\u27e9", "annotated_tactic": ["refine' \u27e8t', t't, _, t'_polish\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\n\u22a2 Continuous f"}, {"tactic": "have : Continuous[t', _] (rangeFactorization f) :=\n  hb.continuous _ fun s hs => t'T \u27e8s, hs\u27e9 _ (Topen \u27e8s, hs\u27e9)", "annotated_tactic": ["have : Continuous[t', _] (<a>rangeFactorization</a> f) :=\n    hb.continuous _ fun s hs => t'T \u27e8s, hs\u27e9 _ (Topen \u27e8s, hs\u27e9)", [{"full_name": "Set.rangeFactorization", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1061, 5], "def_end_pos": [1061, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\n\u22a2 Continuous f", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis\u271d : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\nthis : Continuous (rangeFactorization f)\n\u22a2 Continuous f"}, {"tactic": "exact continuous_subtype_val.comp this", "annotated_tactic": ["exact continuous_subtype_val.comp this", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis\u271d : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\nthis : Continuous (rangeFactorization f)\n\u22a2 Continuous f", "state_after": "no goals"}, {"tactic": "apply MeasurableSet.isClopenable", "annotated_tactic": ["apply <a>MeasurableSet.isClopenable</a>", [{"full_name": "MeasurableSet.isClopenable", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [299, 9], "def_end_pos": [299, 42]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\ns : \u2191b\n\u22a2 IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s)", "state_after": "case hs\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\ns : \u2191b\n\u22a2 MeasurableSet (rangeFactorization f \u207b\u00b9' \u2191s)"}, {"tactic": "exact hf.subtype_mk (hb.isOpen s.2).measurableSet", "annotated_tactic": ["exact hf.subtype_mk (hb.isOpen s.2).<a>measurableSet</a>", [{"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}]], "state_before": "case hs\n\u03b1\u271d : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\ns : \u2191b\n\u22a2 MeasurableSet (rangeFactorization f \u207b\u00b9' \u2191s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "not_primrec\u2082_ack", "start": [392, 1], "end": [393, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.inv_coe_unit", "start": [756, 1], "end": [764, 6], "traced_tactics": [{"tactic": "have := congr_arg ((\u2191) : \u2115 \u2192 ZMod n) (val_coe_unit_coprime u)", "annotated_tactic": ["have := <a>congr_arg</a> ((\u2191) : \u2115 \u2192 <a>ZMod</a> n) (<a>val_coe_unit_coprime</a> u)", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod.val_coe_unit_coprime", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [744, 9], "def_end_pos": [744, 29]}]], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191(Nat.gcd (val \u2191u) n) = \u21911\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "rw [\u2190 mul_inv_eq_gcd, Nat.cast_one] at this", "annotated_tactic": ["rw [\u2190 <a>mul_inv_eq_gcd</a>, <a>Nat.cast_one</a>] at this", [{"full_name": "ZMod.mul_inv_eq_gcd", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [694, 9], "def_end_pos": [694, 23]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}]], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191(Nat.gcd (val \u2191u) n) = \u21911\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "let u' : (ZMod n)\u02e3 := \u27e8u, (u : ZMod n)\u207b\u00b9, this, by rwa [mul_comm]\u27e9", "annotated_tactic": ["let u' : (<a>ZMod</a> n)\u02e3 := \u27e8u, (u : <a>ZMod</a> n)\u207b\u00b9, this, by rwa [<a>mul_comm</a>]\u27e9", [{"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "have h : u = u' := by\n  apply Units.ext\n  rfl", "annotated_tactic": ["have h : u = u' := by\n    apply <a>Units.ext</a>\n    rfl", [{"full_name": "Units.ext", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [139, 9], "def_end_pos": [139, 12]}]], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u')\u207b\u00b9 = \u2191u'\u207b\u00b9"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u')\u207b\u00b9 = \u2191u'\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rwa [mul_comm]", "annotated_tactic": ["rwa [<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": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 * \u2191u = 1", "state_after": "no goals"}, {"tactic": "apply Units.ext", "annotated_tactic": ["apply <a>Units.ext</a>", [{"full_name": "Units.ext", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [139, 9], "def_end_pos": [139, 12]}]], "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 u = u'", "state_after": "case a\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 \u2191u = \u2191u'"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 \u2191u = \u2191u'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.totalVariation_zero", "start": [498, 1], "end": [499, 52], "traced_tactics": [{"tactic": "simp [totalVariation, toJordanDecomposition_zero]", "annotated_tactic": ["simp [<a>totalVariation</a>, <a>toJordanDecomposition_zero</a>]", [{"full_name": "MeasureTheory.SignedMeasure.totalVariation", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [494, 5], "def_end_pos": [494, 19]}, {"full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_zero", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [450, 9], "def_end_pos": [450, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 totalVariation 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "MeasurableSpace.DynkinSystem.ofMeasurableSpace_toMeasurableSpace", "start": [679, 1], "end": [682, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.exists_null_frontier_thickening", "start": [463, 1], "end": [474, 58], "traced_tactics": [{"tactic": "have mbles : \u2200 r : \u211d, MeasurableSet (frontier (Metric.thickening r s)) :=\n  fun r => isClosed_frontier.measurableSet", "annotated_tactic": ["have mbles : \u2200 r : \u211d, <a>MeasurableSet</a> (<a>frontier</a> (<a>Metric.thickening</a> r s)) :=\n    fun r => isClosed_frontier.measurableSet", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "frontier", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [703, 5], "def_end_pos": [703, 13]}, {"full_name": "Metric.thickening", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [898, 5], "def_end_pos": [898, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "have disjs := Metric.frontier_thickening_disjoint s", "annotated_tactic": ["have disjs := <a>Metric.frontier_thickening_disjoint</a> s", [{"full_name": "Metric.frontier_thickening_disjoint", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [961, 9], "def_end_pos": [961, 37]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "have key := Measure.countable_meas_pos_of_disjoint_iUnion (\u03bc := \u03bc) mbles disjs", "annotated_tactic": ["have key := <a>Measure.countable_meas_pos_of_disjoint_iUnion</a> (\u03bc := \u03bc) mbles disjs", [{"full_name": "MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3509, 9], "def_end_pos": [3509, 46]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "have aux := measure_diff_null (s\u2081 := Ioo a b) (Set.Countable.measure_zero key volume)", "annotated_tactic": ["have aux := <a>measure_diff_null</a> (s\u2081 := <a>Ioo</a> a b) (<a>Set.Countable.measure_zero</a> key <a>volume</a>)", [{"full_name": "MeasureTheory.measure_diff_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [239, 9], "def_end_pos": [239, 26]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Set.Countable.measure_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3130, 9], "def_end_pos": [3130, 42]}, {"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\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "have len_pos : 0 < ENNReal.ofReal (b - a) := by simp only [hab, ENNReal.ofReal_pos, sub_pos]", "annotated_tactic": ["have len_pos : 0 < <a>ENNReal.ofReal</a> (b - a) := by simp only [hab, <a>ENNReal.ofReal_pos</a>, <a>sub_pos</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_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 19]}, {"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\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < ENNReal.ofReal (b - a)\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "rw [\u2190 Real.volume_Ioo, \u2190 aux] at len_pos", "annotated_tactic": ["rw [\u2190 <a>Real.volume_Ioo</a>, \u2190 aux] at len_pos", [{"full_name": "Real.volume_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < ENNReal.ofReal (b - a)\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))})\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "rcases nonempty_of_measure_ne_zero len_pos.ne.symm with \u27e8r, \u27e8r_in_Ioo, hr\u27e9\u27e9", "annotated_tactic": ["rcases <a>nonempty_of_measure_ne_zero</a> len_pos.ne.symm with \u27e8r, \u27e8r_in_Ioo, hr\u27e9\u27e9", [{"full_name": "MeasureTheory.nonempty_of_measure_ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [189, 9], "def_end_pos": [189, 36]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))})\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "case intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))})\nr : \u211d\nr_in_Ioo : r \u2208 Ioo a b\nhr : \u00acr \u2208 {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "refine' \u27e8r, r_in_Ioo, _\u27e9", "annotated_tactic": ["refine' \u27e8r, r_in_Ioo, _\u27e9", []], "state_before": "case intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))})\nr : \u211d\nr_in_Ioo : r \u2208 Ioo a b\nhr : \u00acr \u2208 {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\n\u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "case intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))})\nr : \u211d\nr_in_Ioo : r \u2208 Ioo a b\nhr : \u00acr \u2208 {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\n\u22a2 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0"}, {"tactic": "simpa only [mem_setOf_eq, not_lt, le_zero_iff] using hr", "annotated_tactic": ["simpa only [<a>mem_setOf_eq</a>, <a>not_lt</a>, <a>le_zero_iff</a>] using hr", [{"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_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\nlen_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))})\nr : \u211d\nr_in_Ioo : r \u2208 Ioo a b\nhr : \u00acr \u2208 {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\n\u22a2 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0", "state_after": "no goals"}, {"tactic": "simp only [hab, ENNReal.ofReal_pos, sub_pos]", "annotated_tactic": ["simp only [hab, <a>ENNReal.ofReal_pos</a>, <a>sub_pos</a>]", [{"full_name": "ENNReal.ofReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 19]}, {"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\ninst\u271d\u00b3 : PseudoEMetricSpace \u03a9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03a9\na b : \u211d\nhab : a < b\nmbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s))\nkey : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}\naux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b)\n\u22a2 0 < ENNReal.ofReal (b - a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.trim_mkMetric", "start": [414, 1], "end": [419, 7], "traced_tactics": [{"tactic": "simp only [mkMetric, mkMetric'.eq_iSup_nat, trim_iSup]", "annotated_tactic": ["simp only [<a>mkMetric</a>, <a>mkMetric'.eq_iSup_nat</a>, <a>trim_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'.eq_iSup_nat", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [302, 9], "def_end_pos": [302, 20]}, {"full_name": "MeasureTheory.OuterMeasure.trim_iSup", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 18]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 trim (mkMetric m) = mkMetric m", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 i, trim (mkMetric'.pre (fun s => m (diam s)) (\u2191i)\u207b\u00b9) = \u2a06 n, mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9"}, {"tactic": "congr 1 with n : 1", "annotated_tactic": ["congr 1 with n : 1", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 i, trim (mkMetric'.pre (fun s => m (diam s)) (\u2191i)\u207b\u00b9) = \u2a06 n, mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9", "state_after": "case e_s.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nn : \u2115\n\u22a2 trim (mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9) = mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9"}, {"tactic": "refine' mkMetric'.trim_pre _ (fun s => _) _", "annotated_tactic": ["refine' <a>mkMetric'.trim_pre</a> _ (fun s => _) _", [{"full_name": "MeasureTheory.OuterMeasure.mkMetric'.trim_pre", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [311, 9], "def_end_pos": [311, 17]}]], "state_before": "case e_s.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nn : \u2115\n\u22a2 trim (mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9) = mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9", "state_after": "case e_s.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nn : \u2115\ns : Set X\n\u22a2 m (diam (closure s)) = m (diam s)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_s.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nn : \u2115\ns : Set X\n\u22a2 m (diam (closure s)) = m (diam 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.isOrdered_iff", "start": [135, 1], "end": [136, 64], "traced_tactics": [{"tactic": "simp [isOrdered_iff']", "annotated_tactic": ["simp [<a>isOrdered_iff'</a>]", [{"full_name": "Std.RBNode.isOrdered_iff'", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [113, 9], "def_end_pos": [113, 23]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\ninst\u271d : TransCmp cmp\nt : RBNode \u03b1\n\u22a2 isOrdered cmp t none none = true \u2194 Ordered cmp t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_eval", "start": [1721, 1], "end": [1734, 12], "traced_tactics": [{"tactic": "obtain \u27e8i, h\u2081, h\u2082\u27e9 := tr_init c v", "annotated_tactic": ["obtain \u27e8i, h\u2081, h\u2082\u27e9 := <a>tr_init</a> c v", [{"full_name": "Turing.PartrecToTM2.tr_init", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1716, 9], "def_end_pos": [1716, 16]}]], "state_before": "c : Code\nv : List \u2115\n\u22a2 eval (TM2.step tr) (init c v) = halt <$> Code.eval c v", "state_after": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\n\u22a2 eval (TM2.step tr) (init c v) = halt <$> Code.eval c v"}, {"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.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\n\u22a2 eval (TM2.step tr) (init c v) = halt <$> Code.eval c v", "state_after": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 x \u2208 eval (TM2.step tr) (init c v) \u2194 x \u2208 halt <$> Code.eval c v"}, {"tactic": "rw [reaches_eval h\u2082.to_reflTransGen]", "annotated_tactic": ["rw [<a>reaches_eval</a> h\u2082.to_reflTransGen]", [{"full_name": "Turing.reaches_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}]], "state_before": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 x \u2208 eval (TM2.step tr) (init c v) \u2194 x \u2208 halt <$> Code.eval c v", "state_after": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 x \u2208 eval (fun a => TM2.step tr a) i \u2194 x \u2208 halt <$> Code.eval c v"}, {"tactic": "simp [-TM2.step]", "annotated_tactic": ["simp [-<a>TM2.step</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}]], "state_before": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 x \u2208 eval (fun a => TM2.step tr a) i \u2194 x \u2208 halt <$> Code.eval c v", "state_after": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 x \u2208 eval (fun a => TM2.step tr a) i \u2194 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x"}, {"tactic": "refine' \u27e8fun h => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, _\u27e9", []], "state_before": "case intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 x \u2208 eval (fun a => TM2.step tr a) i \u2194 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x", "state_after": "case intro.intro.refine'_1\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\n\u22a2 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x\n\ncase intro.intro.refine'_2\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 (\u2203 a, a \u2208 Code.eval c v \u2227 halt a = x) \u2192 x \u2208 eval (fun a => TM2.step tr a) i"}, {"tactic": "obtain \u27e8c, hc\u2081, hc\u2082\u27e9 := tr_eval_rev tr_respects h\u2081 h", "annotated_tactic": ["obtain \u27e8c, hc\u2081, hc\u2082\u27e9 := <a>tr_eval_rev</a> <a>tr_respects</a> h\u2081 h", [{"full_name": "Turing.tr_eval_rev", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [931, 9], "def_end_pos": [931, 20]}, {"full_name": "Turing.PartrecToTM2.tr_respects", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 20]}]], "state_before": "case intro.intro.refine'_1\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\n\u22a2 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x", "state_after": "case intro.intro.refine'_1.intro.intro\nc\u271d : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c\u271d Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c\u271d v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\nc : Cfg\nhc\u2081 : TrCfg c x\nhc\u2082 : c \u2208 eval step (stepNormal c\u271d Cont.halt v)\n\u22a2 \u2203 a, a \u2208 Code.eval c\u271d v \u2227 halt a = x"}, {"tactic": "simp [stepNormal_eval] at hc\u2082", "annotated_tactic": ["simp [<a>stepNormal_eval</a>] at hc\u2082", [{"full_name": "Turing.ToPartrec.stepNormal_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [718, 9], "def_end_pos": [718, 24]}]], "state_before": "case intro.intro.refine'_1.intro.intro\nc\u271d : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c\u271d Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c\u271d v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\nc : Cfg\nhc\u2081 : TrCfg c x\nhc\u2082 : c \u2208 eval step (stepNormal c\u271d Cont.halt v)\n\u22a2 \u2203 a, a \u2208 Code.eval c\u271d v \u2227 halt a = x", "state_after": "case intro.intro.refine'_1.intro.intro\nc\u271d : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c\u271d Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c\u271d v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\nc : Cfg\nhc\u2081 : TrCfg c x\nhc\u2082 : \u2203 a, a \u2208 Code.eval c\u271d v \u2227 Cfg.halt a = c\n\u22a2 \u2203 a, a \u2208 Code.eval c\u271d v \u2227 halt a = x"}, {"tactic": "obtain \u27e8v', hv, rfl\u27e9 := hc\u2082", "annotated_tactic": ["obtain \u27e8v', hv, rfl\u27e9 := hc\u2082", []], "state_before": "case intro.intro.refine'_1.intro.intro\nc\u271d : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c\u271d Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c\u271d v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\nc : Cfg\nhc\u2081 : TrCfg c x\nhc\u2082 : \u2203 a, a \u2208 Code.eval c\u271d v \u2227 Cfg.halt a = c\n\u22a2 \u2203 a, a \u2208 Code.eval c\u271d v \u2227 halt a = x", "state_after": "case intro.intro.refine'_1.intro.intro.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nhc\u2081 : TrCfg (Cfg.halt v') x\n\u22a2 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x"}, {"tactic": "exact \u27e8_, hv, hc\u2081.symm\u27e9", "annotated_tactic": ["exact \u27e8_, hv, hc\u2081.symm\u27e9", []], "state_before": "case intro.intro.refine'_1.intro.intro.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\nh : x \u2208 eval (fun a => TM2.step tr a) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nhc\u2081 : TrCfg (Cfg.halt v') x\n\u22a2 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x", "state_after": "no goals"}, {"tactic": "rintro \u27e8v', hv, rfl\u27e9", "annotated_tactic": ["rintro \u27e8v', hv, rfl\u27e9", []], "state_before": "case intro.intro.refine'_2\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393')\n\u22a2 (\u2203 a, a \u2208 Code.eval c v \u2227 halt a = x) \u2192 x \u2208 eval (fun a => TM2.step tr a) i", "state_after": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i"}, {"tactic": "have := Turing.tr_eval (b\u2081 := Cfg.halt v') tr_respects h\u2081", "annotated_tactic": ["have := <a>Turing.tr_eval</a> (b\u2081 := <a>Cfg.halt</a> v') <a>tr_respects</a> h\u2081", [{"full_name": "Turing.tr_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [923, 9], "def_end_pos": [923, 16]}, {"full_name": "Turing.ToPartrec.Cfg.halt", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [465, 5], "def_end_pos": [465, 9]}, {"full_name": "Turing.PartrecToTM2.tr_respects", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1706, 9], "def_end_pos": [1706, 20]}]], "state_before": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i", "state_after": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nthis : Cfg.halt v' \u2208 eval step (stepNormal c Cont.halt v) \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i"}, {"tactic": "simp [stepNormal_eval, -TM2.step] at this", "annotated_tactic": ["simp [<a>stepNormal_eval</a>, -<a>TM2.step</a>] at this", [{"full_name": "Turing.ToPartrec.stepNormal_eval", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [718, 9], "def_end_pos": [718, 24]}, {"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}]], "state_before": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nthis : Cfg.halt v' \u2208 eval step (stepNormal c Cont.halt v) \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i", "state_after": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nthis : v' \u2208 Code.eval c v \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i"}, {"tactic": "obtain \u27e8_, \u27e8\u27e9, h\u27e9 := this hv", "annotated_tactic": ["obtain \u27e8_, \u27e8\u27e9, h\u27e9 := this hv", []], "state_before": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nthis : v' \u2208 Code.eval c v \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i", "state_after": "case intro.intro.refine'_2.intro.intro.intro.intro.refl\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nthis : v' \u2208 Code.eval c v \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i\nh : halt v' \u2208 eval (TM2.step tr) i\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case intro.intro.refine'_2.intro.intro.intro.intro.refl\nc : Code\nv : List \u2115\ni : Cfg'\nh\u2081 : TrCfg (stepNormal c Cont.halt v) i\nh\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i\nv' : List \u2115\nhv : v' \u2208 Code.eval c v\nthis : v' \u2208 Code.eval c v \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i\nh : halt v' \u2208 eval (TM2.step tr) i\n\u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) 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.length_push", "start": [32, 9], "end": [34, 6], "traced_tactics": [{"tactic": "rw [push, mk_length, List.length_append, List.length_singleton, Nat.succ.injEq]", "annotated_tactic": ["rw [<a>push</a>, <a>mk_length</a>, <a>List.length_append</a>, <a>List.length_singleton</a>, Nat.succ.injEq]", [{"full_name": "String.push", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}, {"full_name": "String.mk_length", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [26, 17], "def_end_pos": [26, 26]}, {"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_singleton", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [36, 22], "def_end_pos": [36, 38]}]], "state_before": "s : String\nc : Char\n\u22a2 length (push s c) = length s + 1", "state_after": "s : String\nc : Char\n\u22a2 List.length s.data = length s"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "s : String\nc : Char\n\u22a2 List.length s.data = length s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.multiset_prod_mem_multiset_prod", "start": [127, 1], "end": [131, 41], "traced_tactics": [{"tactic": "induction t using Quotient.inductionOn", "annotated_tactic": ["induction t 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": "\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 : Multiset \u03b9\nf : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 (i : \u03b9), i \u2208 t \u2192 g i \u2208 f i\n\u22a2 Multiset.prod (Multiset.map g t) \u2208 Multiset.prod (Multiset.map f t)", "state_after": "case h\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\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 Multiset.prod (Multiset.map g (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2208\n    Multiset.prod (Multiset.map f (Quotient.mk (List.isSetoid \u03b9) a\u271d))"}, {"tactic": "simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod]", "annotated_tactic": ["simp_rw [<a>Multiset.quot_mk_to_coe</a>, <a>Multiset.coe_map</a>, <a>Multiset.coe_prod</a>]", [{"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.coe_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1166, 9], "def_end_pos": [1166, 16]}, {"full_name": "Multiset.coe_prod", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}]], "state_before": "case h\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\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 Multiset.prod (Multiset.map g (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2208\n    Multiset.prod (Multiset.map f (Quotient.mk (List.isSetoid \u03b9) a\u271d))", "state_after": "case h\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\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 List.prod (List.map g a\u271d) \u2208 List.prod (List.map f a\u271d)"}, {"tactic": "exact list_prod_mem_list_prod _ _ _ hg", "annotated_tactic": ["exact <a>list_prod_mem_list_prod</a> _ _ _ hg", [{"full_name": "Set.list_prod_mem_list_prod", "def_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "def_pos": [96, 9], "def_end_pos": [96, 32]}]], "state_before": "case h\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\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 List.prod (List.map g a\u271d) \u2208 List.prod (List.map f a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nonempty_of_ssubset", "start": [485, 1], "end": [486, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/MulAntidiagonal.lean", "full_name": "Finset.swap_mem_mulAntidiagonal", "start": [93, 1], "end": [96, 38], "traced_tactics": [{"tactic": "simp only [mem_mulAntidiagonal, Prod.fst_swap, Prod.snd_swap, Set.swap_mem_mulAntidiagonal_aux,\n           Set.mem_mulAntidiagonal]", "annotated_tactic": ["simp only [<a>mem_mulAntidiagonal</a>, <a>Prod.fst_swap</a>, <a>Prod.snd_swap</a>, <a>Set.swap_mem_mulAntidiagonal_aux</a>,\n             <a>Set.mem_mulAntidiagonal</a>]", [{"full_name": "Finset.mem_mulAntidiagonal", "def_path": "Mathlib/Data/Finset/MulAntidiagonal.lean", "def_pos": [73, 9], "def_end_pos": [73, 28]}, {"full_name": "Prod.fst_swap", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [172, 9], "def_end_pos": [172, 17]}, {"full_name": "Prod.snd_swap", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [177, 9], "def_end_pos": [177, 17]}, {"full_name": "Set.swap_mem_mulAntidiagonal_aux", "def_path": "Mathlib/Data/Set/MulAntidiagonal.lean", "def_pos": [60, 9], "def_end_pos": [60, 37]}, {"full_name": "Set.mem_mulAntidiagonal", "def_path": "Mathlib/Data/Set/MulAntidiagonal.lean", "def_pos": [32, 9], "def_end_pos": [32, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsPwo s\nht : Set.IsPwo t\na : \u03b1\nu : Set \u03b1\nhu : Set.IsPwo u\nx : \u03b1 \u00d7 \u03b1\n\u22a2 Prod.swap x \u2208 mulAntidiagonal hs ht a \u2194 x \u2208 mulAntidiagonal ht hs a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.integral_mul_upcrossingsBefore_le_integral", "start": [655, 1], "end": [667, 65], "traced_tactics": [{"tactic": "rw [\u2190 integral_mul_left]", "annotated_tactic": ["rw [\u2190 <a>integral_mul_left</a>]", [{"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\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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264\n    \u222b (x : \u03a9), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x \u2202\u03bc", "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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 \u222b (a_1 : \u03a9), (b - a) * \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc \u2264\n    \u222b (x : \u03a9), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x \u2202\u03bc"}, {"tactic": "refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _", "annotated_tactic": ["refine' <a>integral_mono_of_nonneg</a> _ ((hf.sum_upcrossingStrat_mul a b N).<a>integrable</a> N) _", [{"full_name": "MeasureTheory.integral_mono_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1356, 9], "def_end_pos": [1356, 32]}, {"full_name": "MeasureTheory.Submartingale.integrable", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [219, 19], "def_end_pos": [219, 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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 \u222b (a_1 : \u03a9), (b - a) * \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc \u2264\n    \u222b (x : \u03a9), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x \u2202\u03bc", "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 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 0 \u2264\u1d50[\u03bc] fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)\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 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 (fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)) \u2264\u1d50[\u03bc] fun x =>\n    Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x"}, {"tactic": "exact eventually_of_forall fun \u03c9 => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun \u03c9 => <a>mul_nonneg</a> (<a>sub_nonneg</a>.2 hab.le) (<a>Nat.cast_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": "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": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "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 : \u03a9\n\u2131 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 0 \u2264\u1d50[\u03bc] fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall fun \u03c9 => _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun \u03c9 => _", [{"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\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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u22a2 (fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)) \u2264\u1d50[\u03bc] fun x =>\n    Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x", "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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u03c9 : \u03a9\n\u22a2 (fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)) \u03c9 \u2264\n    (fun x => Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x) \u03c9"}, {"tactic": "simpa using mul_upcrossingsBefore_le (hfN \u03c9) hab", "annotated_tactic": ["simpa using <a>mul_upcrossingsBefore_le</a> (hfN \u03c9) hab", [{"full_name": "MeasureTheory.mul_upcrossingsBefore_le", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [599, 9], "def_end_pos": [599, 33]}]], "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\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9\nhfzero : 0 \u2264 f 0\nhab : a < b\n\u03c9 : \u03a9\n\u22a2 (fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)) \u03c9 \u2264\n    (fun x => Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x) \u03c9", "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": [274, 1], "end": [281, 27], "traced_tactics": [{"tactic": "rw [pi']", "annotated_tactic": ["rw [<a>pi'</a>]", [{"full_name": "MeasureTheory.Measure.pi'", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [270, 5], "def_end_pos": [270, 8]}]], "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 : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : Encodable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191\u2191(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\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : Encodable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191\u2191(map (TProd.elim' (_ : \u2200 (x : \u03b9), x \u2208 sortedUniv \u03b9)) (Measure.tprod (sortedUniv \u03b9) \u03bc)) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "simp only [TProd.elim']", "annotated_tactic": ["simp only [<a>TProd.elim'</a>]", [{"full_name": "List.TProd.elim'", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [129, 15], "def_end_pos": [129, 20]}]], "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 : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : Encodable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191\u2191(map (TProd.elim' (_ : \u2200 (x : \u03b9), x \u2208 sortedUniv \u03b9)) (Measure.tprod (sortedUniv \u03b9) \u03bc)) (Set.pi univ s) =\n    \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\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : Encodable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191\u2191(map (fun v i => TProd.elim v (_ : i \u2208 sortedUniv \u03b9)) (Measure.tprod (sortedUniv \u03b9) \u03bc)) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)"}, {"tactic": "erw [\u2190 MeasurableEquiv.piMeasurableEquivTProd_symm_apply, MeasurableEquiv.map_apply,\n  MeasurableEquiv.piMeasurableEquivTProd_symm_apply, elim_preimage_pi, tprod_tprod _ \u03bc, \u2190\n  List.prod_toFinset, sortedUniv_toFinset] <;>\nexact sortedUniv_nodup \u03b9", "annotated_tactic": ["erw [\u2190 <a>MeasurableEquiv.piMeasurableEquivTProd_symm_apply</a>, <a>MeasurableEquiv.map_apply</a>,\n    <a>MeasurableEquiv.piMeasurableEquivTProd_symm_apply</a>, <a>elim_preimage_pi</a>, <a>tprod_tprod</a> _ \u03bc, \u2190\n    <a>List.prod_toFinset</a>, <a>sortedUniv_toFinset</a>] <;>\n  exact <a>sortedUniv_nodup</a> \u03b9", [{"full_name": "MeasurableEquiv.piMeasurableEquivTProd_symm_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1621, 3], "def_end_pos": [1621, 47]}, {"full_name": "MeasurableEquiv.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4218, 19], "def_end_pos": [4218, 28]}, {"full_name": "MeasurableEquiv.piMeasurableEquivTProd_symm_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1621, 3], "def_end_pos": [1621, 47]}, {"full_name": "Set.elim_preimage_pi", "def_path": "Mathlib/Data/Prod/TProd.lean", "def_pos": [173, 9], "def_end_pos": [173, 25]}, {"full_name": "MeasureTheory.Measure.tprod_tprod", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [252, 9], "def_end_pos": [252, 20]}, {"full_name": "List.prod_toFinset", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2050, 9], "def_end_pos": [2050, 22]}, {"full_name": "Encodable.sortedUniv_toFinset", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [219, 9], "def_end_pos": [219, 28]}, {"full_name": "Encodable.sortedUniv_nodup", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [214, 9], "def_end_pos": [214, 25]}]], "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 : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b9 : Encodable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 \u2191\u2191(map (fun v i => TProd.elim v (_ : i \u2208 sortedUniv \u03b9)) (Measure.tprod (sortedUniv \u03b9) \u03bc)) (Set.pi univ s) =\n    \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/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le'", "start": [345, 1], "end": [372, 25], "traced_tactics": [{"tactic": "obtain \u27e8M, hMpos, hM\u27e9 := hf.snorm_indicator_norm_ge_pos_le \u03bc hmeas h\u03b5", "annotated_tactic": ["obtain \u27e8M, hMpos, hM\u27e9 := hf.snorm_indicator_norm_ge_pos_le \u03bc hmeas 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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4,\n    \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 2 * ENNReal.ofReal \u03b5", "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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4,\n    \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 2 * ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 :=\n  snorm_indicator_le_of_bound \u03bc (f := { x | \u2016f x\u2016 < M }.indicator f) hp_top h\u03b5 (by\n    intro x\n    rw [norm_indicator_eq_indicator_norm, Set.indicator_apply]\n    split_ifs with h\n    exacts [h, hMpos])", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 :=\n    <a>snorm_indicator_le_of_bound</a> \u03bc (f := { x | \u2016f x\u2016 < M }.<a>indicator</a> f) hp_top h\u03b5 (by\n      intro x\n      rw [<a>norm_indicator_eq_indicator_norm</a>, <a>Set.indicator_apply</a>]\n      split_ifs with h\n      exacts [h, hMpos])", [{"full_name": "MeasureTheory.snorm_indicator_le_of_bound", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [311, 9], "def_end_pos": [311, 36]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"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_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}]], "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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4,\n    \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 2 * ENNReal.ofReal \u03b5", "state_after": "case 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4,\n    \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 2 * ENNReal.ofReal \u03b5"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2200 (x : \u03b1), \u2016Set.indicator {x | \u2016f x\u2016 < M} f x\u2016 < ?m.136721", "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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 \u2016Set.indicator {x | \u2016f x\u2016 < M} f x\u2016 < ?m.136721"}, {"tactic": "rw [norm_indicator_eq_indicator_norm, Set.indicator_apply]", "annotated_tactic": ["rw [<a>norm_indicator_eq_indicator_norm</a>, <a>Set.indicator_apply</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": "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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 \u2016Set.indicator {x | \u2016f x\u2016 < M} f x\u2016 < ?m.136721", "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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 (if x \u2208 {x | \u2016f x\u2016 < M} then \u2016f x\u2016 else 0) < ?m.136721\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d"}, {"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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 (if x \u2208 {x | \u2016f x\u2016 < M} then \u2016f x\u2016 else 0) < ?m.136721\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d", "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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\nh : x \u2208 {x | \u2016f x\u2016 < M}\n\u22a2 \u2016f x\u2016 < ?m.136721\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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\nh : \u00acx \u2208 {x | \u2016f x\u2016 < M}\n\u22a2 0 < ?m.136721\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d"}, {"tactic": "exacts [h, hMpos]", "annotated_tactic": ["exacts [h, hMpos]", []], "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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\nh : x \u2208 {x | \u2016f x\u2016 < M}\n\u22a2 \u2016f x\u2016 < ?m.136721\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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\nh : \u00acx \u2208 {x | \u2016f x\u2016 < M}\n\u22a2 0 < ?m.136721\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u211d", "state_after": "no goals"}, {"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\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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4,\n    \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 2 * ENNReal.ofReal \u03b5", "state_after": "case 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 2 * ENNReal.ofReal \u03b5"}, {"tactic": "rw [(_ : f = { x : \u03b1 | M \u2264 \u2016f x\u2016\u208a }.indicator f + { x : \u03b1 | \u2016f x\u2016 < M }.indicator f)]", "annotated_tactic": ["rw [(_ : f = { x : \u03b1 | M \u2264 \u2016f x\u2016\u208a }.<a>indicator</a> f + { x : \u03b1 | \u2016f x\u2016 < M }.<a>indicator</a> f)]", [{"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 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 2 * ENNReal.ofReal \u03b5", "state_after": "case 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264\n    2 * ENNReal.ofReal \u03b5\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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 f = Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f"}, {"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": "case 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264\n    2 * ENNReal.ofReal \u03b5", "state_after": "case 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264\n    2 * ENNReal.ofReal \u03b5"}, {"tactic": "refine' le_trans (snorm_add_le _ _ hp_one) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>snorm_add_le</a> _ _ hp_one) _", [{"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]}]], "state_before": "case 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\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264\n    2 * ENNReal.ofReal \u03b5", "state_after": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 AEStronglyMeasurable (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) (Measure.restrict \u03bc s)\n\ncase intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 AEStronglyMeasurable (Set.indicator {x | \u2016f x\u2016 < M} f) (Measure.restrict \u03bc s)\n\ncase intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p (Measure.restrict \u03bc s) +\n      snorm (Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264\n    2 * ENNReal.ofReal \u03b5"}, {"tactic": "exact StronglyMeasurable.aestronglyMeasurable\n  (hmeas.indicator (measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe))", "annotated_tactic": ["exact <a>StronglyMeasurable.aestronglyMeasurable</a>\n          (hmeas.indicator (<a>measurableSet_le</a> <a>measurable_const</a> hmeas.nnnorm.measurable.subtype_coe))", [{"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"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.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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 AEStronglyMeasurable (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) (Measure.restrict \u03bc s)", "state_after": "no goals"}, {"tactic": "exact StronglyMeasurable.aestronglyMeasurable\n  (hmeas.indicator (measurableSet_lt hmeas.nnnorm.measurable.subtype_coe measurable_const))", "annotated_tactic": ["exact <a>StronglyMeasurable.aestronglyMeasurable</a>\n          (hmeas.indicator (<a>measurableSet_lt</a> hmeas.nnnorm.measurable.subtype_coe <a>measurable_const</a>))", [{"full_name": "MeasureTheory.StronglyMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}, {"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 intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 AEStronglyMeasurable (Set.indicator {x | \u2016f x\u2016 < M} f) (Measure.restrict \u03bc s)", "state_after": "no goals"}, {"tactic": "rw [two_mul]", "annotated_tactic": ["rw [<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": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p (Measure.restrict \u03bc s) +\n      snorm (Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264\n    2 * ENNReal.ofReal \u03b5", "state_after": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p (Measure.restrict \u03bc s) +\n      snorm (Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264\n    ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b5"}, {"tactic": "refine' add_le_add (le_trans (snorm_mono_measure _ Measure.restrict_le_self) hM) _", "annotated_tactic": ["refine' <a>add_le_add</a> (<a>le_trans</a> (<a>snorm_mono_measure</a> _ <a>Measure.restrict_le_self</a>) hM) _", [{"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_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.snorm_mono_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [612, 9], "def_end_pos": [612, 27]}, {"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": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p (Measure.restrict \u03bc s) +\n      snorm (Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264\n    ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b5", "state_after": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "rw [\u2190 snorm_indicator_eq_snorm_restrict hs]", "annotated_tactic": ["rw [\u2190 <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": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "exact h\u03b4 s hs h\u03bcs", "annotated_tactic": ["exact h\u03b4 s hs h\u03bcs", []], "state_before": "case intro.intro.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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} 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 f = Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f", "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) x"}, {"tactic": "by_cases hx : M \u2264 \u2016f x\u2016", "annotated_tactic": ["by_cases hx : M \u2264 \u2016f x\u2016", []], "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) 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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nx : \u03b1\nhx : M \u2264 \u2016f x\u2016\n\u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) 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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nx : \u03b1\nhx : \u00acM \u2264 \u2016f x\u2016\n\u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) x"}, {"tactic": "rw [Pi.add_apply, Set.indicator_of_mem, Set.indicator_of_not_mem, add_zero] <;> simpa", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>Set.indicator_of_mem</a>, <a>Set.indicator_of_not_mem</a>, <a>add_zero</a>] <;> simpa", [{"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_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 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\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nx : \u03b1\nhx : M \u2264 \u2016f x\u2016\n\u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) x", "state_after": "no goals"}, {"tactic": "rw [Pi.add_apply, Set.indicator_of_not_mem, Set.indicator_of_mem, zero_add] <;>\n  simpa using hx", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>Set.indicator_of_not_mem</a>, <a>Set.indicator_of_mem</a>, <a>zero_add</a>] <;>\n          simpa using hx", [{"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_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"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 : 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\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhMpos : 0 < M\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\nx : \u03b1\nhx : \u00acM \u2264 \u2016f x\u2016\n\u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) 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_ret_respects", "start": [1647, 1], "end": [1703, 28], "traced_tactics": [{"tactic": "induction k generalizing v s", "annotated_tactic": ["induction k generalizing v s", []], "state_before": "k : Cont\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet k v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n        b\u2082", "state_after": "case halt\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet Cont.halt v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont Cont.halt)), var := s, stk := elim (trList v) [] [] (trContStack Cont.halt) } b\u2082\n\ncase cons\u2081\na\u271d\u00b2 : Code\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase cons\u2082\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082"}, {"tactic": "case halt => exact \u27e8_, rfl, TransGen.single rfl\u27e9", "annotated_tactic": ["case halt => exact \u27e8_, <a>rfl</a>, <a>TransGen.single</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": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}, {"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 halt\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet Cont.halt v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont Cont.halt)), var := s, stk := elim (trList v) [] [] (trContStack Cont.halt) } b\u2082\n\ncase cons\u2081\na\u271d\u00b2 : Code\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase cons\u2082\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082", "state_after": "case cons\u2081\na\u271d\u00b2 : Code\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase cons\u2082\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082"}, {"tactic": "case cons\u2082 ns k IH =>\n  obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH (ns.headI :: v) none\n  exact \u27e8c, h\u2081, TransGen.head rfl <| head_stack_ok.trans h\u2082\u27e9", "annotated_tactic": ["case cons\u2082 ns k IH =>\n    obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH (ns.headI :: v) <a>none</a>\n    exact \u27e8c, h\u2081, <a>TransGen.head</a> <a>rfl</a> <| head_stack_ok.trans h\u2082\u27e9", [{"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": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case cons\u2082\na\u271d\u00b9 : List \u2115\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase comp\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082", "state_after": "case comp\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082"}, {"tactic": "case comp f k _ =>\n  obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects f k v s\n  exact \u27e8_, h\u2081, TransGen.head rfl h\u2082\u27e9", "annotated_tactic": ["case comp f k _ =>\n    obtain \u27e8s', h\u2081, h\u2082\u27e9 := <a>trNormal_respects</a> f k v s\n    exact \u27e8_, h\u2081, <a>TransGen.head</a> <a>rfl</a> h\u2082\u27e9", [{"full_name": "Turing.PartrecToTM2.trNormal_respects", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1612, 9], "def_end_pos": [1612, 26]}, {"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case comp\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }\n        b\u2082\n\ncase fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082", "state_after": "case fix\na\u271d\u00b9 : Code\na\u271d : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet a\u271d v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }\n        b\u2082"}, {"tactic": "exact \u27e8_, rfl, TransGen.single rfl\u27e9", "annotated_tactic": ["exact \u27e8_, <a>rfl</a>, <a>TransGen.single</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": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "v : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet Cont.halt v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont Cont.halt)), var := s, stk := elim (trList v) [] [] (trContStack Cont.halt) } b\u2082", "state_after": "no goals"}, {"tactic": "obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects fs (Cont.cons\u2082 v k) as none", "annotated_tactic": ["obtain \u27e8s', h\u2081, h\u2082\u27e9 := <a>trNormal_respects</a> fs (<a>Cont.cons\u2082</a> v k) as <a>none</a>", [{"full_name": "Turing.PartrecToTM2.trNormal_respects", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1612, 9], "def_end_pos": [1612, 26]}, {"full_name": "Turing.ToPartrec.Cont.cons\u2082", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [435, 5], "def_end_pos": [435, 10]}, {"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": "fs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2081 fs as k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2081 fs as k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }\n        b\u2082", "state_after": "case intro.intro\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2081 fs as k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2081 fs as k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }\n        b\u2082"}, {"tactic": "refine' \u27e8s', h\u2081, TransGen.head rfl _\u27e9", "annotated_tactic": ["refine' \u27e8s', h\u2081, <a>TransGen.head</a> <a>rfl</a> _\u27e9", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case intro.intro\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2081 fs as k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2081 fs as k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }\n        b\u2082", "state_after": "case intro.intro\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.cons\u2081 fs as k)))) s\n      (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k))))\n    s'"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro.intro\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.cons\u2081 fs as k)))) s\n      (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k))))\n    s'", "state_after": "case intro.intro\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    {\n      l :=\n        some\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 fs (Cont'.cons\u2082 (trCont k)))))),\n      var := s, stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }\n    s'"}, {"tactic": "refine' (move\u2082_ok (by decide) _ (splitAtPred_false _)).trans _", "annotated_tactic": ["refine' (<a>move\u2082_ok</a> (by decide) _ (<a>splitAtPred_false</a> _)).<a>trans</a> _", [{"full_name": "Turing.PartrecToTM2.move\u2082_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1390, 9], "def_end_pos": [1390, 17]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_false", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1345, 9], "def_end_pos": [1345, 26]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}]], "state_before": "case intro.intro\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    {\n      l :=\n        some\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 fs (Cont'.cons\u2082 (trCont k)))))),\n      var := s, stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }\n    s'", "state_after": "case intro.intro.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) rev = []\n\ncase intro.intro.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    {\n      l :=\n        some\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k))))),\n      var := none,\n      stk :=\n        update\n          (update (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k))) main (Option.elim none id List.cons []))\n          aux\n          (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) main ++\n            elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) aux) }\n    s'"}, {"tactic": "simp only [TM2.step, Option.mem_def, Option.elim, id_eq, elim_update_main, elim_main, elim_aux,\n  List.append_nil, elim_update_aux]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>Option.elim</a>, <a>id_eq</a>, <a>elim_update_main</a>, <a>elim_main</a>, <a>elim_aux</a>,\n      <a>List.append_nil</a>, <a>elim_update_aux</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.elim", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [13, 31], "def_end_pos": [13, 35]}, {"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": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 28]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_aux", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1271, 9], "def_end_pos": [1271, 20]}, {"full_name": "List.append_nil", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [93, 17], "def_end_pos": [93, 27]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_aux", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 27]}]], "state_before": "case intro.intro.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    {\n      l :=\n        some\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k))))),\n      var := none,\n      stk :=\n        update\n          (update (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k))) main (Option.elim none id List.cons []))\n          aux\n          (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) main ++\n            elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) aux) }\n    s'", "state_after": "case intro.intro.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    {\n      l :=\n        some\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k))))),\n      var := none, stk := elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) }\n    s'"}, {"tactic": "refine' (move\u2082_ok (by decide) _ _).trans _", "annotated_tactic": ["refine' (<a>move\u2082_ok</a> (by decide) _ _).<a>trans</a> _", [{"full_name": "Turing.PartrecToTM2.move\u2082_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1390, 9], "def_end_pos": [1390, 17]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}]], "state_before": "case intro.intro.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    {\n      l :=\n        some\n          (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main\n            (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k))))),\n      var := none, stk := elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) }\n    s'", "state_after": "case intro.intro.refine'_2.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 Option \u0393'\n\ncase intro.intro.refine'_2.refine'_3\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_4\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) rev = []\n\ncase intro.intro.refine'_2.refine'_5\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =\n    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)\n\ncase intro.intro.refine'_2.refine'_6\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,\n      stk :=\n        update\n          (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n            (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))\n          main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }\n    s'"}, {"tactic": "pick_goal 4", "annotated_tactic": ["pick_goal 4", []], "state_before": "case intro.intro.refine'_2.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 Option \u0393'\n\ncase intro.intro.refine'_2.refine'_3\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_4\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) rev = []\n\ncase intro.intro.refine'_2.refine'_5\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =\n    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)\n\ncase intro.intro.refine'_2.refine'_6\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,\n      stk :=\n        update\n          (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n            (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))\n          main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }\n    s'", "state_after": "case intro.intro.refine'_2.refine'_4\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) rev = []\n\ncase intro.intro.refine'_2.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 Option \u0393'\n\ncase intro.intro.refine'_2.refine'_3\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_5\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =\n    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)\n\ncase intro.intro.refine'_2.refine'_6\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,\n      stk :=\n        update\n          (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n            (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))\n          main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }\n    s'"}, {"tactic": "pick_goal 4", "annotated_tactic": ["pick_goal 4", []], "state_before": "case intro.intro.refine'_2.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 Option \u0393'\n\ncase intro.intro.refine'_2.refine'_3\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_5\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =\n    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)\n\ncase intro.intro.refine'_2.refine'_6\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,\n      stk :=\n        update\n          (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n            (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))\n          main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }\n    s'", "state_after": "case intro.intro.refine'_2.refine'_5\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =\n    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)\n\ncase intro.intro.refine'_2.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 Option \u0393'\n\ncase intro.intro.refine'_2.refine'_3\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 List \u0393'\n\ncase intro.intro.refine'_2.refine'_6\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,\n      stk :=\n        update\n          (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n            (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))\n          main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }\n    s'"}, {"tactic": "refine' (move\u2082_ok (by decide) _ (splitAtPred_false _)).trans _", "annotated_tactic": ["refine' (<a>move\u2082_ok</a> (by decide) _ (<a>splitAtPred_false</a> _)).<a>trans</a> _", [{"full_name": "Turing.PartrecToTM2.move\u2082_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1390, 9], "def_end_pos": [1390, 17]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_false", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1345, 9], "def_end_pos": [1345, 26]}, {"full_name": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}]], "state_before": "case intro.intro.refine'_2.refine'_6\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,\n      stk :=\n        update\n          (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n            (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n          main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }\n    s'", "state_after": "case intro.intro.refine'_2.refine'_6.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 update\n      (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n        (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n      main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) rev =\n    []\n\ncase intro.intro.refine'_2.refine'_6.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (trNormal fs (Cont'.cons\u2082 (trCont k))), var := none,\n      stk :=\n        update\n          (update\n            (update\n              (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n                (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n              main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main))\n            aux (Option.elim none id List.cons []))\n          stack\n          (update\n              (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n                (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n              main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) aux ++\n            update\n              (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n                (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n              main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) stack) }\n    s'"}, {"tactic": "simp only [TM2.step, Option.mem_def, Option.elim, elim_update_stack, elim_main,\n  List.append_nil, elim_update_main,  id_eq, elim_update_aux, ne_eq, Function.update_noteq,\n  elim_aux, elim_stack]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>Option.elim</a>, <a>elim_update_stack</a>, <a>elim_main</a>,\n      <a>List.append_nil</a>, <a>elim_update_main</a>,  <a>id_eq</a>, <a>elim_update_aux</a>, <a>ne_eq</a>, <a>Function.update_noteq</a>,\n      <a>elim_aux</a>, <a>elim_stack</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2154, 5], "def_end_pos": [2154, 9]}, {"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.elim", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [13, 31], "def_end_pos": [13, 35]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 29]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 21]}, {"full_name": "List.append_nil", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [93, 17], "def_end_pos": [93, 27]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 28]}, {"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": "Turing.PartrecToTM2.K'.elim_update_aux", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 27]}, {"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_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_aux", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1271, 9], "def_end_pos": [1271, 20]}, {"full_name": "Turing.PartrecToTM2.K'.elim_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 22]}]], "state_before": "case intro.intro.refine'_2.refine'_6.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    { l := some (trNormal fs (Cont'.cons\u2082 (trCont k))), var := none,\n      stk :=\n        update\n          (update\n            (update\n              (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n                (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n              main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main))\n            aux (Option.elim none id List.cons []))\n          stack\n          (update\n              (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n                (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n              main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) aux ++\n            update\n              (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n                (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n              main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) stack) }\n    s'", "state_after": "case intro.intro.refine'_2.refine'_6.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (trNormal fs (Cont'.cons\u2082 (trCont k))), var := none,\n      stk := elim (trList as) [] [] (trList v ++ \u0393'.cons\u2097 :: trLList (contStack k)) }\n    s'"}, {"tactic": "exact h\u2082", "annotated_tactic": ["exact h\u2082", []], "state_before": "case intro.intro.refine'_2.refine'_6.refine'_2\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 TransGen\n    (fun a b =>\n      (match a with\n        | { l := none, var := var, stk := stk } => none\n        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =\n        some b)\n    { l := some (trNormal fs (Cont'.cons\u2082 (trCont k))), var := none,\n      stk := elim (trList as) [] [] (trList v ++ \u0393'.cons\u2097 :: trLList (contStack k)) }\n    s'", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "fs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 main \u2260 rev \u2227 aux \u2260 rev \u2227 main \u2260 aux", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) rev = []", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "fs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 stack \u2260 rev \u2227 main \u2260 rev \u2227 stack \u2260 main", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.refine'_2.refine'_4\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) rev = []", "state_after": "no goals"}, {"tactic": "exact\n  splitAtPred_eq _ _ _ (some \u0393'.cons\u2097) _\n    (fun x h => Bool.decide_false (trList_ne_cons\u2097 _ _ h)) \u27e8rfl, rfl\u27e9", "annotated_tactic": ["exact\n        <a>splitAtPred_eq</a> _ _ _ (<a>some</a> <a>\u0393'.cons\u2097</a>) _\n          (fun x h => <a>Bool.decide_false</a> (<a>trList_ne_cons\u2097</a> _ _ h)) \u27e8<a>rfl</a>, <a>rfl</a>\u27e9", [{"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"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.PartrecToTM2.\u0393'.cons\u2097", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [868, 5], "def_end_pos": [868, 10]}, {"full_name": "Bool.decide_false", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [149, 9], "def_end_pos": [149, 21]}, {"full_name": "Turing.PartrecToTM2.trList_ne_cons\u2097", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1469, 9], "def_end_pos": [1469, 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": "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.refine'_5\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =\n    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "fs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 aux \u2260 rev \u2227 stack \u2260 rev \u2227 aux \u2260 stack", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro.intro.refine'_2.refine'_6.refine'_1\nfs : Code\nas : List \u2115\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,\n      stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }\n    s'\n\u22a2 update\n      (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack\n        (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))\n      main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) rev =\n    []", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH (ns.headI :: v) none", "annotated_tactic": ["obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH (ns.headI :: v) <a>none</a>", [{"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": "ns : List \u2115\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 ns k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 ns k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 ns k)) }\n        b\u2082", "state_after": "case intro.intro\nns : List \u2115\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.headI ns :: v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := none, stk := elim (trList (List.headI ns :: v)) [] [] (trContStack k) } c\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 ns k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 ns k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 ns k)) }\n        b\u2082"}, {"tactic": "exact \u27e8c, h\u2081, TransGen.head rfl <| head_stack_ok.trans h\u2082\u27e9", "annotated_tactic": ["exact \u27e8c, h\u2081, <a>TransGen.head</a> <a>rfl</a> <| head_stack_ok.trans h\u2082\u27e9", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case intro.intro\nns : List \u2115\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.headI ns :: v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := none, stk := elim (trList (List.headI ns :: v)) [] [] (trContStack k) } c\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.cons\u2082 ns k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.cons\u2082 ns k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 ns k)) }\n        b\u2082", "state_after": "no goals"}, {"tactic": "obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects f k v s", "annotated_tactic": ["obtain \u27e8s', h\u2081, h\u2082\u27e9 := <a>trNormal_respects</a> f k v s", [{"full_name": "Turing.PartrecToTM2.trNormal_respects", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1612, 9], "def_end_pos": [1612, 26]}]], "state_before": "f : Code\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp f k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp f k)) }\n        b\u2082", "state_after": "case intro.intro\nf : Code\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f k v) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr) { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n    s'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp f k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp f k)) }\n        b\u2082"}, {"tactic": "exact \u27e8_, h\u2081, TransGen.head rfl h\u2082\u27e9", "annotated_tactic": ["exact \u27e8_, h\u2081, <a>TransGen.head</a> <a>rfl</a> h\u2082\u27e9", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case intro.intro\nf : Code\nk : Cont\na_ih\u271d :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f k v) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr) { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n    s'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.comp f k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.comp f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.comp f k)) }\n        b\u2082", "state_after": "no goals"}, {"tactic": "rw [stepRet]", "annotated_tactic": ["rw [<a>stepRet</a>]", [{"full_name": "Turing.ToPartrec.stepRet", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [513, 5], "def_end_pos": [513, 12]}]], "state_before": "f : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet (Cont.fix f k) v) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082", "state_after": "f : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 \u2203 b\u2082,\n    TrCfg (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082"}, {"tactic": "by_cases v.headI = 0 <;> simp only [h, ite_true, ite_false] at this \u22a2", "annotated_tactic": ["by_cases v.headI = 0 <;> simp only [h, <a>ite_true</a>, <a>ite_false</a>] at this \u22a2", [{"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": "ite_false", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [78, 17], "def_end_pos": [78, 26]}]], "state_before": "f : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nthis :\n  if List.headI v = 0 then\n    natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\n  else\n    natEnd (Option.iget (List.head? (trList v))) = false \u2227\n      List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\n\u22a2 \u2203 b\u2082,\n    TrCfg (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082", "state_after": "case pos\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet k (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082\n\ncase neg\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082"}, {"tactic": "cases' v with n", "annotated_tactic": ["cases' v with n", []], "state_before": "f : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\n\u22a2 if List.headI v = 0 then\n    natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\n  else\n    natEnd (Option.iget (List.head? (trList v))) = false \u2227\n      List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)", "state_after": "case nil\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\n\u22a2 if List.headI [] = 0 then\n    natEnd (Option.iget (List.head? (trList []))) = true \u2227 List.tail (trList []) = trList (List.tail [])\n  else\n    natEnd (Option.iget (List.head? (trList []))) = false \u2227\n      List.tail (trList []) = List.tail (trNat (List.headI [])) ++ \u0393'.cons :: trList (List.tail [])\n\ncase cons\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\nn : \u2115\ntail\u271d : List \u2115\n\u22a2 if List.headI (n :: tail\u271d) = 0 then\n    natEnd (Option.iget (List.head? (trList (n :: tail\u271d)))) = true \u2227\n      List.tail (trList (n :: tail\u271d)) = trList (List.tail (n :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trList (n :: tail\u271d)))) = false \u2227\n      List.tail (trList (n :: tail\u271d)) =\n        List.tail (trNat (List.headI (n :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (n :: tail\u271d))"}, {"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "case cons\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\nn : \u2115\ntail\u271d : List \u2115\n\u22a2 if List.headI (n :: tail\u271d) = 0 then\n    natEnd (Option.iget (List.head? (trList (n :: tail\u271d)))) = true \u2227\n      List.tail (trList (n :: tail\u271d)) = trList (List.tail (n :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trList (n :: tail\u271d)))) = false \u2227\n      List.tail (trList (n :: tail\u271d)) =\n        List.tail (trNat (List.headI (n :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (n :: tail\u271d))", "state_after": "case cons.zero\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\ntail\u271d : List \u2115\n\u22a2 if List.headI (Nat.zero :: tail\u271d) = 0 then\n    natEnd (Option.iget (List.head? (trList (Nat.zero :: tail\u271d)))) = true \u2227\n      List.tail (trList (Nat.zero :: tail\u271d)) = trList (List.tail (Nat.zero :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trList (Nat.zero :: tail\u271d)))) = false \u2227\n      List.tail (trList (Nat.zero :: tail\u271d)) =\n        List.tail (trNat (List.headI (Nat.zero :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (Nat.zero :: tail\u271d))\n\ncase cons.succ\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\ntail\u271d : List \u2115\nn : \u2115\n\u22a2 if List.headI (Nat.succ n :: tail\u271d) = 0 then\n    natEnd (Option.iget (List.head? (trList (Nat.succ n :: tail\u271d)))) = true \u2227\n      List.tail (trList (Nat.succ n :: tail\u271d)) = trList (List.tail (Nat.succ n :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trList (Nat.succ n :: tail\u271d)))) = false \u2227\n      List.tail (trList (Nat.succ n :: tail\u271d)) =\n        List.tail (trNat (List.headI (Nat.succ n :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (Nat.succ n :: tail\u271d))"}, {"tactic": "rw [trList, List.headI, trNat, Nat.cast_succ, Num.add_one, Num.succ, List.tail]", "annotated_tactic": ["rw [<a>trList</a>, <a>List.headI</a>, <a>trNat</a>, <a>Nat.cast_succ</a>, <a>Num.add_one</a>, <a>Num.succ</a>, <a>List.tail</a>]", [{"full_name": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"full_name": "List.headI", "def_path": "Mathlib/Init/Data/List/Basic.lean", "def_pos": [39, 5], "def_end_pos": [39, 10]}, {"full_name": "Turing.PartrecToTM2.trNat", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1194, 5], "def_end_pos": [1194, 10]}, {"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", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [243, 5], "def_end_pos": [243, 9]}, {"full_name": "List.tail", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [310, 5], "def_end_pos": [310, 9]}]], "state_before": "case cons.succ\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\ntail\u271d : List \u2115\nn : \u2115\n\u22a2 if List.headI (Nat.succ n :: tail\u271d) = 0 then\n    natEnd (Option.iget (List.head? (trList (Nat.succ n :: tail\u271d)))) = true \u2227\n      List.tail (trList (Nat.succ n :: tail\u271d)) = trList (List.tail (Nat.succ n :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trList (Nat.succ n :: tail\u271d)))) = false \u2227\n      List.tail (trList (Nat.succ n :: tail\u271d)) =\n        List.tail (trNat (List.headI (Nat.succ n :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (Nat.succ n :: tail\u271d))", "state_after": "case cons.succ\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\ntail\u271d : List \u2115\nn : \u2115\n\u22a2 if\n      (match Nat.succ n :: tail\u271d with\n        | [] => default\n        | a :: tail => a) =\n        0 then\n    natEnd (Option.iget (List.head? (trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d))) = true \u2227\n      (match trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d with\n        | [] => []\n        | head :: as => as) =\n        trList (List.tail (Nat.succ n :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d))) = false \u2227\n      (match trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d with\n        | [] => []\n        | head :: as => as) =\n        List.tail (trNum (Num.pos (Num.succ' \u2191n))) ++ \u0393'.cons :: trList (List.tail (Nat.succ n :: tail\u271d))"}, {"tactic": "cases (n : Num).succ' <;> exact \u27e8rfl, rfl\u27e9", "annotated_tactic": ["cases (n : <a>Num</a>).<a>succ'</a> <;> exact \u27e8<a>rfl</a>, <a>rfl</a>\u27e9", [{"full_name": "Num", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [43, 11], "def_end_pos": [43, 14]}, {"full_name": "Num.succ'", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [237, 5], "def_end_pos": [237, 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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case cons.succ\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\ntail\u271d : List \u2115\nn : \u2115\n\u22a2 if\n      (match Nat.succ n :: tail\u271d with\n        | [] => default\n        | a :: tail => a) =\n        0 then\n    natEnd (Option.iget (List.head? (trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d))) = true \u2227\n      (match trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d with\n        | [] => []\n        | head :: as => as) =\n        trList (List.tail (Nat.succ n :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d))) = false \u2227\n      (match trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d with\n        | [] => []\n        | head :: as => as) =\n        List.tail (trNum (Num.pos (Num.succ' \u2191n))) ++ \u0393'.cons :: trList (List.tail (Nat.succ n :: tail\u271d))", "state_after": "no goals"}, {"tactic": "exact \u27e8rfl, rfl\u27e9", "annotated_tactic": ["exact \u27e8<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": "case nil\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\n\u22a2 if List.headI [] = 0 then\n    natEnd (Option.iget (List.head? (trList []))) = true \u2227 List.tail (trList []) = trList (List.tail [])\n  else\n    natEnd (Option.iget (List.head? (trList []))) = false \u2227\n      List.tail (trList []) = List.tail (trNat (List.headI [])) ++ \u0393'.cons :: trList (List.tail [])", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons.zero\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\ns : Option \u0393'\ntail\u271d : List \u2115\n\u22a2 if List.headI (Nat.zero :: tail\u271d) = 0 then\n    natEnd (Option.iget (List.head? (trList (Nat.zero :: tail\u271d)))) = true \u2227\n      List.tail (trList (Nat.zero :: tail\u271d)) = trList (List.tail (Nat.zero :: tail\u271d))\n  else\n    natEnd (Option.iget (List.head? (trList (Nat.zero :: tail\u271d)))) = false \u2227\n      List.tail (trList (Nat.zero :: tail\u271d)) =\n        List.tail (trNat (List.headI (Nat.zero :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (Nat.zero :: tail\u271d))", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH v.tail (trList v).head?", "annotated_tactic": ["obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH v.tail (<a>trList</a> v).<a>head?</a>", [{"full_name": "Turing.PartrecToTM2.trList", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1216, 5], "def_end_pos": [1216, 11]}, {"full_name": "List.head?", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/BasicAux.lean", "def_pos": [34, 5], "def_end_pos": [34, 10]}]], "state_before": "case pos\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet k (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082", "state_after": "case pos.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.tail v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n    c\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet k (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082"}, {"tactic": "refine' \u27e8c, h\u2081, TransGen.head rfl _\u27e9", "annotated_tactic": ["refine' \u27e8c, h\u2081, <a>TransGen.head</a> <a>rfl</a> _\u27e9", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case pos.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.tail v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n    c\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepRet k (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082", "state_after": "case pos.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.tail v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n    c\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.fix f k)))) s (elim (trList v) [] [] (trContStack (Cont.fix f k)))) c"}, {"tactic": "simp only [Option.mem_def, TM2.stepAux, trContStack, contStack, elim_main, this, cond_true,\n  elim_update_main]", "annotated_tactic": ["simp only [<a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>trContStack</a>, <a>contStack</a>, <a>elim_main</a>, this, <a>cond_true</a>,\n        <a>elim_update_main</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.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "Turing.PartrecToTM2.trContStack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1249, 5], "def_end_pos": [1249, 16]}, {"full_name": "Turing.PartrecToTM2.contStack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1239, 5], "def_end_pos": [1239, 14]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 21]}, {"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": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 28]}]], "state_before": "case pos.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.tail v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n    c\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.fix f k)))) s (elim (trList v) [] [] (trContStack (Cont.fix f k)))) c", "state_after": "case pos.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.tail v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n    c\n\u22a2 TransGen (fun a b => TM2.step tr a = some b)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trLList (contStack k)) }\n    c"}, {"tactic": "exact h\u2082", "annotated_tactic": ["exact h\u2082", []], "state_before": "case pos.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : List.headI v = 0\nthis : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)\nc : Cfg'\nh\u2081 : TrCfg (stepRet k (List.tail v)) c\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trContStack k) }\n    c\n\u22a2 TransGen (fun a b => TM2.step tr a = some b)\n    { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),\n      stk := elim (trList (List.tail v)) [] [] (trLList (contStack k)) }\n    c", "state_after": "no goals"}, {"tactic": "obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects f (Cont.fix f k) v.tail (some \u0393'.cons)", "annotated_tactic": ["obtain \u27e8s', h\u2081, h\u2082\u27e9 := <a>trNormal_respects</a> f (<a>Cont.fix</a> f k) v.tail (<a>some</a> <a>\u0393'.cons</a>)", [{"full_name": "Turing.PartrecToTM2.trNormal_respects", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1612, 9], "def_end_pos": [1612, 26]}, {"full_name": "Turing.ToPartrec.Cont.fix", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [437, 5], "def_end_pos": [437, 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": "Turing.PartrecToTM2.\u0393'.cons", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [869, 5], "def_end_pos": [869, 9]}]], "state_before": "case neg\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082"}, {"tactic": "refine' \u27e8_, h\u2081, TransGen.head rfl <| TransGen.trans _ h\u2082\u27e9", "annotated_tactic": ["refine' \u27e8_, h\u2081, <a>TransGen.head</a> <a>rfl</a> <| <a>TransGen.trans</a> _ h\u2082\u27e9", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "Relation.TransGen.trans", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [368, 9], "def_end_pos": [368, 14]}]], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 \u2203 b\u2082,\n    TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227\n      Reaches\u2081 (TM2.step tr)\n        { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,\n          stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }\n        b\u2082", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.fix f k)))) s (elim (trList v) [] [] (trContStack (Cont.fix f k))))\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }"}, {"tactic": "simp only [Option.mem_def, TM2.stepAux, elim_main, this.1, cond_false, elim_update_main,\n  trCont]", "annotated_tactic": ["simp only [<a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>elim_main</a>, this.1, <a>cond_false</a>, <a>elim_update_main</a>,\n        <a>trCont</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.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2142, 5], "def_end_pos": [2142, 12]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 21]}, {"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": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1279, 9], "def_end_pos": [1279, 28]}, {"full_name": "Turing.PartrecToTM2.trCont", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1153, 5], "def_end_pos": [1153, 11]}]], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)\n    (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.fix f k)))) s (elim (trList v) [] [] (trContStack (Cont.fix f k))))\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }", "state_after": "case neg.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 TransGen (fun a b => TM2.step tr a = some b)\n    { l := some (\u039b'.clear natEnd main (trNormal f (Cont'.fix f (trCont k)))), var := List.head? (trList v),\n      stk := elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) }\n    { l := some (trNormal f (Cont'.fix f (trCont k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }"}, {"tactic": "convert clear_ok (splitAtPred_eq _ _ (trNat v.headI).tail (some \u0393'.cons) _ _ _) using 2", "annotated_tactic": ["convert <a>clear_ok</a> (<a>splitAtPred_eq</a> _ _ (<a>trNat</a> v.headI).<a>tail</a> (<a>some</a> <a>\u0393'.cons</a>) _ _ _) using 2", [{"full_name": "Turing.PartrecToTM2.clear_ok", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 17]}, {"full_name": "Turing.PartrecToTM2.splitAtPred_eq", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 23]}, {"full_name": "Turing.PartrecToTM2.trNat", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1194, 5], "def_end_pos": [1194, 10]}, {"full_name": "List.tail", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [310, 5], "def_end_pos": [310, 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.PartrecToTM2.\u0393'.cons", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [869, 5], "def_end_pos": [869, 9]}]], "state_before": "case neg.intro.intro\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 TransGen (fun a b => TM2.step tr a = some b)\n    { l := some (\u039b'.clear natEnd main (trNormal f (Cont'.fix f (trCont k)))), var := List.head? (trList v),\n      stk := elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) }\n    { l := some (trNormal f (Cont'.fix f (trCont k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }", "state_after": "case h.e'_2.h.e'_7\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) =\n    update (elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k))) main ?neg.intro.intro.convert_6\u271d\n\ncase neg.intro.intro.convert_6\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 List \u0393'\n\ncase neg.intro.intro.convert_7\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 \u2200 (x : \u0393'), x \u2208 List.tail (trNat (List.headI v)) \u2192 natEnd x = false\n\ncase neg.intro.intro.convert_8\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 Option.elim'\n    (elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) main = List.tail (trNat (List.headI v)) \u2227\n      ?neg.intro.intro.convert_6\u271d = [])\n    (fun a =>\n      natEnd a = true \u2227\n        elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) main =\n          List.tail (trNat (List.headI v)) ++ a :: ?neg.intro.intro.convert_6\u271d)\n    (some \u0393'.cons)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_7\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) =\n    update (elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k))) main ?neg.intro.intro.convert_6\u271d", "state_after": "case h.e'_2.h.e'_7\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) =\n    elim ?neg.intro.intro.convert_6\u271d [] [] (trContStack (Cont.fix f k))"}, {"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": "case h.e'_2.h.e'_7\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) =\n    elim ?neg.intro.intro.convert_6\u271d [] [] (trContStack (Cont.fix f k))", "state_after": "no goals"}, {"tactic": "exact fun x h => trNat_natEnd _ _ (List.tail_subset _ h)", "annotated_tactic": ["exact fun x h => <a>trNat_natEnd</a> _ _ (<a>List.tail_subset</a> _ h)", [{"full_name": "Turing.PartrecToTM2.trNat_natEnd", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1465, 9], "def_end_pos": [1465, 21]}, {"full_name": "List.tail_subset", "def_path": "Mathlib/Data/List/Infix.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}]], "state_before": "case neg.intro.intro.convert_7\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 \u2200 (x : \u0393'), x \u2208 List.tail (trNat (List.headI v)) \u2192 natEnd x = false", "state_after": "no goals"}, {"tactic": "exact \u27e8rfl, this.2\u27e9", "annotated_tactic": ["exact \u27e8<a>rfl</a>, this.2\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.intro.convert_8\nf : Code\nk : Cont\nIH :\n  \u2200 (v : List \u2115) (s : Option \u0393'),\n    \u2203 b\u2082,\n      TrCfg (stepRet k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }\n          b\u2082\nv : List \u2115\ns : Option \u0393'\nh : \u00acList.headI v = 0\nthis :\n  natEnd (Option.iget (List.head? (trList v))) = false \u2227\n    List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v)\ns' : Cfg'\nh\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s'\nh\u2082 :\n  Reaches\u2081 (TM2.step tr)\n    { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,\n      stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }\n    s'\n\u22a2 Option.elim'\n    (elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) main = List.tail (trNat (List.headI v)) \u2227\n      trList (List.tail v) = [])\n    (fun a =>\n      natEnd a = true \u2227\n        elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) main =\n          List.tail (trNat (List.headI v)) ++ a :: trList (List.tail v))\n    (some \u0393'.cons)", "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_Iio", "start": [97, 1], "end": [99, 58], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Iio</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Iio_subset_Iio</a> <a>le_top</a>)]", [{"full_name": "WithTop.preimage_coe_Iio", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [49, 9], "def_end_pos": [49, 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.Iio_subset_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 23]}, {"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\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Iio a = Iio \u2191a", "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.negOfNat_mul_ofNat", "start": [386, 1], "end": [387, 61], "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 n : Nat\n\u22a2 negOfNat m * \u2191n = negOfNat (m * n)", "state_after": "m n : Nat\n\u22a2 \u2191n * negOfNat m = negOfNat (m * n)"}, {"tactic": "simp [ofNat_mul_negOfNat, Nat.mul_comm]", "annotated_tactic": ["simp [<a>ofNat_mul_negOfNat</a>, <a>Nat.mul_comm</a>]", [{"full_name": "Int.ofNat_mul_negOfNat", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [383, 9], "def_end_pos": [383, 27]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "m n : Nat\n\u22a2 \u2191n * negOfNat m = negOfNat (m * n)", "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_right", "start": [265, 1], "end": [267, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.pred_castSucc_succ", "start": [536, 9], "end": [537, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.divInt_sub_divInt", "start": [244, 1], "end": [247, 66], "traced_tactics": [{"tactic": "simp only [Rat.sub_eq_add_neg, neg_divInt,\n  divInt_add_divInt _ _ z\u2081 z\u2082, Int.neg_mul, Int.sub_eq_add_neg]", "annotated_tactic": ["simp only [<a>Rat.sub_eq_add_neg</a>, <a>neg_divInt</a>,\n    <a>divInt_add_divInt</a> _ _ z\u2081 z\u2082, <a>Int.neg_mul</a>, <a>Int.sub_eq_add_neg</a>]", [{"full_name": "Rat.sub_eq_add_neg", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [241, 19], "def_end_pos": [241, 33]}, {"full_name": "Rat.neg_divInt", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [220, 9], "def_end_pos": [220, 19]}, {"full_name": "Rat.divInt_add_divInt", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [204, 9], "def_end_pos": [204, 26]}, {"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.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": "n\u2081 n\u2082 d\u2081 d\u2082 : Int\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\n\u22a2 n\u2081 /. d\u2081 - n\u2082 /. d\u2082 = (n\u2081 * d\u2082 - n\u2082 * d\u2081) /. (d\u2081 * d\u2082)", "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_of_forall_set_integral_eq'", "start": [93, 1], "end": [114, 15], "traced_tactics": [{"tactic": "suffices h_sub : \u21d1(f - g) =\u1d50[\u03bc] 0", "annotated_tactic": ["suffices h_sub : \u21d1(f - g) =\u1d50[\u03bc] 0", []], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g", "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nh_sub : \u2191\u2191(f - g) =\u1d50[\u03bc] 0\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g\n\ncase h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0"}, {"tactic": "have hfg' : \u2200 s : Set \u03b1, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 := by\n  intro s hs h\u03bcs\n  rw [integral_congr_ae (ae_restrict_of_ae (Lp.coeFn_sub f g))]\n  rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]\n  exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", "annotated_tactic": ["have hfg' : \u2200 s : <a>Set</a> \u03b1, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 := by\n    intro s hs h\u03bcs\n    rw [<a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> (<a>Lp.coeFn_sub</a> f g))]\n    rw [<a>integral_sub'</a> (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]\n    exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"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.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}, {"full_name": "MeasureTheory.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0", "state_after": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0"}, {"tactic": "have hfg_int : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 IntegrableOn (\u21d1(f - g)) s \u03bc := by\n  intro s hs h\u03bcs\n  rw [IntegrableOn, integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub f g))]\n  exact (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs)", "annotated_tactic": ["have hfg_int : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 <a>IntegrableOn</a> (\u21d1(f - g)) s \u03bc := by\n    intro s hs h\u03bcs\n    rw [<a>IntegrableOn</a>, <a>integrable_congr</a> (<a>ae_restrict_of_ae</a> (<a>Lp.coeFn_sub</a> f g))]\n    exact (hf_int_finite s hs h\u03bcs).<a>sub</a> (hg_int_finite s hs h\u03bcs)", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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]}, {"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.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "state_before": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0", "state_after": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0"}, {"tactic": "have hfg_meas : AEStronglyMeasurable' m (\u21d1(f - g)) \u03bc :=\n  AEStronglyMeasurable'.congr (hf_meas.sub hg_meas) (Lp.coeFn_sub f g).symm", "annotated_tactic": ["have hfg_meas : <a>AEStronglyMeasurable'</a> m (\u21d1(f - g)) \u03bc :=\n    <a>AEStronglyMeasurable'.congr</a> (hf_meas.sub hg_meas) (<a>Lp.coeFn_sub</a> f g).<a>symm</a>", [{"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'.congr", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [59, 9], "def_end_pos": [59, 14]}, {"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": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0", "state_after": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s\nhfg_meas : AEStronglyMeasurable' m (\u2191\u2191(f - g)) \u03bc\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0"}, {"tactic": "exact\n  Lp.ae_eq_zero_of_forall_set_integral_eq_zero' \ud835\udd5c hm (f - g) hp_ne_zero hp_ne_top hfg_int hfg'\n    hfg_meas", "annotated_tactic": ["exact\n    <a>Lp.ae_eq_zero_of_forall_set_integral_eq_zero'</a> \ud835\udd5c hm (f - g) hp_ne_zero hp_ne_top hfg_int hfg'\n      hfg_meas", [{"full_name": "MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [72, 9], "def_end_pos": [72, 54]}]], "state_before": "case h_sub\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s\nhfg_meas : AEStronglyMeasurable' m (\u2191\u2191(f - g)) \u03bc\n\u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 sub_ae_eq_zero]", "annotated_tactic": ["rw [\u2190 <a>sub_ae_eq_zero</a>]", [{"full_name": "MeasureTheory.sub_ae_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2654, 9], "def_end_pos": [2654, 23]}]], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nh_sub : \u2191\u2191(f - g) =\u1d50[\u03bc] 0\n\u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g", "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nh_sub : \u2191\u2191(f - g) =\u1d50[\u03bc] 0\n\u22a2 \u2191\u2191f - \u2191\u2191g =\u1d50[\u03bc] 0"}, {"tactic": "exact (Lp.coeFn_sub f g).symm.trans h_sub", "annotated_tactic": ["exact (<a>Lp.coeFn_sub</a> f g).symm.trans h_sub", [{"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}]], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nh_sub : \u2191\u2191(f - g) =\u1d50[\u03bc] 0\n\u22a2 \u2191\u2191f - \u2191\u2191g =\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0"}, {"tactic": "rw [integral_congr_ae (ae_restrict_of_ae (Lp.coeFn_sub f g))]", "annotated_tactic": ["rw [<a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> (<a>Lp.coeFn_sub</a> f g))]", [{"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.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}]], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191f - \u2191\u2191g) a \u2202\u03bc = 0"}, {"tactic": "rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]", "annotated_tactic": ["rw [<a>integral_sub'</a> (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]", [{"full_name": "MeasureTheory.integral_sub'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191f - \u2191\u2191g) a \u2202\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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, \u2191\u2191f a \u2202\u03bc - \u222b (a : \u03b1) in s, \u2191\u2191g a \u2202\u03bc = 0"}, {"tactic": "exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", "annotated_tactic": ["exact sub_eq_zero.mpr (hfg s hs h\u03bcs)", []], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, \u2191\u2191f a \u2202\u03bc - \u222b (a : \u03b1) in s, \u2191\u2191g a \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s", "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191(f - g)) s"}, {"tactic": "rw [IntegrableOn, integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub f g))]", "annotated_tactic": ["rw [<a>IntegrableOn</a>, <a>integrable_congr</a> (<a>ae_restrict_of_ae</a> (<a>Lp.coeFn_sub</a> f g))]", [{"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]}, {"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [236, 9], "def_end_pos": [236, 18]}]], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191(f - g)) s", "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable fun x => (\u2191\u2191f - \u2191\u2191g) x"}, {"tactic": "exact (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs)", "annotated_tactic": ["exact (hf_int_finite s hs h\u03bcs).<a>sub</a> (hg_int_finite s hs h\u03bcs)", [{"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "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 g : { 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\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc\nhf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nhg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable fun x => (\u2191\u2191f - \u2191\u2191g) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "full_name": "MvPolynomial.pderiv_X", "start": [91, 1], "end": [93, 34], "traced_tactics": [{"tactic": "rw [pderiv_def, mkDerivation_X]", "annotated_tactic": ["rw [<a>pderiv_def</a>, <a>mkDerivation_X</a>]", [{"full_name": "MvPolynomial.pderiv_def", "def_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "def_pos": [66, 9], "def_end_pos": [66, 19]}, {"full_name": "MvPolynomial.mkDerivation_X", "def_path": "Mathlib/Data/MvPolynomial/Derivation.lean", "def_pos": [140, 9], "def_end_pos": [140, 23]}]], "state_before": "R : Type u\n\u03c3 : Type v\na a' a\u2081 a\u2082 : R\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : DecidableEq \u03c3\ni j : \u03c3\n\u22a2 \u2191(pderiv i) (X j) = Pi.single i 1 j", "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.singleton_def", "start": [35, 9], "end": [35, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Reaches\u2080.head", "start": [792, 1], "end": [794, 31], "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_modifyNth", "start": [881, 1], "end": [885, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndep.indep", "start": [336, 1], "end": [338, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.pairwiseDisjoint_smul_iff", "start": [868, 1], "end": [870, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "full_name": "Std.RBNode.Ordered.map", "start": [497, 11], "end": [501, 83], "traced_tactics": [{"tactic": "refine \u27e8ax.map ?_, xb.map ?_, ha.map f, hb.map f\u27e9 <;> exact IsMonotone.lt_mono", "annotated_tactic": ["refine \u27e8ax.map ?_, xb.map ?_, ha.map f, hb.map f\u27e9 <;> exact <a>IsMonotone.lt_mono</a>", [{"full_name": "Std.RBNode.IsMonotone.lt_mono", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [488, 3], "def_end_pos": [488, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ncmp\u03b1 : \u03b1 \u2192 \u03b1 \u2192 Ordering\ncmp\u03b2 : \u03b2 \u2192 \u03b2 \u2192 Ordering\nf : \u03b1 \u2192 \u03b2\ninst\u271d : IsMonotone cmp\u03b1 cmp\u03b2 f\nc\u271d : RBColor\na : RBNode \u03b1\nx : \u03b1\nb : RBNode \u03b1\nax : All (fun x_1 => cmpLT cmp\u03b1 x_1 x) a\nxb : All (fun x_1 => cmpLT cmp\u03b1 x x_1) b\nha : Ordered cmp\u03b1 a\nhb : Ordered cmp\u03b1 b\n\u22a2 Ordered cmp\u03b2 (map f (node c\u271d a x b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.biSup_measure_Iic", "start": [2739, 1], "end": [2746, 96], "traced_tactics": [{"tactic": "rw [\u2190 measure_biUnion_eq_iSup hsc]", "annotated_tactic": ["rw [\u2190 <a>measure_biUnion_eq_iSup</a> hsc]", [{"full_name": "MeasureTheory.measure_biUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [499, 9], "def_end_pos": [499, 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\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\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u2a06 x \u2208 s, \u2191\u2191\u03bc (Iic x) = \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\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\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u2191\u2191\u03bc (\u22c3 i \u2208 s, Iic i) = \u2191\u2191\u03bc univ\n\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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 DirectedOn ((fun x x_1 => x \u2286 x_1) on fun x => Iic x) s"}, {"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\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\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u2191\u2191\u03bc (\u22c3 i \u2208 s, Iic i) = \u2191\u2191\u03bc univ", "state_after": "case e_a\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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u22c3 i \u2208 s, Iic i = univ"}, {"tactic": "simp only [\u2190 bex_def] at hst", "annotated_tactic": ["simp only [\u2190 <a>bex_def</a>] at hst", [{"full_name": "bex_def", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 16]}]], "state_before": "case e_a\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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u22c3 i \u2208 s, Iic i = univ", "state_after": "case e_a\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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\nhst : \u2200 (x : \u03b1), \u2203 x_1 x_2, x \u2264 x_1\n\u22a2 \u22c3 i \u2208 s, Iic i = univ"}, {"tactic": "exact iUnion\u2082_eq_univ_iff.2 hst", "annotated_tactic": ["exact <a>iUnion\u2082_eq_univ_iff</a>.2 hst", [{"full_name": "Set.iUnion\u2082_eq_univ_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1331, 9], "def_end_pos": [1331, 28]}]], "state_before": "case e_a\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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\nhst : \u2200 (x : \u03b1), \u2203 x_1 x_2, x \u2264 x_1\n\u22a2 \u22c3 i \u2208 s, Iic i = univ", "state_after": "no goals"}, {"tactic": "exact directedOn_iff_directed.2 (hdir.directed_val.mono_comp _ fun x y => Iic_subset_Iic.2)", "annotated_tactic": ["exact <a>directedOn_iff_directed</a>.2 (hdir.directed_val.mono_comp _ fun x y => <a>Iic_subset_Iic</a>.2)", [{"full_name": "directedOn_iff_directed", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [55, 9], "def_end_pos": [55, 32]}, {"full_name": "Set.Iic_subset_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [419, 9], "def_end_pos": [419, 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\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 DirectedOn ((fun x x_1 => x \u2286 x_1) on fun x => Iic x) s", "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_right", "start": [473, 1], "end": [475, 26], "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 1 = natAbs i", "state_after": "i : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs 1) = natAbs i"}, {"tactic": "apply Nat.lcm_one_right", "annotated_tactic": ["apply <a>Nat.lcm_one_right</a>", [{"full_name": "Nat.lcm_one_right", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [205, 17], "def_end_pos": [205, 30]}]], "state_before": "i : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs 1) = natAbs i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "full_name": "MeasureTheory.condexp_stronglyMeasurable_mul_of_bound", "start": [259, 1], "end": [302, 43], "traced_tactics": [{"tactic": "let fs := hf.approxBounded c", "annotated_tactic": ["let fs := hf.approxBounded c", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "have hfs_tendsto : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) :=\n  hf.tendsto_approxBounded_ae hf_bound", "annotated_tactic": ["have hfs_tendsto : \u2200\u1d50 x \u2202\u03bc, <a>Tendsto</a> (fun n => fs n x) <a>atTop</a> (\ud835\udcdd (f x)) :=\n    hf.tendsto_approxBounded_ae hf_bound", [{"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": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "by_cases h\u03bc : \u03bc = 0", "annotated_tactic": ["by_cases h\u03bc : \u03bc = 0", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u03bc = 0\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "have : \u03bc.ae.NeBot := by simp only [h\u03bc, ae_neBot, Ne.def, not_false_iff]", "annotated_tactic": ["have : \u03bc.ae.NeBot := by simp only [h\u03bc, <a>ae_neBot</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "MeasureTheory.ae_neBot", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2444, 9], "def_end_pos": [2444, 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]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "have hc : 0 \u2264 c :=\n  haveI h_exists : \u2203 x, \u2016f x\u2016 \u2264 c := Eventually.exists hf_bound\n  (norm_nonneg _).trans h_exists.choose_spec", "annotated_tactic": ["have hc : 0 \u2264 c :=\n    haveI h_exists : \u2203 x, \u2016f x\u2016 \u2264 c := <a>Eventually.exists</a> hf_bound\n    (<a>norm_nonneg</a> _).<a>trans</a> h_exists.choose_spec", [{"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}, {"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]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "have hfs_bound : \u2200 n x, \u2016fs n x\u2016 \u2264 c := hf.norm_approxBounded_le hc", "annotated_tactic": ["have hfs_bound : \u2200 n x, \u2016fs n x\u2016 \u2264 c := hf.norm_approxBounded_le hc", []], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "have : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] := by\n  refine' condexp_of_stronglyMeasurable hm (hf.mul stronglyMeasurable_condexp) _\n  exact integrable_condexp.bdd_mul' (hf.mono hm).aestronglyMeasurable hf_bound", "annotated_tactic": ["have : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] := by\n    refine' <a>condexp_of_stronglyMeasurable</a> hm (hf.mul <a>stronglyMeasurable_condexp</a>) _\n    exact integrable_condexp.bdd_mul' (hf.mono hm).<a>aestronglyMeasurable</a> hf_bound", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}, {"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.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [110, 19], "def_end_pos": [110, 58]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] \u03bc[f * \u03bc[g|m]|m]"}, {"tactic": "refine' tendsto_condexp_unique (fun n x => fs n x * g x) (fun n x => fs n x * (\u03bc[g|m]) x) (f * g)\n  (f * \u03bc[g|m]) _ _ _ _ (fun x => c * \u2016g x\u2016) _ (fun x => c * \u2016(\u03bc[g|m]) x\u2016) _ _ _ _", "annotated_tactic": ["refine' <a>tendsto_condexp_unique</a> (fun n x => fs n x * g x) (fun n x => fs n x * (\u03bc[g|m]) x) (f * g)\n    (f * \u03bc[g|m]) _ _ _ _ (fun x => c * \u2016g x\u2016) _ (fun x => c * \u2016(\u03bc[g|m]) x\u2016) _ _ _ _", [{"full_name": "MeasureTheory.tendsto_condexp_unique", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] \u03bc[f * \u03bc[g|m]|m]", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), Integrable ((fun n x => \u2191(fs n) x * g x) n)\n\ncase neg.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), Integrable ((fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n)\n\ncase neg.refine'_3\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => (fun n x => \u2191(fs n) x * g x) n x) atTop (\ud835\udcdd ((f * g) x))\n\ncase neg.refine'_4\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => (fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x) atTop (\ud835\udcdd ((f * \u03bc[g|m]) x))\n\ncase neg.refine'_5\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 Integrable fun x => c * \u2016g x\u2016\n\ncase neg.refine'_6\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 Integrable fun x => c * \u2016(\u03bc[g|m]) x\u2016\n\ncase neg.refine'_7\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016(fun n x => \u2191(fs n) x * g x) n x\u2016 \u2264 (fun x => c * \u2016g x\u2016) x\n\ncase neg.refine'_8\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x\u2016 \u2264 (fun x => c * \u2016(\u03bc[g|m]) x\u2016) x\n\ncase neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), \u03bc[(fun n x => \u2191(fs n) x * g x) n|m] =\u1d50[\u03bc] \u03bc[(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n|m]"}, {"tactic": "simp only [h\u03bc, ae_zero]", "annotated_tactic": ["simp only [h\u03bc, <a>ae_zero</a>]", [{"full_name": "MeasureTheory.ae_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2451, 9], "def_end_pos": [2451, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u03bc = 0\n\u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u03bc = 0\n\u22a2 0[f * g|m] =\u1da0[\u22a5] f * 0[g|m]"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u03bc = 0\n\u22a2 0[f * g|m] =\u1da0[\u22a5] f * 0[g|m]", "state_after": "no goals"}, {"tactic": "simp only [h\u03bc, ae_neBot, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [h\u03bc, <a>ae_neBot</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "MeasureTheory.ae_neBot", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2444, 9], "def_end_pos": [2444, 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]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 NeBot (Measure.ae \u03bc)", "state_after": "no goals"}, {"tactic": "refine' condexp_of_stronglyMeasurable hm (hf.mul stronglyMeasurable_condexp) _", "annotated_tactic": ["refine' <a>condexp_of_stronglyMeasurable</a> hm (hf.mul <a>stronglyMeasurable_condexp</a>) _", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}, {"full_name": "MeasureTheory.stronglyMeasurable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 35]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\n\u22a2 \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\n\u22a2 Integrable (f * \u03bc[g|m])"}, {"tactic": "exact integrable_condexp.bdd_mul' (hf.mono hm).aestronglyMeasurable hf_bound", "annotated_tactic": ["exact integrable_condexp.bdd_mul' (hf.mono hm).<a>aestronglyMeasurable</a> hf_bound", [{"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\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\n\u22a2 Integrable (f * \u03bc[g|m])", "state_after": "no goals"}, {"tactic": "exact fun n => hg.bdd_mul' ((SimpleFunc.stronglyMeasurable (fs n)).mono hm).aestronglyMeasurable\n  (eventually_of_forall (hfs_bound n))", "annotated_tactic": ["exact fun n => hg.bdd_mul' ((<a>SimpleFunc.stronglyMeasurable</a> (fs n)).<a>mono</a> hm).<a>aestronglyMeasurable</a>\n      (<a>eventually_of_forall</a> (hfs_bound n))", [{"full_name": "MeasureTheory.SimpleFunc.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 38]}, {"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": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), Integrable ((fun n x => \u2191(fs n) x * g x) n)", "state_after": "no goals"}, {"tactic": "exact fun n => integrable_condexp.bdd_mul'\n  ((SimpleFunc.stronglyMeasurable (fs n)).mono hm).aestronglyMeasurable\n  (eventually_of_forall (hfs_bound n))", "annotated_tactic": ["exact fun n => integrable_condexp.bdd_mul'\n      ((<a>SimpleFunc.stronglyMeasurable</a> (fs n)).<a>mono</a> hm).<a>aestronglyMeasurable</a>\n      (<a>eventually_of_forall</a> (hfs_bound n))", [{"full_name": "MeasureTheory.SimpleFunc.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 38]}, {"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": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), Integrable ((fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n)", "state_after": "no goals"}, {"tactic": "filter_upwards [hfs_tendsto] with x hx", "annotated_tactic": ["filter_upwards [hfs_tendsto] with x hx", []], "state_before": "case neg.refine'_3\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => (fun n x => \u2191(fs n) x * g x) n x) atTop (\ud835\udcdd ((f * g) x))", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * g x) atTop (\ud835\udcdd ((f * g) x))"}, {"tactic": "rw [Pi.mul_apply]", "annotated_tactic": ["rw [<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\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * g x) atTop (\ud835\udcdd ((f * g) x))", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * g x) atTop (\ud835\udcdd (f x * g x))"}, {"tactic": "exact Tendsto.mul hx tendsto_const_nhds", "annotated_tactic": ["exact <a>Tendsto.mul</a> hx <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": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * g x) atTop (\ud835\udcdd (f x * g x))", "state_after": "no goals"}, {"tactic": "filter_upwards [hfs_tendsto] with x hx", "annotated_tactic": ["filter_upwards [hfs_tendsto] with x hx", []], "state_before": "case neg.refine'_4\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => (fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x) atTop (\ud835\udcdd ((f * \u03bc[g|m]) x))", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * (\u03bc[g|m]) x) atTop (\ud835\udcdd ((f * \u03bc[g|m]) x))"}, {"tactic": "rw [Pi.mul_apply]", "annotated_tactic": ["rw [<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\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * (\u03bc[g|m]) x) atTop (\ud835\udcdd ((f * \u03bc[g|m]) x))", "state_after": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * (\u03bc[g|m]) x) atTop (\ud835\udcdd (f x * (\u03bc[g|m]) x))"}, {"tactic": "exact Tendsto.mul hx tendsto_const_nhds", "annotated_tactic": ["exact <a>Tendsto.mul</a> hx <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": "case h\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nx : \u03b1\nhx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun n => \u2191(fs n) x * (\u03bc[g|m]) x) atTop (\ud835\udcdd (f x * (\u03bc[g|m]) x))", "state_after": "no goals"}, {"tactic": "exact hg.norm.const_mul c", "annotated_tactic": ["exact hg.norm.const_mul c", []], "state_before": "case neg.refine'_5\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 Integrable fun x => c * \u2016g x\u2016", "state_after": "no goals"}, {"tactic": "exact integrable_condexp.norm.const_mul c", "annotated_tactic": ["exact integrable_condexp.norm.const_mul c", []], "state_before": "case neg.refine'_6\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 Integrable fun x => c * \u2016(\u03bc[g|m]) x\u2016", "state_after": "no goals"}, {"tactic": "refine' fun n => eventually_of_forall fun x => _", "annotated_tactic": ["refine' fun n => <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 neg.refine'_7\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016(fun n x => \u2191(fs n) x * g x) n x\u2016 \u2264 (fun x => c * \u2016g x\u2016) x", "state_after": "case neg.refine'_7\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\nx : \u03b1\n\u22a2 \u2016(fun n x => \u2191(fs n) x * g x) n x\u2016 \u2264 (fun x => c * \u2016g x\u2016) x"}, {"tactic": "exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _))", "annotated_tactic": ["exact (<a>norm_mul_le</a> _ _).<a>trans</a> (<a>mul_le_mul_of_nonneg_right</a> (hfs_bound n x) (<a>norm_nonneg</a> _))", [{"full_name": "norm_mul_le", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"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_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case neg.refine'_7\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\nx : \u03b1\n\u22a2 \u2016(fun n x => \u2191(fs n) x * g x) n x\u2016 \u2264 (fun x => c * \u2016g x\u2016) x", "state_after": "no goals"}, {"tactic": "refine' fun n => eventually_of_forall fun x => _", "annotated_tactic": ["refine' fun n => <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 neg.refine'_8\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x\u2016 \u2264 (fun x => c * \u2016(\u03bc[g|m]) x\u2016) x", "state_after": "case neg.refine'_8\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\nx : \u03b1\n\u22a2 \u2016(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x\u2016 \u2264 (fun x => c * \u2016(\u03bc[g|m]) x\u2016) x"}, {"tactic": "exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _))", "annotated_tactic": ["exact (<a>norm_mul_le</a> _ _).<a>trans</a> (<a>mul_le_mul_of_nonneg_right</a> (hfs_bound n x) (<a>norm_nonneg</a> _))", [{"full_name": "norm_mul_le", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"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_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case neg.refine'_8\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\nx : \u03b1\n\u22a2 \u2016(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x\u2016 \u2264 (fun x => c * \u2016(\u03bc[g|m]) x\u2016) x", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\n\u22a2 \u2200 (n : \u2115), \u03bc[(fun n x => \u2191(fs n) x * g x) n|m] =\u1d50[\u03bc] \u03bc[(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n|m]", "state_after": "case neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 \u03bc[(fun n x => \u2191(fs n) x * g x) n|m] =\u1d50[\u03bc] \u03bc[(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n|m]"}, {"tactic": "simp_rw [\u2190 Pi.mul_apply]", "annotated_tactic": ["simp_rw [\u2190 <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 neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 \u03bc[(fun n x => \u2191(fs n) x * g x) n|m] =\u1d50[\u03bc] \u03bc[(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n|m]", "state_after": "case neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * g) x|m] =\u1d50[\u03bc]\n    \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) x|m]"}, {"tactic": "refine' (condexp_stronglyMeasurable_simpleFunc_mul hm _ hg).trans _", "annotated_tactic": ["refine' (<a>condexp_stronglyMeasurable_simpleFunc_mul</a> hm _ hg).<a>trans</a> _", [{"full_name": "MeasureTheory.condexp_stronglyMeasurable_simpleFunc_mul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [231, 9], "def_end_pos": [231, 50]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}]], "state_before": "case neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * g) x|m] =\u1d50[\u03bc]\n    \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) x|m]", "state_after": "case neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 \u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m] =\u1d50[\u03bc]\n    \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) x|m]"}, {"tactic": "rw [condexp_of_stronglyMeasurable hm\n  ((SimpleFunc.stronglyMeasurable _).mul stronglyMeasurable_condexp) _]", "annotated_tactic": ["rw [<a>condexp_of_stronglyMeasurable</a> hm\n      ((<a>SimpleFunc.stronglyMeasurable</a> _).<a>mul</a> <a>stronglyMeasurable_condexp</a>) _]", [{"full_name": "MeasureTheory.condexp_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 38]}, {"full_name": "MeasureTheory.SimpleFunc.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 38]}, {"full_name": "MeasureTheory.StronglyMeasurable.mul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [409, 19], "def_end_pos": [409, 22]}, {"full_name": "MeasureTheory.stronglyMeasurable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 35]}]], "state_before": "case neg.refine'_9\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 \u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m] =\u1d50[\u03bc]\n    \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) x|m]", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 Integrable (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m])"}, {"tactic": "exact integrable_condexp.bdd_mul'\n  ((SimpleFunc.stronglyMeasurable (fs n)).mono hm).aestronglyMeasurable\n  (eventually_of_forall (hfs_bound n))", "annotated_tactic": ["exact integrable_condexp.bdd_mul'\n      ((<a>SimpleFunc.stronglyMeasurable</a> (fs n)).<a>mono</a> hm).<a>aestronglyMeasurable</a>\n      (<a>eventually_of_forall</a> (hfs_bound n))", [{"full_name": "MeasureTheory.SimpleFunc.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 38]}, {"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": "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\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf g : \u03b1 \u2192 \u211d\nhf : StronglyMeasurable f\nhg : Integrable g\nc : \u211d\nhf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c\nfs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c\nhfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))\nh\u03bc : \u00ac\u03bc = 0\nthis\u271d : NeBot (Measure.ae \u03bc)\nhc : 0 \u2264 c\nhfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c\nthis : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m]\nn : \u2115\n\u22a2 Integrable (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m])", "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_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le", "start": [440, 1], "end": [479, 11], "traced_tactics": [{"tactic": "refine' integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ ![f, g]\n  ![f', g'] s hs a b hle _ (fun x hx => _) _ _ Hi", "annotated_tactic": ["refine' <a>integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv</a> e _ _ ![f, g]\n        ![f', g'] s hs a b hle _ (fun x hx => _) _ _ Hi", [{"full_name": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv", "def_path": "Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean", "def_pos": [319, 9], "def_end_pos": [319, 72]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc a b, \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    \u2211 i : Fin 2,\n      ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n        \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x)))", "state_after": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (x y : \u211d \u00d7 \u211d), \u2191e x \u2264 \u2191e y \u2194 x \u2264 y\n\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 MeasurePreserving \u2191e\n\ncase refine'_3\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (i : Fin (1 + 1)), ContinuousOn (Matrix.vecCons f ![g] i) (Set.Icc a b)\n\ncase refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\nhx : x \u2208 interior (Set.Icc a b) \\ s\n\u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x\n\ncase refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (x : \u211d \u00d7 \u211d),\n    \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n      \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))"}, {"tactic": "exact fun x y => (OrderIso.finTwoArrowIso \u211d).symm.le_iff_le", "annotated_tactic": ["exact fun x y => (<a>OrderIso.finTwoArrowIso</a> \u211d).symm.le_iff_le", [{"full_name": "OrderIso.finTwoArrowIso", "def_path": "Mathlib/Logic/Equiv/Fin.lean", "def_pos": [99, 5], "def_end_pos": [99, 28]}]], "state_before": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (x y : \u211d \u00d7 \u211d), \u2191e x \u2264 \u2191e y \u2194 x \u2264 y", "state_after": "no goals"}, {"tactic": "exact (volume_preserving_finTwoArrow \u211d).symm _", "annotated_tactic": ["exact (<a>volume_preserving_finTwoArrow</a> \u211d).<a>symm</a> _", [{"full_name": "MeasureTheory.volume_preserving_finTwoArrow", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [859, 9], "def_end_pos": [859, 38]}, {"full_name": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}]], "state_before": "case refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 MeasurePreserving \u2191e", "state_after": "no goals"}, {"tactic": "exact Fin.forall_fin_two.2 \u27e8Hcf, Hcg\u27e9", "annotated_tactic": ["exact <a>Fin.forall_fin_two</a>.2 \u27e8Hcf, Hcg\u27e9", [{"full_name": "Fin.forall_fin_two", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [640, 9], "def_end_pos": [640, 23]}]], "state_before": "case refine'_3\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (i : Fin (1 + 1)), ContinuousOn (Matrix.vecCons f ![g] i) (Set.Icc a b)", "state_after": "no goals"}, {"tactic": "rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx", "annotated_tactic": ["rw [<a>Icc_prod_eq</a>, <a>interior_prod_eq</a>, <a>interior_Icc</a>, <a>interior_Icc</a>] at hx", [{"full_name": "Set.Icc_prod_eq", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1927, 9], "def_end_pos": [1927, 20]}, {"full_name": "interior_prod_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [747, 9], "def_end_pos": [747, 25]}, {"full_name": "interior_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2371, 9], "def_end_pos": [2371, 21]}, {"full_name": "interior_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2371, 9], "def_end_pos": [2371, 21]}]], "state_before": "case refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\nhx : x \u2208 interior (Set.Icc a b) \\ s\n\u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x", "state_after": "case refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\nhx : x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s\n\u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x"}, {"tactic": "exact Fin.forall_fin_two.2 \u27e8Hdf x hx, Hdg x hx\u27e9", "annotated_tactic": ["exact <a>Fin.forall_fin_two</a>.2 \u27e8Hdf x hx, Hdg x hx\u27e9", [{"full_name": "Fin.forall_fin_two", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [640, 9], "def_end_pos": [640, 23]}]], "state_before": "case refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\nhx : x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s\n\u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (x : \u211d \u00d7 \u211d),\n    \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n      \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))", "state_after": "case refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\n\u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))"}, {"tactic": "rw [Fin.sum_univ_two]", "annotated_tactic": ["rw [<a>Fin.sum_univ_two</a>]", [{"full_name": "Fin.sum_univ_two", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [109, 3], "def_end_pos": [109, 14]}]], "state_before": "case refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\n\u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))", "state_after": "case refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\n\u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    \u2191(Matrix.vecCons f' ![g'] 0 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 0 1)) +\n      \u2191(Matrix.vecCons f' ![g'] 1 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 1 1))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\n\u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n    \u2191(Matrix.vecCons f' ![g'] 0 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 0 1)) +\n      \u2191(Matrix.vecCons f' ![g'] 1 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 1 1))", "state_after": "no goals"}, {"tactic": "have : \u2200 (a b : \u211d\u00b9) (f : \u211d\u00b9 \u2192 E),\n    \u222b x in Icc a b, f x = \u222b x in Icc (a 0) (b 0), f fun _ => x := fun a b f \u21a6 by\n  convert (((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb\n    (MeasurableEquiv.measurableEmbedding _) f _).symm\n  exact ((OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc a b).symm", "annotated_tactic": ["have : \u2200 (a b : \u211d\u00b9) (f : \u211d\u00b9 \u2192 E),\n          \u222b x in <a>Icc</a> a b, f x = \u222b x in <a>Icc</a> (a 0) (b 0), f fun _ => x := fun a b f \u21a6 by\n        convert (((<a>volume_preserving_funUnique</a> (<a>Fin</a> 1) \u211d).<a>symm</a> _).<a>set_integral_preimage_emb</a>\n          (<a>MeasurableEquiv.measurableEmbedding</a> _) f _).<a>symm</a>\n        exact ((<a>OrderIso.funUnique</a> (<a>Fin</a> 1) \u211d).symm.preimage_Icc a b).<a>symm</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.volume_preserving_funUnique", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [825, 9], "def_end_pos": [825, 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": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}, {"full_name": "MeasureTheory.MeasurePreserving.set_integral_preimage_emb", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [531, 9], "def_end_pos": [531, 52]}, {"full_name": "MeasurableEquiv.measurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1451, 19], "def_end_pos": [1451, 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": "OrderIso.funUnique", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [1097, 5], "def_end_pos": [1097, 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": "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\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2211 i : Fin 2,\n      ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n        \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =\n    ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nthis :\n  \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),\n    \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n\u22a2 \u2211 i : Fin 2,\n      ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n        \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =\n    ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2))"}, {"tactic": "simp only [Fin.sum_univ_two, this]", "annotated_tactic": ["simp only [<a>Fin.sum_univ_two</a>, this]", [{"full_name": "Fin.sum_univ_two", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [109, 3], "def_end_pos": [109, 14]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nthis :\n  \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),\n    \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n\u22a2 \u2211 i : Fin 2,\n      ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n        \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n          Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =\n    ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2))", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nthis :\n  \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),\n    \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n\u22a2 ((\u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),\n          Matrix.vecCons f ![g] 0\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 0) fun x_1 =>\n                x))) -\n        \u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),\n          Matrix.vecCons f ![g] 0\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 0) fun x_1 =>\n                x))) +\n      ((\u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),\n          Matrix.vecCons f ![g] 1\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 1) fun x_1 =>\n                x))) -\n        \u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),\n          Matrix.vecCons f ![g] 1\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 1) fun x_1 =>\n                x))) =\n    ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nthis :\n  \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),\n    \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n\u22a2 ((\u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),\n          Matrix.vecCons f ![g] 0\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 0) fun x_1 =>\n                x))) -\n        \u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),\n          Matrix.vecCons f ![g] 0\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 0) fun x_1 =>\n                x))) +\n      ((\u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),\n          Matrix.vecCons f ![g] 1\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 1) fun x_1 =>\n                x))) -\n        \u222b (x : \u211d) in\n          Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)\n            ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),\n          Matrix.vecCons f ![g] 1\n            (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n              (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 1) fun x_1 =>\n                x))) =\n    ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2))", "state_after": "no goals"}, {"tactic": "convert (((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb\n  (MeasurableEquiv.measurableEmbedding _) f _).symm", "annotated_tactic": ["convert (((<a>volume_preserving_funUnique</a> (<a>Fin</a> 1) \u211d).<a>symm</a> _).<a>set_integral_preimage_emb</a>\n          (<a>MeasurableEquiv.measurableEmbedding</a> _) f _).<a>symm</a>", [{"full_name": "MeasureTheory.volume_preserving_funUnique", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [825, 9], "def_end_pos": [825, 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": "MeasureTheory.MeasurePreserving.symm", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [70, 9], "def_end_pos": [70, 13]}, {"full_name": "MeasureTheory.MeasurePreserving.set_integral_preimage_emb", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [531, 9], "def_end_pos": [531, 52]}, {"full_name": "MeasurableEquiv.measurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1451, 19], "def_end_pos": [1451, 38]}, {"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\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u271d b\u271d : \u211d \u00d7 \u211d\nhle : a\u271d \u2264 b\u271d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d)\nHcg : ContinuousOn g (Set.Icc a\u271d b\u271d)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt f\u271d (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\na b : Fin 1 \u2192 \u211d\nf : (Fin 1 \u2192 \u211d) \u2192 E\n\u22a2 \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x", "state_after": "case h.e'_3.h.e'_6.h.e'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u271d b\u271d : \u211d \u00d7 \u211d\nhle : a\u271d \u2264 b\u271d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d)\nHcg : ContinuousOn g (Set.Icc a\u271d b\u271d)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt f\u271d (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\na b : Fin 1 \u2192 \u211d\nf : (Fin 1 \u2192 \u211d) \u2192 E\n\u22a2 Set.Icc (a 0) (b 0) = \u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Set.Icc a b"}, {"tactic": "exact ((OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc a b).symm", "annotated_tactic": ["exact ((<a>OrderIso.funUnique</a> (<a>Fin</a> 1) \u211d).symm.preimage_Icc a b).<a>symm</a>", [{"full_name": "OrderIso.funUnique", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [1097, 5], "def_end_pos": [1097, 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": "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.h.e'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u271d b\u271d : \u211d \u00d7 \u211d\nhle : a\u271d \u2264 b\u271d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d)\nHcg : ContinuousOn g (Set.Icc a\u271d b\u271d)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt f\u271d (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\na b : Fin 1 \u2192 \u211d\nf : (Fin 1 \u2192 \u211d) \u2192 E\n\u22a2 Set.Icc (a 0) (b 0) = \u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Set.Icc a b", "state_after": "no goals"}, {"tactic": "simp only [intervalIntegral.integral_of_le hle.1, intervalIntegral.integral_of_le hle.2,\n  set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]", "annotated_tactic": ["simp only [<a>intervalIntegral.integral_of_le</a> hle.1, <a>intervalIntegral.integral_of_le</a> hle.2,\n        <a>set_integral_congr_set_ae</a> (<a>Ioc_ae_eq_Icc</a> (\u03b1 := \u211d) (\u03bc := <a>volume</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": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"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.Ioc_ae_eq_Icc", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3172, 9], "def_end_pos": [3172, 22]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) =\n    (((\u222b (x : \u211d) in a.1 ..b.1, g (x, b.2)) - \u222b (x : \u211d) in a.1 ..b.1, g (x, a.2)) + \u222b (y : \u211d) in a.2 ..b.2, f (b.1, y)) -\n      \u222b (y : \u211d) in a.2 ..b.2, f (a.1, y)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) =\n    (((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) +\n        \u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) -\n      \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +\n      ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) =\n    (((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) +\n        \u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) -\n      \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.unifIntegrable_fin", "start": [423, 1], "end": [446, 100], "traced_tactics": [{"tactic": "revert f", "annotated_tactic": ["revert f", []], "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nf : Fin n \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : Fin n), Mem\u2112p (f i) p\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\n\u22a2 \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc"}, {"tactic": "induction' n with n h", "annotated_tactic": ["induction' n with n 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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\n\u22a2 \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc", "state_after": "case zero\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\u22a2 \u2200 {f : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\n\ncase succ\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\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\n\u22a2 \u2200 {f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc"}, {"tactic": "intro f hfLp \u03b5 h\u03b5", "annotated_tactic": ["intro f hfLp \u03b5 h\u03b5", []], "state_before": "case succ\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\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\n\u22a2 \u2200 {f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc", "state_after": "case succ\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "let g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f k", "annotated_tactic": ["let g : <a>Fin</a> n \u2192 \u03b1 \u2192 \u03b2 := fun k => f k", [{"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 succ\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 succ\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "have hgLp : \u2200 i, Mem\u2112p (g i) p \u03bc := fun i => hfLp i", "annotated_tactic": ["have hgLp : \u2200 i, <a>Mem\u2112p</a> (g i) p \u03bc := fun i => hfLp i", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}]], "state_before": "case succ\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 succ\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 \u27e8\u03b4\u2081, h\u03b4\u2081pos, h\u03b4\u2081\u27e9 := h hgLp h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4\u2081, h\u03b4\u2081pos, h\u03b4\u2081\u27e9 := h hgLp h\u03b5", []], "state_before": "case succ\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 succ.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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 \u27e8\u03b4\u2082, h\u03b4\u2082pos, h\u03b4\u2082\u27e9 := (hfLp n).snorm_indicator_le \u03bc hp_one hp_top h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4\u2082, h\u03b4\u2082pos, h\u03b4\u2082\u27e9 := (hfLp n).<a>snorm_indicator_le</a> \u03bc hp_one hp_top h\u03b5", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [388, 9], "def_end_pos": [388, 33]}]], "state_before": "case succ.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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 succ.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\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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' \u27e8min \u03b4\u2081 \u03b4\u2082, lt_min h\u03b4\u2081pos h\u03b4\u2082pos, fun i s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>min</a> \u03b4\u2081 \u03b4\u2082, <a>lt_min</a> h\u03b4\u2081pos h\u03b4\u2082pos, fun i s hs h\u03bcs => _\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]}]], "state_before": "case succ.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\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 x,\n    \u2200 (i : Fin (Nat.succ n)) (s : Set \u03b1),\n      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 succ.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\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "by_cases hi : i.val < n", "annotated_tactic": ["by_cases hi : i.val < n", []], "state_before": "case succ.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\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u2191i < n\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "intro f hf", "annotated_tactic": ["intro f hf", []], "state_before": "case zero\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\u22a2 \u2200 {f : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc", "state_after": "case zero\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p\n\u22a2 UnifIntegrable f p \u03bc"}, {"tactic": "have : Subsingleton (Fin Nat.zero) := subsingleton_fin_zero", "annotated_tactic": ["have : <a>Subsingleton</a> (<a>Fin</a> <a>Nat.zero</a>) := <a>subsingleton_fin_zero</a>", [{"full_name": "Subsingleton", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [867, 7], "def_end_pos": [867, 19]}, {"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.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "subsingleton_fin_zero", "def_path": "Mathlib/Logic/Equiv/Fin.lean", "def_pos": [541, 10], "def_end_pos": [541, 31]}]], "state_before": "case zero\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "case zero\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p\nthis : Subsingleton (Fin Nat.zero)\n\u22a2 UnifIntegrable f p \u03bc"}, {"tactic": "exact unifIntegrable_subsingleton \u03bc hp_one hp_top hf", "annotated_tactic": ["exact <a>unifIntegrable_subsingleton</a> \u03bc hp_one hp_top hf", [{"full_name": "MeasureTheory.unifIntegrable_subsingleton", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [410, 9], "def_end_pos": [410, 36]}]], "state_before": "case zero\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p\nthis : Subsingleton (Fin Nat.zero)\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "no goals"}, {"tactic": "rw [(_ : f i = g \u27e8i.val, hi\u27e9)]", "annotated_tactic": ["rw [(_ : f i = g \u27e8i.val, hi\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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u2191i < n\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u2191i < n\n\u22a2 snorm (indicator s (g { val := \u2191i, isLt := hi })) p \u03bc \u2264 ENNReal.ofReal \u03b5\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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u2191i < n\n\u22a2 f i = g { val := \u2191i, isLt := hi }"}, {"tactic": "exact h\u03b4\u2081 _ s hs (le_trans h\u03bcs <| ENNReal.ofReal_le_ofReal <| min_le_left _ _)", "annotated_tactic": ["exact h\u03b4\u2081 _ s hs (<a>le_trans</a> h\u03bcs <| <a>ENNReal.ofReal_le_ofReal</a> <| <a>min_le_left</a> _ _)", [{"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", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u2191i < n\n\u22a2 snorm (indicator s (g { val := \u2191i, isLt := hi })) p \u03bc \u2264 ENNReal.ofReal \u03b5", "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\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u2191i < n\n\u22a2 f i = g { val := \u2191i, isLt := hi }", "state_after": "no goals"}, {"tactic": "rw [(_ : i = n)]", "annotated_tactic": ["rw [(_ : i = n)]", []], "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\n\u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\n\u22a2 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\n\u22a2 i = \u2191n"}, {"tactic": "exact h\u03b4\u2082 _ hs (le_trans h\u03bcs <| ENNReal.ofReal_le_ofReal <| min_le_right _ _)", "annotated_tactic": ["exact h\u03b4\u2082 _ hs (<a>le_trans</a> h\u03bcs <| <a>ENNReal.ofReal_le_ofReal</a> <| <a>min_le_right</a> _ _)", [{"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", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2135, 9], "def_end_pos": [2135, 25]}, {"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\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\n\u22a2 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "have hi' := Fin.is_lt i", "annotated_tactic": ["have hi' := <a>Fin.is_lt</a> i", [{"full_name": "Fin.is_lt", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [25, 17], "def_end_pos": [25, 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 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\n\u22a2 i = \u2191n", "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\nhi' : \u2191i < Nat.succ n\n\u22a2 i = \u2191n"}, {"tactic": "rw [Nat.lt_succ_iff] at hi'", "annotated_tactic": ["rw [<a>Nat.lt_succ_iff</a>] at hi'", [{"full_name": "Nat.lt_succ_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [207, 9], "def_end_pos": [207, 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\u271d : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\nhi' : \u2191i < Nat.succ n\n\u22a2 i = \u2191n", "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\nhi' : \u2191i \u2264 n\n\u22a2 i = \u2191n"}, {"tactic": "rw [not_lt] at hi", "annotated_tactic": ["rw [<a>not_lt</a>] at hi", [{"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\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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : \u00ac\u2191i < n\nhi' : \u2191i \u2264 n\n\u22a2 i = \u2191n", "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : n \u2264 \u2191i\nhi' : \u2191i \u2264 n\n\u22a2 i = \u2191n"}, {"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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : n \u2264 \u2191i\nhi' : \u2191i \u2264 n\n\u22a2 i = \u2191n", "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : n \u2264 \u2191i\nhi' : \u2191i \u2264 n\n\u22a2 \u2191i = \u2191\u2191n"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : n \u2264 \u2191i\nhi' : \u2191i \u2264 n\n\u22a2 \u2191i = \u2191\u2191n", "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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : n \u2264 \u2191i\nhi' : \u2191i \u2264 n\n\u22a2 \u2191\u2191n = \u2191i"}, {"tactic": "rw [Fin.coe_ofNat_eq_mod, le_antisymm hi' hi, Nat.mod_succ_eq_iff_lt, Nat.lt_succ]", "annotated_tactic": ["rw [<a>Fin.coe_ofNat_eq_mod</a>, <a>le_antisymm</a> hi' hi, <a>Nat.mod_succ_eq_iff_lt</a>, <a>Nat.lt_succ</a>]", [{"full_name": "Fin.coe_ofNat_eq_mod", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1779, 9], "def_end_pos": [1779, 25]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.mod_succ_eq_iff_lt", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [727, 9], "def_end_pos": [727, 27]}, {"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 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 : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nn : \u2115\nh : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc\nf : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2\nhfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ng : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k\nhgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p\n\u03b4\u2081 : \u211d\nh\u03b4\u2081pos : 0 < \u03b4\u2081\nh\u03b4\u2081 :\n  \u2200 (i : Fin n) (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u03b4\u2082 : \u211d\nh\u03b4\u2082pos : 0 < \u03b4\u2082\nh\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5\ni : Fin (Nat.succ n)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nhi : n \u2264 \u2191i\nhi' : \u2191i \u2264 n\n\u22a2 \u2191\u2191n = \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjUnion_singleton", "start": [1053, 1], "end": [1055, 43], "traced_tactics": [{"tactic": "rw [disjUnion_comm, singleton_disjUnion]", "annotated_tactic": ["rw [<a>disjUnion_comm</a>, <a>singleton_disjUnion</a>]", [{"full_name": "Finset.disjUnion_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1031, 9], "def_end_pos": [1031, 23]}, {"full_name": "Finset.singleton_disjUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\nh : _root_.Disjoint s {a}\n\u22a2 disjUnion s {a} h = cons a s (_ : \u00aca \u2208 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.indep_limsup_atTop_self", "start": [131, 1], "end": [143, 31], "traced_tactics": [{"tactic": "let ns : \u03b9 \u2192 Set \u03b9 := Set.Iic", "annotated_tactic": ["let ns : \u03b9 \u2192 <a>Set</a> \u03b9 := <a>Set.Iic</a>", [{"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]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\n\u22a2 Indep (limsup s atTop) (limsup s atTop)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\n\u22a2 Indep (limsup s atTop) (limsup s atTop)"}, {"tactic": "have hnsp : \u2200 i, BddAbove (ns i) := fun i => bddAbove_Iic", "annotated_tactic": ["have hnsp : \u2200 i, <a>BddAbove</a> (ns i) := fun i => <a>bddAbove_Iic</a>", [{"full_name": "BddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [56, 5], "def_end_pos": [56, 13]}, {"full_name": "bddAbove_Iic", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 21]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\n\u22a2 Indep (limsup s atTop) (limsup s atTop)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 Indep (limsup s atTop) (limsup s atTop)"}, {"tactic": "refine' indep_limsup_self h_le h_indep _ _ hnsp _", "annotated_tactic": ["refine' <a>indep_limsup_self</a> h_le h_indep _ _ hnsp _", [{"full_name": "ProbabilityTheory.indep_limsup_self", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [110, 9], "def_end_pos": [110, 26]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 Indep (limsup s atTop) (limsup s atTop)", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 \u2200 (t : Set \u03b9), BddAbove t \u2192 t\u1d9c \u2208 atTop\n\ncase refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => ns a\n\ncase refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a"}, {"tactic": "simp only [mem_atTop_sets, ge_iff_le, Set.mem_compl_iff, BddAbove, upperBounds, Set.Nonempty]", "annotated_tactic": ["simp only [<a>mem_atTop_sets</a>, <a>ge_iff_le</a>, <a>Set.mem_compl_iff</a>, <a>BddAbove</a>, <a>upperBounds</a>, <a>Set.Nonempty</a>]", [{"full_name": "Filter.mem_atTop_sets", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [166, 9], "def_end_pos": [166, 23]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "BddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [56, 5], "def_end_pos": [56, 13]}, {"full_name": "upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 16]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 \u2200 (t : Set \u03b9), BddAbove t \u2192 t\u1d9c \u2208 atTop", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 \u2200 (t : Set \u03b9), (\u2203 x, x \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}) \u2192 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t"}, {"tactic": "rintro t \u27e8a, ha\u27e9", "annotated_tactic": ["rintro t \u27e8a, ha\u27e9", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 \u2200 (t : Set \u03b9), (\u2203 x, x \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}) \u2192 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t", "state_after": "case refine'_1.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\n\u22a2 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t"}, {"tactic": "obtain \u27e8b, hb\u27e9 : \u2203 b, a < b := exists_gt a", "annotated_tactic": ["obtain \u27e8b, hb\u27e9 : \u2203 b, a < b := <a>exists_gt</a> a", [{"full_name": "NoMaxOrder.exists_gt", "def_path": "Mathlib/Order/Max.lean", "def_pos": [61, 3], "def_end_pos": [61, 12]}]], "state_before": "case refine'_1.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\n\u22a2 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t", "state_after": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\n\u22a2 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t"}, {"tactic": "refine' \u27e8b, fun c hc hct => _\u27e9", "annotated_tactic": ["refine' \u27e8b, fun c hc hct => _\u27e9", []], "state_before": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\n\u22a2 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t", "state_after": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\n\u22a2 False"}, {"tactic": "suffices : \u2200 i \u2208 t, i < c", "annotated_tactic": ["suffices : \u2200 i \u2208 t, i < c", []], "state_before": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\n\u22a2 False", "state_after": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\nthis : \u2200 (i : \u03b9), i \u2208 t \u2192 i < c\n\u22a2 False\n\ncase this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 i < c"}, {"tactic": "exact lt_irrefl c (this c hct)", "annotated_tactic": ["exact <a>lt_irrefl</a> c (this c hct)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\nthis : \u2200 (i : \u03b9), i \u2208 t \u2192 i < c\n\u22a2 False\n\ncase this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 i < c", "state_after": "case this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 i < c"}, {"tactic": "exact fun i hi => (ha hi).trans_lt (hb.trans_le hc)", "annotated_tactic": ["exact fun i hi => (ha hi).<a>trans_lt</a> (hb.trans_le hc)", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}\nb : \u03b9\nhb : a < b\nc : \u03b9\nhc : b \u2264 c\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 i < c", "state_after": "no goals"}, {"tactic": "exact Monotone.directed_le fun i j hij k hki => le_trans hki hij", "annotated_tactic": ["exact <a>Monotone.directed_le</a> fun i j hij k hki => <a>le_trans</a> hki hij", [{"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [112, 9], "def_end_pos": [112, 29]}, {"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\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => ns a", "state_after": "no goals"}, {"tactic": "exact fun n => \u27e8n, le_rfl\u27e9", "annotated_tactic": ["exact fun n => \u27e8n, <a>le_rfl</a>\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Iic\nhnsp : \u2200 (i : \u03b9), BddAbove (ns i)\n\u22a2 \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.plus_def", "start": [97, 1], "end": [98, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge", "start": [382, 1], "end": [389, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.card_dvd_card_mul_left", "start": [2041, 1], "end": [2044, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_snorm", "start": [94, 1], "end": [136, 87], "traced_tactics": [{"tactic": "by_cases hp_zero : p = 0", "annotated_tactic": ["by_cases hp_zero : 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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : p = 0\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)\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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top", "annotated_tactic": ["have hp : 0 < p.toReal := <a>toReal_pos</a> hp_zero 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": "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "suffices\n    Tendsto (fun n => \u222b\u207b x, (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) atTop\n      (\ud835\udcdd 0) by\n  simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]\n  convert continuous_rpow_const.continuousAt.tendsto.comp this\n  simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]", "annotated_tactic": ["suffices\n      <a>Tendsto</a> (fun n => \u222b\u207b x, (\u2016<a>approxOn</a> f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) <a>atTop</a>\n        (\ud835\udcdd 0) by\n    simp only [<a>snorm_eq_lintegral_rpow_nnnorm</a> hp_zero hp_ne_top]\n    convert continuous_rpow_const.continuousAt.tendsto.comp this\n    simp [<a>zero_rpow_of_pos</a> (_root_.inv_pos.mpr hp)]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"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": "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 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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have hF_meas :\n  \u2200 n, Measurable fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal := by\n  simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n    (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n      (measurable_edist_right.comp hf).pow_const p.toReal", "annotated_tactic": ["have hF_meas :\n    \u2200 n, <a>Measurable</a> fun x => (\u2016<a>approxOn</a> f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal := by\n    simpa only [\u2190 <a>edist_eq_coe_nnnorm_sub</a>] using fun n =>\n      (<a>approxOn</a> f hf s y\u2080 h\u2080 n).<a>measurable_bind</a> (fun y x => <a>edist</a> y (f x) ^ p.toReal) fun y =>\n        (measurable_edist_right.comp hf).<a>pow_const</a> p.toReal", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "MeasureTheory.SimpleFunc.measurable_bind", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [269, 9], "def_end_pos": [269, 24]}, {"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}, {"full_name": "Measurable.pow_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [228, 9], "def_end_pos": [228, 29]}]], "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_bound :\n  \u2200 n, (fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) \u2264\u1d50[\u03bc] fun x =>\n      (\u2016f x - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal :=\n  fun n =>\n  eventually_of_forall fun x =>\n    rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h\u2080 x n)) toReal_nonneg", "annotated_tactic": ["have h_bound :\n    \u2200 n, (fun x => (\u2016<a>approxOn</a> f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) \u2264\u1d50[\u03bc] fun x =>\n        (\u2016f x - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal :=\n    fun n =>\n    <a>eventually_of_forall</a> fun x =>\n      <a>rpow_le_rpow</a> (<a>coe_mono</a> (<a>nnnorm_approxOn_le</a> hf h\u2080 x n)) <a>toReal_nonneg</a>", [{"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 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": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "ENNReal.coe_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [363, 9], "def_end_pos": [363, 17]}, {"full_name": "MeasureTheory.SimpleFunc.nnnorm_approxOn_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}, {"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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_fin : (\u222b\u207b a : \u03b2, (\u2016f a - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) \u2260 \u22a4 :=\n  (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp_zero hp_ne_top hi).ne", "annotated_tactic": ["have h_fin : (\u222b\u207b a : \u03b2, (\u2016f a - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) \u2260 \u22a4 :=\n    (<a>lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top</a> hp_zero hp_ne_top hi).<a>ne</a>", [{"full_name": "MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [149, 9], "def_end_pos": [149, 53]}, {"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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "have h_lim :\n  \u2200\u1d50 a : \u03b2 \u2202\u03bc,\n    Tendsto (fun n => (\u2016approxOn f hf s y\u2080 h\u2080 n a - f a\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop (\ud835\udcdd 0) := by\n  filter_upwards [h\u03bc] with a ha\n  have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n    (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds\n  convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)\n  simp [zero_rpow_of_pos hp]", "annotated_tactic": ["have h_lim :\n    \u2200\u1d50 a : \u03b2 \u2202\u03bc,\n      <a>Tendsto</a> (fun n => (\u2016<a>approxOn</a> f hf s y\u2080 h\u2080 n a - f a\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) <a>atTop</a> (\ud835\udcdd 0) := by\n    filter_upwards [h\u03bc] with a ha\n    have : <a>Tendsto</a> (fun n => (<a>approxOn</a> f hf s y\u2080 h\u2080 n) a - f a) <a>atTop</a> (\ud835\udcdd (f a - f a)) :=\n      (<a>tendsto_approxOn</a> hf h\u2080 ha).<a>sub</a> <a>tendsto_const_nhds</a>\n    convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)\n    simp [<a>zero_rpow_of_pos</a> hp]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"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": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [154, 9], "def_end_pos": [154, 25]}, {"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}, {"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 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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)", "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "simpa using tendsto_lintegral_of_dominated_convergence _ hF_meas h_bound h_fin h_lim", "annotated_tactic": ["simpa using <a>tendsto_lintegral_of_dominated_convergence</a> _ hF_meas h_bound h_fin h_lim", [{"full_name": "MeasureTheory.tendsto_lintegral_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1058, 9], "def_end_pos": [1058, 51]}]], "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds", "annotated_tactic": ["simpa only [hp_zero, <a>snorm_exponent_zero</a>] using <a>tendsto_const_nhds</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": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : p = 0\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]", "annotated_tactic": ["simp only [<a>snorm_eq_lintegral_rpow_nnnorm</a> hp_zero hp_ne_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": "\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) 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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => (\u222b\u207b (x : \u03b2), \u2191\u2016(\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p))\n    atTop (\ud835\udcdd 0)"}, {"tactic": "convert continuous_rpow_const.continuousAt.tendsto.comp this", "annotated_tactic": ["convert continuous_rpow_const.continuousAt.tendsto.comp 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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => (\u222b\u207b (x : \u03b2), \u2191\u2016(\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p))\n    atTop (\ud835\udcdd 0)", "state_after": "case h.e'_5.h.e'_3\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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 0 = 0 ^ (1 / ENNReal.toReal p)"}, {"tactic": "simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]", "annotated_tactic": ["simp [<a>zero_rpow_of_pos</a> (_root_.inv_pos.mpr hp)]", [{"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 h.e'_5.h.e'_3\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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 0 = 0 ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n  (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n    (measurable_edist_right.comp hf).pow_const p.toReal", "annotated_tactic": ["simpa only [\u2190 <a>edist_eq_coe_nnnorm_sub</a>] using fun n =>\n      (<a>approxOn</a> f hf s y\u2080 h\u2080 n).<a>measurable_bind</a> (fun y x => <a>edist</a> y (f x) ^ p.toReal) fun y =>\n        (measurable_edist_right.comp hf).<a>pow_const</a> p.toReal", [{"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": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "MeasureTheory.SimpleFunc.measurable_bind", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [269, 9], "def_end_pos": [269, 24]}, {"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}, {"full_name": "Measurable.pow_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [228, 9], "def_end_pos": [228, 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\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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "filter_upwards [h\u03bc] with a ha", "annotated_tactic": ["filter_upwards [h\u03bc] with a ha", []], "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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)", "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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\n\u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)"}, {"tactic": "have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n  (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds", "annotated_tactic": ["have : <a>Tendsto</a> (fun n => (<a>approxOn</a> f hf s y\u2080 h\u2080 n) a - f a) <a>atTop</a> (\ud835\udcdd (f a - f a)) :=\n      (<a>tendsto_approxOn</a> hf h\u2080 ha).<a>sub</a> <a>tendsto_const_nhds</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.SimpleFunc.approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [127, 19], "def_end_pos": [127, 27]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.SimpleFunc.tendsto_approxOn", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [154, 9], "def_end_pos": [154, 25]}, {"full_name": "Filter.Tendsto.sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1081, 15], "def_end_pos": [1081, 18]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\n\u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)", "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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\nthis : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a))\n\u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)"}, {"tactic": "convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)", "annotated_tactic": ["convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)", []], "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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\nthis : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a))\n\u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)", "state_after": "case h.e'_5.h.e'_3\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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\nthis : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a))\n\u22a2 0 = \u2191\u2016f a - f a\u2016\u208a ^ ENNReal.toReal p"}, {"tactic": "simp [zero_rpow_of_pos hp]", "annotated_tactic": ["simp [<a>zero_rpow_of_pos</a> hp]", [{"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 h.e'_5.h.e'_3\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 \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 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\nthis : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a))\n\u22a2 0 = \u2191\u2016f a - f a\u2016\u208a ^ ENNReal.toReal p", "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.isEmpty", "start": [994, 1], "end": [995, 85], "traced_tactics": [{"tactic": "simp [h.isEmpty, h.toString, ext_iff]", "annotated_tactic": ["simp [h.isEmpty, h.toString, <a>ext_iff</a>]", [{"full_name": "String.ext_iff", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [20, 9], "def_end_pos": [20, 16]}]], "state_before": "x\u271d : Substring\nh\u271d : Valid x\u271d\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 Substring.isEmpty x\u271d = true \u2194 toString x\u271d = \"\"", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.eq_of_forall_toMeasure_apply_eq", "start": [194, 1], "end": [198, 19], "traced_tactics": [{"tactic": "apply toMeasure_injective", "annotated_tactic": ["apply <a>toMeasure_injective</a>", [{"full_name": "MeasureTheory.ProbabilityMeasure.toMeasure_injective", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [140, 9], "def_end_pos": [140, 28]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u03bc = \u03bd", "state_after": "case a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u2191\u03bc = \u2191\u03bd"}, {"tactic": "ext1 s s_mble", "annotated_tactic": ["ext1 s s_mble", []], "state_before": "case a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u2191\u03bc = \u2191\u03bd", "state_after": "case a.h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s"}, {"tactic": "exact h s s_mble", "annotated_tactic": ["exact h s s_mble", []], "state_before": "case a.h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.inv_mul_of_unit", "start": [772, 1], "end": [773, 37], "traced_tactics": [{"tactic": "rw [mul_comm, mul_inv_of_unit a h]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>mul_inv_of_unit</a> a h]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "ZMod.mul_inv_of_unit", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 24]}]], "state_before": "n : \u2115\na : ZMod n\nh : IsUnit a\n\u22a2 a\u207b\u00b9 * a = 1", "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_right_of_mem_uIcc", "start": [492, 1], "end": [493, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inclusion_comp_inclusion", "start": [2822, 1], "end": [2824, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.hasFiniteIntegral_of_bounded", "start": [182, 1], "end": [184, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurableSet.union_null", "start": [189, 11], "end": [191, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.get_ofFn", "start": [148, 1], "end": [151, 35], "traced_tactics": [{"tactic": "conv_rhs => erw [\u2190 List.get_ofFn f \u27e8i, by simp\u27e9]", "annotated_tactic": ["conv_rhs => erw [\u2190 <a>List.get_ofFn</a> f \u27e8i, by simp\u27e9]", [{"full_name": "List.get_ofFn", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [49, 9], "def_end_pos": [49, 17]}]], "state_before": "n\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 get (ofFn f) i = f i", "state_after": "n\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 get (ofFn f) i = List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }"}, {"tactic": "simp only [get_eq_get]", "annotated_tactic": ["simp only [<a>get_eq_get</a>]", [{"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\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 get (ofFn f) i = List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }", "state_after": "n\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 List.get (toList (ofFn f)) (Fin.cast (_ : n = List.length (toList (ofFn f))) i) =\n    List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }"}, {"tactic": "congr <;> simp [Fin.heq_ext_iff]", "annotated_tactic": ["congr <;> simp [<a>Fin.heq_ext_iff</a>]", [{"full_name": "Fin.heq_ext_iff", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [194, 19], "def_end_pos": [194, 30]}]], "state_before": "n\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 List.get (toList (ofFn f)) (Fin.cast (_ : n = List.length (toList (ofFn f))) i) =\n    List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 \u2191i < List.length (List.ofFn f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "MeasureTheory.Integrable.integral_condexpKernel", "start": [151, 1], "end": [157, 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 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9", "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 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9"}, {"tactic": "exact Integrable.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.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.integral_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [183, 9], "def_end_pos": [183, 61]}, {"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 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9", "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": [1004, 1], "end": [1006, 39], "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 a \u2208 [[b, c]] \u2192 b \u2208 [[a, c]] \u2192 a = b", "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 b \u2293 c \u2264 a \u2227 a \u2264 b \u2294 c \u2192 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a = b"}, {"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": [163, 7], "def_end_pos": [163, 33]}]], "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 \u2293 c \u2264 a \u2227 a \u2264 b \u2294 c \u2192 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a = b", "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_Ioo_left", "start": [244, 1], "end": [246, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.uIcc_injective_left", "start": [1019, 1], "end": [1020, 54], "traced_tactics": [{"tactic": "simpa only [uIcc_comm] using uIcc_injective_right a", "annotated_tactic": ["simpa only [<a>uIcc_comm</a>] using <a>uIcc_injective_right</a> a", [{"full_name": "Finset.uIcc_comm", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [928, 9], "def_end_pos": [928, 18]}, {"full_name": "Finset.uIcc_injective_right", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 29]}]], "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 b\u2081 b\u2082 c x a : \u03b1\n\u22a2 Injective (uIcc a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Nat.Primrec'.sub", "start": [1457, 1], "end": [1460, 52], "traced_tactics": [{"tactic": "suffices", "annotated_tactic": ["suffices", []], "state_before": "\u22a2 Primrec' fun v => Vector.head v - Vector.head (Vector.tail v)", "state_after": "this : ?m.396872\n\u22a2 Primrec' fun v => Vector.head v - Vector.head (Vector.tail v)\n\ncase this\n\n\u22a2 ?m.396872"}, {"tactic": "simpa using comp\u2082 (fun a b => b - a) this (tail head) head", "annotated_tactic": ["simpa using <a>comp\u2082</a> (fun a b => b - a) this (<a>tail</a> <a>head</a>) <a>head</a>", [{"full_name": "Nat.Primrec'.comp\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 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]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "this : ?m.396872\n\u22a2 Primrec' fun v => Vector.head v - Vector.head (Vector.tail v)\n\ncase this\n\n\u22a2 ?m.396872", "state_after": "case this\n\n\u22a2 Primrec' fun v => (fun a b => b - a) (Vector.head v) (Vector.head (Vector.tail v))"}, {"tactic": "refine' (prec head (pred.comp\u2081 _ (tail head))).of_eq fun v => _", "annotated_tactic": ["refine' (<a>prec</a> <a>head</a> (pred.comp\u2081 _ (<a>tail</a> <a>head</a>))).<a>of_eq</a> fun v => _", [{"full_name": "Nat.Primrec'.prec", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1367, 5], "def_end_pos": [1367, 9]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"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]}, {"full_name": "Nat.Primrec'.of_eq", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1395, 9], "def_end_pos": [1395, 14]}]], "state_before": "case this\n\n\u22a2 Primrec' fun v => (fun a b => b - a) (Vector.head v) (Vector.head (Vector.tail v))", "state_after": "case this\nv : Vector \u2115 (Nat.succ 0 + 1)\n\u22a2 Nat.rec (Vector.head (Vector.tail v)) (fun y IH => Nat.pred (Vector.head (Vector.tail (y ::\u1d65 IH ::\u1d65 Vector.tail v))))\n      (Vector.head v) =\n    (fun a b => b - a) (Vector.head v) (Vector.head (Vector.tail v))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case this\nv : Vector \u2115 (Nat.succ 0 + 1)\n\u22a2 Nat.rec (Vector.head (Vector.tail v)) (fun y IH => Nat.pred (Vector.head (Vector.tail (y ::\u1d65 IH ::\u1d65 Vector.tail v))))\n      (Vector.head v) =\n    (fun a b => b - a) (Vector.head v) (Vector.head (Vector.tail v))", "state_after": "case this\nv : Vector \u2115 (Nat.succ 0 + 1)\n\u22a2 Nat.rec (Vector.head (Vector.tail v)) (fun y IH => Nat.pred IH) (Vector.head v) =\n    Vector.head (Vector.tail v) - Vector.head v"}, {"tactic": "induction v.head <;> simp [*, Nat.sub_succ]", "annotated_tactic": ["induction v.head <;> simp [*, <a>Nat.sub_succ</a>]", [{"full_name": "Nat.sub_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 17]}]], "state_before": "case this\nv : Vector \u2115 (Nat.succ 0 + 1)\n\u22a2 Nat.rec (Vector.head (Vector.tail v)) (fun y IH => Nat.pred IH) (Vector.head v) =\n    Vector.head (Vector.tail v) - Vector.head v", "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_image_left_comm", "start": [433, 1], "end": [436, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/EpsilonNFA.lean", "full_name": "\u03b5NFA.\u03b5Closure_univ", "start": [65, 1], "end": [66, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_sub", "start": [932, 1], "end": [939, 54], "traced_tactics": [{"tactic": "by_cases hb : b = 0", "annotated_tactic": ["by_cases hb : b = 0", []], "state_before": "n : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\n\u22a2 val (a - b) = val a - val b", "state_after": "case pos\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : b = 0\n\u22a2 val (a - b) = val a - val b\n\ncase neg\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : \u00acb = 0\n\u22a2 val (a - b) = val a - val b"}, {"tactic": "cases hb", "annotated_tactic": ["cases hb", []], "state_before": "case pos\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : b = 0\n\u22a2 val (a - b) = val a - val b", "state_after": "case pos.refl\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : val 0 \u2264 val a\n\u22a2 val (a - 0) = val a - val 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos.refl\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : val 0 \u2264 val a\n\u22a2 val (a - 0) = val a - val 0", "state_after": "no goals"}, {"tactic": "have : NeZero b := \u27e8hb\u27e9", "annotated_tactic": ["have : <a>NeZero</a> b := \u27e8hb\u27e9", [{"full_name": "NeZero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [25, 7], "def_end_pos": [25, 13]}]], "state_before": "case neg\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : \u00acb = 0\n\u22a2 val (a - b) = val a - val b", "state_after": "case neg\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : \u00acb = 0\nthis : NeZero b\n\u22a2 val (a - b) = val a - val b"}, {"tactic": "rw [sub_eq_add_neg, val_add, val_neg_of_ne_zero, \u2190 Nat.add_sub_assoc (le_of_lt (val_lt _)),\n  add_comm, Nat.add_sub_assoc h, Nat.add_mod_left]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>val_add</a>, <a>val_neg_of_ne_zero</a>, \u2190 <a>Nat.add_sub_assoc</a> (<a>le_of_lt</a> (<a>val_lt</a> _)),\n      <a>add_comm</a>, <a>Nat.add_sub_assoc</a> h, <a>Nat.add_mod_left</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": "ZMod.val_add", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 16]}, {"full_name": "ZMod.val_neg_of_ne_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [929, 9], "def_end_pos": [929, 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": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "ZMod.val_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"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.add_mod_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [713, 17], "def_end_pos": [713, 29]}]], "state_before": "case neg\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : \u00acb = 0\nthis : NeZero b\n\u22a2 val (a - b) = val a - val b", "state_after": "case neg\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : \u00acb = 0\nthis : NeZero b\n\u22a2 (val a - val b) % n = val a - val b"}, {"tactic": "apply Nat.mod_eq_of_lt (tsub_lt_of_lt (val_lt _))", "annotated_tactic": ["apply <a>Nat.mod_eq_of_lt</a> (<a>tsub_lt_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": "tsub_lt_of_lt", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}, {"full_name": "ZMod.val_lt", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}]], "state_before": "case neg\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\nh : val b \u2264 val a\nhb : \u00acb = 0\nthis : NeZero b\n\u22a2 (val a - val b) % n = val a - val 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.lintegral_eq_of_subset", "start": [954, 1], "end": [966, 8], "traced_tactics": [{"tactic": "refine' Finset.sum_bij_ne_zero (fun r _ _ => r) _ _ _ _", "annotated_tactic": ["refine' <a>Finset.sum_bij_ne_zero</a> (fun r _ _ => r) _ _ _ _", [{"full_name": "Finset.sum_bij_ne_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1141, 3], "def_end_pos": [1141, 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 lintegral f \u03bc = \u2211 x in s, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x})", "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (a : \u211d\u22650\u221e) (h\u2081 : a \u2208 SimpleFunc.range f) (h\u2082 : a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) \u2260 0), (fun r x x => r) a h\u2081 h\u2082 \u2208 s\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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (a\u2081 a\u2082 : \u211d\u22650\u221e) (h\u2081\u2081 : a\u2081 \u2208 SimpleFunc.range f) (h\u2081\u2082 : a\u2081 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2081}) \u2260 0) (h\u2082\u2081 : a\u2082 \u2208 SimpleFunc.range f)\n    (h\u2082\u2082 : a\u2082 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2082}) \u2260 0), (fun r x x => r) a\u2081 h\u2081\u2081 h\u2081\u2082 = (fun r x x => r) a\u2082 h\u2082\u2081 h\u2082\u2082 \u2192 a\u2081 = a\u2082\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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b \u2208 s \u2192 b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0 \u2192 \u2203 a h\u2081 h\u2082, b = (fun r x x => r) a h\u2081 h\u2082\n\ncase refine'_4\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\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (a : \u211d\u22650\u221e) (h\u2081 : a \u2208 SimpleFunc.range f) (h\u2082 : a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) \u2260 0),\n    a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) = (fun r x x => r) a h\u2081 h\u2082 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {(fun r x x => r) a h\u2081 h\u2082})"}, {"tactic": "simpa only [forall_range_iff, mul_ne_zero_iff, and_imp]", "annotated_tactic": ["simpa only [<a>forall_range_iff</a>, <a>mul_ne_zero_iff</a>, <a>and_imp</a>]", [{"full_name": "MeasureTheory.SimpleFunc.forall_range_iff", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [127, 9], "def_end_pos": [127, 25]}, {"full_name": "mul_ne_zero_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (a : \u211d\u22650\u221e) (h\u2081 : a \u2208 SimpleFunc.range f) (h\u2082 : a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) \u2260 0), (fun r x x => r) a h\u2081 h\u2082 \u2208 s", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (a\u2081 a\u2082 : \u211d\u22650\u221e) (h\u2081\u2081 : a\u2081 \u2208 SimpleFunc.range f) (h\u2081\u2082 : a\u2081 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2081}) \u2260 0) (h\u2082\u2081 : a\u2082 \u2208 SimpleFunc.range f)\n    (h\u2082\u2082 : a\u2082 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2082}) \u2260 0), (fun r x x => r) a\u2081 h\u2081\u2081 h\u2081\u2082 = (fun r x x => r) a\u2082 h\u2082\u2081 h\u2082\u2082 \u2192 a\u2081 = a\u2082", "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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\na\u2081\u271d a\u2082\u271d : \u211d\u22650\u221e\nh\u2081\u2081\u271d : a\u2081\u271d \u2208 SimpleFunc.range f\nh\u2081\u2082\u271d : a\u2081\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2081\u271d}) \u2260 0\nh\u2082\u2081\u271d : a\u2082\u271d \u2208 SimpleFunc.range f\nh\u2082\u2082\u271d : a\u2082\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2082\u271d}) \u2260 0\na\u271d : (fun r x x => r) a\u2081\u271d h\u2081\u2081\u271d h\u2081\u2082\u271d = (fun r x x => r) a\u2082\u271d h\u2082\u2081\u271d h\u2082\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\na\u2081\u271d a\u2082\u271d : \u211d\u22650\u221e\nh\u2081\u2081\u271d : a\u2081\u271d \u2208 SimpleFunc.range f\nh\u2081\u2082\u271d : a\u2081\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2081\u271d}) \u2260 0\nh\u2082\u2081\u271d : a\u2082\u271d \u2208 SimpleFunc.range f\nh\u2082\u2082\u271d : a\u2082\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2082\u271d}) \u2260 0\na\u271d : (fun r x x => r) a\u2081\u271d h\u2081\u2081\u271d h\u2081\u2082\u271d = (fun r x x => r) a\u2082\u271d h\u2082\u2081\u271d h\u2082\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d", "state_after": "no goals"}, {"tactic": "intro b _ hb", "annotated_tactic": ["intro b _ hb", []], "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b \u2208 s \u2192 b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0 \u2192 \u2203 a h\u2081 h\u2082, b = (fun r x x => r) a h\u2081 h\u2082", "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\nb : \u211d\u22650\u221e\na\u271d : b \u2208 s\nhb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0\n\u22a2 \u2203 a h\u2081 h\u2082, b = (fun r x x => r) a h\u2081 h\u2082"}, {"tactic": "refine' \u27e8b, _, hb, rfl\u27e9", "annotated_tactic": ["refine' \u27e8b, _, hb, <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 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\nb : \u211d\u22650\u221e\na\u271d : b \u2208 s\nhb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0\n\u22a2 \u2203 a h\u2081 h\u2082, b = (fun r x x => r) a h\u2081 h\u2082", "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\nb : \u211d\u22650\u221e\na\u271d : b \u2208 s\nhb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0\n\u22a2 b \u2208 SimpleFunc.range f"}, {"tactic": "rw [mem_range, \u2190 preimage_singleton_nonempty]", "annotated_tactic": ["rw [<a>mem_range</a>, \u2190 <a>preimage_singleton_nonempty</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.preimage_singleton_nonempty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 36]}]], "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\nb : \u211d\u22650\u221e\na\u271d : b \u2208 s\nhb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0\n\u22a2 b \u2208 SimpleFunc.range f", "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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\nb : \u211d\u22650\u221e\na\u271d : b \u2208 s\nhb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0\n\u22a2 Set.Nonempty (\u2191f \u207b\u00b9' {b})"}, {"tactic": "exact nonempty_of_measure_ne_zero (mul_ne_zero_iff.1 hb).2", "annotated_tactic": ["exact <a>nonempty_of_measure_ne_zero</a> (<a>mul_ne_zero_iff</a>.1 hb).2", [{"full_name": "MeasureTheory.nonempty_of_measure_ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [189, 9], "def_end_pos": [189, 36]}, {"full_name": "mul_ne_zero_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [253, 9], "def_end_pos": [253, 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\nb : \u211d\u22650\u221e\na\u271d : b \u2208 s\nhb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0\n\u22a2 Set.Nonempty (\u2191f \u207b\u00b9' {b})", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case refine'_4\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\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\n\u22a2 \u2200 (a : \u211d\u22650\u221e) (h\u2081 : a \u2208 SimpleFunc.range f) (h\u2082 : a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) \u2260 0),\n    a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) = (fun r x x => r) a h\u2081 h\u2082 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {(fun r x x => r) a h\u2081 h\u2082})", "state_after": "case refine'_4\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\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\na\u271d : \u211d\u22650\u221e\nh\u2081\u271d : a\u271d \u2208 SimpleFunc.range f\nh\u2082\u271d : a\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u271d}) \u2260 0\n\u22a2 a\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u271d}) = (fun r x x => r) a\u271d h\u2081\u271d h\u2082\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {(fun r x x => r) a\u271d h\u2081\u271d h\u2082\u271d})"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_4\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\u209b \u211d\u22650\u221e\ns : Finset \u211d\u22650\u221e\nhs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s\na\u271d : \u211d\u22650\u221e\nh\u2081\u271d : a\u271d \u2208 SimpleFunc.range f\nh\u2082\u271d : a\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u271d}) \u2260 0\n\u22a2 a\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u271d}) = (fun r x x => r) a\u271d h\u2081\u271d h\u2082\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {(fun r x x => r) a\u271d h\u2081\u271d h\u2082\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.id_eval", "start": [183, 1], "end": [183, 57], "traced_tactics": [{"tactic": "simp [id]", "annotated_tactic": ["simp [<a>id</a>]", [{"full_name": "Turing.ToPartrec.Code.id", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [178, 5], "def_end_pos": [178, 7]}]], "state_before": "v : List \u2115\n\u22a2 eval id v = pure v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.pred'_to_nat", "start": [533, 1], "end": [543, 18], "traced_tactics": [{"tactic": "rw [pred'_to_nat n, Nat.succ_pred_eq_of_pos (to_nat_pos n)]", "annotated_tactic": ["rw [pred'_to_nat n, <a>Nat.succ_pred_eq_of_pos</a> (<a>to_nat_pos</a> n)]", [{"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": "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 Nat.succ \u2191(pred' n) = \u2191n", "state_after": "no goals"}, {"tactic": "rw [\u2190 to_nat_inj.1 h]", "annotated_tactic": ["rw [\u2190 <a>to_nat_inj</a>.1 h]", [{"full_name": "PosNum.to_nat_inj", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [529, 9], "def_end_pos": [529, 19]}]], "state_before": "\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\nh : \u21911 = \u2191n\n\u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u2191n)", "state_after": "\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\nh : \u21911 = \u2191n\n\u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u21911)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\nh : \u21911 = \u2191n\n\u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u21911)", "state_after": "no goals"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\np : PosNum\nh : Nat.succ \u2191p = \u2191n\n\u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 \u2191n)", "state_after": "\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\np : PosNum\nh : Nat.succ \u2191p = \u2191n\n\u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 (Nat.succ \u2191p))"}, {"tactic": "exact (Nat.succ_add p p).symm", "annotated_tactic": ["exact (<a>Nat.succ_add</a> p p).<a>symm</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": "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\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\np : PosNum\nh : Nat.succ \u2191p = \u2191n\n\u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 (Nat.succ \u2191p))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.finset_prod_subset_finset_prod", "start": [162, 1], "end": [164, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_smul_const", "start": [448, 1], "end": [450, 81], "traced_tactics": [{"tactic": "simp only [circleIntegral, intervalIntegral.integral_smul_const, \u2190 smul_assoc]", "annotated_tactic": ["simp only [<a>circleIntegral</a>, <a>intervalIntegral.integral_smul_const</a>, \u2190 <a>smul_assoc</a>]", [{"full_name": "circleIntegral", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [343, 5], "def_end_pos": [343, 19]}, {"full_name": "intervalIntegral.integral_smul_const", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [609, 16], "def_end_pos": [609, 35]}, {"full_name": "smul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 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 \u2102\na : E\nc : \u2102\nR : \u211d\n\u22a2 (\u222e (z : \u2102) in C(c, R), f z \u2022 a) = (\u222e (z : \u2102) in C(c, R), f z) \u2022 a", "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_lpMeas", "start": [483, 1], "end": [504, 45], "traced_tactics": [{"tactic": "let g := lpMeasToLpTrimLie F' \u211d 1 \u03bc hm f", "annotated_tactic": ["let g := <a>lpMeasToLpTrimLie</a> F' \u211d 1 \u03bc hm f", [{"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 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 lpMeas F' \u211d m 1 \u03bc }\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f", "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f"}, {"tactic": "have hfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g := by\n  simp only [LinearIsometryEquiv.symm_apply_apply]", "annotated_tactic": ["have hfg : f = (<a>lpMeasToLpTrimLie</a> F' \u211d 1 \u03bc hm).<a>symm</a> g := by\n    simp only [<a>LinearIsometryEquiv.symm_apply_apply</a>]", [{"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}, {"full_name": "LinearIsometryEquiv.symm_apply_apply", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [747, 9], "def_end_pos": [747, 25]}]], "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f", "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f"}, {"tactic": "rw [hfg]", "annotated_tactic": ["rw [hfg]", []], "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f", "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g)"}, {"tactic": "refine' @Lp.induction \u03b1 F' m _ 1 (\u03bc.trim hm) _ ENNReal.coe_ne_top (fun g : \u03b1 \u2192\u2081[\u03bc.trim hm] F' =>\n  condexpL1Clm F' hm \u03bc ((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g : \u03b1 \u2192\u2081[\u03bc] F') =\n  \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g)) _ _ _ g", "annotated_tactic": ["refine' @<a>Lp.induction</a> \u03b1 F' m _ 1 (\u03bc.trim hm) _ <a>ENNReal.coe_ne_top</a> (fun g : \u03b1 \u2192\u2081[\u03bc.trim hm] F' =>\n    <a>condexpL1Clm</a> F' hm \u03bc ((<a>lpMeasToLpTrimLie</a> F' \u211d 1 \u03bc hm).<a>symm</a> g : \u03b1 \u2192\u2081[\u03bc] F') =\n    \u2191((<a>lpMeasToLpTrimLie</a> F' \u211d 1 \u03bc hm).<a>symm</a> g)) _ _ _ g", [{"full_name": "MeasureTheory.Lp.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [924, 9], "def_end_pos": [924, 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": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}]], "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g)", "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),\n    (fun g =>\n        \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n          \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n      \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) 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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\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 g =>\n            \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n              \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n          (Mem\u2112p.toLp f hf) \u2192\n        (fun g =>\n              \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n                \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n            (Mem\u2112p.toLp g hg) \u2192\n          (fun g =>\n              \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n                \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n            (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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 IsClosed\n    {f |\n      (fun g =>\n          \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n            \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n        f}"}, {"tactic": "simp only [LinearIsometryEquiv.symm_apply_apply]", "annotated_tactic": ["simp only [<a>LinearIsometryEquiv.symm_apply_apply</a>]", [{"full_name": "LinearIsometryEquiv.symm_apply_apply", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [747, 9], "def_end_pos": [747, 25]}]], "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\n\u22a2 f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g", "state_after": "no goals"}, {"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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),\n    (fun g =>\n        \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n          \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n      \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) 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\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm))\n          \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm))\n        \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c))"}, {"tactic": "rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs h\u03bcs.ne c,\n  condexpL1Clm_indicatorConstLp]", "annotated_tactic": ["rw [@<a>Lp.simpleFunc.coe_indicatorConst</a> _ _ m, <a>lpMeasToLpTrimLie_symm_indicator</a> hs h\u03bcs.ne c,\n      <a>condexpL1Clm_indicatorConstLp</a>]", [{"full_name": "MeasureTheory.Lp.simpleFunc.coe_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [700, 9], "def_end_pos": [700, 27]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie_symm_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [535, 9], "def_end_pos": [535, 41]}, {"full_name": "MeasureTheory.condexpL1Clm_indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [393, 9], "def_end_pos": [393, 38]}]], "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\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm))\n          \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm))\n        \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) 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\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 \u2191(condexpInd F' hm \u03bc s) c = indicatorConstLp 1 (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c"}, {"tactic": "exact condexpInd_of_measurable hs ((le_trim hm).trans_lt h\u03bcs).ne c", "annotated_tactic": ["exact <a>condexpInd_of_measurable</a> hs ((<a>le_trim</a> hm).<a>trans_lt</a> h\u03bcs).<a>ne</a> c", [{"full_name": "MeasureTheory.condexpInd_of_measurable", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [346, 9], "def_end_pos": [346, 33]}, {"full_name": "MeasureTheory.le_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [57, 9], "def_end_pos": [57, 16]}, {"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": "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\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4\n\u22a2 \u2191(condexpInd F' hm \u03bc s) c = indicatorConstLp 1 (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c", "state_after": "no goals"}, {"tactic": "intro f g hf hg _ hf_eq hg_eq", "annotated_tactic": ["intro f g hf hg _ hf_eq hg_eq", []], "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\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 g =>\n            \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n              \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n          (Mem\u2112p.toLp f hf) \u2192\n        (fun g =>\n              \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n                \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n            (Mem\u2112p.toLp g hg) \u2192\n          (fun g =>\n              \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n                \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n            (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\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf))\nhg_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))"}, {"tactic": "rw [LinearIsometryEquiv.map_add]", "annotated_tactic": ["rw [<a>LinearIsometryEquiv.map_add</a>]", [{"full_name": "LinearIsometryEquiv.map_add", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [955, 9], "def_end_pos": [955, 16]}]], "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\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf))\nhg_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (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\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf))\nhg_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf) +\n          \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf) +\n        \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "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\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf))\nhg_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf) +\n          \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf) +\n        \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (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\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf))\nhg_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n        \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))"}, {"tactic": "rw [map_add, hf_eq, hg_eq]", "annotated_tactic": ["rw [<a>map_add</a>, hf_eq, hg_eq]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "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\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1\nhg : Mem\u2112p g 1\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf))\nhg_eq :\n  \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 \u2191(condexpL1Clm F' hm \u03bc)\n      (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n        \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))) =\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))", "state_after": "no goals"}, {"tactic": "refine' isClosed_eq _ _", "annotated_tactic": ["refine' <a>isClosed_eq</a> _ _", [{"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}]], "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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 IsClosed\n    {f |\n      (fun g =>\n          \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =\n            \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))\n        f}", "state_after": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f)\n\ncase refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f)"}, {"tactic": "refine' (condexpL1Clm F' hm \u03bc).continuous.comp (continuous_induced_dom.comp _)", "annotated_tactic": ["refine' (<a>condexpL1Clm</a> F' hm \u03bc).continuous.comp (continuous_induced_dom.comp _)", [{"full_name": "MeasureTheory.condexpL1Clm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [380, 5], "def_end_pos": [380, 17]}]], "state_before": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f)", "state_after": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f"}, {"tactic": "exact LinearIsometryEquiv.continuous _", "annotated_tactic": ["exact <a>LinearIsometryEquiv.continuous</a> _", [{"full_name": "LinearIsometryEquiv.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [676, 19], "def_end_pos": [676, 29]}]], "state_before": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f", "state_after": "no goals"}, {"tactic": "refine' continuous_induced_dom.comp _", "annotated_tactic": ["refine' continuous_induced_dom.comp _", []], "state_before": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f)", "state_after": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f"}, {"tactic": "exact LinearIsometryEquiv.continuous _", "annotated_tactic": ["exact <a>LinearIsometryEquiv.continuous</a> _", [{"full_name": "LinearIsometryEquiv.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [676, 19], "def_end_pos": [676, 29]}]], "state_before": "case refine'_3.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\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc }\ng : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\n\u22a2 Continuous fun f => \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "Isometry.hausdorffMeasure_image", "start": [878, 1], "end": [886, 8], "traced_tactics": [{"tactic": "simp only [hausdorffMeasure, \u2190 OuterMeasure.coe_mkMetric, \u2190 OuterMeasure.comap_apply]", "annotated_tactic": ["simp only [<a>hausdorffMeasure</a>, \u2190 <a>OuterMeasure.coe_mkMetric</a>, \u2190 <a>OuterMeasure.comap_apply</a>]", [{"full_name": "MeasureTheory.Measure.hausdorffMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [578, 5], "def_end_pos": [578, 21]}, {"full_name": "MeasureTheory.OuterMeasure.coe_mkMetric", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [471, 9], "def_end_pos": [471, 34]}, {"full_name": "MeasureTheory.OuterMeasure.comap_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [541, 9], "def_end_pos": [541, 20]}]], "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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) = \u2191\u2191\u03bcH[d] s", "state_after": "\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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) \u2191(mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s"}, {"tactic": "simp only [mkMetric_toOuterMeasure]", "annotated_tactic": ["simp only [<a>mkMetric_toOuterMeasure</a>]", [{"full_name": "MeasureTheory.Measure.mkMetric_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [464, 9], "def_end_pos": [464, 32]}]], "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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) \u2191(mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s", "state_after": "\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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s"}, {"tactic": "have : 0 \u2264 d \u2192 Monotone fun r \u21a6 @HPow.hPow \u211d\u22650\u221e \u211d \u211d\u22650\u221e instHPow r d := by\n  exact fun hd x y hxy => ENNReal.rpow_le_rpow hxy hd", "annotated_tactic": ["have : 0 \u2264 d \u2192 <a>Monotone</a> fun r \u21a6 @<a>HPow.hPow</a> \u211d\u22650\u221e \u211d \u211d\u22650\u221e <a>instHPow</a> r d := by\n    exact fun hd x y hxy => <a>ENNReal.rpow_le_rpow</a> hxy hd", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"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": "instHPow", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1385, 1], "def_end_pos": [1385, 9]}, {"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": "\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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s", "state_after": "\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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\nthis : 0 \u2264 d \u2192 Monotone fun r => r ^ d\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s"}, {"tactic": "have := OuterMeasure.isometry_comap_mkMetric (fun (r : \u211d\u22650\u221e) => r ^ d) hf (hd.imp_left this)", "annotated_tactic": ["have := <a>OuterMeasure.isometry_comap_mkMetric</a> (fun (r : \u211d\u22650\u221e) => r ^ d) hf (hd.imp_left this)", [{"full_name": "MeasureTheory.OuterMeasure.isometry_comap_mkMetric", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [371, 9], "def_end_pos": [371, 32]}]], "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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\nthis : 0 \u2264 d \u2192 Monotone fun r => r ^ d\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s", "state_after": "\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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\nthis\u271d : 0 \u2264 d \u2192 Monotone fun r => r ^ d\nthis : \u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d) = OuterMeasure.mkMetric fun r => r ^ d\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\nthis\u271d : 0 \u2264 d \u2192 Monotone fun r => r ^ d\nthis : \u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d) = OuterMeasure.mkMetric fun r => r ^ d\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s", "state_after": "no goals"}, {"tactic": "exact fun hd x y hxy => ENNReal.rpow_le_rpow hxy hd", "annotated_tactic": ["exact fun hd x y hxy => <a>ENNReal.rpow_le_rpow</a> hxy hd", [{"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": "\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\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 0 \u2264 d \u2192 Monotone fun r => r ^ d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.Countable.of_subsingleton", "start": [240, 1], "end": [241, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Set.iInter_eq_iInter_finset'", "start": [1966, 1], "end": [1968, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.Fix.rec_eq", "start": [278, 1], "end": [293, 25], "traced_tactics": [{"tactic": "have : recF g \u2218 fixToW = Fix.rec g := by\n  apply funext\n  apply Quotient.ind\n  intro x\n  apply recF_eq_of_Wequiv\n  rw [fixToW]\n  apply Wrepr_equiv", "annotated_tactic": ["have : <a>recF</a> g \u2218 <a>fixToW</a> = <a>Fix.rec</a> g := by\n    apply <a>funext</a>\n    apply <a>Quotient.ind</a>\n    intro x\n    apply <a>recF_eq_of_Wequiv</a>\n    rw [<a>fixToW</a>]\n    apply <a>Wrepr_equiv</a>", [{"full_name": "QPF.recF", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [164, 5], "def_end_pos": [164, 9]}, {"full_name": "QPF.fixToW", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [263, 5], "def_end_pos": [263, 11]}, {"full_name": "QPF.Fix.rec", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [258, 5], "def_end_pos": [258, 12]}, {"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "Quotient.ind", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1355, 19], "def_end_pos": [1355, 22]}, {"full_name": "QPF.recF_eq_of_Wequiv", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 26]}, {"full_name": "QPF.fixToW", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [263, 5], "def_end_pos": [263, 11]}, {"full_name": "QPF.Wrepr_equiv", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 20]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 rec g (mk x) = g (rec g <$> x)", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 rec g (mk x) = g (rec g <$> x)"}, {"tactic": "conv =>\n  lhs\n  rw [Fix.rec, Fix.mk]\n  dsimp", "annotated_tactic": ["conv =>\n    lhs\n    rw [<a>Fix.rec</a>, <a>Fix.mk</a>]\n    dsimp", [{"full_name": "QPF.Fix.rec", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [258, 5], "def_end_pos": [258, 12]}, {"full_name": "QPF.Fix.mk", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [269, 5], "def_end_pos": [269, 11]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 rec g (mk x) = g (rec g <$> x)", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 recF g (PFunctor.W.mk (fixToW <$> repr x)) = g (rec g <$> x)"}, {"tactic": "cases' h : repr x with a f", "annotated_tactic": ["cases' h : <a>repr</a> x with a f", [{"full_name": "QPF.repr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [54, 3], "def_end_pos": [54, 7]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 recF g (PFunctor.W.mk (fixToW <$> repr x)) = g (rec g <$> x)", "state_after": "case mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Fix F\nh : repr x = { fst := a, snd := f }\n\u22a2 recF g (PFunctor.W.mk (fixToW <$> { fst := a, snd := f })) = g (rec g <$> x)"}, {"tactic": "rw [PFunctor.map_eq, recF_eq, \u2190 PFunctor.map_eq, PFunctor.W.dest_mk, \u2190 PFunctor.comp_map, abs_map,\n  \u2190 h, abs_repr, this]", "annotated_tactic": ["rw [<a>PFunctor.map_eq</a>, <a>recF_eq</a>, \u2190 <a>PFunctor.map_eq</a>, <a>PFunctor.W.dest_mk</a>, \u2190 <a>PFunctor.comp_map</a>, <a>abs_map</a>,\n    \u2190 h, <a>abs_repr</a>, this]", [{"full_name": "PFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [65, 19], "def_end_pos": [65, 25]}, {"full_name": "QPF.recF_eq", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 16]}, {"full_name": "PFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [65, 19], "def_end_pos": [65, 25]}, {"full_name": "PFunctor.W.dest_mk", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 18]}, {"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"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]}]], "state_before": "case mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Fix F\nh : repr x = { fst := a, snd := f }\n\u22a2 recF g (PFunctor.W.mk (fixToW <$> { fst := a, snd := f })) = g (rec g <$> x)", "state_after": "no goals"}, {"tactic": "apply funext", "annotated_tactic": ["apply <a>funext</a>", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}]], "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 recF g \u2218 fixToW = rec g", "state_after": "case h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 \u2200 (x : Fix F), (recF g \u2218 fixToW) x = rec g x"}, {"tactic": "apply Quotient.ind", "annotated_tactic": ["apply <a>Quotient.ind</a>", [{"full_name": "Quotient.ind", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1355, 19], "def_end_pos": [1355, 22]}]], "state_before": "case h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 \u2200 (x : Fix F), (recF g \u2218 fixToW) x = rec g x", "state_after": "case h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 \u2200 (a : PFunctor.W (P F)), (recF g \u2218 fixToW) (Quotient.mk Wsetoid a) = rec g (Quotient.mk Wsetoid a)"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 \u2200 (a : PFunctor.W (P F)), (recF g \u2218 fixToW) (Quotient.mk Wsetoid a) = rec g (Quotient.mk Wsetoid a)", "state_after": "case h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 (recF g \u2218 fixToW) (Quotient.mk Wsetoid x) = rec g (Quotient.mk Wsetoid x)"}, {"tactic": "apply recF_eq_of_Wequiv", "annotated_tactic": ["apply <a>recF_eq_of_Wequiv</a>", [{"full_name": "QPF.recF_eq_of_Wequiv", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 26]}]], "state_before": "case h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 (recF g \u2218 fixToW) (Quotient.mk Wsetoid x) = rec g (Quotient.mk Wsetoid x)", "state_after": "case h.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 Wequiv (fixToW (Quotient.mk Wsetoid x)) x"}, {"tactic": "rw [fixToW]", "annotated_tactic": ["rw [<a>fixToW</a>]", [{"full_name": "QPF.fixToW", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [263, 5], "def_end_pos": [263, 11]}]], "state_before": "case h.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 Wequiv (fixToW (Quotient.mk Wsetoid x)) x", "state_after": "case h.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 Wequiv\n    (Quotient.lift Wrepr\n      (_ :\n        \u2200 (x y : PFunctor.W (P F)),\n          Wequiv x y \u2192 recF (fun x => PFunctor.W.mk (repr x)) x = recF (fun x => PFunctor.W.mk (repr x)) y)\n      (Quotient.mk Wsetoid x))\n    x"}, {"tactic": "apply Wrepr_equiv", "annotated_tactic": ["apply <a>Wrepr_equiv</a>", [{"full_name": "QPF.Wrepr_equiv", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 20]}]], "state_before": "case h.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 Wequiv\n    (Quotient.lift Wrepr\n      (_ :\n        \u2200 (x y : PFunctor.W (P F)),\n          Wequiv x y \u2192 recF (fun x => PFunctor.W.mk (repr x)) x = recF (fun x => PFunctor.W.mk (repr x)) y)\n      (Quotient.mk Wsetoid x))\n    x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.ordConnectedComponent_empty", "start": [67, 1], "end": [68, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_zero_measure", "start": [567, 1], "end": [569, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.isUnit_singleton", "start": [1131, 1], "end": [1132, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.inv_smul_integral_comp_div_sub", "start": [831, 1], "end": [833, 59], "traced_tactics": [{"tactic": "by_cases hc : c = 0 <;> simp [hc, integral_comp_div_sub]", "annotated_tactic": ["by_cases hc : c = 0 <;> simp [hc, <a>integral_comp_div_sub</a>]", [{"full_name": "intervalIntegral.integral_comp_div_sub", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [825, 9], "def_end_pos": [825, 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 (x / c - d) = \u222b (x : \u211d) in a / c - d..b / c - d, 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.sum_cond", "start": [2055, 1], "end": [2056, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mul_get_eq", "start": [736, 1], "end": [737, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left", "start": [100, 1], "end": [108, 66], "traced_tactics": [{"tactic": "rw [\u2190 kernel.kernel_sum_seq \u03ba]", "annotated_tactic": ["rw [\u2190 <a>kernel.kernel_sum_seq</a> \u03ba]", [{"full_name": "ProbabilityTheory.kernel.kernel_sum_seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 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 : \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a => \u2191\u2191(\u2191\u03ba a) (Prod.mk a \u207b\u00b9' t)", "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 : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t)"}, {"tactic": "have : \u2200 a, kernel.sum (kernel.seq \u03ba) a (Prod.mk a \u207b\u00b9' t) =\n    \u2211' n, kernel.seq \u03ba n a (Prod.mk a \u207b\u00b9' t) := fun a =>\n  kernel.sum_apply' _ _ (measurable_prod_mk_left ht)", "annotated_tactic": ["have : \u2200 a, <a>kernel.sum</a> (<a>kernel.seq</a> \u03ba) a (<a>Prod.mk</a> a \u207b\u00b9' t) =\n      \u2211' n, <a>kernel.seq</a> \u03ba n a (<a>Prod.mk</a> a \u207b\u00b9' t) := fun a =>\n    <a>kernel.sum_apply'</a> _ _ (<a>measurable_prod_mk_left</a> ht)", [{"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.seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [294, 19], "def_end_pos": [294, 22]}, {"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": "ProbabilityTheory.kernel.seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [294, 19], "def_end_pos": [294, 22]}, {"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": "ProbabilityTheory.kernel.sum_apply'", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"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\ninst\u271d : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t)", "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 : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nthis : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t)"}, {"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 : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nthis : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t)", "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 : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nthis : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)\n\u22a2 Measurable fun a => \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)"}, {"tactic": "refine' Measurable.ennreal_tsum fun n => _", "annotated_tactic": ["refine' <a>Measurable.ennreal_tsum</a> fun n => _", [{"full_name": "Measurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [2136, 9], "def_end_pos": [2136, 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\ninst\u271d : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nthis : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)\n\u22a2 Measurable fun a => \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)", "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 : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nthis : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)\nn : \u2115\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)"}, {"tactic": "exact measurable_kernel_prod_mk_left_of_finite ht inferInstance", "annotated_tactic": ["exact <a>measurable_kernel_prod_mk_left_of_finite</a> ht <a>inferInstance</a>", [{"full_name": "ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite", "def_path": "Mathlib/Probability/Kernel/MeasurableIntegral.lean", "def_pos": [40, 9], "def_end_pos": [40, 49]}, {"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\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 : IsSFiniteKernel \u03ba\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nthis : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)\nn : \u2115\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coeFn_add", "start": [232, 1], "end": [233, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.posOf_eq", "start": [328, 1], "end": [328, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.inv_smul_integral_comp_div_add", "start": [781, 1], "end": [783, 59], "traced_tactics": [{"tactic": "by_cases hc : c = 0 <;> simp [hc, integral_comp_div_add]", "annotated_tactic": ["by_cases hc : c = 0 <;> simp [hc, <a>integral_comp_div_add</a>]", [{"full_name": "intervalIntegral.integral_comp_div_add", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [775, 9], "def_end_pos": [775, 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 (x / c + d) = \u222b (x : \u211d) in a / c + d..b / c + d, f x", "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_deleteMin", "start": [248, 1], "end": [258, 96], "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\na : \u03b1\ns' s : Heap \u03b1\neq : deleteMin le s = some (a, s')\n\u22a2 realSize s = realSize s' + 1", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n          (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +\n      1"}, {"tactic": "have : (s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9).HasSize (c.realSize + s.realSize + 1) :=\n  Heap.realSize_findMin (c.realSize + 1) (by simp) (Nat.add_right_comm ..) \u27e80, by simp\u27e9", "annotated_tactic": ["have : (s.findMin le (<a>cons</a> r a c) \u27e8<a>id</a>, a, c, s\u27e9).<a>HasSize</a> (c.realSize + s.realSize + 1) :=\n    <a>Heap.realSize_findMin</a> (c.realSize + 1) (by simp) (<a>Nat.add_right_comm</a> ..) \u27e80, by simp\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": "_private.\u00ablake-packages\u00bb.std.Std.Data.BinomialHeap.Basic.0.Std.BinomialHeap.Imp.FindMin.HasSize", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [224, 13], "def_end_pos": [224, 28]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.Data.BinomialHeap.Basic.0.Std.BinomialHeap.Imp.Heap.realSize_findMin", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [229, 17], "def_end_pos": [229, 38]}, {"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 cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n          (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +\n      1", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nthis :\n  Std.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n    (HeapNode.realSize c + realSize s + 1)\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n          (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +\n      1"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nthis :\n  Std.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n    (HeapNode.realSize c + realSize s + 1)\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n          (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +\n      1", "state_after": "case cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 Std.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n      (HeapNode.realSize c + realSize s + 1) \u2192\n    realSize (cons r a c s) =\n      realSize\n          (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n            (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n              (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +\n        1"}, {"tactic": "match s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9 with\n| { before, val, node, next } =>\n  intro \u27e8m, ih\u2081, ih\u2082\u27e9; dsimp only at ih\u2081 ih\u2082\n  rw [realSize, Nat.add_right_comm, ih\u2082]\n  simp only [realSize_merge, HeapNode.realSize_toHeap, ih\u2081, Nat.add_assoc, Nat.add_left_comm]", "annotated_tactic": ["match s.findMin le (<a>cons</a> r a c) \u27e8<a>id</a>, a, c, s\u27e9 with\n  | { before, val, node, next } =>\n    intro \u27e8m, ih\u2081, ih\u2082\u27e9; dsimp only at ih\u2081 ih\u2082\n    rw [<a>realSize</a>, <a>Nat.add_right_comm</a>, ih\u2082]\n    simp only [<a>realSize_merge</a>, <a>HeapNode.realSize_toHeap</a>, ih\u2081, <a>Nat.add_assoc</a>, <a>Nat.add_left_comm</a>]", [{"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.Heap.realSize", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [85, 13], "def_end_pos": [85, 26]}, {"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.BinomialHeap.Imp.Heap.realSize_merge", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 28]}, {"full_name": "Std.BinomialHeap.Imp.HeapNode.realSize_toHeap", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 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]}, {"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 cons.refl\n\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 Std.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n      (HeapNode.realSize c + realSize s + 1) \u2192\n    realSize (cons r a c s) =\n      realSize\n          (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n            (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n              (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +\n        1", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 \u2200 (s : Heap \u03b1), realSize (cons r a c s) = HeapNode.realSize c + 1 + realSize s", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\n\u22a2 (\u2200 (s_1 : Heap \u03b1),\n      realSize (FindMin.before { before := id, val := a, node := c, next := s } s_1) = 0 + realSize s_1) \u2227\n    HeapNode.realSize c + realSize s + 1 =\n      0 + HeapNode.realSize { before := id, val := a, node := c, next := s }.node +\n          realSize { before := id, val := a, node := c, next := s }.next +\n        1", "state_after": "no goals"}, {"tactic": "intro \u27e8m, ih\u2081, ih\u2082\u27e9", "annotated_tactic": ["intro \u27e8m, ih\u2081, ih\u2082\u27e9", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\n\u22a2 Std.BinomialHeap.Imp.FindMin.HasSize { before := before, val := val, node := node, next := next }\n      (HeapNode.realSize c + realSize s + 1) \u2192\n    realSize (cons r a c s) =\n      realSize\n          (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)) +\n        1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nm : Nat\nih\u2081 :\n  \u2200 (s : Heap \u03b1),\n    realSize (FindMin.before { before := before, val := val, node := node, next := next } s) = m + realSize s\nih\u2082 :\n  HeapNode.realSize c + realSize s + 1 =\n    m + HeapNode.realSize { before := before, val := val, node := node, next := next }.node +\n        realSize { before := before, val := val, node := node, next := next }.next +\n      1\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (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)) +\n      1"}, {"tactic": "dsimp only at ih\u2081 ih\u2082", "annotated_tactic": ["dsimp only at ih\u2081 ih\u2082", []], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nm : Nat\nih\u2081 :\n  \u2200 (s : Heap \u03b1),\n    realSize (FindMin.before { before := before, val := val, node := node, next := next } s) = m + realSize s\nih\u2082 :\n  HeapNode.realSize c + realSize s + 1 =\n    m + HeapNode.realSize { before := before, val := val, node := node, next := next }.node +\n        realSize { before := before, val := val, node := node, next := next }.next +\n      1\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (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)) +\n      1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nm : Nat\nih\u2081 : \u2200 (s : Heap \u03b1), realSize (before s) = m + realSize s\nih\u2082 : HeapNode.realSize c + realSize s + 1 = m + HeapNode.realSize node + realSize next + 1\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (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)) +\n      1"}, {"tactic": "rw [realSize, Nat.add_right_comm, ih\u2082]", "annotated_tactic": ["rw [<a>realSize</a>, <a>Nat.add_right_comm</a>, ih\u2082]", [{"full_name": "Std.BinomialHeap.Imp.Heap.realSize", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [85, 13], "def_end_pos": [85, 26]}, {"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\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nm : Nat\nih\u2081 : \u2200 (s : Heap \u03b1), realSize (before s) = m + realSize s\nih\u2082 : HeapNode.realSize c + realSize s + 1 = m + HeapNode.realSize node + realSize next + 1\n\u22a2 realSize (cons r a c s) =\n    realSize\n        (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)) +\n      1", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nm : Nat\nih\u2081 : \u2200 (s : Heap \u03b1), realSize (before s) = m + realSize s\nih\u2082 : HeapNode.realSize c + realSize s + 1 = m + HeapNode.realSize node + realSize next + 1\n\u22a2 m + HeapNode.realSize node + realSize next + 1 =\n    realSize\n        (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)) +\n      1"}, {"tactic": "simp only [realSize_merge, HeapNode.realSize_toHeap, ih\u2081, Nat.add_assoc, Nat.add_left_comm]", "annotated_tactic": ["simp only [<a>realSize_merge</a>, <a>HeapNode.realSize_toHeap</a>, ih\u2081, <a>Nat.add_assoc</a>, <a>Nat.add_left_comm</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.realSize_merge", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 28]}, {"full_name": "Std.BinomialHeap.Imp.HeapNode.realSize_toHeap", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 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]}, {"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\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nr : Nat\na : \u03b1\nc : HeapNode \u03b1\ns : Heap \u03b1\nbefore : Heap \u03b1 \u2192 Heap \u03b1\nval : \u03b1\nnode : HeapNode \u03b1\nnext : Heap \u03b1\nm : Nat\nih\u2081 : \u2200 (s : Heap \u03b1), realSize (before s) = m + realSize s\nih\u2082 : HeapNode.realSize c + realSize s + 1 = m + HeapNode.realSize node + realSize next + 1\n\u22a2 m + HeapNode.realSize node + realSize next + 1 =\n    realSize\n        (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)) +\n      1", "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.neg_add_lt_right_of_lt_add", "start": [1102, 11], "end": [1104, 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 < b + c\n\u22a2 -c + a < b", "state_after": "a b c : Int\nh : a < c + b\n\u22a2 -c + a < b"}, {"tactic": "exact Int.neg_add_lt_left_of_lt_add h", "annotated_tactic": ["exact <a>Int.neg_add_lt_left_of_lt_add</a> h", [{"full_name": "Int.neg_add_lt_left_of_lt_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1094, 19], "def_end_pos": [1094, 44]}]], "state_before": "a b c : Int\nh : a < c + b\n\u22a2 -c + a < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "ProbabilityTheory.condexp_ae_eq_integral_condexpKernel'", "start": [189, 1], "end": [195, 55], "traced_tactics": [{"tactic": "have hX : @Measurable \u03a9 \u03a9 m\u03a9 (m \u2293 m\u03a9) id := measurable_id.mono le_rfl (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)", "annotated_tactic": ["have hX : @<a>Measurable</a> \u03a9 \u03a9 m\u03a9 (m \u2293 m\u03a9) <a>id</a> := measurable_id.mono <a>le_rfl</a> (<a>inf_le_right</a> : m \u2293 m\u03a9 \u2264 m\u03a9)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9", "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\nhX : Measurable id\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9"}, {"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_int : Integrable f\nhX : Measurable id\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9", "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\nhX : Measurable id\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)"}, {"tactic": "have h := condexp_ae_eq_integral_condDistrib_id hX hf_int", "annotated_tactic": ["have h := <a>condexp_ae_eq_integral_condDistrib_id</a> hX hf_int", [{"full_name": "ProbabilityTheory.condexp_ae_eq_integral_condDistrib_id", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [338, 9], "def_end_pos": [338, 46]}]], "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\nhX : Measurable id\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)", "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\nhX : Measurable id\nh : \u03bc[f|MeasurableSpace.comap id (m \u2293 m\u03a9)] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id a)\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)"}, {"tactic": "simpa only [MeasurableSpace.comap_id, id_eq] using h", "annotated_tactic": ["simpa only [<a>MeasurableSpace.comap_id</a>, <a>id_eq</a>] using h", [{"full_name": "MeasurableSpace.comap_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [117, 9], "def_end_pos": [117, 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": "\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\nhX : Measurable id\nh : \u03bc[f|MeasurableSpace.comap id (m \u2293 m\u03a9)] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id a)\n\u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.fiber_card_ne_zero_iff_mem_image", "start": [255, 1], "end": [257, 65], "traced_tactics": [{"tactic": "rw [\u2190 pos_iff_ne_zero, card_pos, fiber_nonempty_iff_mem_image]", "annotated_tactic": ["rw [\u2190 <a>pos_iff_ne_zero</a>, <a>card_pos</a>, <a>fiber_nonempty_iff_mem_image</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": "Finset.card_pos", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}, {"full_name": "Finset.fiber_nonempty_iff_mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [387, 9], "def_end_pos": [387, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b2\ny : \u03b2\n\u22a2 card (filter (fun x => f x = y) s) \u2260 0 \u2194 y \u2208 image f s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.withDensity_preCdf", "start": [302, 1], "end": [304, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.mem\u2112p_const_iff", "start": [364, 1], "end": [367, 65], "traced_tactics": [{"tactic": "rw [\u2190 snorm_const_lt_top_iff hp_ne_zero hp_ne_top]", "annotated_tactic": ["rw [\u2190 <a>snorm_const_lt_top_iff</a> hp_ne_zero hp_ne_top]", [{"full_name": "MeasureTheory.snorm_const_lt_top_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [325, 9], "def_end_pos": [325, 31]}]], "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\nc : E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => c) p \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4", "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\nc : E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => c) p \u2194 snorm (fun x => c) p \u03bc < \u22a4"}, {"tactic": "exact \u27e8fun h => h.2, fun h => \u27e8aestronglyMeasurable_const, h\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun h => h.2, fun h => \u27e8<a>aestronglyMeasurable_const</a>, h\u27e9\u27e9", [{"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\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\nc : E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => c) p \u2194 snorm (fun x => c) p \u03bc < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "full_name": "MeasureTheory.Submartingale.exists_tendsto_of_abs_bddAbove_aux", "start": [148, 1], "end": [169, 26], "traced_tactics": [{"tactic": "have ht :\n  \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f i n) \u03c9) atTop (\ud835\udcdd c) := by\n  rw [ae_all_iff]\n  exact fun i => Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)\n    (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd)", "annotated_tactic": ["have ht :\n    \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, \u2203 c, <a>Tendsto</a> (fun n => <a>stoppedValue</a> f (<a>leastGE</a> f i n) \u03c9) <a>atTop</a> (\ud835\udcdd c) := by\n    rw [<a>ae_all_iff</a>]\n    exact fun i => <a>Submartingale.exists_ae_tendsto_of_bdd</a> (hf.stoppedValue_leastGE i)\n      (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"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.Submartingale.exists_ae_tendsto_of_bdd", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [196, 9], "def_end_pos": [196, 47]}]], "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\nhf0 : f 0 = 0\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) \u2192 \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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2192 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "filter_upwards [ht] with \u03c9 h\u03c9 h\u03c9b", "annotated_tactic": ["filter_upwards [ht] with \u03c9 h\u03c9 h\u03c9b", []], "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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2192 \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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : BddAbove (Set.range fun n => f n \u03c9)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "rw [BddAbove] at h\u03c9b", "annotated_tactic": ["rw [<a>BddAbove</a>] at h\u03c9b", [{"full_name": "BddAbove", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [56, 5], "def_end_pos": [56, 13]}]], "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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : BddAbove (Set.range fun n => f n \u03c9)\n\u22a2 \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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "obtain \u27e8i, hi\u27e9 := exists_nat_gt h\u03c9b.some", "annotated_tactic": ["obtain \u27e8i, hi\u27e9 := <a>exists_nat_gt</a> h\u03c9b.some", [{"full_name": "exists_nat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}]], "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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h.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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "have hib : \u2200 n, f n \u03c9 < i := by\n  intro n\n  exact lt_of_le_of_lt ((mem_upperBounds.1 h\u03c9b.some_mem) _ \u27e8n, rfl\u27e9) hi", "annotated_tactic": ["have hib : \u2200 n, f n \u03c9 < i := by\n    intro n\n    exact <a>lt_of_le_of_lt</a> ((<a>mem_upperBounds</a>.1 h\u03c9b.some_mem) _ \u27e8n, <a>rfl</a>\u27e9) hi", [{"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": "mem_upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 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 h.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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h.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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "have heq : \u2200 n, stoppedValue f (leastGE f i n) \u03c9 = f n \u03c9 := by\n  intro n\n  rw [leastGE]; unfold hitting; rw [stoppedValue]\n  rw [if_neg]\n  simp only [Set.mem_Icc, Set.mem_union, Set.mem_Ici]\n  push_neg\n  exact fun j _ => hib j", "annotated_tactic": ["have heq : \u2200 n, <a>stoppedValue</a> f (<a>leastGE</a> f i n) \u03c9 = f n \u03c9 := by\n    intro n\n    rw [<a>leastGE</a>]; unfold <a>hitting</a>; rw [<a>stoppedValue</a>]\n    rw [<a>if_neg</a>]\n    simp only [<a>Set.mem_Icc</a>, <a>Set.mem_union</a>, <a>Set.mem_Ici</a>]\n    push_neg\n    exact fun j _ => hib j", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 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": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"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_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}]], "state_before": "case h.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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "case h.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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nheq : \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)"}, {"tactic": "simp only [\u2190 heq, h\u03c9 i]", "annotated_tactic": ["simp only [\u2190 heq, h\u03c9 i]", []], "state_before": "case h.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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nheq : \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)", "state_after": "no goals"}, {"tactic": "rw [ae_all_iff]", "annotated_tactic": ["rw [<a>ae_all_iff</a>]", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "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\nhf0 : f 0 = 0\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), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) 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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) a) atTop (\ud835\udcdd c)"}, {"tactic": "exact fun i => Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)\n  (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd)", "annotated_tactic": ["exact fun i => <a>Submartingale.exists_ae_tendsto_of_bdd</a> (hf.stoppedValue_leastGE i)\n      (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd)", [{"full_name": "MeasureTheory.Submartingale.exists_ae_tendsto_of_bdd", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [196, 9], "def_end_pos": [196, 47]}]], "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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) a) atTop (\ud835\udcdd c)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\n\u22a2 \u2200 (n : \u2115), f n \u03c9 < \u2191i", "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nn : \u2115\n\u22a2 f n \u03c9 < \u2191i"}, {"tactic": "exact lt_of_le_of_lt ((mem_upperBounds.1 h\u03c9b.some_mem) _ \u27e8n, rfl\u27e9) hi", "annotated_tactic": ["exact <a>lt_of_le_of_lt</a> ((<a>mem_upperBounds</a>.1 h\u03c9b.some_mem) _ \u27e8n, <a>rfl</a>\u27e9) hi", [{"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": "mem_upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 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": "\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nn : \u2115\n\u22a2 f n \u03c9 < \u2191i", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\n\u22a2 \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9", "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9"}, {"tactic": "rw [leastGE]", "annotated_tactic": ["rw [<a>leastGE</a>]", [{"full_name": "MeasureTheory.leastGE", "def_path": "Mathlib/Probability/Martingale/BorelCantelli.lean", "def_pos": [50, 19], "def_end_pos": [50, 26]}]], "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9", "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f (hitting f (Set.Ici \u2191i) 0 n) \u03c9 = f n \u03c9"}, {"tactic": "unfold hitting", "annotated_tactic": ["unfold <a>hitting</a>", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}]], "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f (hitting f (Set.Ici \u2191i) 0 n) \u03c9 = f n \u03c9", "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f\n      (fun x =>\n        if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 x \u2208 Set.Ici \u2191i}) else n)\n      \u03c9 =\n    f n \u03c9"}, {"tactic": "rw [stoppedValue]", "annotated_tactic": ["rw [<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": "\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f\n      (fun x =>\n        if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 x \u2208 Set.Ici \u2191i}) else n)\n      \u03c9 =\n    f n \u03c9", "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\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 f (if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 \u03c9 \u2208 Set.Ici \u2191i}) else n) \u03c9 =\n    f n \u03c9"}, {"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": "\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 f (if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 \u03c9 \u2208 Set.Ici \u2191i}) else n) \u03c9 =\n    f n \u03c9", "state_after": "case hnc\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u00ac\u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i"}, {"tactic": "simp only [Set.mem_Icc, Set.mem_union, Set.mem_Ici]", "annotated_tactic": ["simp only [<a>Set.mem_Icc</a>, <a>Set.mem_union</a>, <a>Set.mem_Ici</a>]", [{"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_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [767, 9], "def_end_pos": [767, 18]}, {"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 hnc\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u00ac\u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i", "state_after": "case hnc\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u00ac\u2203 j, (0 \u2264 j \u2227 j \u2264 n) \u2227 \u2191i \u2264 f j \u03c9"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "case hnc\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u00ac\u2203 j, (0 \u2264 j \u2227 j \u2264 n) \u2227 \u2191i \u2264 f j \u03c9", "state_after": "case hnc\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u2200 (j : \u2115), 0 \u2264 j \u2227 j \u2264 n \u2192 f j \u03c9 < \u2191i"}, {"tactic": "exact fun j _ => hib j", "annotated_tactic": ["exact fun j _ => hib j", []], "state_before": "case hnc\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\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u2200 (j : \u2115), 0 \u2264 j \u2227 j \u2264 n \u2192 f j \u03c9 < \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero", "start": [520, 1], "end": [530, 27], "traced_tactics": [{"tactic": "have hf_Lp : Mem\u2112p f 1 \u03bc := mem\u2112p_one_iff_integrable.mpr hf", "annotated_tactic": ["have hf_Lp : <a>Mem\u2112p</a> f 1 \u03bc := mem\u2112p_one_iff_integrable.mpr hf", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}]], "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 : \u03b1 \u2192 E\nhf : Integrable f\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\n\u22a2 f =\u1d50[\u03bc] 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "let f_Lp := hf_Lp.toLp f", "annotated_tactic": ["let f_Lp := hf_Lp.toLp f", []], "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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\n\u22a2 f =\u1d50[\u03bc] 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have hf_f_Lp : f =\u1d50[\u03bc] f_Lp := (Mem\u2112p.coeFn_toLp hf_Lp).symm", "annotated_tactic": ["have hf_f_Lp : f =\u1d50[\u03bc] f_Lp := (<a>Mem\u2112p.coeFn_toLp</a> hf_Lp).<a>symm</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": "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\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\n\u22a2 f =\u1d50[\u03bc] 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "refine' hf_f_Lp.trans _", "annotated_tactic": ["refine' hf_f_Lp.trans _", []], "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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 f =\u1d50[\u03bc] 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2191\u2191f_Lp =\u1d50[\u03bc] 0"}, {"tactic": "refine' Lp.ae_eq_zero_of_forall_set_integral_eq_zero f_Lp one_ne_zero ENNReal.coe_ne_top _ _", "annotated_tactic": ["refine' <a>Lp.ae_eq_zero_of_forall_set_integral_eq_zero</a> f_Lp <a>one_ne_zero</a> <a>ENNReal.coe_ne_top</a> _ _", [{"full_name": "MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [472, 9], "def_end_pos": [472, 53]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"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\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2191\u2191f_Lp =\u1d50[\u03bc] 0", "state_after": "case refine'_1\n\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f_Lp) s\n\ncase refine'_2\n\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0"}, {"tactic": "exact fun s _ _ => Integrable.integrableOn (L1.integrable_coeFn _)", "annotated_tactic": ["exact fun s _ _ => <a>Integrable.integrableOn</a> (<a>L1.integrable_coeFn</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.L1.integrable_coeFn", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 25]}]], "state_before": "case refine'_1\n\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f_Lp) s", "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\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0", "state_after": "case refine'_2\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0"}, {"tactic": "rw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)]", "annotated_tactic": ["rw [<a>integral_congr_ae</a> (<a>ae_restrict_of_ae</a> hf_f_Lp.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.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\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0", "state_after": "case refine'_2\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f 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\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 : \u03b1 \u2192 E\nhf : Integrable f\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\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.inter_get_eq", "start": [836, 1], "end": [838, 29], "traced_tactics": [{"tactic": "simp [inter_def]", "annotated_tactic": ["simp [<a>inter_def</a>]", [{"full_name": "Part.inter_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [707, 9], "def_end_pos": [707, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inter \u03b1\na b : Part \u03b1\nhab : (a \u2229 b).Dom\n\u22a2 get (a \u2229 b) hab = get a (_ : a.Dom) \u2229 get b (_ : b.Dom)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inter \u03b1\na b : Part \u03b1\nhab : (a \u2229 b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \u2229 x) b) (_ : (Part.bind a fun y => map (fun x => y \u2229 x) b).Dom) =\n    get a (_ : a.Dom) \u2229 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 : Inter \u03b1\na b : Part \u03b1\nhab : (a \u2229 b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \u2229 x) b) (_ : (Part.bind a fun y => map (fun x => y \u2229 x) b).Dom) =\n    get a (_ : a.Dom) \u2229 get b (_ : b.Dom)", "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_left", "start": [1896, 1], "end": [1897, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/BinomialHeap/Lemmas.lean", "full_name": "Std.BinomialHeap.Imp.Heap.WF.size_eq", "start": [28, 9], "end": [33, 72], "traced_tactics": [{"tactic": "simp [size, Nat.shiftLeft, size_eq h\u2082, Nat.pow_succ, Nat.mul_succ]", "annotated_tactic": ["simp [<a>size</a>, <a>Nat.shiftLeft</a>, size_eq h\u2082, <a>Nat.pow_succ</a>, <a>Nat.mul_succ</a>]", [{"full_name": "Std.BinomialHeap.Imp.Heap.size", "def_path": "lake-packages/std/Std/Data/BinomialHeap/Basic.lean", "def_pos": [90, 5], "def_end_pos": [90, 14]}, {"full_name": "Nat.shiftLeft", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Bitwise.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"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_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn a\u271d\u00b3 : Nat\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : HeapNode \u03b1\na\u271d : Heap \u03b1\nleft\u271d : n \u2264 a\u271d\u00b3\nh\u2081 : HeapNode.WF le a\u271d\u00b2 a\u271d\u00b9 a\u271d\u00b3\nh\u2082 : WF le (a\u271d\u00b3 + 1) a\u271d\n\u22a2 size (cons a\u271d\u00b3 a\u271d\u00b2 a\u271d\u00b9 a\u271d) = realSize (cons a\u271d\u00b3 a\u271d\u00b2 a\u271d\u00b9 a\u271d)", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn a\u271d\u00b3 : Nat\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : HeapNode \u03b1\na\u271d : Heap \u03b1\nleft\u271d : n \u2264 a\u271d\u00b3\nh\u2081 : HeapNode.WF le a\u271d\u00b2 a\u271d\u00b9 a\u271d\u00b3\nh\u2082 : WF le (a\u271d\u00b3 + 1) a\u271d\n\u22a2 2 ^ a\u271d\u00b3 + realSize a\u271d = HeapNode.realSize a\u271d\u00b9 + 1 + realSize a\u271d"}, {"tactic": "simp [Nat.add_assoc, Nat.one_shiftLeft, h\u2081.realSize_eq, h\u2082.size_eq]", "annotated_tactic": ["simp [<a>Nat.add_assoc</a>, <a>Nat.one_shiftLeft</a>, h\u2081.realSize_eq, h\u2082.size_eq]", [{"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.one_shiftLeft", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [797, 9], "def_end_pos": [797, 22]}]], "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\nn a\u271d\u00b3 : Nat\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : HeapNode \u03b1\na\u271d : Heap \u03b1\nleft\u271d : n \u2264 a\u271d\u00b3\nh\u2081 : HeapNode.WF le a\u271d\u00b2 a\u271d\u00b9 a\u271d\u00b3\nh\u2082 : WF le (a\u271d\u00b3 + 1) a\u271d\n\u22a2 2 ^ a\u271d\u00b3 + realSize a\u271d = HeapNode.realSize a\u271d\u00b9 + 1 + realSize a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae_aux", "start": [831, 1], "end": [842, 44], "traced_tactics": [{"tactic": "rcases exists_absolutelyContinuous_isFiniteMeasure \u03bc with \u27e8\u03bd, h\u03bd, h\u03bc\u03bd\u27e9", "annotated_tactic": ["rcases <a>exists_absolutelyContinuous_isFiniteMeasure</a> \u03bc with \u27e8\u03bd, h\u03bd, h\u03bc\u03bd\u27e9", [{"full_name": "MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [456, 9], "def_end_pos": [456, 52]}]], "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 : SigmaFinite \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\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 : SigmaFinite \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\u03bd : Measure \u03b1\nh\u03bd : IsFiniteMeasure \u03bd\nh\u03bc\u03bd : \u03bc \u226a \u03bd\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": "rcases exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux \u03bd f s hf with\n  \u27e8t, t_count, ts, tr, t\u03bd, tdisj\u27e9", "annotated_tactic": ["rcases <a>exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux</a> \u03bd f s hf with\n    \u27e8t, t_count, ts, tr, t\u03bd, tdisj\u27e9", [{"full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [686, 9], "def_end_pos": [686, 68]}]], "state_before": "case 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 : SigmaFinite \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\u03bd : Measure \u03b1\nh\u03bd : IsFiniteMeasure \u03bd\nh\u03bc\u03bd : \u03bc \u226a \u03bd\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.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 : SigmaFinite \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\u03bd : Measure \u03b1\nh\u03bd : IsFiniteMeasure \u03bd\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set (\u03b1 \u00d7 \u211d)\nt_count : Set.Countable t\nts : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s\ntr : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt\u03bd : \u2191\u2191\u03bd (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0\ntdisj : PairwiseDisjoint t fun p => 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": "exact \u27e8t, t_count, ts, tr, h\u03bc\u03bd t\u03bd, tdisj\u27e9", "annotated_tactic": ["exact \u27e8t, t_count, ts, tr, h\u03bc\u03bd t\u03bd, tdisj\u27e9", []], "state_before": "case intro.intro.intro.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 : SigmaFinite \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\u03bd : Measure \u03b1\nh\u03bd : IsFiniteMeasure \u03bd\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set (\u03b1 \u00d7 \u211d)\nt_count : Set.Countable t\nts : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s\ntr : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1\nt\u03bd : \u2191\u2191\u03bd (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0\ntdisj : PairwiseDisjoint t fun p => 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": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.gcd_least_linear", "start": [432, 1], "end": [438, 53], "traced_tactics": [{"tactic": "simp_rw [\u2190 gcd_dvd_iff]", "annotated_tactic": ["simp_rw [\u2190 <a>gcd_dvd_iff</a>]", [{"full_name": "Int.gcd_dvd_iff", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [392, 9], "def_end_pos": [392, 20]}]], "state_before": "a b : \u2124\nha : a \u2260 0\n\u22a2 IsLeast {n | 0 < n \u2227 \u2203 x y, \u2191n = a * x + b * y} (gcd a b)", "state_after": "a b : \u2124\nha : a \u2260 0\n\u22a2 IsLeast {n | 0 < n \u2227 gcd a b \u2223 n} (gcd a b)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "a b : \u2124\nha : a \u2260 0\n\u22a2 IsLeast {n | 0 < n \u2227 gcd a b \u2223 n} (gcd a b)", "state_after": "case left\na b : \u2124\nha : a \u2260 0\n\u22a2 gcd a b \u2208 {n | 0 < n \u2227 gcd a b \u2223 n}\n\ncase right\na b : \u2124\nha : a \u2260 0\n\u22a2 gcd a b \u2208 lowerBounds {n | 0 < n \u2227 gcd a b \u2223 n}"}, {"tactic": "simpa [and_true_iff, dvd_refl, Set.mem_setOf_eq] using gcd_pos_of_ne_zero_left b ha", "annotated_tactic": ["simpa [<a>and_true_iff</a>, <a>dvd_refl</a>, <a>Set.mem_setOf_eq</a>] using <a>gcd_pos_of_ne_zero_left</a> b ha", [{"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}, {"full_name": "dvd_refl", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [121, 9], "def_end_pos": [121, 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": "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 left\na b : \u2124\nha : a \u2260 0\n\u22a2 gcd a b \u2208 {n | 0 < n \u2227 gcd a b \u2223 n}", "state_after": "no goals"}, {"tactic": "simp only [lowerBounds, and_imp, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>lowerBounds</a>, <a>and_imp</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}, {"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_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case right\na b : \u2124\nha : a \u2260 0\n\u22a2 gcd a b \u2208 lowerBounds {n | 0 < n \u2227 gcd a b \u2223 n}", "state_after": "case right\na b : \u2124\nha : a \u2260 0\n\u22a2 \u2200 \u2983a_1 : \u2115\u2984, 0 < a_1 \u2192 gcd a b \u2223 a_1 \u2192 gcd a b \u2264 a_1"}, {"tactic": "exact fun n hn_pos hn => Nat.le_of_dvd hn_pos hn", "annotated_tactic": ["exact fun n hn_pos hn => <a>Nat.le_of_dvd</a> hn_pos hn", [{"full_name": "Nat.le_of_dvd", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [907, 9], "def_end_pos": [907, 18]}]], "state_before": "case right\na b : \u2124\nha : a \u2260 0\n\u22a2 \u2200 \u2983a_1 : \u2115\u2984, 0 < a_1 \u2192 gcd a b \u2223 a_1 \u2192 gcd a b \u2264 a_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "full_name": "MeasureTheory.Martingale.stoppedValue_ae_eq_restrict_eq", "start": [83, 1], "end": [91, 29], "traced_tactics": [{"tactic": "refine' Filter.EventuallyEq.trans _\n  (condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const h h\u03c4 h\u03c4_le i).symm", "annotated_tactic": ["refine' <a>Filter.EventuallyEq.trans</a> _\n    (<a>condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const</a> h h\u03c4 h\u03c4_le i).<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.Martingale.condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const", "def_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "def_pos": [67, 9], "def_end_pos": [67, 66]}, {"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\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] f i"}, {"tactic": "rw [Filter.EventuallyEq, ae_restrict_iff' (\u2131.le _ _ (h\u03c4.measurableSet_eq i))]", "annotated_tactic": ["rw [<a>Filter.EventuallyEq</a>, <a>ae_restrict_iff'</a> (\u2131.le _ _ (h\u03c4.measurableSet_eq i))]", [{"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": [2572, 9], "def_end_pos": [2572, 25]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] f i", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 {\u03c9 | \u03c4 \u03c9 = i} \u2192 stoppedValue f \u03c4 x = f i x"}, {"tactic": "refine' Filter.eventually_of_forall fun x hx => _", "annotated_tactic": ["refine' <a>Filter.eventually_of_forall</a> fun x hx => _", [{"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\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 {\u03c9 | \u03c4 \u03c9 = i} \u2192 stoppedValue f \u03c4 x = f i x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\nx : \u03a9\nhx : x \u2208 {\u03c9 | \u03c4 \u03c9 = i}\n\u22a2 stoppedValue f \u03c4 x = f i x"}, {"tactic": "rw [Set.mem_setOf_eq] at hx", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>] at hx", [{"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\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\nx : \u03a9\nhx : x \u2208 {\u03c9 | \u03c4 \u03c9 = i}\n\u22a2 stoppedValue f \u03c4 x = f i x", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\nx : \u03a9\nhx : \u03c4 x = i\n\u22a2 stoppedValue f \u03c4 x = f i x"}, {"tactic": "simp_rw [stoppedValue, hx]", "annotated_tactic": ["simp_rw [<a>stoppedValue</a>, hx]", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [768, 5], "def_end_pos": [768, 17]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nf : \u03b9 \u2192 \u03a9 \u2192 E\ni\u271d n : \u03b9\nh : Martingale f \u2131 \u03bc\nh\u03c4 : IsStoppingTime \u2131 \u03c4\nh\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\ni : \u03b9\nx : \u03a9\nhx : \u03c4 x = i\n\u22a2 stoppedValue f \u03c4 x = f i x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.mkMetric_top", "start": [490, 1], "end": [492, 78], "traced_tactics": [{"tactic": "apply toOuterMeasure_injective", "annotated_tactic": ["apply <a>toOuterMeasure_injective</a>", [{"full_name": "MeasureTheory.Measure.toOuterMeasure_injective", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [129, 9], "def_end_pos": [129, 33]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\n\u22a2 (mkMetric fun x => \u22a4) = \u22a4", "state_after": "case a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\n\u22a2 \u2191(mkMetric fun x => \u22a4) = \u2191\u22a4"}, {"tactic": "rw [mkMetric_toOuterMeasure, OuterMeasure.mkMetric_top, toOuterMeasure_top]", "annotated_tactic": ["rw [<a>mkMetric_toOuterMeasure</a>, <a>OuterMeasure.mkMetric_top</a>, <a>toOuterMeasure_top</a>]", [{"full_name": "MeasureTheory.Measure.mkMetric_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [464, 9], "def_end_pos": [464, 32]}, {"full_name": "MeasureTheory.OuterMeasure.mkMetric_top", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [357, 9], "def_end_pos": [357, 21]}, {"full_name": "MeasureTheory.Measure.toOuterMeasure_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1076, 9], "def_end_pos": [1076, 27]}]], "state_before": "case a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\n\u22a2 \u2191(mkMetric fun x => \u22a4) = \u2191\u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.indepFun_finset", "start": [584, 1], "end": [588, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.decode\u2082", "start": [766, 11], "end": [768, 95], "traced_tactics": [{"tactic": "exact encode_iff.mpr snd", "annotated_tactic": ["exact encode_iff.mpr <a>snd</a>", [{"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\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\n\u22a2 Primrec\u2082 fun p a => encode a", "state_after": "no goals"}, {"tactic": "exact fst.comp fst", "annotated_tactic": ["exact fst.comp <a>fst</a>", [{"full_name": "Primrec.fst", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [340, 9], "def_end_pos": [340, 12]}]], "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\n\u22a2 Primrec\u2082 fun p a => p.1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Funext.lean", "full_name": "MvPolynomial.funext", "start": [46, 1], "end": [59, 63], "traced_tactics": [{"tactic": "suffices \u2200 p, (\u2200 x : \u03c3 \u2192 R, eval x p = 0) \u2192 p = 0 by\n  rw [\u2190 sub_eq_zero, this (p - q)]\n  simp only [h, RingHom.map_sub, forall_const, sub_self]", "annotated_tactic": ["suffices \u2200 p, (\u2200 x : \u03c3 \u2192 R, <a>eval</a> x p = 0) \u2192 p = 0 by\n    rw [\u2190 <a>sub_eq_zero</a>, this (p - q)]\n    simp only [h, <a>RingHom.map_sub</a>, <a>forall_const</a>, <a>sub_self</a>]", [{"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [618, 19], "def_end_pos": [618, 26]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np q : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q\n\u22a2 p = q", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np q : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0"}, {"tactic": "clear h p q", "annotated_tactic": ["clear h p q", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np q : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0"}, {"tactic": "intro p h", "annotated_tactic": ["intro p h", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0"}, {"tactic": "obtain \u27e8n, f, hf, p, rfl\u27e9 := exists_fin_rename p", "annotated_tactic": ["obtain \u27e8n, f, hf, p, rfl\u27e9 := <a>exists_fin_rename</a> p", [{"full_name": "MvPolynomial.exists_fin_rename", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [272, 9], "def_end_pos": [272, 26]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0\n\u22a2 p = 0", "state_after": "case intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\n\u22a2 \u2191(rename f) p = 0"}, {"tactic": "suffices p = 0 by rw [this, AlgHom.map_zero]", "annotated_tactic": ["suffices p = 0 by rw [this, <a>AlgHom.map_zero</a>]", [{"full_name": "AlgHom.map_zero", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [248, 19], "def_end_pos": [248, 27]}]], "state_before": "case intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\n\u22a2 \u2191(rename f) p = 0", "state_after": "case intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\n\u22a2 p = 0"}, {"tactic": "apply funext_fin", "annotated_tactic": ["apply <a>funext_fin</a>", [{"full_name": "_private.Mathlib.Data.MvPolynomial.Funext.0.MvPolynomial.funext_fin", "def_path": "Mathlib/Data/MvPolynomial/Funext.lean", "def_pos": [30, 17], "def_end_pos": [30, 27]}]], "state_before": "case intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\n\u22a2 p = 0", "state_after": "case intro.intro.intro.intro.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\n\u22a2 \u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case intro.intro.intro.intro.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\n\u22a2 \u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0", "state_after": "case intro.intro.intro.intro.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) p = 0"}, {"tactic": "classical\n  convert h (Function.extend f x 0)\n  simp only [eval, eval\u2082Hom_rename, Function.extend_comp hf]", "annotated_tactic": ["classical\n    convert h (<a>Function.extend</a> f x 0)\n    simp only [<a>eval</a>, <a>eval\u2082Hom_rename</a>, <a>Function.extend_comp</a> hf]", [{"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [711, 5], "def_end_pos": [711, 11]}, {"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}, {"full_name": "MvPolynomial.eval\u2082Hom_rename", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [191, 9], "def_end_pos": [191, 24]}, {"full_name": "Function.extend_comp", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [790, 9], "def_end_pos": [790, 20]}]], "state_before": "case intro.intro.intro.intro.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) p = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 sub_eq_zero, this (p - q)]", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>, this (p - q)]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np q : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q\nthis : \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\n\u22a2 p = q", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np q : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q\nthis : \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\n\u22a2 \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (p - q) = 0"}, {"tactic": "simp only [h, RingHom.map_sub, forall_const, sub_self]", "annotated_tactic": ["simp only [h, <a>RingHom.map_sub</a>, <a>forall_const</a>, <a>sub_self</a>]", [{"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [618, 19], "def_end_pos": [618, 26]}, {"full_name": "forall_const", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [435, 17], "def_end_pos": [435, 29]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\np q : MvPolynomial \u03c3 R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q\nthis : \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0\n\u22a2 \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (p - q) = 0", "state_after": "no goals"}, {"tactic": "rw [this, AlgHom.map_zero]", "annotated_tactic": ["rw [this, <a>AlgHom.map_zero</a>]", [{"full_name": "AlgHom.map_zero", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [248, 19], "def_end_pos": [248, 27]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\nthis : p = 0\n\u22a2 \u2191(rename f) p = 0", "state_after": "no goals"}, {"tactic": "convert h (Function.extend f x 0)", "annotated_tactic": ["convert h (<a>Function.extend</a> f x 0)", [{"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [711, 5], "def_end_pos": [711, 11]}]], "state_before": "case intro.intro.intro.intro.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) p = 0", "state_after": "case h.e'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) p = \u2191(eval (Function.extend f x 0)) (\u2191(rename f) p)"}, {"tactic": "simp only [eval, eval\u2082Hom_rename, Function.extend_comp hf]", "annotated_tactic": ["simp only [<a>eval</a>, <a>eval\u2082Hom_rename</a>, <a>Function.extend_comp</a> hf]", [{"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}, {"full_name": "MvPolynomial.eval\u2082Hom_rename", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [191, 9], "def_end_pos": [191, 24]}, {"full_name": "Function.extend_comp", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [790, 9], "def_end_pos": [790, 20]}]], "state_before": "case h.e'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Infinite R\n\u03c3 : Type u_2\nn : \u2115\nf : Fin n \u2192 \u03c3\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0\nx : Fin n \u2192 R\n\u22a2 \u2191(eval x) p = \u2191(eval (Function.extend f x 0)) (\u2191(rename f) 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.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on", "start": [906, 1], "end": [940, 35], "traced_tactics": [{"tactic": "have hs_m\u2082 : MeasurableSet[m\u2082] s := by\n  rw [\u2190 Set.inter_univ s]\n  refine' hs Set.univ _\n  rwa [Set.inter_univ]", "annotated_tactic": ["have hs_m\u2082 : MeasurableSet[m\u2082] s := by\n    rw [\u2190 <a>Set.inter_univ</a> s]\n    refine' hs <a>Set.univ</a> _\n    rwa [<a>Set.inter_univ</a>]", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\n\u22a2 StronglyMeasurable f", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\n\u22a2 StronglyMeasurable f"}, {"tactic": "obtain \u27e8g_seq_s, hg_seq_tendsto, hg_seq_zero\u27e9 := stronglyMeasurable_in_set hs_m hf hf_zero", "annotated_tactic": ["obtain \u27e8g_seq_s, hg_seq_tendsto, hg_seq_zero\u27e9 := <a>stronglyMeasurable_in_set</a> hs_m hf hf_zero", [{"full_name": "MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [883, 9], "def_end_pos": [883, 34]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\n\u22a2 StronglyMeasurable f", "state_after": "case intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\n\u22a2 StronglyMeasurable f"}, {"tactic": "exact \u27e8g_seq_s\u2082, hg_seq_tendsto\u27e9", "annotated_tactic": ["exact \u27e8g_seq_s\u2082, hg_seq_tendsto\u27e9", []], "state_before": "case intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\ng_seq_s\u2082 : \u2115 \u2192 \u03b1 \u2192\u209b E :=\n  fun n =>\n    { toFun := \u2191(g_seq_s n), measurableSet_fiber' := (_ : \u2200 (x : E), MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x})),\n      finite_range' := (_ : Set.Finite (range \u2191(g_seq_s n))) }\n\u22a2 StronglyMeasurable f", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.inter_univ s]", "annotated_tactic": ["rw [\u2190 <a>Set.inter_univ</a> s]", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\n\u22a2 MeasurableSet s", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\n\u22a2 MeasurableSet (s \u2229 univ)"}, {"tactic": "rwa [Set.inter_univ]", "annotated_tactic": ["rwa [<a>Set.inter_univ</a>]", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\n\u22a2 MeasurableSet (s \u2229 univ)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.inter_univ (g_seq_s n \u207b\u00b9' {x}), \u2190 Set.union_compl_self s,\n  Set.inter_union_distrib_left, Set.inter_comm (g_seq_s n \u207b\u00b9' {x})]", "annotated_tactic": ["rw [\u2190 <a>Set.inter_univ</a> (g_seq_s n \u207b\u00b9' {x}), \u2190 <a>Set.union_compl_self</a> s,\n          <a>Set.inter_union_distrib_left</a>, <a>Set.inter_comm</a> (g_seq_s n \u207b\u00b9' {x})]", [{"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_compl_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1739, 9], "def_end_pos": [1739, 25]}, {"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.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x})", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (s \u2229 \u2191(g_seq_s n) \u207b\u00b9' {x} \u222a \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)"}, {"tactic": "refine' MeasurableSet.union (hs _ (hs_m.inter _)) _", "annotated_tactic": ["refine' <a>MeasurableSet.union</a> (hs _ (hs_m.inter _)) _", [{"full_name": "MeasurableSet.union", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [191, 19], "def_end_pos": [191, 38]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (s \u2229 \u2191(g_seq_s n) \u207b\u00b9' {x} \u222a \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)", "state_after": "case refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x})\n\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)"}, {"tactic": "by_cases hx : x = 0", "annotated_tactic": ["by_cases hx : x = 0", []], "state_before": "case refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)"}, {"tactic": "exact @SimpleFunc.measurableSet_fiber _ _ m _ _", "annotated_tactic": ["exact @<a>SimpleFunc.measurableSet_fiber</a> _ _ m _ _", [{"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 refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x})", "state_after": "no goals"}, {"tactic": "suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c by\n  rw [this]\n  exact hs_m\u2082.compl", "annotated_tactic": ["suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c by\n            rw [this]\n            exact hs_m\u2082.compl", []], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\n\u22a2 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c"}, {"tactic": "ext1 y", "annotated_tactic": ["ext1 y", []], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\n\u22a2 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c", "state_after": "case pos.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\ny : \u03b1\n\u22a2 y \u2208 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c \u2194 y \u2208 s\u1d9c"}, {"tactic": "rw [hx, Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff]", "annotated_tactic": ["rw [hx, <a>Set.mem_inter_iff</a>, <a>Set.mem_preimage</a>, <a>Set.mem_singleton_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_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 pos.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\ny : \u03b1\n\u22a2 y \u2208 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c \u2194 y \u2208 s\u1d9c", "state_after": "case pos.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\ny : \u03b1\n\u22a2 \u2191(g_seq_s n) y = 0 \u2227 y \u2208 s\u1d9c \u2194 y \u2208 s\u1d9c"}, {"tactic": "exact \u27e8fun h => h.2, fun h => \u27e8hg_seq_zero y h n, h\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun h => h.2, fun h => \u27e8hg_seq_zero y h n, h\u27e9\u27e9", []], "state_before": "case pos.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\ny : \u03b1\n\u22a2 \u2191(g_seq_s n) y = 0 \u2227 y \u2208 s\u1d9c \u2194 y \u2208 s\u1d9c", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\nthis : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\nthis : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c\n\u22a2 MeasurableSet s\u1d9c"}, {"tactic": "exact hs_m\u2082.compl", "annotated_tactic": ["exact hs_m\u2082.compl", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : x = 0\nthis : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c\n\u22a2 MeasurableSet s\u1d9c", "state_after": "no goals"}, {"tactic": "suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 by\n  rw [this]\n  exact MeasurableSet.empty", "annotated_tactic": ["suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 by\n            rw [this]\n            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 neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\n\u22a2 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205"}, {"tactic": "ext1 y", "annotated_tactic": ["ext1 y", []], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\n\u22a2 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205", "state_after": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\n\u22a2 y \u2208 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c \u2194 y \u2208 \u2205"}, {"tactic": "simp only [mem_inter_iff, mem_preimage, mem_singleton_iff, mem_compl_iff,\n  mem_empty_iff_false, iff_false_iff, not_and, not_not_mem]", "annotated_tactic": ["simp only [<a>mem_inter_iff</a>, <a>mem_preimage</a>, <a>mem_singleton_iff</a>, <a>mem_compl_iff</a>,\n            <a>mem_empty_iff_false</a>, <a>iff_false_iff</a>, <a>not_and</a>, <a>not_not_mem</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_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_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 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": "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": "Set.not_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [429, 9], "def_end_pos": [429, 20]}]], "state_before": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\n\u22a2 y \u2208 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c \u2194 y \u2208 \u2205", "state_after": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\n\u22a2 \u2191(g_seq_s n) y = x \u2192 y \u2208 s"}, {"tactic": "refine' Function.mtr fun hys => _", "annotated_tactic": ["refine' <a>Function.mtr</a> fun hys => _", [{"full_name": "Function.mtr", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [391, 19], "def_end_pos": [391, 31]}]], "state_before": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\n\u22a2 \u2191(g_seq_s n) y = x \u2192 y \u2208 s", "state_after": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\nhys : \u00acy \u2208 s\n\u22a2 \u00ac\u2191(g_seq_s n) y = x"}, {"tactic": "rw [hg_seq_zero y hys n]", "annotated_tactic": ["rw [hg_seq_zero y hys n]", []], "state_before": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\nhys : \u00acy \u2208 s\n\u22a2 \u00ac\u2191(g_seq_s n) y = x", "state_after": "case neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\nhys : \u00acy \u2208 s\n\u22a2 \u00ac0 = x"}, {"tactic": "exact Ne.symm hx", "annotated_tactic": ["exact <a>Ne.symm</a> hx", [{"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.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\ny : \u03b1\nhys : \u00acy \u2208 s\n\u22a2 \u00ac0 = x", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\nthis : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205\n\u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c)", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\nthis : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \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": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : Countable \u03b9\nf\u271d g : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_5\nE : Type u_6\nm m\u2082 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : StronglyMeasurable f\nhf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0\nhs_m\u2082 : MeasurableSet s\ng_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E\nhg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x))\nhg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0\nn : \u2115\nx : E\nhx : \u00acx = 0\nthis : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205\n\u22a2 MeasurableSet \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.codeSupp_supports", "start": [2076, 1], "end": [2077, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.eq_singleton_or_nontrivial", "start": [819, 1], "end": [820, 67], "traced_tactics": [{"tactic": "rw [\u2190 coe_eq_singleton]", "annotated_tactic": ["rw [\u2190 <a>coe_eq_singleton</a>]", [{"full_name": "Finset.coe_eq_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [731, 9], "def_end_pos": [731, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\n\u22a2 s = {a} \u2228 Finset.Nontrivial s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\n\u22a2 \u2191s = {a} \u2228 Finset.Nontrivial s"}, {"tactic": "exact Set.eq_singleton_or_nontrivial ha", "annotated_tactic": ["exact <a>Set.eq_singleton_or_nontrivial</a> ha", [{"full_name": "Set.eq_singleton_or_nontrivial", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2648, 7], "def_end_pos": [2648, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\n\u22a2 \u2191s = {a} \u2228 Finset.Nontrivial s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_natAbs_eq_min", "start": [1102, 1], "end": [1118, 27], "traced_tactics": [{"tactic": "rw [valMinAbs_def_pos]", "annotated_tactic": ["rw [<a>valMinAbs_def_pos</a>]", [{"full_name": "ZMod.valMinAbs_def_pos", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [963, 9], 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["apply\n      <a>min_eq_left</a> (<a>le_trans</a> h (<a>le_trans</a> (<a>Nat.half_le_of_sub_le_half</a> _) (<a>Nat.sub_le_sub_left</a> n h)))", [{"full_name": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}, {"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": "Nat.half_le_of_sub_le_half", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 31]}, {"full_name": "Nat.sub_le_sub_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [394, 19], "def_end_pos": [394, 34]}]], "state_before": "case pos\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 min (val a) (n - val a) = val a", "state_after": "n\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 n - (n - n / 2) \u2264 n / 2"}, {"tactic": "rw [Nat.sub_sub_self (Nat.div_le_self _ _)]", "annotated_tactic": ["rw [<a>Nat.sub_sub_self</a> (<a>Nat.div_le_self</a> _ _)]", [{"full_name": "Nat.sub_sub_self", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [408, 19], "def_end_pos": [408, 31]}, {"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]}]], "state_before": "n\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 n - (n - n / 2) \u2264 n / 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 Int.natAbs_neg, neg_sub, \u2190 Nat.cast_sub a.val_le]", "annotated_tactic": ["rw [\u2190 <a>Int.natAbs_neg</a>, <a>neg_sub</a>, \u2190 <a>Nat.cast_sub</a> a.val_le]", [{"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": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "Nat.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 17]}]], "state_before": "case neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 Int.natAbs (\u2191(val a) - \u2191n) = min (val a) (n - val a)", "state_after": "case neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 Int.natAbs \u2191(n - val a) = min (val a) (n - val a)"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 Int.natAbs \u2191(n - val a) = min (val a) (n - val a)", "state_after": "case neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 min (val a) (n - val a) = Int.natAbs \u2191(n - val a)"}, {"tactic": "apply\n  min_eq_right\n    (le_trans (le_trans (Nat.sub_le_sub_left n (lt_of_not_ge h)) (Nat.le_half_of_half_lt_sub _))\n      (le_of_not_ge h))", "annotated_tactic": ["apply\n      <a>min_eq_right</a>\n        (<a>le_trans</a> (<a>le_trans</a> (<a>Nat.sub_le_sub_left</a> n (<a>lt_of_not_ge</a> h)) (<a>Nat.le_half_of_half_lt_sub</a> _))\n          (<a>le_of_not_ge</a> h))", [{"full_name": "min_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}, {"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": "Nat.sub_le_sub_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [394, 19], "def_end_pos": [394, 34]}, {"full_name": "lt_of_not_ge", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 21]}, {"full_name": "Nat.le_half_of_half_lt_sub", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 31]}, {"full_name": "le_of_not_ge", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}]], "state_before": "case neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 min (val a) (n - val a) = Int.natAbs \u2191(n - val a)", "state_after": "n\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 n / 2 < n - (n - Nat.succ (n / 2))"}, {"tactic": "rw [Nat.sub_sub_self (Nat.div_lt_self (lt_of_le_of_ne' (Nat.zero_le _) hpos.1) one_lt_two)]", "annotated_tactic": ["rw [<a>Nat.sub_sub_self</a> (<a>Nat.div_lt_self</a> (<a>lt_of_le_of_ne'</a> (<a>Nat.zero_le</a> _) hpos.1) <a>one_lt_two</a>)]", [{"full_name": "Nat.sub_sub_self", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [408, 19], "def_end_pos": [408, 31]}, {"full_name": "Nat.div_lt_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [56, 9], "def_end_pos": [56, 20]}, {"full_name": "lt_of_le_of_ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 24]}, {"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": "one_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [106, 7], "def_end_pos": [106, 17]}]], "state_before": "n\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 n / 2 < n - (n - Nat.succ (n / 2))", "state_after": "n\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 n / 2 < Nat.succ (n / 2)"}, {"tactic": "apply Nat.lt_succ_self", "annotated_tactic": ["apply <a>Nat.lt_succ_self</a>", [{"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 a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 n / 2 < Nat.succ (n / 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.coe_coeMonoidHom", "start": [832, 1], "end": [833, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.nonempty_product", "start": [209, 1], "end": [210, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Count.lean", "full_name": "MeasureTheory.Measure.count_ne_zero'", "start": [143, 1], "end": [145, 21], "traced_tactics": [{"tactic": "rw [Ne.def, count_eq_zero_iff' s_mble]", "annotated_tactic": ["rw [<a>Ne.def</a>, <a>count_eq_zero_iff'</a> s_mble]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "MeasureTheory.Measure.count_eq_zero_iff'", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [134, 9], "def_end_pos": [134, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.20372\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs' : Set.Nonempty s\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191count s \u2260 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.20372\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs' : Set.Nonempty s\ns_mble : MeasurableSet s\n\u22a2 \u00acs = \u2205"}, {"tactic": "exact hs'.ne_empty", "annotated_tactic": ["exact hs'.ne_empty", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.20372\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\nhs' : Set.Nonempty s\ns_mble : MeasurableSet s\n\u22a2 \u00acs = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.card_image_of_injective", "start": [1234, 1], "end": [1236, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "ProbabilityTheory.set_lintegral_condKernelReal_prod", "start": [98, 1], "end": [133, 55], "traced_tactics": [{"tactic": "apply MeasurableSpace.induction_on_inter (borel_eq_generateFrom_Iic \u211d) isPiSystem_Iic _ _ _ _ ht", "annotated_tactic": ["apply <a>MeasurableSpace.induction_on_inter</a> (<a>borel_eq_generateFrom_Iic</a> \u211d) <a>isPiSystem_Iic</a> _ _ _ _ ht", [{"full_name": "MeasurableSpace.induction_on_inter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [745, 9], "def_end_pos": [745, 27]}, {"full_name": "borel_eq_generateFrom_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 34]}, {"full_name": "isPiSystem_Iic", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [156, 9], "def_end_pos": [156, 23]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) \u2205 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205)\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u211d),\n    MeasurableSet t \u2192\n      \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2192\n        \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)) \u2192\n          \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 i, f i)"}, {"tactic": "simp only [measure_empty, lintegral_const, zero_mul, prod_empty]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>lintegral_const</a>, <a>zero_mul</a>, <a>prod_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": "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": "Set.prod_empty", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) \u2205 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205)", "state_after": "no goals"}, {"tactic": "rintro t \u27e8q, rfl\u27e9", "annotated_tactic": ["rintro t \u27e8q, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)", "state_after": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nq : \u211d\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic q) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic q)"}, {"tactic": "exact set_lintegral_condKernelReal_Iic \u03c1 q hs", "annotated_tactic": ["exact <a>set_lintegral_condKernelReal_Iic</a> \u03c1 q hs", [{"full_name": "ProbabilityTheory.set_lintegral_condKernelReal_Iic", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [78, 9], "def_end_pos": [78, 41]}]], "state_before": "case intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nq : \u211d\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic q) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic q)", "state_after": "no goals"}, {"tactic": "intro t ht ht_lintegral", "annotated_tactic": ["intro t ht ht_lintegral", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u211d),\n    MeasurableSet t \u2192\n      \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2192\n        \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c)"}, {"tactic": "congr with a", "annotated_tactic": ["congr with a", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) 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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c = \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t"}, {"tactic": "rw [measure_compl ht (measure_ne_top (condKernelReal \u03c1 a) _)]", "annotated_tactic": ["rw [<a>measure_compl</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": "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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c = \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t", "state_after": "no goals"}, {"tactic": "rw [lintegral_sub (kernel.measurable_coe (condKernelReal \u03c1) ht)]", "annotated_tactic": ["rw [<a>lintegral_sub</a> (<a>kernel.measurable_coe</a> (<a>condKernelReal</a> \u03c1) ht)]", [{"full_name": "MeasureTheory.lintegral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [933, 9], "def_end_pos": [933, 22]}, {"full_name": "ProbabilityTheory.kernel.measurable_coe", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [208, 19], "def_end_pos": [208, 33]}, {"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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 =\n    \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n      \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1", "state_after": "case hg_fin\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 \u2260 \u22a4\n\ncase h_le\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) t) \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ"}, {"tactic": "rw [ht_lintegral]", "annotated_tactic": ["rw [ht_lintegral]", []], "state_before": "case hg_fin\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 \u2260 \u22a4", "state_after": "case hg_fin\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2260 \u22a4"}, {"tactic": "exact measure_ne_top \u03c1 _", "annotated_tactic": ["exact <a>measure_ne_top</a> \u03c1 _", [{"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 hg_fin\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun a => measure_mono (subset_univ _)", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun a => <a>measure_mono</a> (<a>subset_univ</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": "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 h_le\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) t) \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ", "state_after": "no goals"}, {"tactic": "rw [set_lintegral_condKernelReal_univ \u03c1 hs, ht_lintegral]", "annotated_tactic": ["rw [<a>set_lintegral_condKernelReal_univ</a> \u03c1 hs, ht_lintegral]", [{"full_name": "ProbabilityTheory.set_lintegral_condKernelReal_univ", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [84, 9], "def_end_pos": [84, 42]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n      \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 =\n    \u2191\u2191\u03c1 (s \u00d7\u02e2 univ) - \u2191\u2191\u03c1 (s \u00d7\u02e2 t)", "state_after": "no goals"}, {"tactic": "rw [\u2190 measure_diff _ (hs.prod ht) (measure_ne_top \u03c1 _)]", "annotated_tactic": ["rw [\u2190 <a>measure_diff</a> _ (hs.prod ht) (<a>measure_ne_top</a> \u03c1 _)]", [{"full_name": "MeasureTheory.measure_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [252, 9], "def_end_pos": [252, 21]}, {"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\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ) - \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ \\ s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t \u2286 s \u00d7\u02e2 univ"}, {"tactic": "rw [prod_diff_prod, compl_eq_univ_diff]", "annotated_tactic": ["rw [<a>prod_diff_prod</a>, <a>compl_eq_univ_diff</a>]", [{"full_name": "Set.prod_diff_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [403, 9], "def_end_pos": [403, 23]}, {"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\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ \\ s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t) \u222a (s \\ s) \u00d7\u02e2 univ) = \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t))"}, {"tactic": "simp only [diff_self, empty_prod, union_empty]", "annotated_tactic": ["simp only [<a>diff_self</a>, <a>empty_prod</a>, <a>union_empty</a>]", [{"full_name": "Set.diff_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2089, 9], "def_end_pos": [2089, 18]}, {"full_name": "Set.empty_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [119, 9], "def_end_pos": [119, 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\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t) \u222a (s \\ s) \u00d7\u02e2 univ) = \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t))", "state_after": "no goals"}, {"tactic": "rw [prod_subset_prod_iff]", "annotated_tactic": ["rw [<a>prod_subset_prod_iff</a>]", [{"full_name": "Set.prod_subset_prod_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [410, 9], "def_end_pos": [410, 29]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t \u2286 s \u00d7\u02e2 univ", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 s \u2286 s \u2227 t \u2286 univ \u2228 s = \u2205 \u2228 t = \u2205"}, {"tactic": "exact Or.inl \u27e8subset_rfl, subset_univ t\u27e9", "annotated_tactic": ["exact <a>Or.inl</a> \u27e8<a>subset_rfl</a>, <a>subset_univ</a> t\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]}, {"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\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 s \u2286 s \u2227 t \u2286 univ \u2228 s = \u2205 \u2228 t = \u2205", "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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n        (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)) \u2192\n          \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 i, f i)"}, {"tactic": "simp_rw [measure_iUnion hf_disj hf_meas]", "annotated_tactic": ["simp_rw [<a>measure_iUnion</a> hf_disj hf_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\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 i, f i)"}, {"tactic": "rw [lintegral_tsum fun i => (kernel.measurable_coe _ (hf_meas i)).aemeasurable.restrict,\n  prod_iUnion, measure_iUnion]", "annotated_tactic": ["rw [<a>lintegral_tsum</a> fun i => (<a>kernel.measurable_coe</a> _ (hf_meas i)).aemeasurable.restrict,\n      <a>prod_iUnion</a>, <a>measure_iUnion</a>]", [{"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_coe", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [208, 19], "def_end_pos": [208, 33]}, {"full_name": "Set.prod_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 20]}, {"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\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\ncase hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 Pairwise (Disjoint on fun i => s \u00d7\u02e2 f i)\n\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u00d7\u02e2 f i)"}, {"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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)", "state_after": "no goals"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "case hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 Pairwise (Disjoint on fun i => s \u00d7\u02e2 f i)", "state_after": "case hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => s \u00d7\u02e2 f i) i j"}, {"tactic": "rw [Function.onFun, disjoint_prod]", "annotated_tactic": ["rw [<a>Function.onFun</a>, <a>disjoint_prod</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.disjoint_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}]], "state_before": "case hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => s \u00d7\u02e2 f i) i j", "state_after": "case hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint s s \u2228 Disjoint (f i) (f j)"}, {"tactic": "exact Or.inr (hf_disj hij)", "annotated_tactic": ["exact <a>Or.inr</a> (hf_disj hij)", [{"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 hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint s s \u2228 Disjoint (f i) (f j)", "state_after": "no goals"}, {"tactic": "exact fun i => MeasurableSet.prod hs (hf_meas i)", "annotated_tactic": ["exact fun i => <a>MeasurableSet.prod</a> hs (hf_meas i)", [{"full_name": "MeasurableSet.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [770, 19], "def_end_pos": [770, 37]}]], "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\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u00d7\u02e2 f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.exists_mem_not_mem_of_ncard_lt_ncard", "start": [896, 1], "end": [898, 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_one_iff", "start": [851, 1], "end": [851, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_zero_left", "start": [1037, 1], "end": [1042, 97], "traced_tactics": [{"tactic": "suffices setToL1 hT = 0 by rw [this]; simp", "annotated_tactic": ["suffices <a>setToL1</a> hT = 0 by rw [this]; simp", [{"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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc 0 C\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 hT) f = 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\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 0 C\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = 0"}, {"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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc 0 C\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = 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\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 0 C\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp 0 (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 0 C\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp 0 (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 0 C\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp 0 (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "rw [setToL1SCLM_zero_left hT f, ContinuousLinearMap.zero_comp, ContinuousLinearMap.zero_apply]", "annotated_tactic": ["rw [<a>setToL1SCLM_zero_left</a> hT f, <a>ContinuousLinearMap.zero_comp</a>, <a>ContinuousLinearMap.zero_apply</a>]", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [890, 9], "def_end_pos": [890, 30]}, {"full_name": "ContinuousLinearMap.zero_comp", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 18]}, {"full_name": "ContinuousLinearMap.zero_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [644, 9], "def_end_pos": [644, 19]}]], "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 0 C\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp 0 (coeToLp \u03b1 E \u211d)) 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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nhT : DominatedFinMeasAdditive \u03bc 0 C\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = 0\n\u22a2 \u2191(setToL1 hT) f = 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\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 0 C\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = 0\n\u22a2 \u21910 f = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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 0 C\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = 0\n\u22a2 \u21910 f = 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.eq_zero_iff_ae_eq_zero", "start": [353, 1], "end": [354, 79], "traced_tactics": [{"tactic": "rw [\u2190 (Lp.mem\u2112p f).toLp_eq_toLp_iff zero_mem\u2112p, Mem\u2112p.toLp_zero, toLp_coeFn]", "annotated_tactic": ["rw [\u2190 (<a>Lp.mem\u2112p</a> f).<a>toLp_eq_toLp_iff</a> <a>zero_mem\u2112p</a>, <a>Mem\u2112p.toLp_zero</a>, <a>toLp_coeFn</a>]", [{"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.toLp_eq_toLp_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [128, 9], "def_end_pos": [128, 25]}, {"full_name": "MeasureTheory.zero_mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [214, 9], "def_end_pos": [214, 19]}, {"full_name": "MeasureTheory.Mem\u2112p.toLp_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [133, 9], "def_end_pos": [133, 18]}, {"full_name": "MeasureTheory.Lp.toLp_coeFn", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [193, 9], "def_end_pos": [193, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : { x // x \u2208 Lp E p }\n\u22a2 f = 0 \u2194 \u2191\u2191f =\u1d50[\u03bc] 0", "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.bisim", "start": [487, 1], "end": [495, 11], "traced_tactics": [{"tactic": "apply Cofix.bisim_rel", "annotated_tactic": ["apply <a>Cofix.bisim_rel</a>", [{"full_name": "QPF.Cofix.bisim_rel", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [467, 9], "def_end_pos": [467, 24]}]], "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 Liftr r (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 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 Liftr r (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y"}, {"tactic": "intro x y rxy", "annotated_tactic": ["intro x y rxy", []], "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 Liftr r (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> 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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\n\u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y"}, {"tactic": "rcases (liftr_iff r _ _).mp (h x y rxy) with \u27e8a, f\u2080, f\u2081, dxeq, dyeq, h'\u27e9", "annotated_tactic": ["rcases (<a>liftr_iff</a> r _ _).<a>mp</a> (h x y rxy) with \u27e8a, f\u2080, f\u2081, dxeq, dyeq, h'\u27e9", [{"full_name": "QPF.liftr_iff", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [134, 9], "def_end_pos": [134, 18]}, {"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 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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\n\u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y", "state_after": "case h.intro.intro.intro.intro.intro\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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y"}, {"tactic": "rw [dxeq, dyeq, \u2190 abs_map, \u2190 abs_map, PFunctor.map_eq, PFunctor.map_eq]", "annotated_tactic": ["rw [dxeq, dyeq, \u2190 <a>abs_map</a>, \u2190 <a>abs_map</a>, <a>PFunctor.map_eq</a>, <a>PFunctor.map_eq</a>]", [{"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_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"full_name": "PFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [65, 19], "def_end_pos": [65, 25]}, {"full_name": "PFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [65, 19], "def_end_pos": [65, 25]}]], "state_before": "case h.intro.intro.intro.intro.intro\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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y", "state_after": "case h.intro.intro.intro.intro.intro\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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2080 } = abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2081 }"}, {"tactic": "congr 2 with i", "annotated_tactic": ["congr 2 with i", []], "state_before": "case h.intro.intro.intro.intro.intro\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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2080 } = abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2081 }", "state_after": "case h.intro.intro.intro.intro.intro.e_a.e_snd.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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\ni : PFunctor.B (P F) a\n\u22a2 ((Quot.mk fun x y => r x y) \u2218 f\u2080) i = ((Quot.mk fun x y => r x y) \u2218 f\u2081) i"}, {"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 h.intro.intro.intro.intro.intro.e_a.e_snd.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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\ni : PFunctor.B (P F) a\n\u22a2 ((Quot.mk fun x y => r x y) \u2218 f\u2080) i = ((Quot.mk fun x y => r x y) \u2218 f\u2081) i", "state_after": "case h.intro.intro.intro.intro.intro.e_a.e_snd.h.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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\ni : PFunctor.B (P F) a\n\u22a2 r (f\u2080 i) (f\u2081 i)"}, {"tactic": "apply h'", "annotated_tactic": ["apply h'", []], "state_before": "case h.intro.intro.intro.intro.intro.e_a.e_snd.h.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 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\ni : PFunctor.B (P F) a\n\u22a2 r (f\u2080 i) (f\u2081 i)", "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_log", "start": [125, 1], "end": [127, 56], "traced_tactics": []}, {"url": 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_ _ hp \ud835\udd5c).<a>mono</a> ?_", [{"full_name": "BoundedContinuousFunction.toLp_denseRange", "def_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "def_pos": [368, 9], "def_end_pos": [368, 24]}, {"full_name": "Dense.mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [682, 9], "def_end_pos": [682, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : T4Space \u03b1\ninst\u271d\u2078 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompactSpace \u03b1\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : T4Space \u03b1\ninst\u271d\u2078 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompactSpace \u03b1\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 range \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) \u2286 range \u2191(toLp p \u03bc \ud835\udd5c)"}, {"tactic": "refine range_subset_iff.2 fun f \u21a6 ?_", "annotated_tactic": ["refine <a>range_subset_iff</a>.2 fun f \u21a6 ?_", [{"full_name": "Set.range_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [749, 9], "def_end_pos": [749, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : T4Space \u03b1\ninst\u271d\u2078 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompactSpace \u03b1\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 range \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) \u2286 range \u2191(toLp p \u03bc \ud835\udd5c)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : T4Space \u03b1\ninst\u271d\u2078 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompactSpace \u03b1\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) f \u2208 range \u2191(toLp p \u03bc \ud835\udd5c)"}, {"tactic": "exact \u27e8f.toContinuousMap, rfl\u27e9", "annotated_tactic": ["exact \u27e8f.toContinuousMap, <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": "\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : T4Space \u03b1\ninst\u271d\u2078 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompactSpace \u03b1\ninst\u271d\u00b9 : Measure.WeaklyRegular \u03bc\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) f \u2208 range \u2191(toLp p \u03bc \ud835\udd5c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.Measure.map_mul_right_inv_eq_self", "start": [521, 1], "end": [523, 47], "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.extract", "start": [612, 1], "end": [616, 60], "traced_tactics": [{"tactic": "cases h\u2081.out", "annotated_tactic": ["cases h\u2081.out", []], "state_before": "l m r : List Char\nit\u2081 it\u2082 : Iterator\nh\u2081 : ValidFor l (m ++ r) it\u2081\nh\u2082 : ValidFor (List.reverse m ++ l) r it\u2082\n\u22a2 Iterator.extract it\u2081 it\u2082 = { data := m }", "state_after": "case refl\nl m r : List Char\nit\u2082 : Iterator\nh\u2082 : ValidFor (List.reverse m ++ l) r it\u2082\nh\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n\u22a2 Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } it\u2082 = { data := m }"}, {"tactic": "cases h\u2082.out", "annotated_tactic": ["cases h\u2082.out", []], "state_before": "case refl\nl m r : List Char\nit\u2082 : Iterator\nh\u2082 : ValidFor (List.reverse m ++ l) r it\u2082\nh\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n\u22a2 Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } it\u2082 = { data := m }", "state_after": "case refl.refl\nl m r : List Char\nh\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh\u2082 :\n  ValidFor (List.reverse m ++ l) r\n    { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n\u22a2 Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n      { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } =\n    { data := m }"}, {"tactic": "simp [Iterator.extract, List.reverseAux_eq, Nat.not_lt.2 (Nat.le_add_left ..)]", "annotated_tactic": ["simp [<a>Iterator.extract</a>, <a>List.reverseAux_eq</a>, <a>Nat.not_lt</a>.2 (<a>Nat.le_add_left</a> ..)]", [{"full_name": "String.Iterator.extract", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [344, 5], "def_end_pos": [344, 12]}, {"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}, {"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_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 refl.refl\nl m r : List Char\nh\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh\u2082 :\n  ValidFor (List.reverse m ++ l) r\n    { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n\u22a2 Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n      { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } =\n    { data := m }", "state_after": "case refl.refl\nl m r : List Char\nh\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh\u2082 :\n  ValidFor (List.reverse m ++ l) r\n    { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n\u22a2 String.extract { data := List.reverse l ++ (m ++ r) } { byteIdx := utf8Len l } { byteIdx := utf8Len m + utf8Len l } =\n    { data := m }"}, {"tactic": "simpa [Nat.add_comm] using extract_of_valid l.reverse m r", "annotated_tactic": ["simpa [<a>Nat.add_comm</a>] using <a>extract_of_valid</a> l.reverse m r", [{"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": "String.extract_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [457, 9], "def_end_pos": [457, 25]}]], "state_before": "case refl.refl\nl m r : List Char\nh\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh\u2082 :\n  ValidFor (List.reverse m ++ l) r\n    { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n\u22a2 String.extract { data := List.reverse l ++ (m ++ r) } { byteIdx := utf8Len l } { byteIdx := utf8Len m + utf8Len l } =\n    { data := m }", "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_smul", "start": [309, 1], "end": [315, 72], "traced_tactics": [{"tactic": "refine' (integrable_prod_iff _).2 \u27e8_, _\u27e9", "annotated_tactic": ["refine' (<a>integrable_prod_iff</a> _).2 \u27e8_, _\u27e9", [{"full_name": "MeasureTheory.integrable_prod_iff", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [272, 9], "def_end_pos": [272, 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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable fun z => f z.1 \u2022 g z.2", "state_after": "case refine'_1\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 AEStronglyMeasurable (fun z => f z.1 \u2022 g z.2) (Measure.prod \u03bc \u03bd)\n\ncase refine'_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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y).1 \u2022 g (x, y).2\n\ncase refine'_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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y).1 \u2022 g (x, y).2\u2016 \u2202\u03bd"}, {"tactic": "exact hf.1.fst.smul hg.1.snd", "annotated_tactic": ["exact hf.1.fst.smul hg.1.<a>snd</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.snd", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [191, 9], "def_end_pos": [191, 47]}]], "state_before": "case refine'_1\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 AEStronglyMeasurable (fun z => f z.1 \u2022 g z.2) (Measure.prod \u03bc \u03bd)", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun x => hg.smul (f x)", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => hg.smul (f 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'_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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y).1 \u2022 g (x, y).2", "state_after": "no goals"}, {"tactic": "simpa only [norm_smul, integral_mul_left] using hf.norm.mul_const _", "annotated_tactic": ["simpa only [<a>norm_smul</a>, <a>integral_mul_left</a>] using hf.norm.mul_const _", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}]], "state_before": "case refine'_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\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\ninst\u271d\u00b2 : SigmaFinite \u03bd\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b2 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y).1 \u2022 g (x, y).2\u2016 \u2202\u03bd", "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_Iic", "start": [91, 1], "end": [97, 90], "traced_tactics": [{"tactic": "have A : MeasurableEmbedding fun x : \u211d => -x :=\n  (Homeomorph.neg \u211d).closedEmbedding.measurableEmbedding", "annotated_tactic": ["have A : <a>MeasurableEmbedding</a> fun x : \u211d => -x :=\n    (<a>Homeomorph.neg</a> \u211d).closedEmbedding.measurableEmbedding", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1178, 11], "def_end_pos": [1178, 30]}, {"full_name": "Homeomorph.neg", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [330, 3], "def_end_pos": [330, 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 Ioi (-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\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "have := MeasurableEmbedding.set_integral_map (\u03bc := volume) A f (Ici (-c))", "annotated_tactic": ["have := <a>MeasurableEmbedding.set_integral_map</a> (\u03bc := <a>volume</a>) A f (<a>Ici</a> (-c))", [{"full_name": "MeasurableEmbedding.set_integral_map", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [519, 9], "def_end_pos": [519, 52]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-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\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "rw [Measure.map_neg_eq_self (volume : Measure \u211d)] at this", "annotated_tactic": ["rw [<a>Measure.map_neg_eq_self</a> (<a>volume</a> : <a>Measure</a> \u211d)] at this", [{"full_name": "MeasureTheory.Measure.map_neg_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [421, 3], "def_end_pos": [421, 14]}, {"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]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-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\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "simp_rw [\u2190 integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg_Ici, neg_neg]", "annotated_tactic": ["simp_rw [\u2190 <a>integral_Ici_eq_integral_Ioi</a>, this, <a>neg_preimage</a>, <a>preimage_neg_Ici</a>, <a>neg_neg</a>]", [{"full_name": "MeasureTheory.integral_Ici_eq_integral_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [699, 9], "def_end_pos": [699, 37]}, {"full_name": "Set.neg_preimage", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [185, 3], "def_end_pos": [185, 14]}, {"full_name": "Set.preimage_neg_Ici", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [130, 9], "def_end_pos": [130, 25]}, {"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\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.jordanDecomposition_add_withDensity_mutuallySingular", "start": [955, 1], "end": [968, 50], "traced_tactics": [{"tactic": "rw [mutuallySingular_ennreal_iff, totalVariation_mutuallySingular_iff] at ht\u03bc", "annotated_tactic": ["rw [<a>mutuallySingular_ennreal_iff</a>, <a>totalVariation_mutuallySingular_iff</a>] at ht\u03bc", [{"full_name": "MeasureTheory.SignedMeasure.mutuallySingular_ennreal_iff", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [569, 9], "def_end_pos": [569, 37]}, {"full_name": "MeasureTheory.SignedMeasure.totalVariation_mutuallySingular_iff", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [590, 9], "def_end_pos": [590, 44]}]], "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\nht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc\n\u22a2 ((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\nht\u03bc :\n  (toJordanDecomposition t).posPart \u27c2\u2098 VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc) \u2227\n    (toJordanDecomposition t).negPart \u27c2\u2098 VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc)\n\u22a2 ((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": "change\n  _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) \u2227\n    _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) at ht\u03bc", "annotated_tactic": ["change\n    _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) \u2227\n      _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) at ht\u03bc", []], "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\nht\u03bc :\n  (toJordanDecomposition t).posPart \u27c2\u2098 VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc) \u2227\n    (toJordanDecomposition t).negPart \u27c2\u2098 VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc)\n\u22a2 ((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\nht\u03bc :\n  (toJordanDecomposition t).posPart \u27c2\u2098\n      Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc) \u2227\n    (toJordanDecomposition t).negPart \u27c2\u2098\n      Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc)\n\u22a2 ((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": "rw [VectorMeasure.equivMeasure.right_inv] at ht\u03bc", "annotated_tactic": ["rw [VectorMeasure.equivMeasure.right_inv] at ht\u03bc", []], "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\nht\u03bc :\n  (toJordanDecomposition t).posPart \u27c2\u2098\n      Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc) \u2227\n    (toJordanDecomposition t).negPart \u27c2\u2098\n      Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc)\n\u22a2 ((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\nht\u03bc : (toJordanDecomposition t).posPart \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition t).negPart \u27c2\u2098 \u03bc\n\u22a2 ((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": "exact\n  ((JordanDecomposition.mutuallySingular _).add_right\n        (ht\u03bc.1.mono_ac (refl _) (withDensity_absolutelyContinuous _ _))).add_left\n    ((ht\u03bc.2.symm.mono_ac (withDensity_absolutelyContinuous _ _) (refl _)).add_right\n      (withDensity_ofReal_mutuallySingular hf))", "annotated_tactic": ["exact\n    ((<a>JordanDecomposition.mutuallySingular</a> _).<a>add_right</a>\n          (ht\u03bc.1.<a>mono_ac</a> (<a>refl</a> _) (<a>withDensity_absolutelyContinuous</a> _ _))).<a>add_left</a>\n      ((ht\u03bc.2.symm.mono_ac (<a>withDensity_absolutelyContinuous</a> _ _) (<a>refl</a> _)).<a>add_right</a>\n        (<a>withDensity_ofReal_mutuallySingular</a> hf))", [{"full_name": "MeasureTheory.JordanDecomposition.mutuallySingular", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [60, 3], "def_end_pos": [60, 19]}, {"full_name": "MeasureTheory.Measure.MutuallySingular.add_right", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "MeasureTheory.Measure.MutuallySingular.mono_ac", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [76, 9], "def_end_pos": [76, 16]}, {"full_name": "refl", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [304, 9], "def_end_pos": [304, 13]}, {"full_name": "MeasureTheory.withDensity_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [117, 9], "def_end_pos": [117, 41]}, {"full_name": "MeasureTheory.Measure.MutuallySingular.add_left", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "MeasureTheory.withDensity_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [117, 9], "def_end_pos": [117, 41]}, {"full_name": "refl", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [304, 9], "def_end_pos": [304, 13]}, {"full_name": "MeasureTheory.Measure.MutuallySingular.add_right", "def_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "MeasureTheory.withDensity_ofReal_mutuallySingular", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [162, 9], "def_end_pos": [162, 44]}]], "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\nht\u03bc : (toJordanDecomposition t).posPart \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition t).negPart \u27c2\u2098 \u03bc\n\u22a2 ((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": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mem_pow", "start": [919, 1], "end": [921, 41], "traced_tactics": [{"tactic": "simp [\u2190 mem_coe, coe_pow, Set.mem_pow]", "annotated_tactic": ["simp [\u2190 <a>mem_coe</a>, <a>coe_pow</a>, <a>Set.mem_pow</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_pow", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [857, 9], "def_end_pos": [857, 16]}, {"full_name": "Set.mem_pow", "def_path": "Mathlib/Data/Set/Pointwise/ListOfFn.lean", "def_pos": [52, 9], "def_end_pos": [52, 16]}]], "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\u271d : \u03b1\nm n\u271d : \u2115\na : \u03b1\nn : \u2115\n\u22a2 a \u2208 s ^ n \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/IntervalIntegral.lean", "full_name": "intervalIntegral.continuousWithinAt_primitive", "start": [1125, 1], "end": [1190, 22], "traced_tactics": [{"tactic": "by_cases h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082", "annotated_tactic": ["by_cases h\u2080 : b\u2080 \u2208 <a>Icc</a> b\u2081 b\u2082", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080\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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : \u00acb\u2080 \u2208 Icc b\u2081 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "have h\u2081\u2082 : b\u2081 \u2264 b\u2082 := h\u2080.1.trans h\u2080.2", "annotated_tactic": ["have h\u2081\u2082 : b\u2081 \u2264 b\u2082 := h\u2080.1.<a>trans</a> h\u2080.2", [{"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\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "have min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 := min_eq_left h\u2081\u2082", "annotated_tactic": ["have min\u2081\u2082 : <a>min</a> b\u2081 b\u2082 = b\u2081 := <a>min_eq_left</a> h\u2081\u2082", [{"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": "min_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [100, 9], "def_end_pos": [100, 20]}]], "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "apply ContinuousWithinAt.congr _ this (this _ h\u2080)", "annotated_tactic": ["apply <a>ContinuousWithinAt.congr</a> _ this (this _ h\u2080)", [{"full_name": "ContinuousWithinAt.congr", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 33]}]], "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis : \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis : \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "clear this", "annotated_tactic": ["clear this", []], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis : \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "refine' continuousWithinAt_const.add _", "annotated_tactic": ["refine' continuousWithinAt_const.add _", []], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "have :\n  (fun b => \u222b x in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b =>\n    \u222b x in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc := by\n  apply eventuallyEq_of_mem self_mem_nhdsWithin\n  exact fun b b_in => (integral_indicator b_in).symm", "annotated_tactic": ["have :\n      (fun b => \u222b x in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[<a>Icc</a> b\u2081 b\u2082] b\u2080] fun b =>\n        \u222b x in b\u2081..b\u2082, <a>indicator</a> {x | x \u2264 b} f x \u2202\u03bc := by\n      apply <a>eventuallyEq_of_mem</a> <a>self_mem_nhdsWithin</a>\n      exact fun b b_in => (<a>integral_indicator</a> b_in).<a>symm</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.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Filter.eventuallyEq_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1473, 9], "def_end_pos": [1473, 28]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}, {"full_name": "intervalIntegral.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1001, 16], "def_end_pos": [1001, 34]}, {"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\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "apply ContinuousWithinAt.congr_of_eventuallyEq _ this (integral_indicator h\u2080).symm", "annotated_tactic": ["apply <a>ContinuousWithinAt.congr_of_eventuallyEq</a> _ this (<a>integral_indicator</a> h\u2080).<a>symm</a>", [{"full_name": "ContinuousWithinAt.congr_of_eventuallyEq", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1008, 9], "def_end_pos": [1008, 49]}, {"full_name": "intervalIntegral.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1001, 16], "def_end_pos": [1001, 34]}, {"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\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "have : IntervalIntegrable (fun x => \u2016f x\u2016) \u03bc b\u2081 b\u2082 :=\n  IntervalIntegrable.norm (h_int' <| right_mem_Icc.mpr h\u2081\u2082)", "annotated_tactic": ["have : <a>IntervalIntegrable</a> (fun x => \u2016f x\u2016) \u03bc b\u2081 b\u2082 :=\n      <a>IntervalIntegrable.norm</a> (h_int' <| right_mem_Icc.mpr h\u2081\u2082)", [{"full_name": "IntervalIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [70, 5], "def_end_pos": [70, 23]}, {"full_name": "IntervalIntegrable.norm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [174, 9], "def_end_pos": [174, 13]}]], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : IntervalIntegrable (fun x => \u2016f x\u2016) \u03bc b\u2081 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "refine' continuousWithinAt_of_dominated_interval _ _ this _ <;> clear this", "annotated_tactic": ["refine' <a>continuousWithinAt_of_dominated_interval</a> _ _ this _ <;> clear this", [{"full_name": "intervalIntegral.continuousWithinAt_of_dominated_interval", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1076, 9], "def_end_pos": [1076, 49]}]], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : IntervalIntegrable (fun x => \u2016f x\u2016) \u03bc b\u2081 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080,\n    AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082))\n\ncase refine'_2\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016indicator {x_1 | x_1 \u2264 x} f t\u2016 \u2264 \u2016f t\u2016\n\ncase refine'_3\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f t) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "rintro x \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["rintro x \u27e8h\u2081, h\u2082\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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\n\u22a2 \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x", "state_after": "case 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 IntervalIntegrable f \u03bc b\u2081 x"}, {"tactic": "apply h_int.mono_set", "annotated_tactic": ["apply h_int.mono_set", []], "state_before": "case 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 IntervalIntegrable f \u03bc b\u2081 x", "state_after": "case 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 [[b\u2081, x]] \u2286 [[min a b\u2081, max a b\u2082]]"}, {"tactic": "apply uIcc_subset_uIcc", "annotated_tactic": ["apply <a>uIcc_subset_uIcc</a>", [{"full_name": "Set.uIcc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [107, 7], "def_end_pos": [107, 23]}]], "state_before": "case 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 [[b\u2081, x]] \u2286 [[min a b\u2081, max a b\u2082]]", "state_after": "case intro.h\u2081\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 b\u2081 \u2208 [[min a b\u2081, max a b\u2082]]\n\ncase intro.h\u2082\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 x \u2208 [[min a b\u2081, max a b\u2082]]"}, {"tactic": "exact \u27e8min_le_of_left_le (min_le_right a b\u2081),\n  h\u2081.trans (h\u2082.trans <| le_max_of_le_right <| le_max_right _ _)\u27e9", "annotated_tactic": ["exact \u27e8<a>min_le_of_left_le</a> (<a>min_le_right</a> a b\u2081),\n          h\u2081.trans (h\u2082.trans <| <a>le_max_of_le_right</a> <| <a>le_max_right</a> _ _)\u27e9", [{"full_name": "min_le_of_left_le", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "le_max_of_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case intro.h\u2081\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 b\u2081 \u2208 [[min a b\u2081, max a b\u2082]]", "state_after": "no goals"}, {"tactic": "exact \u27e8min_le_of_left_le <| (min_le_right _ _).trans h\u2081,\n  le_max_of_le_right <| h\u2082.trans <| le_max_right _ _\u27e9", "annotated_tactic": ["exact \u27e8<a>min_le_of_left_le</a> <| (<a>min_le_right</a> _ _).<a>trans</a> h\u2081,\n          <a>le_max_of_le_right</a> <| h\u2082.trans <| <a>le_max_right</a> _ _\u27e9", [{"full_name": "min_le_of_left_le", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_max_of_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case intro.h\u2082\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nx : \u211d\nh\u2081 : b\u2081 \u2264 x\nh\u2082 : x \u2264 b\u2082\n\u22a2 x \u2208 [[min a b\u2081, max a b\u2082]]", "state_after": "no goals"}, {"tactic": "rintro b \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["rintro b \u27e8h\u2081, h\u2082\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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc", "state_after": "case 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\na b\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_add_adjacent_intervals _ (h_int' \u27e8h\u2081, h\u2082\u27e9)]", "annotated_tactic": ["rw [\u2190 <a>integral_add_adjacent_intervals</a> _ (h_int' \u27e8h\u2081, h\u2082\u27e9)]", [{"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": "case 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\na b\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc", "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\na b\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 IntervalIntegrable f \u03bc a b\u2081"}, {"tactic": "apply h_int.mono_set", "annotated_tactic": ["apply h_int.mono_set", []], "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\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 IntervalIntegrable f \u03bc a b\u2081", "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\na b\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 [[a, b\u2081]] \u2286 [[min a b\u2081, max a b\u2082]]"}, {"tactic": "apply uIcc_subset_uIcc", "annotated_tactic": ["apply <a>uIcc_subset_uIcc</a>", [{"full_name": "Set.uIcc_subset_uIcc", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [107, 7], "def_end_pos": [107, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 [[a, b\u2081]] \u2286 [[min a b\u2081, max a b\u2082]]", "state_after": "case h\u2081\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\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 a \u2208 [[min a b\u2081, max a b\u2082]]\n\ncase h\u2082\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\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 b\u2081 \u2208 [[min a b\u2081, max a b\u2082]]"}, {"tactic": "exact \u27e8min_le_of_left_le (min_le_left a b\u2081), le_max_of_le_right (le_max_left _ _)\u27e9", "annotated_tactic": ["exact \u27e8<a>min_le_of_left_le</a> (<a>min_le_left</a> a b\u2081), <a>le_max_of_le_right</a> (<a>le_max_left</a> _ _)\u27e9", [{"full_name": "min_le_of_left_le", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "le_max_of_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}, {"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\u2081\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\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 a \u2208 [[min a b\u2081, max a b\u2082]]", "state_after": "no goals"}, {"tactic": "exact \u27e8min_le_of_left_le (min_le_right _ _),\n  le_max_of_le_right (h\u2081.trans <| h\u2082.trans (le_max_right a b\u2082))\u27e9", "annotated_tactic": ["exact \u27e8<a>min_le_of_left_le</a> (<a>min_le_right</a> _ _),\n          <a>le_max_of_le_right</a> (h\u2081.trans <| h\u2082.trans (<a>le_max_right</a> a b\u2082))\u27e9", [{"full_name": "min_le_of_left_le", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "le_max_of_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}, {"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\u2082\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\u271d b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nb : \u211d\nh\u2081 : b\u2081 \u2264 b\nh\u2082 : b \u2264 b\u2082\n\u22a2 b\u2081 \u2208 [[min a b\u2081, max a b\u2082]]", "state_after": "no goals"}, {"tactic": "apply eventuallyEq_of_mem self_mem_nhdsWithin", "annotated_tactic": ["apply <a>eventuallyEq_of_mem</a> <a>self_mem_nhdsWithin</a>", [{"full_name": "Filter.eventuallyEq_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1473, 9], "def_end_pos": [1473, 28]}, {"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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 EqOn (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082)"}, {"tactic": "exact fun b b_in => (integral_indicator b_in).symm", "annotated_tactic": ["exact fun b b_in => (<a>integral_indicator</a> b_in).<a>symm</a>", [{"full_name": "intervalIntegral.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1001, 16], "def_end_pos": [1001, 34]}, {"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\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\n\u22a2 EqOn (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082)", "state_after": "no goals"}, {"tactic": "apply Eventually.mono self_mem_nhdsWithin", "annotated_tactic": ["apply <a>Eventually.mono</a> <a>self_mem_nhdsWithin</a>", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [151, 9], "def_end_pos": [151, 28]}]], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080,\n    AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082))", "state_after": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200 (x : \u211d),\n    b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u2192 AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082))"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200 (x : \u211d),\n    b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u2192 AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082))", "state_after": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082))"}, {"tactic": "erw [aestronglyMeasurable_indicator_iff, Measure.restrict_restrict, Iic_inter_Ioc_of_le]", "annotated_tactic": ["erw [<a>aestronglyMeasurable_indicator_iff</a>, <a>Measure.restrict_restrict</a>, <a>Iic_inter_Ioc_of_le</a>]", [{"full_name": "aestronglyMeasurable_indicator_iff", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1509, 9], "def_end_pos": [1509, 50]}, {"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.Iic_inter_Ioc_of_le", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [710, 9], "def_end_pos": [710, 28]}]], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082))", "state_after": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc (min b\u2081 b\u2082) x))\n\ncase refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 x \u2264 max b\u2081 b\u2082\n\ncase refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 MeasurableSet {x_1 | x_1 \u2264 x}\n\ncase refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 MeasurableSet {x_1 | x_1 \u2264 x}"}, {"tactic": "exacts [measurableSet_Iic, measurableSet_Iic]", "annotated_tactic": ["exacts [<a>measurableSet_Iic</a>, <a>measurableSet_Iic</a>]", [{"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}, {"full_name": "measurableSet_Iic", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 26]}]], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 MeasurableSet {x_1 | x_1 \u2264 x}\n\ncase refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 MeasurableSet {x_1 | x_1 \u2264 x}", "state_after": "no goals"}, {"tactic": "rw [min\u2081\u2082]", "annotated_tactic": ["rw [min\u2081\u2082]", []], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc (min b\u2081 b\u2082) x))", "state_after": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc b\u2081 x))"}, {"tactic": "exact (h_int' hx).1.aestronglyMeasurable", "annotated_tactic": ["exact (h_int' hx).1.<a>aestronglyMeasurable</a>", [{"full_name": "MeasureTheory.Integrable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [457, 9], "def_end_pos": [457, 40]}]], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc b\u2081 x))", "state_after": "no goals"}, {"tactic": "exact le_max_of_le_right hx.2", "annotated_tactic": ["exact <a>le_max_of_le_right</a> hx.2", [{"full_name": "le_max_of_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}]], "state_before": "case refine'_1\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082\n\u22a2 x \u2264 max b\u2081 b\u2082", "state_after": "no goals"}, {"tactic": "refine' eventually_of_forall fun x => eventually_of_forall fun t => _", "annotated_tactic": ["refine' <a>eventually_of_forall</a> fun x => <a>eventually_of_forall</a> fun t => _", [{"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": "case refine'_2\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016indicator {x_1 | x_1 \u2264 x} f t\u2016 \u2264 \u2016f t\u2016", "state_after": "case refine'_2\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx t : \u211d\n\u22a2 t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016indicator {x_1 | x_1 \u2264 x} f t\u2016 \u2264 \u2016f t\u2016"}, {"tactic": "dsimp [indicator]", "annotated_tactic": ["dsimp [<a>indicator</a>]", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}]], "state_before": "case refine'_2\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx t : \u211d\n\u22a2 t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016indicator {x_1 | x_1 \u2264 x} f t\u2016 \u2264 \u2016f t\u2016", "state_after": "case refine'_2\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx t : \u211d\n\u22a2 t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016if t \u2264 x then f t else 0\u2016 \u2264 \u2016f t\u2016"}, {"tactic": "split_ifs <;> simp", "annotated_tactic": ["split_ifs <;> simp", []], "state_before": "case refine'_2\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx t : \u211d\n\u22a2 t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016if t \u2264 x then f t else 0\u2016 \u2264 \u2016f t\u2016", "state_after": "no goals"}, {"tactic": "have : \u2200\u1d50 t \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t := by\n  apply Eventually.mono (compl_mem_ae_iff.mpr hb\u2080)\n  intro x hx\n  exact Ne.lt_or_lt hx", "annotated_tactic": ["have : \u2200\u1d50 t \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t := by\n        apply <a>Eventually.mono</a> (compl_mem_ae_iff.mpr hb\u2080)\n        intro x hx\n        exact <a>Ne.lt_or_lt</a> hx", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Ne.lt_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [496, 9], "def_end_pos": [496, 20]}]], "state_before": "case refine'_3\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f t) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f t) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "apply this.mono", "annotated_tactic": ["apply this.mono", []], "state_before": "case refine'_3\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f t) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\n\u22a2 \u2200 (x : \u211d), x < b\u2080 \u2228 b\u2080 < x \u2192 x \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x_1 => indicator {x | x \u2264 x_1} f x) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "rintro x\u2080 (hx\u2080 | hx\u2080) -", "annotated_tactic": ["rintro x\u2080 (hx\u2080 | hx\u2080) -", []], "state_before": "case refine'_3\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\n\u22a2 \u2200 (x : \u211d), x < b\u2080 \u2228 b\u2080 < x \u2192 x \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x_1 => indicator {x | x \u2264 x_1} f x) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3.inl\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080\n\ncase refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "apply Eventually.mono (compl_mem_ae_iff.mpr hb\u2080)", "annotated_tactic": ["apply <a>Eventually.mono</a> (compl_mem_ae_iff.mpr hb\u2080)", [{"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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200 (x : \u211d), \u00acx \u2208 {b\u2080} \u2192 x < b\u2080 \u2228 b\u2080 < x"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\n\u22a2 \u2200 (x : \u211d), \u00acx \u2208 {b\u2080} \u2192 x < b\u2080 \u2228 b\u2080 < x", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : \u00acx \u2208 {b\u2080}\n\u22a2 x < b\u2080 \u2228 b\u2080 < x"}, {"tactic": "exact Ne.lt_or_lt hx", "annotated_tactic": ["exact <a>Ne.lt_or_lt</a> hx", [{"full_name": "Ne.lt_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [496, 9], "def_end_pos": [496, 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\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nx : \u211d\nhx : \u00acx \u2208 {b\u2080}\n\u22a2 x < b\u2080 \u2228 b\u2080 < x", "state_after": "no goals"}, {"tactic": "have : \u2200\u1da0 x in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, {t : \u211d | t \u2264 x}.indicator f x\u2080 = f x\u2080 := by\n  apply mem_nhdsWithin_of_mem_nhds\n  apply Eventually.mono (Ioi_mem_nhds hx\u2080)\n  intro x hx\n  simp [hx.le]", "annotated_tactic": ["have : \u2200\u1da0 x in \ud835\udcdd[<a>Icc</a> b\u2081 b\u2082] b\u2080, {t : \u211d | t \u2264 x}.<a>indicator</a> f x\u2080 = f x\u2080 := by\n          apply <a>mem_nhdsWithin_of_mem_nhds</a>\n          apply <a>Eventually.mono</a> (<a>Ioi_mem_nhds</a> hx\u2080)\n          intro x hx\n          simp [hx.le]", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Ioi_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 21]}]], "state_before": "case refine'_3.inl\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3.inl\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "apply continuousWithinAt_const.congr_of_eventuallyEq this", "annotated_tactic": ["apply continuousWithinAt_const.congr_of_eventuallyEq this", []], "state_before": "case refine'_3.inl\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3.inl\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080\n\u22a2 indicator {t | t \u2264 b\u2080} f x\u2080 = f x\u2080"}, {"tactic": "simp [hx\u2080.le]", "annotated_tactic": ["simp [hx\u2080.le]", []], "state_before": "case refine'_3.inl\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080\n\u22a2 indicator {t | t \u2264 b\u2080} f x\u2080 = f x\u2080", "state_after": "no goals"}, {"tactic": "apply mem_nhdsWithin_of_mem_nhds", "annotated_tactic": ["apply <a>mem_nhdsWithin_of_mem_nhds</a>", [{"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}]], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 {x | (fun x => indicator {t | t \u2264 x} f x\u2080 = f x\u2080) x} \u2208 \ud835\udcdd b\u2080"}, {"tactic": "apply Eventually.mono (Ioi_mem_nhds hx\u2080)", "annotated_tactic": ["apply <a>Eventually.mono</a> (<a>Ioi_mem_nhds</a> hx\u2080)", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Ioi_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 {x | (fun x => indicator {t | t \u2264 x} f x\u2080 = f x\u2080) x} \u2208 \ud835\udcdd b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 \u2200 (x : \u211d), x\u2080 < x \u2192 indicator {t | t \u2264 x} f x\u2080 = f x\u2080"}, {"tactic": "intro x hx", "annotated_tactic": ["intro 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\n\u22a2 \u2200 (x : \u211d), x\u2080 < x \u2192 indicator {t | t \u2264 x} f x\u2080 = f x\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\nx : \u211d\nhx : x\u2080 < x\n\u22a2 indicator {t | t \u2264 x} f x\u2080 = f x\u2080"}, {"tactic": "simp [hx.le]", "annotated_tactic": ["simp [hx.le]", []], "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : x\u2080 < b\u2080\nx : \u211d\nhx : x\u2080 < x\n\u22a2 indicator {t | t \u2264 x} f x\u2080 = f x\u2080", "state_after": "no goals"}, {"tactic": "have : \u2200\u1da0 x in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, {t : \u211d | t \u2264 x}.indicator f x\u2080 = 0 := by\n  apply mem_nhdsWithin_of_mem_nhds\n  apply Eventually.mono (Iio_mem_nhds hx\u2080)\n  intro x hx\n  simp [hx]", "annotated_tactic": ["have : \u2200\u1da0 x in \ud835\udcdd[<a>Icc</a> b\u2081 b\u2082] b\u2080, {t : \u211d | t \u2264 x}.<a>indicator</a> f x\u2080 = 0 := by\n          apply <a>mem_nhdsWithin_of_mem_nhds</a>\n          apply <a>Eventually.mono</a> (<a>Iio_mem_nhds</a> hx\u2080)\n          intro x hx\n          simp [hx]", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Iio_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 21]}]], "state_before": "case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080"}, {"tactic": "apply continuousWithinAt_const.congr_of_eventuallyEq this", "annotated_tactic": ["apply continuousWithinAt_const.congr_of_eventuallyEq this", []], "state_before": "case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0\n\u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0\n\u22a2 indicator {t | t \u2264 b\u2080} f x\u2080 = 0"}, {"tactic": "simp [hx\u2080]", "annotated_tactic": ["simp [hx\u2080]", []], "state_before": "case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d\u00b9 :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0\n\u22a2 indicator {t | t \u2264 b\u2080} f x\u2080 = 0", "state_after": "no goals"}, {"tactic": "apply mem_nhdsWithin_of_mem_nhds", "annotated_tactic": ["apply <a>mem_nhdsWithin_of_mem_nhds</a>", [{"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}]], "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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 {x | (fun x => indicator {t | t \u2264 x} f x\u2080 = 0) x} \u2208 \ud835\udcdd b\u2080"}, {"tactic": "apply Eventually.mono (Iio_mem_nhds hx\u2080)", "annotated_tactic": ["apply <a>Eventually.mono</a> (<a>Iio_mem_nhds</a> hx\u2080)", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Iio_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 {x | (fun x => indicator {t | t \u2264 x} f x\u2080 = 0) x} \u2208 \ud835\udcdd b\u2080", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 \u2200 (x : \u211d), x < x\u2080 \u2192 indicator {t | t \u2264 x} f x\u2080 = 0"}, {"tactic": "intro x hx", "annotated_tactic": ["intro 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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\n\u22a2 \u2200 (x : \u211d), x < x\u2080 \u2192 indicator {t | t \u2264 x} f x\u2080 = 0", "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\nx : \u211d\nhx : x < x\u2080\n\u22a2 indicator {t | t \u2264 x} f x\u2080 = 0"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082\nh\u2081\u2082 : b\u2081 \u2264 b\u2082\nmin\u2081\u2082 : min b\u2081 b\u2082 = b\u2081\nh_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x\nthis\u271d :\n  (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc\nthis : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t\nx\u2080 : \u211d\nhx\u2080 : b\u2080 < x\u2080\nx : \u211d\nhx : x < x\u2080\n\u22a2 indicator {t | t \u2264 x} f x\u2080 = 0", "state_after": "no goals"}, {"tactic": "apply continuousWithinAt_of_not_mem_closure", "annotated_tactic": ["apply <a>continuousWithinAt_of_not_mem_closure</a>", [{"full_name": "continuousWithinAt_of_not_mem_closure", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1146, 9], "def_end_pos": [1146, 46]}]], "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\na b b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : \u00acb\u2080 \u2208 Icc b\u2081 b\u2082\n\u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080", "state_after": "case neg.a\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : \u00acb\u2080 \u2208 Icc b\u2081 b\u2082\n\u22a2 \u00acb\u2080 \u2208 closure (Icc b\u2081 b\u2082)"}, {"tactic": "rwa [closure_Icc]", "annotated_tactic": ["rwa [<a>closure_Icc</a>]", [{"full_name": "closure_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [219, 9], "def_end_pos": [219, 20]}]], "state_before": "case neg.a\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 b\u2080 b\u2081 b\u2082 : \u211d\n\u03bc : Measure \u211d\nf g : \u211d \u2192 E\nhb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0\nh_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082)\nh\u2080 : \u00acb\u2080 \u2208 Icc b\u2081 b\u2082\n\u22a2 \u00acb\u2080 \u2208 closure (Icc b\u2081 b\u2082)", "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_map_eq_of_ncard_lt_of_maps_to", "start": [770, 1], "end": [775, 55], "traced_tactics": [{"tactic": "by_contra h'", "annotated_tactic": ["by_contra h'", []], "state_before": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\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_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u00ac\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y\n\u22a2 False"}, {"tactic": "simp only [Ne.def, exists_prop, not_exists, not_and, not_imp_not] at h'", "annotated_tactic": ["simp only [<a>Ne.def</a>, <a>exists_prop</a>, <a>not_exists</a>, <a>not_and</a>, <a>not_imp_not</a>] at h'", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"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_imp_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "state_before": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u00ac\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y\n\u22a2 False", "state_after": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 f x = f x_1 \u2192 x = x_1\n\u22a2 False"}, {"tactic": "exact (ncard_le_ncard_of_injOn f hf h' ht).not_lt hc", "annotated_tactic": ["exact (<a>ncard_le_ncard_of_injOn</a> f hf h' ht).<a>not_lt</a> hc", [{"full_name": "Set.ncard_le_ncard_of_injOn", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [763, 9], "def_end_pos": [763, 32]}, {"full_name": "LE.le.not_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [379, 7], "def_end_pos": [379, 19]}]], "state_before": "\u03b1 : Type u_2\ns t\u271d : Set \u03b1\n\u03b2 : Type u_1\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 f x = f x_1 \u2192 x = x_1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.exists_le_integral", "start": [578, 1], "end": [579, 95], "traced_tactics": [{"tactic": "simpa only [average_eq_integral] using exists_le_average (IsProbabilityMeasure.ne_zero \u03bc) hf", "annotated_tactic": ["simpa only [<a>average_eq_integral</a>] using <a>exists_le_average</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hf", [{"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_le_average", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [529, 9], "def_end_pos": [529, 26]}, {"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\n\u22a2 \u2203 x, f x \u2264 \u222b (a : \u03b1), f a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.exists_intermediate_Set", "start": [934, 1], "end": [945, 40], "traced_tactics": [{"tactic": "cases' t.finite_or_infinite with ht ht", "annotated_tactic": ["cases' t.finite_or_infinite with ht ht", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 ncard t\nh\u2082 : s \u2286 t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 ncard t\nh\u2082 : s \u2286 t\nht : Set.Finite t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s\n\ncase inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 ncard t\nh\u2082 : s \u2286 t\nht : Set.Infinite t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s"}, {"tactic": "rw [ht.ncard] at h\u2081", "annotated_tactic": ["rw [ht.ncard] at h\u2081", []], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 ncard t\nh\u2082 : s \u2286 t\nht : Set.Infinite t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s", "state_after": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s"}, {"tactic": "have h\u2081' := Nat.eq_zero_of_le_zero h\u2081", "annotated_tactic": ["have h\u2081' := <a>Nat.eq_zero_of_le_zero</a> h\u2081", [{"full_name": "Nat.eq_zero_of_le_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [301, 9], "def_end_pos": [301, 27]}]], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s", "state_after": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\nh\u2081' : i + ncard s = 0\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s"}, {"tactic": "rw [add_eq_zero_iff] at h\u2081'", "annotated_tactic": ["rw [<a>add_eq_zero_iff</a>] at h\u2081'", [{"full_name": "add_eq_zero_iff", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}]], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\nh\u2081' : i + ncard s = 0\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s", "state_after": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\nh\u2081' : i = 0 \u2227 ncard s = 0\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s"}, {"tactic": "refine' \u27e8t, h\u2082, rfl.subset, _\u27e9", "annotated_tactic": ["refine' \u27e8t, h\u2082, rfl.subset, _\u27e9", []], "state_before": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\nh\u2081' : i = 0 \u2227 ncard s = 0\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s", "state_after": "case inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\nh\u2081' : i = 0 \u2227 ncard s = 0\n\u22a2 ncard t = i + ncard s"}, {"tactic": "rw [h\u2081'.2, h\u2081'.1, ht.ncard, add_zero]", "annotated_tactic": ["rw [h\u2081'.2, h\u2081'.1, ht.ncard, <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 inr\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 0\nh\u2082 : s \u2286 t\nht : Set.Infinite t\nh\u2081' : i = 0 \u2227 ncard s = 0\n\u22a2 ncard t = i + ncard s", "state_after": "no goals"}, {"tactic": "rw [ncard_eq_toFinset_card _ (ht.subset h\u2082)] at h\u2081 \u22a2", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a> _ (ht.subset h\u2082)] at h\u2081 \u22a2", [{"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": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2081 : i + ncard s \u2264 ncard t\nh\u2082 : s \u2286 t\nht : Set.Finite t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 ncard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))"}, {"tactic": "rw [ncard_eq_toFinset_card t ht] at h\u2081", "annotated_tactic": ["rw [<a>ncard_eq_toFinset_card</a> t ht] at h\u2081", [{"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": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 ncard t\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))", "state_after": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))"}, {"tactic": "obtain \u27e8r', hsr', hr't, hr'\u27e9 := Finset.exists_intermediate_set _ h\u2081 (by simpa)", "annotated_tactic": ["obtain \u27e8r', hsr', hr't, hr'\u27e9 := <a>Finset.exists_intermediate_set</a> _ h\u2081 (by simpa)", [{"full_name": "Finset.exists_intermediate_set", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [478, 9], "def_end_pos": [478, 32]}]], "state_before": "case inl\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))", "state_after": "case inl.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\nr' : Finset \u03b1\nhsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r'\nhr't : r' \u2286 Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))"}, {"tactic": "exact \u27e8r', by simpa using hsr', by simpa using hr't, by rw [\u2190 hr', ncard_coe_Finset]\u27e9", "annotated_tactic": ["exact \u27e8r', by simpa using hsr', by simpa using hr't, by rw [\u2190 hr', <a>ncard_coe_Finset</a>]\u27e9", [{"full_name": "Set.ncard_coe_Finset", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [522, 17], "def_end_pos": [522, 33]}]], "state_before": "case inl.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\nr' : Finset \u03b1\nhsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r'\nhr't : r' \u2286 Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n\u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\n\u22a2 Finite.toFinset (_ : Set.Finite s) \u2286 Finite.toFinset ht", "state_after": "no goals"}, {"tactic": "simpa using hsr'", "annotated_tactic": ["simpa using hsr'", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\nr' : Finset \u03b1\nhsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r'\nhr't : r' \u2286 Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n\u22a2 s \u2286 \u2191r'", "state_after": "no goals"}, {"tactic": "simpa using hr't", "annotated_tactic": ["simpa using hr't", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\nr' : Finset \u03b1\nhsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r'\nhr't : r' \u2286 Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n\u22a2 \u2191r' \u2286 t", "state_after": "no goals"}, {"tactic": "rw [\u2190 hr', ncard_coe_Finset]", "annotated_tactic": ["rw [\u2190 hr', <a>ncard_coe_Finset</a>]", [{"full_name": "Set.ncard_coe_Finset", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [522, 17], "def_end_pos": [522, 33]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\ni : \u2115\nh\u2082 : s \u2286 t\nht : Set.Finite t\nh\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht)\nr' : Finset \u03b1\nhsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r'\nhr't : r' \u2286 Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n\u22a2 ncard \u2191r' = i + Finset.card (Finite.toFinset (_ : Set.Finite 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.condexpIndL1Fin_disjoint_union", "start": [147, 1], "end": [167, 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x =\n    condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 \u2191\u2191(condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x)"}, {"tactic": "have h\u03bcst :=\n  ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr \u27e8h\u03bcs, h\u03bct\u27e9))).ne", "annotated_tactic": ["have h\u03bcst :=\n    ((<a>measure_union_le</a> s t).<a>trans_lt</a> (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr \u27e8h\u03bcs, h\u03bct\u27e9))).<a>ne</a>", [{"full_name": "MeasureTheory.measure_union_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}, {"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": "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\n\u22a2 \u2191\u2191(condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 \u2191\u2191(condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x)"}, {"tactic": "refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) h\u03bcst x).trans _", "annotated_tactic": ["refine' (<a>condexpIndL1Fin_ae_eq_condexpIndSMul</a> hm (hs.union ht) h\u03bcst x).<a>trans</a> _", [{"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.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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 \u2191\u2191(condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x)"}, {"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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x)"}, {"tactic": "have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x", "annotated_tactic": ["have hs_eq := <a>condexpIndL1Fin_ae_eq_condexpIndSMul</a> hm hs h\u03bcs x", [{"full_name": "MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [78, 9], "def_end_pos": [78, 45]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x)"}, {"tactic": "have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht h\u03bct x", "annotated_tactic": ["have ht_eq := <a>condexpIndL1Fin_ae_eq_condexpIndSMul</a> hm ht h\u03bct x", [{"full_name": "MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [78, 9], "def_end_pos": [78, 45]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x)"}, {"tactic": "refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)", "annotated_tactic": ["refine' <a>EventuallyEq.trans</a> _ (<a>EventuallyEq.add</a> hs_eq.symm ht_eq.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.EventuallyEq.add", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 3], "def_end_pos": [1530, 14]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]\n    \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc] fun x_1 =>\n    \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"tactic": "rw [condexpIndSMul]", "annotated_tactic": ["rw [<a>condexpIndSMul</a>]", [{"full_name": "MeasureTheory.condexpIndSMul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [384, 19], "def_end_pos": [384, 33]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc] fun x_1 =>\n    \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 (_ : MeasurableSet (s \u222a t)) h\u03bcst 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"tactic": "rw [indicatorConstLp_disjoint_union hs ht h\u03bcs h\u03bct hst (1 : \u211d)]", "annotated_tactic": ["rw [<a>indicatorConstLp_disjoint_union</a> hs ht h\u03bcs h\u03bct hst (1 : \u211d)]", [{"full_name": "MeasureTheory.indicatorConstLp_disjoint_union", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [820, 9], "def_end_pos": [820, 40]}]], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 (_ : MeasurableSet (s \u222a t)) h\u03bcst 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1 + indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"tactic": "rw [(condexpL2 \u211d \u211d hm).map_add]", "annotated_tactic": ["rw [(<a>condexpL2</a> \u211d \u211d hm).<a>map_add</a>]", [{"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"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 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1 + indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1) +\n              \u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1) +\n              \u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          (\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n            \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1)))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"tactic": "rw [((toSpanSingleton \u211d x).compLpL 2 \u03bc).map_add]", "annotated_tactic": ["rw [((<a>toSpanSingleton</a> \u211d x).<a>compLpL</a> 2 \u03bc).<a>map_add</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.compLpL", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1176, 5], "def_end_pos": [1176, 12]}, {"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 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))\n          (\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n            \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1)))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n          \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"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\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n          \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) +\n      \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1"}, {"tactic": "refine' eventually_of_forall fun y => _", "annotated_tactic": ["refine' <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\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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\n\u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) +\n      \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]\n    fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) 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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\ny : \u03b1\n\u22a2 (\u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) +\n        \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))))\n      y =\n    (fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1) y"}, {"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\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : s \u2229 t = \u2205\nx : G\nh\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4\nhs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\nht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x)\ny : \u03b1\n\u22a2 (\u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) +\n        \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))))\n      y =\n    (fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEquiv.measurable_comp_iff", "start": [1471, 11], "end": [1477, 33], "traced_tactics": [{"tactic": "have : Measurable (f \u2218 (e.symm.trans e).toEquiv) := hfe.comp e.symm.measurable", "annotated_tactic": ["have : <a>Measurable</a> (f \u2218 (e.symm.trans e).toEquiv) := hfe.comp e.symm.measurable", [{"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\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b2 \u2192 \u03b3\ne : \u03b1 \u2243\u1d50 \u03b2\nhfe : Measurable (f \u2218 \u2191e)\n\u22a2 Measurable f", "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 t u : Set \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b2 \u2192 \u03b3\ne : \u03b1 \u2243\u1d50 \u03b2\nhfe : Measurable (f \u2218 \u2191e)\nthis : Measurable (f \u2218 \u2191(trans (symm e) e).toEquiv)\n\u22a2 Measurable f"}, {"tactic": "rwa [coe_toEquiv, symm_trans_self] at this", "annotated_tactic": ["rwa [<a>coe_toEquiv</a>, <a>symm_trans_self</a>] at this", [{"full_name": "MeasurableEquiv.coe_toEquiv", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1293, 9], "def_end_pos": [1293, 20]}, {"full_name": "MeasurableEquiv.symm_trans_self", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1387, 9], "def_end_pos": [1387, 24]}]], "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\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b2 \u2192 \u03b3\ne : \u03b1 \u2243\u1d50 \u03b2\nhfe : Measurable (f \u2218 \u2191e)\nthis : Measurable (f \u2218 \u2191(trans (symm e) e).toEquiv)\n\u22a2 Measurable f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_liftCover", "start": [834, 1], "end": [840, 60], "traced_tactics": [{"tactic": "rw [preimage_liftCover]", "annotated_tactic": ["rw [<a>preimage_liftCover</a>]", [{"full_name": "Set.preimage_liftCover", "def_path": "Mathlib/Data/Set/UnionLift.lean", "def_pos": [176, 9], "def_end_pos": [176, 27]}]], "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\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nf : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2\nhfm : \u2200 (i : \u03b9), Measurable (f i)\nhf :\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 }\nhtU : \u22c3 i, t i = univ\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (liftCover t f hf htU \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\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nf : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2\nhfm : \u2200 (i : \u03b9), Measurable (f i)\nhf :\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 }\nhtU : \u22c3 i, t i = univ\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 i, Subtype.val '' (f i \u207b\u00b9' s))"}, {"tactic": "exact .iUnion fun i => .subtype_image (htm i) <| hfm i hs", "annotated_tactic": ["exact .iUnion fun i => .subtype_image (htm i) <| hfm i hs", []], "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\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nf : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2\nhfm : \u2200 (i : \u03b9), Measurable (f i)\nhf :\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 }\nhtU : \u22c3 i, t i = univ\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 i, Subtype.val '' (f i \u207b\u00b9' s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "lipschitzWith_circleMap", "start": [208, 1], "end": [210, 35], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR \u03b8 : \u211d\n\u22a2 \u2191\u2016deriv (circleMap c R) \u03b8\u2016\u208a \u2264 \u2191(\u2191Real.nnabs R)", "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_center", "start": [426, 1], "end": [429, 34], "traced_tactics": [{"tactic": "rw [this, measure_preimage_add]", "annotated_tactic": ["rw [this, <a>measure_preimage_add</a>]", [{"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\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\nthis : ball 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' ball x r\n\u22a2 \u2191\u2191\u03bc (ball x r) = \u2191\u2191\u03bc (ball 0 r)", "state_after": "no goals"}, {"tactic": "simp [preimage_add_ball]", "annotated_tactic": ["simp [<a>preimage_add_ball</a>]", [{"full_name": "preimage_add_ball", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1578, 3], "def_end_pos": [1578, 14]}]], "state_before": "E\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\n\u22a2 ball 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' ball x r", "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_and_zero_of_ae_le_mul_neg", "start": [1311, 1], "end": [1317, 48], "traced_tactics": [{"tactic": "simp_rw [le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg (norm_nonneg _) hc (norm_nonneg _),\n  norm_eq_zero, eventually_and] at h", "annotated_tactic": ["simp_rw [<a>le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg</a> (<a>norm_nonneg</a> _) hc (<a>norm_nonneg</a> _),\n    <a>norm_eq_zero</a>, <a>eventually_and</a>] at h", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSeminorm.0.MeasureTheory.le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1299, 17], "def_end_pos": [1299, 62]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "norm_eq_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2018, 30], "def_end_pos": [2018, 42]}, {"full_name": "Filter.eventually_and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1151, 9], "def_end_pos": [1151, 23]}]], "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\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhc : c < 0\np : \u211d\u22650\u221e\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 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\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nhc : c < 0\np : \u211d\u22650\u221e\nh : (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = 0) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x = 0\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0"}, {"tactic": "change f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0 at h", "annotated_tactic": ["change f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0 at h", []], "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\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nhc : c < 0\np : \u211d\u22650\u221e\nh : (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = 0) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x = 0\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 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\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nhc : c < 0\np : \u211d\u22650\u221e\nh : f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0"}, {"tactic": "simp [snorm_congr_ae h.1, snorm_congr_ae h.2]", "annotated_tactic": ["simp [<a>snorm_congr_ae</a> h.1, <a>snorm_congr_ae</a> h.2]", [{"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_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}]], "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\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nhc : c < 0\np : \u211d\u22650\u221e\nh : f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.continuous_integral_integral", "start": [421, 1], "end": [447, 61], "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\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'\n\u22a2 Continuous fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc", "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\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'\n\u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) x"}, {"tactic": "intro g", "annotated_tactic": ["intro g", []], "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'\n\u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) 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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) g"}, {"tactic": "refine'\n  tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left\n    (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_prod_left) _", "annotated_tactic": ["refine'\n    <a>tendsto_integral_of_L1</a> _ (<a>L1.integrable_coeFn</a> g).<a>integral_prod_left</a>\n      (<a>eventually_of_forall</a> fun h => (<a>L1.integrable_coeFn</a> h).<a>integral_prod_left</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_prod_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [326, 9], "def_end_pos": [326, 38]}, {"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_prod_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [326, 9], "def_end_pos": [326, 38]}]], "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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) g", "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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) \u2202\u03bd - \u222b (y : \u03b2), \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "simp_rw [\u2190\n  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>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": "MeasureTheory.lintegral_fn_integral_sub", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [379, 9], "def_end_pos": [379, 34]}, {"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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) \u2202\u03bd - \u222b (y : \u03b2), \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_1\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'\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\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'\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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 ?refine'_1"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine'_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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n\ncase refine'_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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc", "state_after": "case refine'_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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc\n\ncase refine'_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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "have : \u2200 i : \u03b1 \u00d7 \u03b2 \u2192\u2081[\u03bc.prod \u03bd] E, 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 : \u03b1 \u00d7 \u03b2 \u2192\u2081[\u03bc.prod \u03bd] E, <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": "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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_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'\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 : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "conv =>\n  congr\n  ext\n  rw [\u2190 lintegral_prod_of_measurable _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral]", "annotated_tactic": ["conv =>\n    congr\n    ext\n    rw [\u2190 <a>lintegral_prod_of_measurable</a> _ (this _), \u2190 <a>L1.ofReal_norm_sub_eq_lintegral</a>]", [{"full_name": "MeasureTheory.lintegral_prod_of_measurable", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [821, 9], "def_end_pos": [821, 37]}, {"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]}]], "state_before": "case refine'_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'\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 : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_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'\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 x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 ofReal_zero]", "annotated_tactic": ["rw [\u2190 <a>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": "case refine'_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'\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 x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_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'\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 x => ENNReal.ofReal \u2016x - 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\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'\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 x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd (ENNReal.ofReal 0))", "state_after": "case refine'_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'\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 x => \u2016x - 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\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'\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 x => \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)", "state_after": "case refine'_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'\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 x => x) (\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\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'\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 x => x) (\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\u03bd \u2202\u03bc", "annotated_tactic": ["exact fun i => \u222b\u207b x, \u222b\u207b y, \u2016i (x, y) - g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc", []], "state_before": "case refine'_1\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'\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\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'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>\n    \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \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.RBNode.zoom_toList", "start": [520, 1], "end": [521, 83], "traced_tactics": [{"tactic": "rw [\u2190 fill_toList, \u2190 zoom_fill eq]", "annotated_tactic": ["rw [\u2190 <a>fill_toList</a>, \u2190 <a>zoom_fill</a> eq]", [{"full_name": "Std.RBNode.Path.fill_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [517, 17], "def_end_pos": [517, 28]}, {"full_name": "Std.RBNode.Path.zoom_fill", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [54, 9], "def_end_pos": [54, 18]}]], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\np' : Path \u03b1\nt : RBNode \u03b1\neq : zoom cut t root = (t', p')\n\u22a2 withList p' (toList t') = toList t", "state_after": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\np' : Path \u03b1\nt : RBNode \u03b1\neq : zoom cut t root = (t', p')\n\u22a2 toList (fill root t) = toList t"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\nt' : RBNode \u03b1\np' : Path \u03b1\nt : RBNode \u03b1\neq : zoom cut t root = (t', p')\n\u22a2 toList (fill root t) = toList t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.getOrElse_some", "start": [285, 1], "end": [286, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.bind\u2081_rename", "start": [268, 1], "end": [270, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr_eq_map", "start": [234, 1], "end": [241, 58], "traced_tactics": [{"tactic": "rw[Vector.map_eq_mapAccumr]", "annotated_tactic": ["rw[<a>Vector.map_eq_mapAccumr</a>]", [{"full_name": "Vector.map_eq_mapAccumr", "def_path": "Mathlib/Data/Vector/MapLemmas.lean", "def_pos": [224, 19], "def_end_pos": [224, 35]}]], "state_before": "\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 (mapAccumr f xs s\u2080).2 = map (fun x => (f x s\u2080).2) xs", "state_after": "\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 (mapAccumr f xs s\u2080).2 = (mapAccumr (fun x x_1 => ((), (f x s\u2080).2)) xs ()).2"}, {"tactic": "apply mapAccumr_bisim_tail", "annotated_tactic": ["apply <a>mapAccumr_bisim_tail</a>", [{"full_name": "Vector.mapAccumr_bisim_tail", "def_path": "Mathlib/Data/Vector/MapLemmas.lean", "def_pos": [185, 9], "def_end_pos": [185, 29]}]], "state_before": "\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 (mapAccumr f xs s\u2080).2 = (mapAccumr (fun x x_1 => ((), (f x s\u2080).2)) xs ()).2", "state_after": "case h\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 \u2203 R, R s\u2080 () \u2227 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), R s q \u2192 R (f a s).1 ((), (f a s\u2080).2).1 \u2227 (f a s).2 = ((), (f a 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 u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 \u2203 R, R s\u2080 () \u2227 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), R s q \u2192 R (f a s).1 ((), (f a s\u2080).2).1 \u2227 (f a s).2 = ((), (f a s\u2080).2).2", "state_after": "case right\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), s \u2208 S \u2192 (f a s).1 \u2208 S \u2227 (f a s).2 = ((), (f a s\u2080).2).2"}, {"tactic": "exact @fun s _q a h => \u27e8closure a s h, out a s s\u2080 h h\u2080\u27e9", "annotated_tactic": ["exact @fun s _q a h => \u27e8closure a s h, out a s s\u2080 h h\u2080\u27e9", []], "state_before": "case right\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2\n\u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), s \u2208 S \u2192 (f a s).1 \u2208 S \u2227 (f a s).2 = ((), (f a s\u2080).2).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Accumulate.lean", "full_name": "Set.mem_accumulate", "start": [31, 1], "end": [32, 53], "traced_tactics": [{"tactic": "simp_rw [accumulate_def, mem_iUnion\u2082, exists_prop]", "annotated_tactic": ["simp_rw [<a>accumulate_def</a>, <a>mem_iUnion\u2082</a>, <a>exists_prop</a>]", [{"full_name": "Set.accumulate_def", "def_path": "Mathlib/Data/Set/Accumulate.lean", "def_pos": [26, 9], "def_end_pos": [26, 23]}, {"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}, {"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\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : LE \u03b1\nx : \u03b1\nz : \u03b2\n\u22a2 z \u2208 Accumulate s x \u2194 \u2203 y, y \u2264 x \u2227 z \u2208 s y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.measurable_of_not_restrict_le_zero", "start": [1010, 1], "end": [1011, 57], "traced_tactics": []}, {"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", "start": [59, 1], "end": [64, 74], "traced_tactics": [{"tactic": "have : arr = #[] \u2228 0 < arr.size :=\n  match arr with | \u27e8[]\u27e9 => .inl rfl | \u27e8a::l\u27e9 => .inr (Nat.zero_lt_succ _)", "annotated_tactic": ["have : arr = #[] \u2228 0 < arr.size :=\n    match arr with | \u27e8[]\u27e9 => .inl <a>rfl</a> | \u27e8a::l\u27e9 => .inr (<a>Nat.zero_lt_succ</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": "Nat.zero_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1586, 9], "def_end_pos": [1586, 25]}]], "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\ninit : \u03b2\narr : Array \u03b1\n\u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data)", "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\ninit : \u03b2\narr : Array \u03b1\nthis : arr = #[] \u2228 0 < size arr\n\u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data)"}, {"tactic": "match arr, this with | _, .inl rfl => rfl | arr, .inr h => ?_", "annotated_tactic": ["match arr, this with | _, .inl <a>rfl</a> => rfl | arr, .inr 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": "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\ninit : \u03b2\narr : Array \u03b1\nthis : arr = #[] \u2228 0 < size arr\n\u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data)", "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\ninit : \u03b2\narr\u271d : Array \u03b1\nthis : arr\u271d = #[] \u2228 0 < size arr\u271d\narr : Array \u03b1\nh : 0 < size arr\n\u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data)"}, {"tactic": "simp [foldrM, h, \u2190 foldrM_eq_reverse_foldlM_data.aux, List.take_length]", "annotated_tactic": ["simp [<a>foldrM</a>, h, \u2190 <a>foldrM_eq_reverse_foldlM_data.aux</a>, <a>List.take_length</a>]", [{"full_name": "Array.foldrM", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [233, 5], "def_end_pos": [233, 11]}, {"full_name": "Array.foldrM_eq_reverse_foldlM_data.aux", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [51, 9], "def_end_pos": [51, 42]}, {"full_name": "List.take_length", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [169, 17], "def_end_pos": [169, 28]}]], "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\ninit : \u03b2\narr\u271d : Array \u03b1\nthis : arr\u271d = #[] \u2228 0 < size arr\u271d\narr : Array \u03b1\nh : 0 < size arr\n\u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data)", "state_after": "no goals"}, {"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\ninit : \u03b2\narr : Array \u03b1\nthis : arr = #[] \u2228 0 < size arr\n\u22a2 foldrM f init #[] (size #[]) = List.foldlM (fun x y => f y x) init (List.reverse #[].data)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.bind\u2081_bind\u2081", "start": [224, 1], "end": [226, 29], "traced_tactics": [{"tactic": "simp [bind\u2081, \u2190 comp_aeval]", "annotated_tactic": ["simp [<a>bind\u2081</a>, \u2190 <a>comp_aeval</a>]", [{"full_name": "MvPolynomial.bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [69, 5], "def_end_pos": [69, 10]}, {"full_name": "MvPolynomial.comp_aeval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1496, 9], "def_end_pos": [1496, 19]}]], "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\n\u03c5 : Type u_6\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ng : \u03c4 \u2192 MvPolynomial \u03c5 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(bind\u2081 g) (\u2191(bind\u2081 f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(bind\u2081 g) (f i)) \u03c6", "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.haarMeasure_apply", "start": [611, 1], "end": [614, 66], "traced_tactics": [{"tactic": "change ((haarContent K\u2080).outerMeasure K\u2080)\u207b\u00b9 * (haarContent K\u2080).measure s = _", "annotated_tactic": ["change ((<a>haarContent</a> K\u2080).<a>outerMeasure</a> K\u2080)\u207b\u00b9 * (<a>haarContent</a> K\u2080).<a>measure</a> s = _", [{"full_name": "MeasureTheory.Measure.haar.haarContent", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [540, 19], "def_end_pos": [540, 30]}, {"full_name": "MeasureTheory.Content.outerMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [248, 15], "def_end_pos": [248, 27]}, {"full_name": "MeasureTheory.Measure.haar.haarContent", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [540, 19], "def_end_pos": [540, 30]}, {"full_name": "MeasureTheory.Content.measure", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [380, 15], "def_end_pos": [380, 22]}]], "state_before": "G : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(haarMeasure K\u2080) s = \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080", "state_after": "G : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ns : Set G\nhs : MeasurableSet s\n\u22a2 (\u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080)\u207b\u00b9 * \u2191\u2191(Content.measure (haarContent K\u2080)) s =\n    \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080"}, {"tactic": "simp only [hs, div_eq_mul_inv, mul_comm, Content.measure_apply]", "annotated_tactic": ["simp only [hs, <a>div_eq_mul_inv</a>, <a>mul_comm</a>, <a>Content.measure_apply</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_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.Content.measure_apply", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [384, 9], "def_end_pos": [384, 22]}]], "state_before": "G : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ns : Set G\nhs : MeasurableSet s\n\u22a2 (\u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080)\u207b\u00b9 * \u2191\u2191(Content.measure (haarContent K\u2080)) s =\n    \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.extend_comp_eq", "start": [379, 1], "end": [381, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.dlookup_list_toFinmap", "start": [273, 1], "end": [274, 55], "traced_tactics": [{"tactic": "rw [List.toFinmap, lookup_toFinmap, lookup_to_alist]", "annotated_tactic": ["rw [<a>List.toFinmap</a>, <a>lookup_toFinmap</a>, <a>lookup_to_alist</a>]", [{"full_name": "List.toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [89, 5], "def_end_pos": [89, 18]}, {"full_name": "Finmap.lookup_toFinmap", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [267, 9], "def_end_pos": [267, 24]}, {"full_name": "AList.lookup_to_alist", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [321, 9], "def_end_pos": [321, 24]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : List (Sigma \u03b2)\n\u22a2 lookup a (List.toFinmap s) = dlookup a s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to0.tr_respects", "start": [1499, 1], "end": [1513, 100], "traced_tactics": [{"tactic": "cases' l\u2081 with l\u2081", "annotated_tactic": ["cases' l\u2081 with l\u2081", []], "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\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := l\u2081, var := v, Tape := T })\n    (TM1.step M { l := l\u2081, var := v, Tape := T })", "state_after": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := none, var := v, Tape := T })\n    (TM1.step M { l := none, var := v, Tape := T })\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\nl\u2081 : \u039b\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := some l\u2081, var := v, Tape := T })\n    (TM1.step M { l := some l\u2081, var := v, Tape := T })"}, {"tactic": "simp only [trCfg, TM1.step, FRespects, Option.map]", "annotated_tactic": ["simp only [<a>trCfg</a>, <a>TM1.step</a>, <a>FRespects</a>, <a>Option.map</a>]", [{"full_name": "Turing.TM1to0.trCfg", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1495, 5], "def_end_pos": [1495, 10]}, {"full_name": "Turing.TM1.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1291, 5], "def_end_pos": [1291, 9]}, {"full_name": "Turing.FRespects", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [957, 5], "def_end_pos": [957, 14]}, {"full_name": "Option.map", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2167, 25], "def_end_pos": [2167, 35]}]], "state_before": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\nl\u2081 : \u039b\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := some l\u2081, var := v, Tape := T })\n    (TM1.step M { l := some l\u2081, var := v, Tape := T })", "state_after": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\nl\u2081 : \u039b\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (M l\u2081), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (M l\u2081) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (M l\u2081) v T).var),\n      Tape := (TM1.stepAux (M l\u2081) v T).Tape }"}, {"tactic": "induction' M l\u2081 with _ q IH _ q IH _ q IH generalizing v T", "annotated_tactic": ["induction' M l\u2081 with _ q IH _ q IH _ q IH generalizing v T", []], "state_before": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\nl\u2081 : \u039b\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (M l\u2081), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (M l\u2081) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (M l\u2081) v T).var),\n      Tape := (TM1.stepAux (M l\u2081) v T).Tape }", "state_after": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : Dir\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }"}, {"tactic": "case move d q IH => exact TransGen.head rfl (IH _ _)", "annotated_tactic": ["case move d q IH => exact <a>TransGen.head</a> <a>rfl</a> (IH _ _)", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : Dir\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }", "state_after": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }"}, {"tactic": "case write a q IH => exact TransGen.head rfl (IH _ _)", "annotated_tactic": ["case write a q IH => exact <a>TransGen.head</a> <a>rfl</a> (IH _ _)", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }", "state_after": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }"}, {"tactic": "case load a q IH => exact (reaches\u2081_eq (by rfl)).2 (IH _ _)", "annotated_tactic": ["case load a q IH => exact (<a>reaches\u2081_eq</a> (by rfl)).2 (IH _ _)", [{"full_name": "Turing.reaches\u2081_eq", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [755, 9], "def_end_pos": [755, 20]}]], "state_before": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }", "state_after": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d\u00b9 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d\u00b9 v T).var),\n        Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux a\u271d v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux a\u271d v T).var),\n        Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }"}, {"tactic": "iterate 2\n  exact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T)))", "annotated_tactic": ["iterate 2\n      exact <a>TransGen.single</a> (<a>congr_arg</a> <a>some</a> (<a>congr</a> (<a>congr_arg</a> <a>TM0.Cfg.mk</a> <a>rfl</a>) (<a>Tape.write_self</a> T)))", [{"full_name": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"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": "congr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [359, 9], "def_end_pos": [359, 14]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Turing.TM0.Cfg.mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}, {"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.Tape.write_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\n\ncase 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }", "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 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := none, var := v, Tape := T })\n    (TM1.step M { l := none, var := v, Tape := T })", "state_after": "no goals"}, {"tactic": "exact TransGen.head rfl (IH _ _)", "annotated_tactic": ["exact <a>TransGen.head</a> <a>rfl</a> (IH _ _)", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nd : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : Dir\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move IH\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).Tape }", "state_after": "no goals"}, {"tactic": "exact TransGen.head rfl (IH _ _)", "annotated_tactic": ["exact <a>TransGen.head</a> <a>rfl</a> (IH _ _)", [{"full_name": "Relation.TransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [385, 9], "def_end_pos": [385, 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write IH\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).Tape }", "state_after": "no goals"}, {"tactic": "exact (reaches\u2081_eq (by rfl)).2 (IH _ _)", "annotated_tactic": ["exact (<a>reaches\u2081_eq</a> (by rfl)).2 (IH _ _)", [{"full_name": "Turing.reaches\u2081_eq", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [755, 9], "def_end_pos": [755, 20]}]], "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\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).Tape }", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q v T).var),\n        Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T } =\n    TM0.step (tr M) { q := (some q, IH\u271d T.head v), Tape := T }", "state_after": "no goals"}, {"tactic": "unfold TM1.stepAux", "annotated_tactic": ["unfold <a>TM1.stepAux</a>", [{"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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).var),\n      Tape := (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).Tape }", "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n          | some x => some (M x)\n          | none => none,\n          (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n      Tape := (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }"}, {"tactic": "cases e : p T.1 v", "annotated_tactic": ["cases e : p T.1 v", []], "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n          | some x => some (M x)\n          | none => none,\n          (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n      Tape := (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }", "state_after": "case false\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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n          | some x => some (M x)\n          | none => none,\n          (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n      Tape := (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }\n\ncase true\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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n          | some x => some (M x)\n          | none => none,\n          (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n      Tape := (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }"}, {"tactic": "exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2082 _ _)", "annotated_tactic": ["exact (<a>reaches\u2081_eq</a> (by simp only [<a>TM0.step</a>, <a>tr</a>, <a>trAux</a>, e]; rfl)).2 (IH\u2082 _ _)", [{"full_name": "Turing.reaches\u2081_eq", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [755, 9], "def_end_pos": [755, 20]}, {"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM1to0.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1489, 5], "def_end_pos": [1489, 7]}, {"full_name": "Turing.TM1to0.trAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1477, 5], "def_end_pos": [1477, 10]}]], "state_before": "case false\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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n          | some x => some (M x)\n          | none => none,\n          (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n      Tape := (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }", "state_after": "no goals"}, {"tactic": "simp only [TM0.step, tr, trAux, e]", "annotated_tactic": ["simp only [<a>TM0.step</a>, <a>tr</a>, <a>trAux</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.TM1to0.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1489, 5], "def_end_pos": [1489, 7]}, {"full_name": "Turing.TM1to0.trAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1477, 5], "def_end_pos": [1477, 10]}]], "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T } =\n    TM0.step (tr M) { q := (some q\u2082, v), Tape := T }", "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (bif false then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (trAux M T.head q\u2082 v))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (bif false then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (trAux M T.head q\u2082 v))", "state_after": "no goals"}, {"tactic": "exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2081 _ _)", "annotated_tactic": ["exact (<a>reaches\u2081_eq</a> (by simp only [<a>TM0.step</a>, <a>tr</a>, <a>trAux</a>, e]; rfl)).2 (IH\u2081 _ _)", [{"full_name": "Turing.reaches\u2081_eq", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [755, 9], "def_end_pos": [755, 20]}, {"full_name": "Turing.TM0.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1076, 5], "def_end_pos": [1076, 9]}, {"full_name": "Turing.TM1to0.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1489, 5], "def_end_pos": [1489, 7]}, {"full_name": "Turing.TM1to0.trAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1477, 5], "def_end_pos": [1477, 10]}]], "state_before": "case true\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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n    {\n      q :=\n        (match (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n          | some x => some (M x)\n          | none => none,\n          (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n      Tape := (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }", "state_after": "no goals"}, {"tactic": "simp only [TM0.step, tr, trAux, e]", "annotated_tactic": ["simp only [<a>TM0.step</a>, <a>tr</a>, <a>trAux</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.TM1to0.tr", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1489, 5], "def_end_pos": [1489, 7]}, {"full_name": "Turing.TM1to0.trAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1477, 5], "def_end_pos": [1477, 10]}]], "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T } =\n    TM0.step (tr M) { q := (some q\u2081, v), Tape := T }", "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (bif true then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (trAux M T.head q\u2081 v))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "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\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2081 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2081 v T).var),\n        Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n  \u2200 (v : \u03c3) (T : Tape \u0393),\n    Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n      {\n        q :=\n          (match (TM1.stepAux q\u2082 v T).l with\n            | some x => some (M x)\n            | none => none,\n            (TM1.stepAux q\u2082 v T).var),\n        Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (bif true then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =\n    Option.map\n      (fun x =>\n        { q := x.1,\n          Tape :=\n            match x.2 with\n            | move d => Tape.move d T\n            | write a => Tape.write a T })\n      (some (trAux M T.head q\u2081 v))", "state_after": "no goals"}, {"tactic": "exact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T)))", "annotated_tactic": ["exact <a>TransGen.single</a> (<a>congr_arg</a> <a>some</a> (<a>congr</a> (<a>congr_arg</a> <a>TM0.Cfg.mk</a> <a>rfl</a>) (<a>Tape.write_self</a> T)))", [{"full_name": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [242, 5], "def_end_pos": [242, 11]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"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": "congr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [359, 9], "def_end_pos": [359, 14]}, {"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Turing.TM0.Cfg.mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}, {"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.Tape.write_self", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case 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\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n    {\n      q :=\n        (match (TM1.stepAux TM1.Stmt.halt v T).l with\n          | some x => some (M x)\n          | none => none,\n          (TM1.stepAux TM1.Stmt.halt v T).var),\n      Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }", "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.bsize", "start": [828, 1], "end": [829, 64], "traced_tactics": [{"tactic": "simp [Substring.bsize, Nat.add_sub_cancel_left]", "annotated_tactic": ["simp [<a>Substring.bsize</a>, <a>Nat.add_sub_cancel_left</a>]", [{"full_name": "Substring.bsize", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2331, 15], "def_end_pos": [2331, 30]}, {"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 m r : List Char\n\u22a2 Substring.bsize\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n        stopPos := { byteIdx := utf8Len l + utf8Len m } } =\n    utf8Len m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.hasPDF_iff", "start": [221, 1], "end": [229, 33], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\n\u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc", "state_after": "case mp\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\n\u22a2 HasPDF X \u2119 \u2192 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc\n\ncase mpr\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\n\u22a2 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc \u2192 HasPDF X \u2119"}, {"tactic": "intro hX'", "annotated_tactic": ["intro hX'", []], "state_before": "case mp\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\n\u22a2 HasPDF X \u2119 \u2192 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc", "state_after": "case mp\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX' : HasPDF X \u2119\n\u22a2 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc"}, {"tactic": "exact \u27e8hX'.pdf'.1, haveLebesgueDecomposition_of_hasPDF, map_absolutelyContinuous\u27e9", "annotated_tactic": ["exact \u27e8hX'.pdf'.1, <a>haveLebesgueDecomposition_of_hasPDF</a>, <a>map_absolutelyContinuous</a>\u27e9", [{"full_name": "MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [214, 9], "def_end_pos": [214, 44]}, {"full_name": "MeasureTheory.pdf.map_absolutelyContinuous", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [204, 9], "def_end_pos": [204, 33]}]], "state_before": "case mp\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX' : HasPDF X \u2119\n\u22a2 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc", "state_after": "no goals"}, {"tactic": "rintro \u27e8hX, h_decomp, h\u27e9", "annotated_tactic": ["rintro \u27e8hX, h_decomp, h\u27e9", []], "state_before": "case mpr\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\n\u22a2 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc \u2192 HasPDF X \u2119", "state_after": "case mpr.intro.intro\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX : Measurable X\nh_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc\nh : map X \u2119 \u226a \u03bc\n\u22a2 HasPDF X \u2119"}, {"tactic": "haveI := h_decomp", "annotated_tactic": ["haveI := h_decomp", []], "state_before": "case mpr.intro.intro\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX : Measurable X\nh_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc\nh : map X \u2119 \u226a \u03bc\n\u22a2 HasPDF X \u2119", "state_after": "case mpr.intro.intro\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX : Measurable X\nh_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc\nh : map X \u2119 \u226a \u03bc\nthis : HaveLebesgueDecomposition (map X \u2119) \u03bc\n\u22a2 HasPDF X \u2119"}, {"tactic": "refine' \u27e8\u27e8hX, (Measure.map X \u2119).rnDeriv \u03bc, measurable_rnDeriv _ _, _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8hX, (<a>Measure.map</a> X \u2119).<a>rnDeriv</a> \u03bc, <a>measurable_rnDeriv</a> _ _, _\u27e9\u27e9", [{"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.rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [95, 17], "def_end_pos": [95, 24]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "case mpr.intro.intro\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX : Measurable X\nh_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc\nh : map X \u2119 \u226a \u03bc\nthis : HaveLebesgueDecomposition (map X \u2119) \u03bc\n\u22a2 HasPDF X \u2119", "state_after": "case mpr.intro.intro\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX : Measurable X\nh_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc\nh : map X \u2119 \u226a \u03bc\nthis : HaveLebesgueDecomposition (map X \u2119) \u03bc\n\u22a2 map X \u2119 = withDensity \u03bc (rnDeriv (map X \u2119) \u03bc)"}, {"tactic": "rwa [withDensity_rnDeriv_eq]", "annotated_tactic": ["rwa [<a>withDensity_rnDeriv_eq</a>]", [{"full_name": "MeasureTheory.Measure.withDensity_rnDeriv_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "def_pos": [46, 9], "def_end_pos": [46, 31]}]], "state_before": "case mpr.intro.intro\n\u03a9 : Type u_1\nE : Type u_2\ninst\u271d : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\nhX : Measurable X\nh_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc\nh : map X \u2119 \u226a \u03bc\nthis : HaveLebesgueDecomposition (map X \u2119) \u03bc\n\u22a2 map X \u2119 = withDensity \u03bc (rnDeriv (map X \u2119) \u03bc)", "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_closedBall'", "start": [481, 1], "end": [483, 60], "traced_tactics": [{"tactic": "rw [\u2190 addHaar_closedBall_mul \u03bc x hr zero_le_one, mul_one]", "annotated_tactic": ["rw [\u2190 <a>addHaar_closedBall_mul</a> \u03bc x hr <a>zero_le_one</a>, <a>mul_one</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_closedBall_mul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [471, 9], "def_end_pos": [471, 31]}, {"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]}]], "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\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)", "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.dropSlice_eq_dropSliceTR", "start": [1312, 10], "end": [1321, 30], "traced_tactics": [{"tactic": "funext \u03b1 n m l", "annotated_tactic": ["funext \u03b1 n m l", []], "state_before": "\u22a2 @dropSlice = @dropSliceTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nn m : Nat\nl : List \u03b1\n\u22a2 dropSlice n m l = dropSliceTR n m l"}, {"tactic": "simp [dropSliceTR]", "annotated_tactic": ["simp [<a>dropSliceTR</a>]", [{"full_name": "List.dropSliceTR", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1296, 15], "def_end_pos": [1296, 26]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nn m : Nat\nl : List \u03b1\n\u22a2 dropSlice n m l = dropSliceTR n m l", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nn m : Nat\nl : List \u03b1\n\u22a2 dropSlice n m l =\n    match m with\n    | 0 => l\n    | succ m => dropSliceTR.go l m l n #[]"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nn m : Nat\nl : List \u03b1\n\u22a2 dropSlice n m l =\n    match m with\n    | 0 => l\n    | succ m => dropSliceTR.go l m l n #[]", "state_after": "case h.h.h.h.h_1\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d : Nat\n\u22a2 dropSlice n 0 l = l\n\ncase h.h.h.h.h_2\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d\u00b9 m\u271d : Nat\n\u22a2 dropSlice n (succ m\u271d) l = dropSliceTR.go l m\u271d l n #[]"}, {"tactic": "{ rw [dropSlice_zero\u2082] }", "annotated_tactic": ["{ rw [<a>dropSlice_zero\u2082</a>] }", [{"full_name": "List.dropSlice_zero\u2082", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 24]}]], "state_before": "case h.h.h.h.h_1\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d : Nat\n\u22a2 dropSlice n 0 l = l\n\ncase h.h.h.h.h_2\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d\u00b9 m\u271d : Nat\n\u22a2 dropSlice n (succ m\u271d) l = dropSliceTR.go l m\u271d l n #[]", "state_after": "case h.h.h.h.h_2\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d\u00b9 m\u271d : Nat\n\u22a2 dropSlice n (succ m\u271d) l = dropSliceTR.go l m\u271d l n #[]"}, {"tactic": "rename_i m", "annotated_tactic": ["rename_i m", []], "state_before": "case h.h.h.h.h_2\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d\u00b9 m\u271d : Nat\n\u22a2 dropSlice n (succ m\u271d) l = dropSliceTR.go l m\u271d l n #[]", "state_after": "case h.h.h.h.h_2\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d m : Nat\n\u22a2 dropSlice n (succ m) l = dropSliceTR.go l m l n #[]"}, {"tactic": "exact (go #[] _ _ rfl).symm", "annotated_tactic": ["exact (go #[] _ _ <a>rfl</a>).<a>symm</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": "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_2\n\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d m : Nat\n\u22a2 dropSlice n (succ m) l = dropSliceTR.go l m l n #[]", "state_after": "no goals"}, {"tactic": "simp [dropSliceTR.go, dropSlice, h]", "annotated_tactic": ["simp [<a>dropSliceTR.go</a>, <a>dropSlice</a>, h]", [{"full_name": "List.dropSliceTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1303, 3], "def_end_pos": [1303, 5]}, {"full_name": "List.dropSlice", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1290, 13], "def_end_pos": [1290, 22]}]], "state_before": "\u03b1 : Type u_1\nn : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nhead\u271d : \u03b1\nxs : List \u03b1\nh : l = acc.data ++ head\u271d :: xs\n\u22a2 dropSliceTR.go l m (head\u271d :: xs) 0 acc = acc.data ++ dropSlice 0 (m + 1) (head\u271d :: xs)", "state_after": "no goals"}, {"tactic": "simp [dropSliceTR.go, dropSlice]", "annotated_tactic": ["simp [<a>dropSliceTR.go</a>, <a>dropSlice</a>]", [{"full_name": "List.dropSliceTR.go", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1303, 3], "def_end_pos": [1303, 5]}, {"full_name": "List.dropSlice", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1290, 13], "def_end_pos": [1290, 22]}]], "state_before": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\n\u22a2 l = acc.data ++ x :: xs \u2192 dropSliceTR.go l m (x :: xs) (n + 1) acc = acc.data ++ dropSlice (n + 1) (m + 1) (x :: xs)", "state_after": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\n\u22a2 l = acc.data ++ x :: xs \u2192 dropSliceTR.go l m xs n (Array.push acc x) = acc.data ++ x :: dropSlice n (m + 1) xs"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\n\u22a2 l = acc.data ++ x :: xs \u2192 dropSliceTR.go l m xs n (Array.push acc x) = acc.data ++ x :: dropSlice n (m + 1) xs", "state_after": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 dropSliceTR.go l m xs n (Array.push acc x) = acc.data ++ x :: dropSlice n (m + 1) xs"}, {"tactic": "rw [go _ xs]", "annotated_tactic": ["rw [go _ xs]", []], "state_before": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 dropSliceTR.go l m xs n (Array.push acc x) = acc.data ++ x :: dropSlice n (m + 1) xs", "state_after": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 (Array.push acc x).data ++ dropSlice n (m + 1) xs = acc.data ++ x :: dropSlice n (m + 1) xs\n\ncase a\n\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs"}, {"tactic": "{simp}", "annotated_tactic": ["{simp}", []], "state_before": "\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 (Array.push acc x).data ++ dropSlice n (m + 1) xs = acc.data ++ x :: dropSlice n (m + 1) xs\n\ncase a\n\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs", "state_after": "case a\n\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case a\n\u03b1 : Type u_1\nn\u271d : Nat\nl : List \u03b1\nm\u271d m : Nat\nacc : Array \u03b1\nx : \u03b1\nxs : List \u03b1\nn : Nat\nh : l = acc.data ++ x :: xs\n\u22a2 l = (Array.push acc x).data ++ xs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.ulower_down", "start": [1257, 1], "end": [1259, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.toIterator", "start": [837, 1], "end": [840, 80], "traced_tactics": [{"tactic": "simp [Substring.toIterator]", "annotated_tactic": ["simp [<a>Substring.toIterator</a>]", [{"full_name": "Substring.toIterator", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [528, 15], "def_end_pos": [528, 25]}]], "state_before": "l m r : List Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 Iterator.ValidFor (List.reverse l) (m ++ r) (Substring.toIterator x\u271d)", "state_after": "l m r : List Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 Iterator.ValidFor (List.reverse l) (m ++ r) { s := x\u271d.str, i := x\u271d.startPos }"}, {"tactic": "exact .of_eq _ (by simp [h.str, List.reverseAux_eq]) (by simp [h.startPos])", "annotated_tactic": ["exact .of_eq _ (by simp [h.str, <a>List.reverseAux_eq</a>]) (by simp [h.startPos])", [{"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}]], "state_before": "l m r : List Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 Iterator.ValidFor (List.reverse l) (m ++ r) { s := x\u271d.str, i := x\u271d.startPos }", "state_after": "no goals"}, {"tactic": "simp [h.str, List.reverseAux_eq]", "annotated_tactic": ["simp [h.str, <a>List.reverseAux_eq</a>]", [{"full_name": "List.reverseAux_eq", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}]], "state_before": "l m r : List Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 { s := x\u271d.str, i := x\u271d.startPos }.s.data = List.reverseAux (List.reverse l) (m ++ r)", "state_after": "no goals"}, {"tactic": "simp [h.startPos]", "annotated_tactic": ["simp [h.startPos]", []], "state_before": "l m r : List Char\nx\u271d : Substring\nh : ValidFor l m r x\u271d\n\u22a2 { s := x\u271d.str, i := x\u271d.startPos }.i.byteIdx = utf8Len (List.reverse l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_mem", "start": [1018, 1], "end": [1020, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.smul_testAgainstNN_apply", "start": [364, 1], "end": [367, 28], "traced_tactics": [{"tactic": "simp only [testAgainstNN, toMeasure_smul, smul_eq_mul, \u2190 ENNReal.smul_toNNReal, ENNReal.smul_def,\n  lintegral_smul_measure]", "annotated_tactic": ["simp only [<a>testAgainstNN</a>, <a>toMeasure_smul</a>, <a>smul_eq_mul</a>, \u2190 <a>ENNReal.smul_toNNReal</a>, <a>ENNReal.smul_def</a>,\n    <a>lintegral_smul_measure</a>]", [{"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.toMeasure_smul", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [235, 9], "def_end_pos": [235, 23]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "ENNReal.smul_toNNReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2265, 9], "def_end_pos": [2265, 22]}, {"full_name": "ENNReal.smul_def", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [503, 9], "def_end_pos": [503, 17]}, {"full_name": "MeasureTheory.lintegral_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [587, 9], "def_end_pos": [587, 31]}]], "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 : TopologicalSpace \u03a9\nc : \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN (c \u2022 \u03bc) f = c \u2022 testAgainstNN \u03bc f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.of_part", "start": [392, 1], "end": [407, 62], "traced_tactics": [{"tactic": "let g := fun n\u2081 =>\n  (Part.ofOption (decode (\u03b1 := Vector \u2115 n) n\u2081)).bind (fun a => Part.map encode (f a))", "annotated_tactic": ["let g := fun n\u2081 =>\n        (<a>Part.ofOption</a> (<a>decode</a> (\u03b1 := <a>Vector</a> \u2115 n) n\u2081)).<a>bind</a> (fun a => <a>Part.map</a> <a>encode</a> (f a))", [{"full_name": "Part.ofOption", "def_path": "Mathlib/Data/Part.lean", "def_pos": [324, 5], "def_end_pos": [324, 13]}, {"full_name": "Encodable.decode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [51, 3], "def_end_pos": [51, 9]}, {"full_name": "Vector", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [22, 5], "def_end_pos": [22, 11]}, {"full_name": "Part.bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [428, 15], "def_end_pos": [428, 19]}, {"full_name": "Part.map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [434, 5], "def_end_pos": [434, 8]}, {"full_name": "Encodable.encode", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 9]}]], "state_before": "this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\n\u22a2 Partrec' f", "state_after": "this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\n\u22a2 Partrec' f"}, {"tactic": "exact\n  (comp\u2081 g (this g hf) (prim Nat.Primrec'.encode)).of_eq fun i => by\n    dsimp only; simp [encodek, Part.map_id']", "annotated_tactic": ["exact\n        (<a>comp\u2081</a> g (this g hf) (<a>prim</a> <a>Nat.Primrec'.encode</a>)).<a>of_eq</a> fun i => by\n          dsimp only; simp [<a>encodek</a>, <a>Part.map_id'</a>]", [{"full_name": "Nat.Partrec'.comp\u2081", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [362, 9], "def_end_pos": [362, 14]}, {"full_name": "Nat.Partrec'.prim", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [278, 5], "def_end_pos": [278, 9]}, {"full_name": "Nat.Primrec'.encode", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1481, 19], "def_end_pos": [1481, 25]}, {"full_name": "Nat.Partrec'.of_eq", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [307, 9], "def_end_pos": [307, 14]}, {"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}, {"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "state_before": "this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\n\u22a2 Partrec' f", "state_after": "no goals"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\ni : Vector \u2115 n\n\u22a2 g (encode i) = f i", "state_after": "this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\ni : Vector \u2115 n\n\u22a2 (Part.bind \u2191(decode (encode i)) fun a => Part.map encode (f a)) = f i"}, {"tactic": "simp [encodek, Part.map_id']", "annotated_tactic": ["simp [<a>encodek</a>, <a>Part.map_id'</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}, {"full_name": "Part.map_id'", "def_path": "Mathlib/Data/Part.lean", "def_pos": [589, 9], "def_end_pos": [589, 16]}]], "state_before": "this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\ni : Vector \u2115 n\n\u22a2 (Part.bind \u2191(decode (encode i)) fun a => Part.map encode (f a)) = f i", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, rfl\u27e9 := exists_code.1 hf", "annotated_tactic": ["obtain \u27e8c, rfl\u27e9 := <a>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 : \u2115 \u2192. \u2115\nhf : Partrec f\n\u22a2 Partrec' fun v => f (Vector.head v)", "state_after": "case intro\nc : Code\nhf : Partrec (eval c)\n\u22a2 Partrec' fun v => eval c (Vector.head v)"}, {"tactic": "simpa [eval_eq_rfindOpt] using\n  rfindOpt <|\n    of_prim <|\n      Primrec.encode_iff.2 <|\n        evaln_prim.comp <|\n          (Primrec.vector_head.pair (_root_.Primrec.const c)).pair <|\n            Primrec.vector_head.comp Primrec.vector_tail", "annotated_tactic": ["simpa [<a>eval_eq_rfindOpt</a>] using\n      <a>rfindOpt</a> <|\n        <a>of_prim</a> <|\n          <a>Primrec.encode_iff</a>.2 <|\n            evaln_prim.comp <|\n              (Primrec.vector_head.pair (<a>_root_.Primrec.const</a> c)).<a>pair</a> <|\n                Primrec.vector_head.comp <a>Primrec.vector_tail</a>", [{"full_name": "Nat.Partrec.Code.eval_eq_rfindOpt", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [1154, 9], "def_end_pos": [1154, 25]}, {"full_name": "Nat.Partrec'.rfindOpt", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [367, 9], "def_end_pos": [367, 17]}, {"full_name": "Nat.Partrec'.of_prim", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [311, 9], "def_end_pos": [311, 16]}, {"full_name": "Primrec.encode_iff", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [272, 9], "def_end_pos": [272, 19]}, {"full_name": "Primrec.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [250, 9], "def_end_pos": [250, 14]}, {"full_name": "Primrec.pair", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [362, 9], "def_end_pos": [362, 13]}, {"full_name": "Primrec.vector_tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1302, 9], "def_end_pos": [1302, 20]}]], "state_before": "case intro\nc : Code\nhf : Partrec (eval c)\n\u22a2 Partrec' fun v => eval c (Vector.head v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.out_injective", "start": [403, 1], "end": [404, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_add_uIcc", "start": [426, 1], "end": [427, 90], "traced_tactics": [{"tactic": "simp only [\u2190 Icc_min_max, preimage_const_add_Icc, min_sub_sub_right, max_sub_sub_right]", "annotated_tactic": ["simp only [\u2190 <a>Icc_min_max</a>, <a>preimage_const_add_Icc</a>, <a>min_sub_sub_right</a>, <a>max_sub_sub_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": "Set.preimage_const_add_Icc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [60, 9], "def_end_pos": [60, 31]}, {"full_name": "min_sub_sub_right", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [51, 15], "def_end_pos": [51, 32]}, {"full_name": "max_sub_sub_right", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [57, 15], "def_end_pos": [57, 32]}]], "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]] = [[b - a, c - a]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.mem_degrees", "start": [187, 1], "end": [190, 100], "traced_tactics": [{"tactic": "classical\nsimp only [degrees_def, Multiset.mem_sup, \u2190 mem_support_iff, Finsupp.mem_toMultiset, exists_prop]", "annotated_tactic": ["classical\n  simp only [<a>degrees_def</a>, <a>Multiset.mem_sup</a>, \u2190 <a>mem_support_iff</a>, <a>Finsupp.mem_toMultiset</a>, <a>exists_prop</a>]", [{"full_name": "MvPolynomial.degrees_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [92, 9], "def_end_pos": [92, 20]}, {"full_name": "Multiset.mem_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 16]}, {"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "Finsupp.mem_toMultiset", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [129, 9], "def_end_pos": [129, 23]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "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 p : MvPolynomial \u03c3 R\ni : \u03c3\n\u22a2 i \u2208 degrees p \u2194 \u2203 d, coeff d p \u2260 0 \u2227 i \u2208 d.support", "state_after": "no goals"}, {"tactic": "simp only [degrees_def, Multiset.mem_sup, \u2190 mem_support_iff, Finsupp.mem_toMultiset, exists_prop]", "annotated_tactic": ["simp only [<a>degrees_def</a>, <a>Multiset.mem_sup</a>, \u2190 <a>mem_support_iff</a>, <a>Finsupp.mem_toMultiset</a>, <a>exists_prop</a>]", [{"full_name": "MvPolynomial.degrees_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [92, 9], "def_end_pos": [92, 20]}, {"full_name": "Multiset.mem_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 16]}, {"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "Finsupp.mem_toMultiset", "def_path": "Mathlib/Data/Finsupp/Multiset.lean", "def_pos": [129, 9], "def_end_pos": [129, 23]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "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 p : MvPolynomial \u03c3 R\ni : \u03c3\n\u22a2 i \u2208 degrees p \u2194 \u2203 d, coeff d p \u2260 0 \u2227 i \u2208 d.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable", "start": [231, 1], "end": [260, 87], "traced_tactics": [{"tactic": "have : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).ne'", "annotated_tactic": ["have : \u03b5 / 2 \u2260 0 := (<a>ENNReal.half_pos</a> \u03b50).<a>ne'</a>", [{"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": "\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "rcases exists_lt_lowerSemicontinuous_lintegral_ge \u03bc (fmeas.mk f) fmeas.measurable_mk this with\n  \u27e8g0, f_lt_g0, g0_cont, g0_int\u27e9", "annotated_tactic": ["rcases <a>exists_lt_lowerSemicontinuous_lintegral_ge</a> \u03bc (fmeas.mk f) fmeas.measurable_mk this with\n    \u27e8g0, f_lt_g0, g0_cont, g0_int\u27e9", [{"full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [203, 9], "def_end_pos": [203, 51]}]], "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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "case 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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with \u27e8s, hs, smeas, \u03bcs\u27e9", "annotated_tactic": ["rcases <a>exists_measurable_superset_of_null</a> fmeas.ae_eq_mk with \u27e8s, hs, smeas, \u03bcs\u27e9", [{"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": "case 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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "case 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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "rcases exists_le_lowerSemicontinuous_lintegral_ge \u03bc (s.indicator fun _x => \u221e)\n    (measurable_const.indicator smeas) this with\n  \u27e8g1, le_g1, g1_cont, g1_int\u27e9", "annotated_tactic": ["rcases <a>exists_le_lowerSemicontinuous_lintegral_ge</a> \u03bc (s.indicator fun _x => \u221e)\n      (measurable_const.indicator smeas) this with\n    \u27e8g1, le_g1, g1_cont, g1_int\u27e9", [{"full_name": "MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [164, 9], "def_end_pos": [164, 51]}]], "state_before": "case 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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "refine' \u27e8fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _\u27e9", "annotated_tactic": ["refine' \u27e8fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "case intro.intro.intro.intro.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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n\ncase intro.intro.intro.intro.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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), (fun x => g0 x + g1 x) x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "by_cases h : x \u2208 s", "annotated_tactic": ["by_cases h : x \u2208 s", []], "state_before": "case intro.intro.intro.intro.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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x", "state_after": "case pos\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n\ncase neg\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x"}, {"tactic": "have := le_g1 x", "annotated_tactic": ["have := le_g1 x", []], "state_before": "case pos\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x", "state_after": "case pos\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\nthis : Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x"}, {"tactic": "simp only [h, Set.indicator_of_mem, top_le_iff] at this", "annotated_tactic": ["simp only [h, <a>Set.indicator_of_mem</a>, <a>top_le_iff</a>] at this", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"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\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\nthis : Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x", "state_after": "case pos\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\nthis : g1 x = \u22a4\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case pos\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\nthis : g1 x = \u22a4\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x", "state_after": "no goals"}, {"tactic": "have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h", "annotated_tactic": ["have : f x = fmeas.mk f x := by rw [<a>Set.compl_subset_comm</a>] at hs; exact hs h", [{"full_name": "Set.compl_subset_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1747, 9], "def_end_pos": [1747, 26]}]], "state_before": "case neg\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x", "state_after": "case neg\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\nthis : f x = AEMeasurable.mk f fmeas x\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case neg\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\nthis : f x = AEMeasurable.mk f fmeas x\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x", "state_after": "case neg\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\nthis : f x = AEMeasurable.mk f fmeas x\n\u22a2 \u2191(AEMeasurable.mk f fmeas x) < (fun x => g0 x + g1 x) x"}, {"tactic": "exact (f_lt_g0 x).trans_le le_self_add", "annotated_tactic": ["exact (f_lt_g0 x).<a>trans_le</a> <a>le_self_add</a>", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "le_self_add", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [154, 3], "def_end_pos": [154, 14]}]], "state_before": "case neg\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\nthis : f x = AEMeasurable.mk f fmeas x\n\u22a2 \u2191(AEMeasurable.mk f fmeas x) < (fun x => g0 x + g1 x) x", "state_after": "no goals"}, {"tactic": "rw [Set.compl_subset_comm] at hs", "annotated_tactic": ["rw [<a>Set.compl_subset_comm</a>] at hs", [{"full_name": "Set.compl_subset_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1747, 9], "def_end_pos": [1747, 26]}]], "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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 f x = AEMeasurable.mk f fmeas 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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : s\u1d9c \u2286 {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 f x = AEMeasurable.mk f fmeas x"}, {"tactic": "exact hs h", "annotated_tactic": ["exact hs h", []], "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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : s\u1d9c \u2286 {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 f x = AEMeasurable.mk f fmeas x", "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\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc + \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + (0 + \u03b5 / 2)", "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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 0 + \u03b5 / 2"}, {"tactic": "convert g0_int using 2", "annotated_tactic": ["convert g0_int using 2", []], "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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2", "state_after": "case h.e'_4.h.e'_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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc"}, {"tactic": "exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)", "annotated_tactic": ["exact <a>lintegral_congr_ae</a> (fmeas.ae_eq_mk.fun_comp _)", [{"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'_4.h.e'_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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "convert g1_int", "annotated_tactic": ["convert g1_int", []], "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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 0 + \u03b5 / 2", "state_after": "case h.e'_4.h.e'_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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 0 = \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc"}, {"tactic": "simp only [smeas, \u03bcs, lintegral_const, Set.univ_inter, MeasurableSet.univ,\n  lintegral_indicator, mul_zero, restrict_apply]", "annotated_tactic": ["simp only [smeas, \u03bcs, <a>lintegral_const</a>, <a>Set.univ_inter</a>, <a>MeasurableSet.univ</a>,\n            <a>lintegral_indicator</a>, <a>mul_zero</a>, <a>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.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [762, 9], "def_end_pos": [762, 28]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"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.e'_4.h.e'_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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 0 = \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [add_assoc, ENNReal.add_halves, zero_add]", "annotated_tactic": ["simp only [<a>add_assoc</a>, <a>ENNReal.add_halves</a>, <a>zero_add</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": "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\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\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + (0 + \u03b5 / 2) = \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.smul_eval", "start": [1174, 1], "end": [1175, 47], "traced_tactics": [{"tactic": "rw [smul_eq_C_mul, (eval x).map_mul, eval_C]", "annotated_tactic": ["rw [<a>smul_eq_C_mul</a>, (<a>eval</a> x).<a>map_mul</a>, <a>eval_C</a>]", [{"full_name": "MvPolynomial.smul_eq_C_mul", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [280, 9], "def_end_pos": [280, 22]}, {"full_name": "MvPolynomial.eval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1145, 5], "def_end_pos": [1145, 9]}, {"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.eval_C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1164, 9], "def_end_pos": [1164, 15]}]], "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\nf x : \u03c3 \u2192 R\np : MvPolynomial \u03c3 R\ns : R\n\u22a2 \u2191(eval x) (s \u2022 p) = s * \u2191(eval x) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_coe", "start": [441, 1], "end": [445, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.evariance_eq_lintegral_ofReal", "start": [108, 1], "end": [115, 64], "traced_tactics": [{"tactic": "rw [evariance]", "annotated_tactic": ["rw [<a>evariance</a>]", [{"full_name": "ProbabilityTheory.evariance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [55, 5], "def_end_pos": [55, 14]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 evariance X \u03bc = \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc = \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) \u2202\u03bc"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc = \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) \u2202\u03bc", "state_after": "case e_f\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) = fun \u03c9 => ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "case e_f\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) = fun \u03c9 => ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)", "state_after": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)"}, {"tactic": "rw [pow_two, \u2190 ENNReal.coe_mul, \u2190 nnnorm_mul, \u2190 pow_two]", "annotated_tactic": ["rw [<a>pow_two</a>, \u2190 <a>ENNReal.coe_mul</a>, \u2190 <a>nnnorm_mul</a>, \u2190 <a>pow_two</a>]", [{"full_name": "pow_two", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 16]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "nnnorm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}, {"full_name": "pow_two", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 16]}]], "state_before": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)", "state_after": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016(X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\u2016\u208a = ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016(X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\u2016\u208a = ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016(X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\u2016\u208a = Real.toNNReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2)"}, {"tactic": "exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm", "annotated_tactic": ["exact (<a>Real.toNNReal_eq_nnnorm_of_nonneg</a> <| <a>sq_nonneg</a> _).<a>symm</a>", [{"full_name": "Real.toNNReal_eq_nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1816, 9], "def_end_pos": [1816, 37]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 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": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016(X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\u2016\u208a = Real.toNNReal ((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/Function/SpecialFunctions/IsROrC.lean", "full_name": "aemeasurable_of_re_im", "start": [79, 1], "end": [83, 34], "traced_tactics": [{"tactic": "convert AEMeasurable.add (M := \ud835\udd5c) (IsROrC.measurable_ofReal.comp_aemeasurable hre)\n    ((IsROrC.measurable_ofReal.comp_aemeasurable him).mul_const IsROrC.I)", "annotated_tactic": ["convert <a>AEMeasurable.add</a> (M := \ud835\udd5c) (IsROrC.measurable_ofReal.comp_aemeasurable hre)\n      ((IsROrC.measurable_ofReal.comp_aemeasurable him).<a>mul_const</a> <a>IsROrC.I</a>)", [{"full_name": "AEMeasurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [154, 3], "def_end_pos": [154, 14]}, {"full_name": "AEMeasurable.mul_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [127, 9], "def_end_pos": [127, 31]}, {"full_name": "IsROrC.I", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [61, 3], "def_end_pos": [61, 4]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\n\u03bc : MeasureTheory.Measure \u03b1\nhre : AEMeasurable fun x => \u2191IsROrC.re (f x)\nhim : AEMeasurable fun x => \u2191IsROrC.im (f x)\n\u22a2 AEMeasurable f", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\n\u03bc : MeasureTheory.Measure \u03b1\nhre : AEMeasurable fun x => \u2191IsROrC.re (f x)\nhim : AEMeasurable fun x => \u2191IsROrC.im (f x)\nx\u271d : \u03b1\n\u22a2 f x\u271d = (IsROrC.ofReal \u2218 fun x => \u2191IsROrC.re (f x)) x\u271d + (IsROrC.ofReal \u2218 fun x => \u2191IsROrC.im (f x)) x\u271d * IsROrC.I"}, {"tactic": "exact (IsROrC.re_add_im _).symm", "annotated_tactic": ["exact (<a>IsROrC.re_add_im</a> _).<a>symm</a>", [{"full_name": "IsROrC.re_add_im", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 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": "case h.e'_5.h\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\n\u03bc : MeasureTheory.Measure \u03b1\nhre : AEMeasurable fun x => \u2191IsROrC.re (f x)\nhim : AEMeasurable fun x => \u2191IsROrC.im (f x)\nx\u271d : \u03b1\n\u22a2 f x\u271d = (IsROrC.ofReal \u2218 fun x => \u2191IsROrC.re (f x)) x\u271d + (IsROrC.ofReal \u2218 fun x => \u2191IsROrC.im (f x)) x\u271d * IsROrC.I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousMap.range_toLp", "start": [1837, 1], "end": [1844, 25], "traced_tactics": [{"tactic": "refine' SetLike.ext' _", "annotated_tactic": ["refine' <a>SetLike.ext'</a> _", [{"full_name": "SetLike.ext'", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 13]}]], "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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c)) = 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\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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2191(Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u2191(Lp.boundedContinuousFunction E p \u03bc)"}, {"tactic": "have := (linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c).surjective", "annotated_tactic": ["have := (<a>linearIsometryBoundedOfCompact</a> \u03b1 E \ud835\udd5c).<a>surjective</a>", [{"full_name": "ContinuousMap.linearIsometryBoundedOfCompact", "def_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "def_pos": [281, 5], "def_end_pos": [281, 35]}, {"full_name": "LinearIsometryEquiv.surjective", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [997, 19], "def_end_pos": [997, 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\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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2191(Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u2191(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\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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nthis : Function.Surjective \u2191(linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c)\n\u22a2 \u2191(Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u2191(Lp.boundedContinuousFunction E p \u03bc)"}, {"tactic": "convert Function.Surjective.range_comp this (BoundedContinuousFunction.toLp (E := E) p \u03bc \ud835\udd5c)", "annotated_tactic": ["convert <a>Function.Surjective.range_comp</a> this (<a>BoundedContinuousFunction.toLp</a> (E := E) p \u03bc \ud835\udd5c)", [{"full_name": "Function.Surjective.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 30]}, {"full_name": "BoundedContinuousFunction.toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1778, 5], "def_end_pos": [1778, 9]}]], "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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nthis : Function.Surjective \u2191(linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c)\n\u22a2 \u2191(Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u2191(Lp.boundedContinuousFunction E p \u03bc)", "state_after": "case h.e'_3\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\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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nthis : Function.Surjective \u2191(linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c)\n\u22a2 \u2191(Lp.boundedContinuousFunction E p \u03bc) = Set.range \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c)"}, {"tactic": "rw [\u2190 BoundedContinuousFunction.range_toLp p \u03bc (\ud835\udd5c := \ud835\udd5c), Submodule.coe_toAddSubgroup,\n  LinearMap.range_coe]", "annotated_tactic": ["rw [\u2190 <a>BoundedContinuousFunction.range_toLp</a> p \u03bc (\ud835\udd5c := \ud835\udd5c), <a>Submodule.coe_toAddSubgroup</a>,\n    <a>LinearMap.range_coe</a>]", [{"full_name": "BoundedContinuousFunction.range_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1792, 9], "def_end_pos": [1792, 19]}, {"full_name": "Submodule.coe_toAddSubgroup", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [525, 9], "def_end_pos": [525, 26]}, {"full_name": "LinearMap.range_coe", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1084, 9], "def_end_pos": [1084, 18]}]], "state_before": "case h.e'_3\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\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 : NormedField \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nthis : Function.Surjective \u2191(linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c)\n\u22a2 \u2191(Lp.boundedContinuousFunction E p \u03bc) = Set.range \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c)", "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_Ioc", "start": [1107, 1], "end": [1110, 38], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_map, coe_Ioc, coe_Ioc]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_map</a>, <a>coe_Ioc</a>, <a>coe_Ioc</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_Ioc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [356, 9], "def_end_pos": [356, 16]}, {"full_name": "Finset.coe_Ioc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [356, 9], "def_end_pos": [356, 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) (Ioc a b) = Ioc (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.Ioc a b = Set.Ioc (c + a) (c + b)"}, {"tactic": "exact Set.image_const_add_Ioc _ _ _", "annotated_tactic": ["exact <a>Set.image_const_add_Ioc</a> _ _ _", [{"full_name": "Set.image_const_add_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [134, 9], "def_end_pos": [134, 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.Ioc a b = Set.Ioc (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.prod_mk", "start": [797, 1], "end": [804, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.exists_pos_ball", "start": [3932, 1], "end": [3934, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.map_add_left_Ioo", "start": [1121, 1], "end": [1124, 38], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_map, coe_Ioo, coe_Ioo]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_map</a>, <a>coe_Ioo</a>, <a>coe_Ioo</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_Ioo", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [361, 9], "def_end_pos": [361, 16]}, {"full_name": "Finset.coe_Ioo", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [361, 9], "def_end_pos": [361, 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) (Ioo a b) = Ioo (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.Ioo a b = Set.Ioo (c + a) (c + b)"}, {"tactic": "exact Set.image_const_add_Ioo _ _ _", "annotated_tactic": ["exact <a>Set.image_const_add_Ioo</a> _ _ _", [{"full_name": "Set.image_const_add_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "def_pos": [139, 9], "def_end_pos": [139, 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.Ioo a b = Set.Ioo (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/ExpDecay.lean", "full_name": "integrable_of_isBigO_exp_neg", "start": [40, 1], "end": [66, 79], "traced_tactics": [{"tactic": "cases' h2.isBigOWith with c h3", "annotated_tactic": ["cases' h2.isBigOWith with c h3", []], "state_before": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "case intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc : \u211d\nh3 : Asymptotics.IsBigOWith c atTop f fun x => rexp (-b * x)\n\u22a2 IntegrableOn f (Ioi a)"}, {"tactic": "rw [Asymptotics.isBigOWith_iff, eventually_atTop] at h3", "annotated_tactic": ["rw [<a>Asymptotics.isBigOWith_iff</a>, <a>eventually_atTop</a>] at h3", [{"full_name": "Asymptotics.isBigOWith_iff", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [93, 9], "def_end_pos": [93, 23]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [178, 9], "def_end_pos": [178, 25]}]], "state_before": "case intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc : \u211d\nh3 : Asymptotics.IsBigOWith c atTop f fun x => rexp (-b * x)\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "case intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc : \u211d\nh3 : \u2203 a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\n\u22a2 IntegrableOn f (Ioi a)"}, {"tactic": "cases' h3 with r bdr", "annotated_tactic": ["cases' h3 with r bdr", []], "state_before": "case intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc : \u211d\nh3 : \u2203 a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\n\u22a2 IntegrableOn f (Ioi a)"}, {"tactic": "let v := max a r", "annotated_tactic": ["let v := <a>max</a> a r", [{"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": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntegrableOn f (Ioi a)"}, {"tactic": "have int_left : IntegrableOn f (Ioc a v) := by\n  rw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le (le_max_left a r)]\n  have u : Icc a v \u2286 Ici a := Icc_subset_Ici_self\n  exact (h1.mono u).intervalIntegrable_of_Icc (le_max_left a r)", "annotated_tactic": ["have int_left : <a>IntegrableOn</a> f (<a>Ioc</a> a v) := by\n    rw [\u2190 <a>intervalIntegrable_iff_integrable_Ioc_of_le</a> (<a>le_max_left</a> a r)]\n    have u : <a>Icc</a> a v \u2286 <a>Ici</a> a := <a>Icc_subset_Ici_self</a>\n    exact (h1.mono u).<a>intervalIntegrable_of_Icc</a> (<a>le_max_left</a> a r)", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 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.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}, {"full_name": "ContinuousOn.intervalIntegrable_of_Icc", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [359, 9], "def_end_pos": [359, 47]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 IntegrableOn f (Ioi a)"}, {"tactic": "suffices IntegrableOn f (Ioi v) by\n  have t := integrableOn_union.mpr \u27e8int_left, this\u27e9\n  simpa only [Ioc_union_Ioi_eq_Ioi, le_max_iff, le_refl, true_or_iff] using t", "annotated_tactic": ["suffices <a>IntegrableOn</a> f (<a>Ioi</a> v) by\n    have t := integrableOn_union.mpr \u27e8int_left, this\u27e9\n    simpa only [<a>Ioc_union_Ioi_eq_Ioi</a>, <a>le_max_iff</a>, <a>le_refl</a>, <a>true_or_iff</a>] using t", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"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": "le_max_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [38, 9], "def_end_pos": [38, 19]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 IntegrableOn f (Ioi v)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 IntegrableOn f (Ioi v)", "state_after": "case intro.intro.left\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 AEStronglyMeasurable f (Measure.restrict volume (Ioi v))\n\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 HasFiniteIntegral f"}, {"tactic": "have : HasFiniteIntegral (fun x : \u211d => c * exp (-b * x)) (volume.restrict (Ioi v)) :=\n  (exp_neg_integrableOn_Ioi v h0).hasFiniteIntegral.const_mul c", "annotated_tactic": ["have : <a>HasFiniteIntegral</a> (fun x : \u211d => c * <a>exp</a> (-b * x)) (volume.restrict (<a>Ioi</a> v)) :=\n    (<a>exp_neg_integrableOn_Ioi</a> v h0).hasFiniteIntegral.const_mul c", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "exp_neg_integrableOn_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/ExpDecay.lean", "def_pos": [29, 9], "def_end_pos": [29, 33]}]], "state_before": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 HasFiniteIntegral f", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\n\u22a2 HasFiniteIntegral f"}, {"tactic": "apply this.mono", "annotated_tactic": ["apply this.mono", []], "state_before": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\n\u22a2 HasFiniteIntegral f", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioi v), \u2016f a\u2016 \u2264 \u2016c * rexp (-b * a)\u2016"}, {"tactic": "refine' (ae_restrict_iff' measurableSet_Ioi).mpr _", "annotated_tactic": ["refine' (<a>ae_restrict_iff'</a> <a>measurableSet_Ioi</a>).<a>mpr</a> _", [{"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_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}, {"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 intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioi v), \u2016f a\u2016 \u2264 \u2016c * rexp (-b * a)\u2016", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 Ioi v \u2192 \u2016f x\u2016 \u2264 \u2016c * rexp (-b * x)\u2016"}, {"tactic": "refine' ae_of_all _ fun x h1x => _", "annotated_tactic": ["refine' <a>ae_of_all</a> _ fun x h1x => _", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [407, 9], "def_end_pos": [407, 18]}]], "state_before": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 Ioi v \u2192 \u2016f x\u2016 \u2264 \u2016c * rexp (-b * x)\u2016", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : x \u2208 Ioi v\n\u22a2 \u2016f x\u2016 \u2264 \u2016c * rexp (-b * x)\u2016"}, {"tactic": "rw [norm_mul, norm_eq_abs]", "annotated_tactic": ["rw [<a>norm_mul</a>, <a>norm_eq_abs</a>]", [{"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"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 intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : x \u2208 Ioi v\n\u22a2 \u2016f x\u2016 \u2264 \u2016c * rexp (-b * x)\u2016", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : x \u2208 Ioi v\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016"}, {"tactic": "rw [mem_Ioi] at h1x", "annotated_tactic": ["rw [<a>mem_Ioi</a>] at h1x", [{"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.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : x \u2208 Ioi v\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : v < x\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016"}, {"tactic": "specialize bdr x ((le_max_right a r).trans h1x.le)", "annotated_tactic": ["specialize bdr x ((<a>le_max_right</a> a r).<a>trans</a> h1x.le)", [{"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"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.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : v < x\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016", "state_after": "case intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : v < x\nbdr : \u2016f x\u2016 \u2264 c * \u2016rexp (-b * x)\u2016\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016"}, {"tactic": "exact bdr.trans (mul_le_mul_of_nonneg_right (le_abs_self c) (norm_nonneg _))", "annotated_tactic": ["exact bdr.trans (<a>mul_le_mul_of_nonneg_right</a> (<a>le_abs_self</a> c) (<a>norm_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": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 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.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * rexp (-b * x)\nx : \u211d\nh1x : v < x\nbdr : \u2016f x\u2016 \u2264 c * \u2016rexp (-b * x)\u2016\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016", "state_after": "no goals"}, {"tactic": "rw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le (le_max_left a r)]", "annotated_tactic": ["rw [\u2190 <a>intervalIntegrable_iff_integrable_Ioc_of_le</a> (<a>le_max_left</a> a r)]", [{"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": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntegrableOn f (Ioc a v)", "state_after": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntervalIntegrable f volume a (max a r)"}, {"tactic": "have u : Icc a v \u2286 Ici a := Icc_subset_Ici_self", "annotated_tactic": ["have u : <a>Icc</a> a v \u2286 <a>Ici</a> a := <a>Icc_subset_Ici_self</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.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "state_before": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntervalIntegrable f volume a (max a r)", "state_after": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nu : Icc a v \u2286 Ici a\n\u22a2 IntervalIntegrable f volume a (max a r)"}, {"tactic": "exact (h1.mono u).intervalIntegrable_of_Icc (le_max_left a r)", "annotated_tactic": ["exact (h1.mono u).<a>intervalIntegrable_of_Icc</a> (<a>le_max_left</a> a r)", [{"full_name": "ContinuousOn.intervalIntegrable_of_Icc", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [359, 9], "def_end_pos": [359, 47]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nu : Icc a v \u2286 Ici a\n\u22a2 IntervalIntegrable f volume a (max a r)", "state_after": "no goals"}, {"tactic": "have t := integrableOn_union.mpr \u27e8int_left, this\u27e9", "annotated_tactic": ["have t := integrableOn_union.mpr \u27e8int_left, this\u27e9", []], "state_before": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : IntegrableOn f (Ioi v)\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : IntegrableOn f (Ioi v)\nt : IntegrableOn f (Ioc a v \u222a Ioi v)\n\u22a2 IntegrableOn f (Ioi a)"}, {"tactic": "simpa only [Ioc_union_Ioi_eq_Ioi, le_max_iff, le_refl, true_or_iff] using t", "annotated_tactic": ["simpa only [<a>Ioc_union_Ioi_eq_Ioi</a>, <a>le_max_iff</a>, <a>le_refl</a>, <a>true_or_iff</a>] using t", [{"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": "le_max_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [38, 9], "def_end_pos": [38, 19]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "true_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [182, 9], "def_end_pos": [182, 20]}]], "state_before": "f : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : IntegrableOn f (Ioi v)\nt : IntegrableOn f (Ioc a v \u222a Ioi v)\n\u22a2 IntegrableOn f (Ioi a)", "state_after": "no goals"}, {"tactic": "exact (h1.mono <| Ioi_subset_Ici <| le_max_left a r).aestronglyMeasurable measurableSet_Ioi", "annotated_tactic": ["exact (h1.mono <| <a>Ioi_subset_Ici</a> <| <a>le_max_left</a> a r).<a>aestronglyMeasurable</a> <a>measurableSet_Ioi</a>", [{"full_name": "Set.Ioi_subset_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [603, 9], "def_end_pos": [603, 23]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "ContinuousOn.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [542, 9], "def_end_pos": [542, 42]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "case intro.intro.left\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => rexp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 AEStronglyMeasurable f (Measure.restrict volume (Ioi v))", "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_lintegral_le", "start": [758, 1], "end": [761, 77], "traced_tactics": [{"tactic": "simpa only [laverage_eq_lintegral] using\n  exists_not_mem_null_laverage_le (IsProbabilityMeasure.ne_zero \u03bc) hint hN", "annotated_tactic": ["simpa only [<a>laverage_eq_lintegral</a>] using\n    <a>exists_not_mem_null_laverage_le</a> (<a>IsProbabilityMeasure.ne_zero</a> \u03bc) hint hN", [{"full_name": "MeasureTheory.laverage_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [95, 9], "def_end_pos": [95, 30]}, {"full_name": "MeasureTheory.exists_not_mem_null_laverage_le", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [682, 9], "def_end_pos": [682, 40]}, {"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\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u222b\u207b (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/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.degrees_rename", "start": [235, 1], "end": [249, 39], "traced_tactics": [{"tactic": "classical\nintro i\nrw [mem_degrees, Multiset.mem_map]\nrintro \u27e8d, hd, hi\u27e9\nobtain \u27e8x, rfl, hx\u27e9 := coeff_rename_ne_zero _ _ _ hd\nsimp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi\nrw [sum_apply, Finsupp.sum] at hi\ncontrapose! hi\nrw [Finset.sum_eq_zero]\nintro j hj\nsimp only [exists_prop, mem_degrees] at hi\nspecialize hi j \u27e8x, hx, hj\u27e9\nrw [Finsupp.single_apply, if_neg hi]", "annotated_tactic": ["classical\n  intro i\n  rw [<a>mem_degrees</a>, <a>Multiset.mem_map</a>]\n  rintro \u27e8d, hd, hi\u27e9\n  obtain \u27e8x, rfl, hx\u27e9 := <a>coeff_rename_ne_zero</a> _ _ _ hd\n  simp only [<a>Finsupp.mapDomain</a>, <a>Finsupp.mem_support_iff</a>] at hi\n  rw [<a>sum_apply</a>, <a>Finsupp.sum</a>] at hi\n  contrapose! hi\n  rw [<a>Finset.sum_eq_zero</a>]\n  intro j hj\n  simp only [<a>exists_prop</a>, <a>mem_degrees</a>] at hi\n  specialize hi j \u27e8x, hx, hj\u27e9\n  rw [<a>Finsupp.single_apply</a>, <a>if_neg</a> hi]", [{"full_name": "MvPolynomial.mem_degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}, {"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 16]}, {"full_name": "MvPolynomial.coeff_rename_ne_zero", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [319, 9], "def_end_pos": [319, 29]}, {"full_name": "Finsupp.mapDomain", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [448, 5], "def_end_pos": [448, 14]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}, {"full_name": "Finsupp.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [321, 9], "def_end_pos": [321, 18]}, {"full_name": "Finsupp.sum", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [51, 3], "def_end_pos": [51, 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": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "MvPolynomial.mem_degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}, {"full_name": "Finsupp.single_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [307, 9], "def_end_pos": [307, 21]}, {"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": "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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 degrees (\u2191(rename f) \u03c6) \u2286 Multiset.map f (degrees \u03c6)", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 degrees (\u2191(rename f) \u03c6) \u2286 Multiset.map f (degrees \u03c6)", "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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\n\u22a2 i \u2208 degrees (\u2191(rename f) \u03c6) \u2192 i \u2208 Multiset.map f (degrees \u03c6)"}, {"tactic": "rw [mem_degrees, Multiset.mem_map]", "annotated_tactic": ["rw [<a>mem_degrees</a>, <a>Multiset.mem_map</a>]", [{"full_name": "MvPolynomial.mem_degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}, {"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\n\u22a2 i \u2208 degrees (\u2191(rename f) \u03c6) \u2192 i \u2208 Multiset.map f (degrees \u03c6)", "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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\n\u22a2 (\u2203 d, coeff d (\u2191(rename f) \u03c6) \u2260 0 \u2227 i \u2208 d.support) \u2192 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i"}, {"tactic": "rintro \u27e8d, hd, hi\u27e9", "annotated_tactic": ["rintro \u27e8d, hd, hi\u27e9", []], "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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\n\u22a2 (\u2203 d, coeff d (\u2191(rename f) \u03c6) \u2260 0 \u2227 i \u2208 d.support) \u2192 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i", "state_after": "case intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nd : \u03c4 \u2192\u2080 \u2115\nhd : coeff d (\u2191(rename f) \u03c6) \u2260 0\nhi : i \u2208 d.support\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i"}, {"tactic": "obtain \u27e8x, rfl, hx\u27e9 := coeff_rename_ne_zero _ _ _ hd", "annotated_tactic": ["obtain \u27e8x, rfl, hx\u27e9 := <a>coeff_rename_ne_zero</a> _ _ _ hd", [{"full_name": "MvPolynomial.coeff_rename_ne_zero", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [319, 9], "def_end_pos": [319, 29]}]], "state_before": "case intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nd : \u03c4 \u2192\u2080 \u2115\nhd : coeff d (\u2191(rename f) \u03c6) \u2260 0\nhi : i \u2208 d.support\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : i \u2208 (Finsupp.mapDomain f x).support\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i"}, {"tactic": "simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi", "annotated_tactic": ["simp only [<a>Finsupp.mapDomain</a>, <a>Finsupp.mem_support_iff</a>] at hi", [{"full_name": "Finsupp.mapDomain", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [448, 5], "def_end_pos": [448, 14]}, {"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 intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : i \u2208 (Finsupp.mapDomain f x).support\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2191(sum x fun a => Finsupp.single (f a)) i \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i"}, {"tactic": "rw [sum_apply, Finsupp.sum] at hi", "annotated_tactic": ["rw [<a>sum_apply</a>, <a>Finsupp.sum</a>] at hi", [{"full_name": "Finsupp.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [321, 9], "def_end_pos": [321, 18]}, {"full_name": "Finsupp.sum", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [51, 3], "def_end_pos": [51, 14]}]], "state_before": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2191(sum x fun a => Finsupp.single (f a)) i \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2211 a in x.support, (\u2191fun\u2080 | f a => \u2191x a) i \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i"}, {"tactic": "contrapose! hi", "annotated_tactic": ["contrapose! hi", []], "state_before": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2211 a in x.support, (\u2191fun\u2080 | f a => \u2191x a) i \u2260 0\n\u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i\n\u22a2 \u2211 x_1 in x.support, (\u2191fun\u2080 | f x_1 => \u2191x x_1) i = 0"}, {"tactic": "rw [Finset.sum_eq_zero]", "annotated_tactic": ["rw [<a>Finset.sum_eq_zero</a>]", [{"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 intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i\n\u22a2 \u2211 x_1 in x.support, (\u2191fun\u2080 | f x_1 => \u2191x x_1) i = 0", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i\n\u22a2 \u2200 (x_1 : \u03c3), x_1 \u2208 x.support \u2192 (\u2191fun\u2080 | f x_1 => \u2191x x_1) i = 0"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i\n\u22a2 \u2200 (x_1 : \u03c3), x_1 \u2208 x.support \u2192 (\u2191fun\u2080 | f x_1 => \u2191x x_1) i = 0", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i\nj : \u03c3\nhj : j \u2208 x.support\n\u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0"}, {"tactic": "simp only [exists_prop, mem_degrees] at hi", "annotated_tactic": ["simp only [<a>exists_prop</a>, <a>mem_degrees</a>] at hi", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "MvPolynomial.mem_degrees", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [187, 9], "def_end_pos": [187, 20]}]], "state_before": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nhi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i\nj : \u03c3\nhj : j \u2208 x.support\n\u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nj : \u03c3\nhj : j \u2208 x.support\nhi : \u2200 (a : \u03c3), (\u2203 d, coeff d \u03c6 \u2260 0 \u2227 a \u2208 d.support) \u2192 f a \u2260 i\n\u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0"}, {"tactic": "specialize hi j \u27e8x, hx, hj\u27e9", "annotated_tactic": ["specialize hi j \u27e8x, hx, hj\u27e9", []], "state_before": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nj : \u03c3\nhj : j \u2208 x.support\nhi : \u2200 (a : \u03c3), (\u2203 d, coeff d \u03c6 \u2260 0 \u2227 a \u2208 d.support) \u2192 f a \u2260 i\n\u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0", "state_after": "case intro.intro.intro.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 q : MvPolynomial \u03c3 R\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nj : \u03c3\nhj : j \u2208 x.support\nhi : f j \u2260 i\n\u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0"}, {"tactic": "rw [Finsupp.single_apply, if_neg hi]", "annotated_tactic": ["rw [<a>Finsupp.single_apply</a>, <a>if_neg</a> hi]", [{"full_name": "Finsupp.single_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [307, 9], "def_end_pos": [307, 21]}, {"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\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\nf : \u03c3 \u2192 \u03c4\n\u03c6 : MvPolynomial \u03c3 R\ni : \u03c4\nx : \u03c3 \u2192\u2080 \u2115\nhx : coeff x \u03c6 \u2260 0\nhd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0\nj : \u03c3\nhj : j \u2208 x.support\nhi : f j \u2260 i\n\u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0", "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", "start": [957, 1], "end": [961, 60], "traced_tactics": [{"tactic": "have : (fun x => g x * (f x).toReal) = fun x => (f x).toReal \u2022 g x := by simp [mul_comm]", "annotated_tactic": ["have : (fun x => g x * (f x).<a>toReal</a>) = fun x => (f x).<a>toReal</a> \u2022 g x := by simp [<a>mul_comm</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": "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\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\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\n\u22a2 Integrable g \u2194 Integrable fun x => g x * ENNReal.toReal (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\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\nthis : (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x\n\u22a2 Integrable g \u2194 Integrable fun x => g x * ENNReal.toReal (f x)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "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\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\nthis : (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x\n\u22a2 Integrable g \u2194 Integrable fun x => g x * ENNReal.toReal (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\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\nthis : (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x\n\u22a2 Integrable g \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x"}, {"tactic": "exact integrable_withDensity_iff_integrable_smul' hf hflt", "annotated_tactic": ["exact <a>integrable_withDensity_iff_integrable_smul'</a> hf hflt", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [923, 9], "def_end_pos": [923, 52]}]], "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\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\nthis : (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x\n\u22a2 Integrable g \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x", "state_after": "no goals"}, {"tactic": "simp [mul_comm]", "annotated_tactic": ["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": "\u03b1 : Type u_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\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\n\u22a2 (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x", "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_le", "start": [647, 1], "end": [648, 90], "traced_tactics": [{"tactic": "to_encard_tac", "annotated_tactic": ["to_encard_tac", []], "state_before": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 ncard (f '' s) \u2264 ncard s", "state_after": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2191(ncard (f '' s)) \u2264 \u2191(ncard s)"}, {"tactic": "rw [hs.cast_ncard_eq, (hs.image _).cast_ncard_eq]", "annotated_tactic": ["rw [hs.cast_ncard_eq, (hs.image _).<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 : Set \u03b1\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 \u2191(ncard (f '' s)) \u2264 \u2191(ncard s)", "state_after": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 encard (f '' s) \u2264 encard s"}, {"tactic": "apply encard_image_le", "annotated_tactic": ["apply <a>encard_image_le</a>", [{"full_name": "Set.encard_image_le", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [394, 9], "def_end_pos": [394, 24]}]], "state_before": "\u03b1 : Type u_2\ns t : Set \u03b1\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nhs : autoParam (Set.Finite s) _auto\u271d\n\u22a2 encard (f '' s) \u2264 encard s", "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_boundedBy", "start": [892, 1], "end": [895, 61], "traced_tactics": [{"tactic": "simp only [boundedBy , smul_ofFunction hc]", "annotated_tactic": ["simp only [<a>boundedBy</a> , <a>smul_ofFunction</a> hc]", [{"full_name": "MeasureTheory.OuterMeasure.boundedBy", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [832, 5], "def_end_pos": [832, 14]}, {"full_name": "MeasureTheory.OuterMeasure.smul_ofFunction", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [814, 9], "def_end_pos": [814, 24]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 c \u2022 boundedBy m = boundedBy (c \u2022 m)", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 OuterMeasure.ofFunction (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) (_ : c \u2022 \u2a06 (_ : Set.Nonempty \u2205), m \u2205 = 0) =\n    OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s) (_ : \u2a06 (_ : Set.Nonempty \u2205), (c \u2022 m) \u2205 = 0)"}, {"tactic": "congr 1 with s : 1", "annotated_tactic": ["congr 1 with s : 1", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 OuterMeasure.ofFunction (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) (_ : c \u2022 \u2a06 (_ : Set.Nonempty \u2205), m \u2205 = 0) =\n    OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s) (_ : \u2a06 (_ : Set.Nonempty \u2205), (c \u2022 m) \u2205 = 0)", "state_after": "case e_m.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\n\u22a2 (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) s = \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s"}, {"tactic": "rcases s.eq_empty_or_nonempty with (rfl | hs) <;> simp [*]", "annotated_tactic": ["rcases s.eq_empty_or_nonempty with (rfl | hs) <;> simp [*]", []], "state_before": "case e_m.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\n\u22a2 (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) s = \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.fixInduction_spec", "start": [340, 1], "end": [347, 6], "traced_tactics": [{"tactic": "unfold fixInduction", "annotated_tactic": ["unfold <a>fixInduction</a>", [{"full_name": "PFun.fixInduction", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [328, 5], "def_end_pos": [328, 17]}]], "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\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 fixInduction h H = H a h fun a' h' => fixInduction (_ : b \u2208 fix f a') H", "state_after": "\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\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 (let_fun h\u2082 :=\n      (_ :\n        b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));\n    Acc.rec (motive := fun {a} h\u2081 =>\n      b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a h\u2081 \u2192\n        C a)\n      (fun a ha IH h\u2082 =>\n        let_fun h :=\n          (_ :\n            b \u2208\n              Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a h);\n        H a h fun a' fa' =>\n          IH a' fa'\n            (_ :\n              b \u2208\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n      (_ : Acc (fun x y => Sum.inr x \u2208 f y) a) h\u2082) =\n    H a h fun a' h' =>\n      let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          (fun x => C x) a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082"}, {"tactic": "generalize (Part.mem_assert_iff.1 h).fst = ha", "annotated_tactic": ["generalize (<a>Part.mem_assert_iff</a>.1 h).<a>fst</a> = ha", [{"full_name": "Part.mem_assert_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [466, 9], "def_end_pos": [466, 23]}, {"full_name": "Exists.fst", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [864, 9], "def_end_pos": [864, 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\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 (let_fun h\u2082 :=\n      (_ :\n        b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));\n    Acc.rec (motive := fun {a} h\u2081 =>\n      b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a h\u2081 \u2192\n        C a)\n      (fun a ha IH h\u2082 =>\n        let_fun h :=\n          (_ :\n            b \u2208\n              Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a h);\n        H a h fun a' fa' =>\n          IH a' fa'\n            (_ :\n              b \u2208\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n      (_ : Acc (fun x y => Sum.inr x \u2208 f y) a) h\u2082) =\n    H a h fun a' h' =>\n      let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          (fun x => C x) a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082", "state_after": "\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\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nha : Acc (fun x y => Sum.inr x \u2208 f y) a\n\u22a2 (let_fun h\u2082 :=\n      (_ :\n        b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));\n    Acc.rec (motive := fun {a} h\u2081 =>\n      b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a h\u2081 \u2192\n        C a)\n      (fun a ha IH h\u2082 =>\n        let_fun h :=\n          (_ :\n            b \u2208\n              Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a h);\n        H a h fun a' fa' =>\n          IH a' fa'\n            (_ :\n              b \u2208\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n      ha h\u2082) =\n    H a h fun a' h' =>\n      let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          (fun x => C x) a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082"}, {"tactic": "induction ha", "annotated_tactic": ["induction ha", []], "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\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nha : Acc (fun x y => Sum.inr x \u2208 f y) a\n\u22a2 (let_fun h\u2082 :=\n      (_ :\n        b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));\n    Acc.rec (motive := fun {a} h\u2081 =>\n      b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a h\u2081 \u2192\n        C a)\n      (fun a ha IH h\u2082 =>\n        let_fun h :=\n          (_ :\n            b \u2208\n              Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a h);\n        H a h fun a' fa' =>\n          IH a' fa'\n            (_ :\n              b \u2208\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n      ha h\u2082) =\n    H a h fun a' h' =>\n      let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          (fun x => C x) a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082", "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\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nx\u271d : \u03b1\nh\u271d : \u2200 (y : \u03b1), Sum.inr y \u2208 f x\u271d \u2192 Acc (fun x y => Sum.inr x \u2208 f y) y\nh_ih\u271d :\n  \u2200 (y : \u03b1) (a : Sum.inr y \u2208 f x\u271d) (h : b \u2208 fix f y),\n    (let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              y (_ : Acc (fun x y => Sum.inr x \u2208 f y) y));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          C a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc ?m.25674 y) h\u2082) =\n      H y h fun a' h' =>\n        let_fun h\u2082 :=\n          (_ :\n            b \u2208\n              WellFounded.fixF\n                (fun a IH =>\n                  Part.assert (f a).Dom fun hf =>\n                    match e : Part.get (f a) hf with\n                    | Sum.inl b => Part.some b\n                    | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n        Acc.rec (motive := fun {a} h\u2081 =>\n          b \u2208\n              WellFounded.fixF\n                (fun a IH =>\n                  Part.assert (f a).Dom fun hf =>\n                    match e : Part.get (f a) hf with\n                    | Sum.inl b => Part.some b\n                    | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                a h\u2081 \u2192\n            (fun x => C x) a)\n          (fun a ha IH h\u2082 =>\n            let_fun h :=\n              (_ :\n                b \u2208\n                  Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                    WellFounded.fixF\n                      (fun a IH =>\n                        Part.assert (f a).Dom fun hf =>\n                          match e : Part.get (f a) hf with\n                          | Sum.inl b => Part.some b\n                          | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                      a h);\n            H a h fun a' fa' =>\n              IH a' fa'\n                (_ :\n                  b \u2208\n                    WellFounded.fixF\n                      (fun a IH =>\n                        Part.assert (f a).Dom fun hf =>\n                          match e : Part.get (f a) hf with\n                          | Sum.inl b => Part.some b\n                          | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                      a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n          (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082\nh : b \u2208 fix f x\u271d\n\u22a2 (let_fun h\u2082 :=\n      (_ :\n        b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            x\u271d (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d));\n    Acc.rec (motive := fun {a} h\u2081 =>\n      b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a h\u2081 \u2192\n        C a)\n      (fun a ha IH h\u2082 =>\n        let_fun h :=\n          (_ :\n            b \u2208\n              Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a h);\n        H a h fun a' fa' =>\n          IH a' fa'\n            (_ :\n              b \u2208\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n      (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d) h\u2082) =\n    H x\u271d h fun a' h' =>\n      let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          (fun x => C x) a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082"}, {"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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nx\u271d : \u03b1\nh\u271d : \u2200 (y : \u03b1), Sum.inr y \u2208 f x\u271d \u2192 Acc (fun x y => Sum.inr x \u2208 f y) y\nh_ih\u271d :\n  \u2200 (y : \u03b1) (a : Sum.inr y \u2208 f x\u271d) (h : b \u2208 fix f y),\n    (let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              y (_ : Acc (fun x y => Sum.inr x \u2208 f y) y));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          C a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc ?m.25674 y) h\u2082) =\n      H y h fun a' h' =>\n        let_fun h\u2082 :=\n          (_ :\n            b \u2208\n              WellFounded.fixF\n                (fun a IH =>\n                  Part.assert (f a).Dom fun hf =>\n                    match e : Part.get (f a) hf with\n                    | Sum.inl b => Part.some b\n                    | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n        Acc.rec (motive := fun {a} h\u2081 =>\n          b \u2208\n              WellFounded.fixF\n                (fun a IH =>\n                  Part.assert (f a).Dom fun hf =>\n                    match e : Part.get (f a) hf with\n                    | Sum.inl b => Part.some b\n                    | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                a h\u2081 \u2192\n            (fun x => C x) a)\n          (fun a ha IH h\u2082 =>\n            let_fun h :=\n              (_ :\n                b \u2208\n                  Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                    WellFounded.fixF\n                      (fun a IH =>\n                        Part.assert (f a).Dom fun hf =>\n                          match e : Part.get (f a) hf with\n                          | Sum.inl b => Part.some b\n                          | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                      a h);\n            H a h fun a' fa' =>\n              IH a' fa'\n                (_ :\n                  b \u2208\n                    WellFounded.fixF\n                      (fun a IH =>\n                        Part.assert (f a).Dom fun hf =>\n                          match e : Part.get (f a) hf with\n                          | Sum.inl b => Part.some b\n                          | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                      a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n          (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082\nh : b \u2208 fix f x\u271d\n\u22a2 (let_fun h\u2082 :=\n      (_ :\n        b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            x\u271d (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d));\n    Acc.rec (motive := fun {a} h\u2081 =>\n      b \u2208\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : Part.get (f a) hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n            a h\u2081 \u2192\n        C a)\n      (fun a ha IH h\u2082 =>\n        let_fun h :=\n          (_ :\n            b \u2208\n              Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a h);\n        H a h fun a' fa' =>\n          IH a' fa'\n            (_ :\n              b \u2208\n                WellFounded.fixF\n                  (fun a IH =>\n                    Part.assert (f a).Dom fun hf =>\n                      match e : Part.get (f a) hf with\n                      | Sum.inl b => Part.some b\n                      | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                  a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n      (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d) h\u2082) =\n    H x\u271d h fun a' h' =>\n      let_fun h\u2082 :=\n        (_ :\n          b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n      Acc.rec (motive := fun {a} h\u2081 =>\n        b \u2208\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : Part.get (f a) hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n              a h\u2081 \u2192\n          (fun x => C x) a)\n        (fun a ha IH h\u2082 =>\n          let_fun h :=\n            (_ :\n              b \u2208\n                Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a h);\n          H a h fun a' fa' =>\n            IH a' fa'\n              (_ :\n                b \u2208\n                  WellFounded.fixF\n                    (fun a IH =>\n                      Part.assert (f a).Dom fun hf =>\n                        match e : Part.get (f a) hf with\n                        | Sum.inl b => Part.some b\n                        | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n                    a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n        (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082", "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_eq_iff_eq_mul_left", "start": [764, 11], "end": [766, 60], "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": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 div a b = c \u2194 a = c * b", "state_after": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 div a b = c \u2194 a = b * c"}, {"tactic": "exact Int.div_eq_iff_eq_mul_right H H'", "annotated_tactic": ["exact <a>Int.div_eq_iff_eq_mul_right</a> H H'", [{"full_name": "Int.div_eq_iff_eq_mul_right", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [756, 19], "def_end_pos": [756, 42]}]], "state_before": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 div a b = c \u2194 a = b * c", "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_zero", "start": [450, 1], "end": [452, 30], "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\n\u22a2 toJordanDecomposition 0 = 0", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 toSignedMeasure (toJordanDecomposition 0) = toSignedMeasure 0"}, {"tactic": "simp [toSignedMeasure_zero]", "annotated_tactic": ["simp [<a>toSignedMeasure_zero</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_zero", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [173, 9], "def_end_pos": [173, 29]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 toSignedMeasure (toJordanDecomposition 0) = toSignedMeasure 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.sum_mul_sub", "start": [516, 1], "end": [539, 55], "traced_tactics": [{"tactic": "have h\u03bebdd : \u2200 i, \u2203 C, \u2200 \u03c9, |\u03be i \u03c9| \u2264 C := fun i =>\n  \u27e8R, fun \u03c9 => (abs_of_nonneg (hnonneg i \u03c9)).trans_le (hbdd i \u03c9)\u27e9", "annotated_tactic": ["have h\u03bebdd : \u2200 i, \u2203 C, \u2200 \u03c9, |\u03be i \u03c9| \u2264 C := fun i =>\n    \u27e8R, fun \u03c9 => (<a>abs_of_nonneg</a> (hnonneg i \u03c9)).<a>trans_le</a> (hbdd i \u03c9)\u27e9", [{"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 18]}]], "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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc"}, {"tactic": "have hint : \u2200 m, Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) \u03bc := fun m =>\n  integrable_finset_sum' _ fun i _ => Integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _))\n    h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _)", "annotated_tactic": ["have hint : \u2200 m, <a>Integrable</a> (\u2211 k in <a>Finset.range</a> m, \u03be k * (f (k + 1) - f k)) \u03bc := fun m =>\n    <a>integrable_finset_sum'</a> _ fun i _ => <a>Integrable.bdd_mul</a> ((hf.integrable _).<a>sub</a> (hf.integrable _))\n      h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "MeasureTheory.integrable_finset_sum'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [682, 9], "def_end_pos": [682, 31]}, {"full_name": "MeasureTheory.Integrable.bdd_mul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [728, 9], "def_end_pos": [728, 27]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc"}, {"tactic": "have hadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) := by\n  intro m\n  refine' Finset.stronglyMeasurable_sum' _ fun i hi => _\n  rw [Finset.mem_range] at hi\n  exact (h\u03be.stronglyMeasurable_le hi.le).mul\n    ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub\n      (hf.adapted.stronglyMeasurable_le hi.le))", "annotated_tactic": ["have hadp : <a>Adapted</a> \ud835\udca2 fun n => \u2211 k in <a>Finset.range</a> n, \u03be k * (f (k + 1) - f k) := by\n    intro m\n    refine' <a>Finset.stronglyMeasurable_sum'</a> _ fun i hi => _\n    rw [<a>Finset.mem_range</a>] at hi\n    exact (h\u03be.stronglyMeasurable_le hi.le).<a>mul</a>\n      ((hf.adapted.stronglyMeasurable_le (<a>Nat.succ_le_of_lt</a> hi)).<a>sub</a>\n        (hf.adapted.stronglyMeasurable_le hi.le))", [{"full_name": "MeasureTheory.Adapted", "def_path": "Mathlib/Probability/Process/Adapted.lean", "def_pos": [49, 5], "def_end_pos": [49, 12]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.stronglyMeasurable_sum'", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [611, 3], "def_end_pos": [611, 14]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}, {"full_name": "MeasureTheory.StronglyMeasurable.mul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [409, 19], "def_end_pos": [409, 22]}, {"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": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc"}, {"tactic": "refine' submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _", "annotated_tactic": ["refine' <a>submartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => _", [{"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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u2211 k in Finset.range (i + 1), \u03be k * (f (k + 1) - f k) - \u2211 k in Finset.range i, \u03be k * (f (k + 1) - f k)|\u2191\ud835\udca2 i]"}, {"tactic": "simp only [\u2190 Finset.sum_Ico_eq_sub _ (Nat.le_succ _), Finset.sum_apply, Pi.mul_apply,\n  Pi.sub_apply, Nat.Ico_succ_singleton, Finset.sum_singleton]", "annotated_tactic": ["simp only [\u2190 <a>Finset.sum_Ico_eq_sub</a> _ (<a>Nat.le_succ</a> _), <a>Finset.sum_apply</a>, <a>Pi.mul_apply</a>,\n    <a>Pi.sub_apply</a>, <a>Nat.Ico_succ_singleton</a>, <a>Finset.sum_singleton</a>]", [{"full_name": "Finset.sum_Ico_eq_sub", "def_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "def_pos": [102, 3], "def_end_pos": [102, 14]}, {"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": "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]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "Nat.Ico_succ_singleton", "def_path": "Mathlib/Data/Nat/Interval.lean", "def_pos": [201, 9], "def_end_pos": [201, 27]}, {"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u2211 k in Finset.range (i + 1), \u03be k * (f (k + 1) - f k) - \u2211 k in Finset.range i, \u03be k * (f (k + 1) - f k)|\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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u03be i * (f (i + 1) - f i)|\u2191\ud835\udca2 i]"}, {"tactic": "exact EventuallyLE.trans (EventuallyLE.mul_nonneg (eventually_of_forall (hnonneg _))\n  (hf.condexp_sub_nonneg (Nat.le_succ _))) (condexp_stronglyMeasurable_mul (h\u03be _)\n  (((hf.integrable _).sub (hf.integrable _)).bdd_mul\n    h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _))\n  ((hf.integrable _).sub (hf.integrable _))).symm.le", "annotated_tactic": ["exact <a>EventuallyLE.trans</a> (<a>EventuallyLE.mul_nonneg</a> (<a>eventually_of_forall</a> (hnonneg _))\n    (hf.condexp_sub_nonneg (<a>Nat.le_succ</a> _))) (<a>condexp_stronglyMeasurable_mul</a> (h\u03be _)\n    (((hf.integrable _).<a>sub</a> (hf.integrable _)).<a>bdd_mul</a>\n      h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _))\n    ((hf.integrable _).<a>sub</a> (hf.integrable _))).symm.le", [{"full_name": "Filter.EventuallyLE.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 27]}, {"full_name": "Filter.EventuallyLE.mul_nonneg", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1792, 9], "def_end_pos": [1792, 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": "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.condexp_stronglyMeasurable_mul", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [320, 9], "def_end_pos": [320, 39]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}, {"full_name": "MeasureTheory.Integrable.bdd_mul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [728, 9], "def_end_pos": [728, 27]}, {"full_name": "MeasureTheory.Integrable.sub", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [702, 9], "def_end_pos": [702, 23]}]], "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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u03be i * (f (i + 1) - f i)|\u2191\ud835\udca2 i]", "state_after": "no goals"}, {"tactic": "intro m", "annotated_tactic": ["intro m", []], "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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\n\u22a2 Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)", "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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm : \u2115\n\u22a2 StronglyMeasurable ((fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) m)"}, {"tactic": "refine' Finset.stronglyMeasurable_sum' _ fun i hi => _", "annotated_tactic": ["refine' <a>Finset.stronglyMeasurable_sum'</a> _ fun i hi => _", [{"full_name": "Finset.stronglyMeasurable_sum'", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [611, 3], "def_end_pos": [611, 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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm : \u2115\n\u22a2 StronglyMeasurable ((fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) m)", "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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm i : \u2115\nhi : i \u2208 Finset.range m\n\u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i))"}, {"tactic": "rw [Finset.mem_range] at hi", "annotated_tactic": ["rw [<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\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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm i : \u2115\nhi : i \u2208 Finset.range m\n\u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f 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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm i : \u2115\nhi : i < m\n\u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i))"}, {"tactic": "exact (h\u03be.stronglyMeasurable_le hi.le).mul\n  ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub\n    (hf.adapted.stronglyMeasurable_le hi.le))", "annotated_tactic": ["exact (h\u03be.stronglyMeasurable_le hi.le).<a>mul</a>\n      ((hf.adapted.stronglyMeasurable_le (<a>Nat.succ_le_of_lt</a> hi)).<a>sub</a>\n        (hf.adapted.stronglyMeasurable_le hi.le))", [{"full_name": "MeasureTheory.StronglyMeasurable.mul", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [409, 19], "def_end_pos": [409, 22]}, {"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": "MeasureTheory.StronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [436, 3], "def_end_pos": [436, 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\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm i : \u2115\nhi : i < m\n\u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.ext_s", "start": [47, 1], "end": [50, 72], "traced_tactics": [{"tactic": "refine' \u27e8congr_arg _, fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>congr_arg</a> _, fun h => _\u27e9", [{"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\n\u03b2 : Type u_2\nq\u2081 q\u2082 : Semiquot \u03b1\n\u22a2 q\u2081 = q\u2082 \u2194 q\u2081.s = q\u2082.s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq\u2081 q\u2082 : Semiquot \u03b1\nh : q\u2081.s = q\u2082.s\n\u22a2 q\u2081 = q\u2082"}, {"tactic": "cases' q\u2081 with _ v\u2081", "annotated_tactic": ["cases' q\u2081 with _ v\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nq\u2081 q\u2082 : Semiquot \u03b1\nh : q\u2081.s = q\u2082.s\n\u22a2 q\u2081 = q\u2082", "state_after": "case mk'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nq\u2082 : Semiquot \u03b1\ns\u271d : Set \u03b1\nv\u2081 : Trunc \u2191s\u271d\nh : { s := s\u271d, val := v\u2081 }.s = q\u2082.s\n\u22a2 { s := s\u271d, val := v\u2081 } = q\u2082"}, {"tactic": "cases' q\u2082 with _ v\u2082", "annotated_tactic": ["cases' q\u2082 with _ v\u2082", []], "state_before": "case mk'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nq\u2082 : Semiquot \u03b1\ns\u271d : Set \u03b1\nv\u2081 : Trunc \u2191s\u271d\nh : { s := s\u271d, val := v\u2081 }.s = q\u2082.s\n\u22a2 { s := s\u271d, val := v\u2081 } = q\u2082", "state_after": "case mk'.mk'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d\u00b9 : Set \u03b1\nv\u2081 : Trunc \u2191s\u271d\u00b9\ns\u271d : Set \u03b1\nv\u2082 : Trunc \u2191s\u271d\nh : { s := s\u271d\u00b9, val := v\u2081 }.s = { s := s\u271d, val := v\u2082 }.s\n\u22a2 { s := s\u271d\u00b9, val := v\u2081 } = { s := s\u271d, val := v\u2082 }"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk'.mk'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d\u00b9 : Set \u03b1\nv\u2081 : Trunc \u2191s\u271d\u00b9\ns\u271d : Set \u03b1\nv\u2082 : Trunc \u2191s\u271d\nh : { s := s\u271d\u00b9, val := v\u2081 }.s = { s := s\u271d, val := v\u2082 }.s\n\u22a2 { s := s\u271d\u00b9, val := v\u2081 } = { s := s\u271d, val := v\u2082 }", "state_after": "case mk'.mk'.h.e_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d\u00b9 : Set \u03b1\nv\u2081 : Trunc \u2191s\u271d\u00b9\ns\u271d : Set \u03b1\nv\u2082 : Trunc \u2191s\u271d\nh : { s := s\u271d\u00b9, val := v\u2081 }.s = { s := s\u271d, val := v\u2082 }.s\n\u22a2 HEq v\u2081 v\u2082"}, {"tactic": "exact Subsingleton.helim (congrArg Trunc (congrArg Set.Elem h)) v\u2081 v\u2082", "annotated_tactic": ["exact <a>Subsingleton.helim</a> (<a>congrArg</a> <a>Trunc</a> (<a>congrArg</a> <a>Set.Elem</a> h)) v\u2081 v\u2082", [{"full_name": "Subsingleton.helim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [876, 19], "def_end_pos": [876, 37]}, {"full_name": "congrArg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [349, 9], "def_end_pos": [349, 17]}, {"full_name": "Trunc", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [467, 5], "def_end_pos": [467, 10]}, {"full_name": "congrArg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [349, 9], "def_end_pos": [349, 17]}, {"full_name": "Set.Elem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [151, 23], "def_end_pos": [151, 27]}]], "state_before": "case mk'.mk'.h.e_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d\u00b9 : Set \u03b1\nv\u2081 : Trunc \u2191s\u271d\u00b9\ns\u271d : Set \u03b1\nv\u2082 : Trunc \u2191s\u271d\nh : { s := s\u271d\u00b9, val := v\u2081 }.s = { s := s\u271d, val := v\u2082 }.s\n\u22a2 HEq v\u2081 v\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ManyOneDegree.of_le_of", "start": [436, 1], "end": [437, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.tendsto_filterAt", "start": [1137, 1], "end": [1149, 44], "traced_tactics": [{"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "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 : SigmaFinite \u03bc\nx : \u03b1\n\u22a2 Tendsto (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x)", "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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u22a2 s \u2208 map (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0)"}, {"tactic": "simp only [mem_map]", "annotated_tactic": ["simp only [<a>mem_map</a>]", [{"full_name": "Filter.mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1855, 9], "def_end_pos": [1855, 16]}]], "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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u22a2 s \u2208 map (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0)", "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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0"}, {"tactic": "obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 :\n  \u2203 (\u03b5 : \u211d), \u03b5 > 0 \u2227\n    \u2200 a : Set \u03b1, a \u2208 (Besicovitch.vitaliFamily \u03bc).setsAt x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s :=\n  (VitaliFamily.mem_filterAt_iff _).1 hs", "annotated_tactic": ["obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 :\n    \u2203 (\u03b5 : \u211d), \u03b5 > 0 \u2227\n      \u2200 a : <a>Set</a> \u03b1, a \u2208 (<a>Besicovitch.vitaliFamily</a> \u03bc).<a>setsAt</a> x \u2192 a \u2286 <a>closedBall</a> x \u03b5 \u2192 a \u2208 s :=\n    (<a>VitaliFamily.mem_filterAt_iff</a> _).1 hs", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Besicovitch.vitaliFamily", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [1089, 15], "def_end_pos": [1089, 27]}, {"full_name": "VitaliFamily.setsAt", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "VitaliFamily.mem_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [222, 9], "def_end_pos": [222, 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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0", "state_after": "case 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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0"}, {"tactic": "have : Ioc (0 : \u211d) \u03b5 \u2208 \ud835\udcdd[>] (0 : \u211d) := Ioc_mem_nhdsWithin_Ioi \u27e8le_rfl, \u03b5pos\u27e9", "annotated_tactic": ["have : <a>Ioc</a> (0 : \u211d) \u03b5 \u2208 \ud835\udcdd[>] (0 : \u211d) := <a>Ioc_mem_nhdsWithin_Ioi</a> \u27e8<a>le_rfl</a>, \u03b5pos\u27e9", [{"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Ioc_mem_nhdsWithin_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 31]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case 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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0", "state_after": "case 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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0"}, {"tactic": "filter_upwards [this] with _ hr", "annotated_tactic": ["filter_upwards [this] with _ hr", []], "state_before": "case 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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0", "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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 a\u271d \u2208 (fun r => closedBall x r) \u207b\u00b9' s"}, {"tactic": "apply h\u03b5", "annotated_tactic": ["apply h\u03b5", []], "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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 a\u271d \u2208 (fun r => closedBall x r) \u207b\u00b9' s", "state_after": "case h.a\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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 (fun r => closedBall x r) a\u271d \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x\n\ncase h.a\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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 (fun r => closedBall x r) a\u271d \u2286 closedBall x \u03b5"}, {"tactic": "exact mem_image_of_mem _ hr.1", "annotated_tactic": ["exact <a>mem_image_of_mem</a> _ hr.1", [{"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 h.a\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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 (fun r => closedBall x r) a\u271d \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x", "state_after": "no goals"}, {"tactic": "exact closedBall_subset_closedBall hr.2", "annotated_tactic": ["exact <a>closedBall_subset_closedBall</a> hr.2", [{"full_name": "Metric.closedBall_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}]], "state_before": "case h.a\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 : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 (fun r => closedBall x r) a\u271d \u2286 closedBall x \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Convergence.lean", "full_name": "MeasureTheory.Integrable.tendsto_ae_condexp", "start": [373, 1], "end": [420, 57], "traced_tactics": [{"tactic": "have hle : \u2a06 n, \u2131 n \u2264 m0 := sSup_le fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _", "annotated_tactic": ["have hle : \u2a06 n, \u2131 n \u2264 m0 := <a>sSup_le</a> fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _", [{"full_name": "sSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [200, 9], "def_end_pos": [200, 16]}]], "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\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "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\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "have hunif : UniformIntegrable (fun n => \u03bc[g|\u2131 n]) 1 \u03bc :=\n  hg.uniformIntegrable_condexp_filtration", "annotated_tactic": ["have hunif : <a>UniformIntegrable</a> (fun n => \u03bc[g|\u2131 n]) 1 \u03bc :=\n    hg.uniformIntegrable_condexp_filtration", [{"full_name": "MeasureTheory.UniformIntegrable", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [72, 5], "def_end_pos": [72, 22]}]], "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\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "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\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "obtain \u27e8R, hR\u27e9 := hunif.2.2", "annotated_tactic": ["obtain \u27e8R, hR\u27e9 := hunif.2.2", []], "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\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "have hlimint : Integrable (\u2131.limitProcess (fun n => \u03bc[g|\u2131 n]) \u03bc) \u03bc :=\n  (mem\u2112p_limitProcess_of_snorm_bdd hunif.1 hR).integrable le_rfl", "annotated_tactic": ["have hlimint : <a>Integrable</a> (\u2131.limitProcess (fun n => \u03bc[g|\u2131 n]) \u03bc) \u03bc :=\n    (<a>mem\u2112p_limitProcess_of_snorm_bdd</a> hunif.1 hR).<a>integrable</a> <a>le_rfl</a>", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.Filtration.mem\u2112p_limitProcess_of_snorm_bdd", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [338, 9], "def_end_pos": [338, 40]}, {"full_name": "MeasureTheory.Mem\u2112p.integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [807, 9], "def_end_pos": [807, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "suffices g =\u1d50[\u03bc] \u2131.limitProcess (fun n x => (\u03bc[g|\u2131 n]) x) \u03bc by\n  filter_upwards [this, (martingale_condexp g \u2131 \u03bc).submartingale.ae_tendsto_limitProcess hR] with\n    x heq ht\n  rwa [heq]", "annotated_tactic": ["suffices g =\u1d50[\u03bc] \u2131.limitProcess (fun n x => (\u03bc[g|\u2131 n]) x) \u03bc by\n    filter_upwards [this, (<a>martingale_condexp</a> g \u2131 \u03bc).submartingale.ae_tendsto_limitProcess hR] with\n      x heq ht\n    rwa [heq]", [{"full_name": "MeasureTheory.martingale_condexp", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 27]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\n\u22a2 g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc"}, {"tactic": "have : \u2200 n s, MeasurableSet[\u2131 n] s \u2192\n    \u222b x in s, g x \u2202\u03bc = \u222b x in s, \u2131.limitProcess (fun n x => (\u03bc[g|\u2131 n]) x) \u03bc x \u2202\u03bc := by\n  intro n s hs\n  rw [\u2190 set_integral_condexp (\u2131.le n) hg hs, \u2190 set_integral_condexp (\u2131.le n) hlimint hs]\n  refine' set_integral_congr_ae (\u2131.le _ _ hs) _\n  filter_upwards [(martingale_condexp g \u2131 \u03bc).ae_eq_condexp_limitProcess hunif n] with x hx _\n  rw [hx]", "annotated_tactic": ["have : \u2200 n s, MeasurableSet[\u2131 n] s \u2192\n      \u222b x in s, g x \u2202\u03bc = \u222b x in s, \u2131.limitProcess (fun n x => (\u03bc[g|\u2131 n]) x) \u03bc x \u2202\u03bc := by\n    intro n s hs\n    rw [\u2190 <a>set_integral_condexp</a> (\u2131.le n) hg hs, \u2190 <a>set_integral_condexp</a> (\u2131.le n) hlimint hs]\n    refine' <a>set_integral_congr_ae</a> (\u2131.le _ _ hs) _\n    filter_upwards [(<a>martingale_condexp</a> g \u2131 \u03bc).<a>ae_eq_condexp_limitProcess</a> hunif n] with x hx _\n    rw [hx]", [{"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.set_integral_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 29]}, {"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.martingale_condexp", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 27]}, {"full_name": "MeasureTheory.Martingale.ae_eq_condexp_limitProcess", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [360, 9], "def_end_pos": [360, 46]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\n\u22a2 g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc"}, {"tactic": "refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hle (fun s _ _ => hg.integrableOn)\n  (fun s _ _ => hlimint.integrableOn) (fun s hs => _) hgmeas.aeStronglyMeasurable'\n  stronglyMeasurable_limitProcess.aeStronglyMeasurable'", "annotated_tactic": ["refine' <a>ae_eq_of_forall_set_integral_eq_of_sigmaFinite'</a> hle (fun s _ _ => hg.integrableOn)\n    (fun s _ _ => hlimint.integrableOn) (fun s hs => _) hgmeas.aeStronglyMeasurable'\n    stronglyMeasurable_limitProcess.aeStronglyMeasurable'", [{"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 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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "apply @MeasurableSpace.induction_on_inter _ _ _ (\u2a06 n, \u2131 n)\n  (MeasurableSpace.measurableSpace_iSup_eq \u2131) _ _ _ _ _ _ hs", "annotated_tactic": ["apply @<a>MeasurableSpace.induction_on_inter</a> _ _ _ (\u2a06 n, \u2131 n)\n    (<a>MeasurableSpace.measurableSpace_iSup_eq</a> \u2131) _ _ _ _ _ _ hs", [{"full_name": "MeasurableSpace.induction_on_inter", "def_path": "Mathlib/MeasureTheory/PiSystem.lean", "def_pos": [745, 9], "def_end_pos": [745, 27]}, {"full_name": "MeasurableSpace.measurableSpace_iSup_eq", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [517, 9], "def_end_pos": [517, 32]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 IsPiSystem {s | \u2203 n, MeasurableSet s}\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4 \u2192 \u222b (x : \u03a9) in \u2205, g x \u2202\u03bc = \u222b (x : \u03a9) in \u2205, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u03a9),\n    t \u2208 {s | \u2203 n, MeasurableSet s} \u2192\n      \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u03a9),\n    MeasurableSet t \u2192\n      (\u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc) \u2192\n        \u2191\u2191\u03bc t\u1d9c < \u22a4 \u2192 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : 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),\n            \u2191\u2191\u03bc (f i) < \u22a4 \u2192\n              \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc) \u2192\n          \u2191\u2191\u03bc (\u22c3 i, f i) < \u22a4 \u2192\n            \u222b (x : \u03a9) in \u22c3 i, f i, g x \u2202\u03bc = \u222b (x : \u03a9) in \u22c3 i, f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "filter_upwards [this, (martingale_condexp g \u2131 \u03bc).submartingale.ae_tendsto_limitProcess hR] with\n  x heq ht", "annotated_tactic": ["filter_upwards [this, (<a>martingale_condexp</a> g \u2131 \u03bc).submartingale.ae_tendsto_limitProcess hR] with\n      x heq ht", [{"full_name": "MeasureTheory.martingale_condexp", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 27]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis : g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "state_after": "case h\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis : g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc\nx : \u03a9\nheq : g x = limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x\nht : Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (limitProcess (fun i => \u03bc[g|\u2191\u2131 i]) \u2131 \u03bc x))\n\u22a2 Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "rwa [heq]", "annotated_tactic": ["rwa [heq]", []], "state_before": "case h\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis : g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc\nx : \u03a9\nheq : g x = limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x\nht : Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (limitProcess (fun i => \u03bc[g|\u2191\u2131 i]) \u2131 \u03bc x))\n\u22a2 Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x))", "state_after": "no goals"}, {"tactic": "intro n s hs", "annotated_tactic": ["intro n s hs", []], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\n\u22a2 \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "rw [\u2190 set_integral_condexp (\u2131.le n) hg hs, \u2190 set_integral_condexp (\u2131.le n) hlimint hs]", "annotated_tactic": ["rw [\u2190 <a>set_integral_condexp</a> (\u2131.le n) hg hs, \u2190 <a>set_integral_condexp</a> (\u2131.le n) hlimint hs]", [{"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.set_integral_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 29]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03a9) in s, (\u03bc[g|\u2191\u2131 n]) x \u2202\u03bc = \u222b (x : \u03a9) in s, (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x \u2202\u03bc"}, {"tactic": "refine' set_integral_congr_ae (\u2131.le _ _ hs) _", "annotated_tactic": ["refine' <a>set_integral_congr_ae</a> (\u2131.le _ _ hs) _", [{"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": "\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03a9) in s, (\u03bc[g|\u2191\u2131 n]) x \u2202\u03bc = \u222b (x : \u03a9) in s, (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x"}, {"tactic": "filter_upwards [(martingale_condexp g \u2131 \u03bc).ae_eq_condexp_limitProcess hunif n] with x hx _", "annotated_tactic": ["filter_upwards [(<a>martingale_condexp</a> g \u2131 \u03bc).<a>ae_eq_condexp_limitProcess</a> hunif n] with x hx _", [{"full_name": "MeasureTheory.martingale_condexp", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 27]}, {"full_name": "MeasureTheory.Martingale.ae_eq_condexp_limitProcess", "def_path": "Mathlib/Probability/Martingale/Convergence.lean", "def_pos": [360, 9], "def_end_pos": [360, 46]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x", "state_after": "case h\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\nx : \u03a9\nhx : (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun i => \u03bc[g|\u2191\u2131 i]) \u2131 \u03bc|\u2191\u2131 n]) x\na\u271d : x \u2208 s\n\u22a2 (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x"}, {"tactic": "rw [hx]", "annotated_tactic": ["rw [hx]", []], "state_before": "case h\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nn : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\nx : \u03a9\nhx : (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun i => \u03bc[g|\u2191\u2131 i]) \u2131 \u03bc|\u2191\u2131 n]) x\na\u271d : x \u2208 s\n\u22a2 (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x", "state_after": "no goals"}, {"tactic": "rintro s \u27e8n, hs\u27e9 t \u27e8m, ht\u27e9 -", "annotated_tactic": ["rintro s \u27e8n, hs\u27e9 t \u27e8m, ht\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 IsPiSystem {s | \u2203 n, MeasurableSet s}", "state_after": "case 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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns\u271d : Set \u03a9\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03a9\nn : \u2115\nhs : MeasurableSet s\nt : Set \u03a9\nm : \u2115\nht : MeasurableSet t\n\u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s}"}, {"tactic": "by_cases hnm : n \u2264 m", "annotated_tactic": ["by_cases hnm : n \u2264 m", []], "state_before": "case 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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns\u271d : Set \u03a9\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03a9\nn : \u2115\nhs : MeasurableSet s\nt : Set \u03a9\nm : \u2115\nht : MeasurableSet t\n\u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s}", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns\u271d : Set \u03a9\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03a9\nn : \u2115\nhs : MeasurableSet s\nt : Set \u03a9\nm : \u2115\nht : MeasurableSet t\nhnm : n \u2264 m\n\u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s}\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 : \u03a9\nR\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns\u271d : Set \u03a9\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03a9\nn : \u2115\nhs : MeasurableSet s\nt : Set \u03a9\nm : \u2115\nht : MeasurableSet t\nhnm : \u00acn \u2264 m\n\u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s}"}, {"tactic": "exact \u27e8m, (\u2131.mono hnm _ hs).inter ht\u27e9", "annotated_tactic": ["exact \u27e8m, (\u2131.mono hnm _ hs).<a>inter</a> ht\u27e9", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns\u271d : Set \u03a9\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03a9\nn : \u2115\nhs : MeasurableSet s\nt : Set \u03a9\nm : \u2115\nht : MeasurableSet t\nhnm : n \u2264 m\n\u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s}", "state_after": "no goals"}, {"tactic": "exact \u27e8n, hs.inter (\u2131.mono (not_le.1 hnm).le _ ht)\u27e9", "annotated_tactic": ["exact \u27e8n, hs.inter (\u2131.mono (<a>not_le</a>.1 hnm).<a>le</a> _ ht)\u27e9", [{"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 : \u03a9\nR\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns\u271d : Set \u03a9\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03a9\nn : \u2115\nhs : MeasurableSet s\nt : Set \u03a9\nm : \u2115\nht : MeasurableSet t\nhnm : \u00acn \u2264 m\n\u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s}", "state_after": "no goals"}, {"tactic": "simp only [measure_empty, WithTop.zero_lt_top, Measure.restrict_empty, integral_zero_measure,\n  forall_true_left]", "annotated_tactic": ["simp only [<a>measure_empty</a>, <a>WithTop.zero_lt_top</a>, <a>Measure.restrict_empty</a>, <a>integral_zero_measure</a>,\n      <a>forall_true_left</a>]", [{"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"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": "MeasureTheory.Measure.restrict_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1696, 9], "def_end_pos": [1696, 23]}, {"full_name": "MeasureTheory.integral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 30]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4 \u2192 \u222b (x : \u03a9) in \u2205, g x \u2202\u03bc = \u222b (x : \u03a9) in \u2205, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rintro t \u27e8n, ht\u27e9 -", "annotated_tactic": ["rintro t \u27e8n, ht\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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u03a9),\n    t \u2208 {s | \u2203 n, MeasurableSet s} \u2192\n      \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nn : \u2115\nht : MeasurableSet t\n\u22a2 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "exact this n _ ht", "annotated_tactic": ["exact this n _ ht", []], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nn : \u2115\nht : MeasurableSet t\n\u22a2 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rintro t htmeas ht -", "annotated_tactic": ["rintro t htmeas ht -", []], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u03a9),\n    MeasurableSet t \u2192\n      (\u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc) \u2192\n        \u2191\u2191\u03bc t\u1d9c < \u22a4 \u2192 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "have hgeq := @integral_add_compl _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ htmeas (hg.trim hle hgmeas)", "annotated_tactic": ["have hgeq := @<a>integral_add_compl</a> _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ htmeas (hg.trim hle hgmeas)", [{"full_name": "MeasureTheory.integral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [162, 9], "def_end_pos": [162, 27]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\nhgeq :\n  \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle + \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "have hheq := @integral_add_compl _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ htmeas\n  (hlimint.trim hle stronglyMeasurable_limitProcess)", "annotated_tactic": ["have hheq := @<a>integral_add_compl</a> _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ htmeas\n      (hlimint.trim hle <a>stronglyMeasurable_limitProcess</a>)", [{"full_name": "MeasureTheory.integral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [162, 9], "def_end_pos": [162, 27]}, {"full_name": "MeasureTheory.Filtration.stronglyMeasurable_limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [328, 9], "def_end_pos": [328, 40]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\nhgeq :\n  \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle + \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\nhgeq :\n  \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle + \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle\nhheq :\n  \u222b (x : \u03a9) in t, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle +\n      \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "rw [add_comm, \u2190 eq_sub_iff_add_eq] at hgeq hheq", "annotated_tactic": ["rw [<a>add_comm</a>, \u2190 <a>eq_sub_iff_add_eq</a>] at hgeq hheq", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "eq_sub_iff_add_eq", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [824, 15], "def_end_pos": [824, 32]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\nhgeq :\n  \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle + \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle\nhheq :\n  \u222b (x : \u03a9) in t, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle +\n      \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\nhgeq :\n  \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle - \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle\nhheq :\n  \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle -\n      \u222b (x : \u03a9) in t, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "rw [set_integral_trim hle hgmeas htmeas.compl,\n  set_integral_trim hle stronglyMeasurable_limitProcess htmeas.compl, hgeq, hheq, \u2190\n  set_integral_trim hle hgmeas htmeas, \u2190\n  set_integral_trim hle stronglyMeasurable_limitProcess htmeas, \u2190 integral_trim hle hgmeas, \u2190\n  integral_trim hle stronglyMeasurable_limitProcess, \u2190 integral_univ,\n  this 0 _ MeasurableSet.univ, integral_univ, ht (measure_lt_top _ _)]", "annotated_tactic": ["rw [<a>set_integral_trim</a> hle hgmeas htmeas.compl,\n      <a>set_integral_trim</a> hle <a>stronglyMeasurable_limitProcess</a> htmeas.compl, hgeq, hheq, \u2190\n      <a>set_integral_trim</a> hle hgmeas htmeas, \u2190\n      <a>set_integral_trim</a> hle <a>stronglyMeasurable_limitProcess</a> htmeas, \u2190 <a>integral_trim</a> hle hgmeas, \u2190\n      <a>integral_trim</a> hle <a>stronglyMeasurable_limitProcess</a>, \u2190 <a>integral_univ</a>,\n      this 0 _ <a>MeasurableSet.univ</a>, <a>integral_univ</a>, ht (<a>measure_lt_top</a> _ _)]", [{"full_name": "MeasureTheory.set_integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [621, 9], "def_end_pos": [621, 26]}, {"full_name": "MeasureTheory.set_integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [621, 9], "def_end_pos": [621, 26]}, {"full_name": "MeasureTheory.Filtration.stronglyMeasurable_limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [328, 9], "def_end_pos": [328, 40]}, {"full_name": "MeasureTheory.set_integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [621, 9], "def_end_pos": [621, 26]}, {"full_name": "MeasureTheory.set_integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [621, 9], "def_end_pos": [621, 26]}, {"full_name": "MeasureTheory.Filtration.stronglyMeasurable_limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [328, 9], "def_end_pos": [328, 40]}, {"full_name": "MeasureTheory.integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 22]}, {"full_name": "MeasureTheory.integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1869, 9], "def_end_pos": [1869, 22]}, {"full_name": "MeasureTheory.Filtration.stronglyMeasurable_limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [328, 9], "def_end_pos": [328, 40]}, {"full_name": "MeasureTheory.integral_univ", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [152, 9], "def_end_pos": [152, 22]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.integral_univ", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [152, 9], "def_end_pos": [152, 22]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nt : Set \u03a9\nhtmeas : MeasurableSet t\nht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\nhgeq :\n  \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle - \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle\nhheq :\n  \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle =\n    \u222b (x : \u03a9), limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle -\n      \u222b (x : \u03a9) in t, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle\n\u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rintro f hf hfmeas heq -", "annotated_tactic": ["rintro f hf hfmeas heq -", []], "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\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : 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),\n            \u2191\u2191\u03bc (f i) < \u22a4 \u2192\n              \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc) \u2192\n          \u2191\u2191\u03bc (\u22c3 i, f i) < \u22a4 \u2192\n            \u222b (x : \u03a9) in \u22c3 i, f i, g x \u2202\u03bc = \u222b (x : \u03a9) in \u22c3 i, f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u03a9\nhf : Pairwise (Disjoint on f)\nhfmeas : \u2200 (i : \u2115), MeasurableSet (f i)\nheq :\n  \u2200 (i : \u2115),\n    \u2191\u2191\u03bc (f i) < \u22a4 \u2192 \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 \u222b (x : \u03a9) in \u22c3 i, f i, g x \u2202\u03bc = \u222b (x : \u03a9) in \u22c3 i, f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc"}, {"tactic": "rw [integral_iUnion (fun n => hle _ (hfmeas n)) hf hg.integrableOn,\n  integral_iUnion (fun n => hle _ (hfmeas n)) hf hlimint.integrableOn]", "annotated_tactic": ["rw [<a>integral_iUnion</a> (fun n => hle _ (hfmeas n)) hf hg.integrableOn,\n      <a>integral_iUnion</a> (fun n => hle _ (hfmeas n)) hf hlimint.integrableOn]", [{"full_name": "MeasureTheory.integral_iUnion", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [251, 9], "def_end_pos": [251, 24]}, {"full_name": "MeasureTheory.integral_iUnion", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [251, 9], "def_end_pos": [251, 24]}]], "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\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u03a9\nhf : Pairwise (Disjoint on f)\nhfmeas : \u2200 (i : \u2115), MeasurableSet (f i)\nheq :\n  \u2200 (i : \u2115),\n    \u2191\u2191\u03bc (f i) < \u22a4 \u2192 \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 \u222b (x : \u03a9) in \u22c3 i, f i, g x \u2202\u03bc = \u222b (x : \u03a9) in \u22c3 i, f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc", "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\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u03a9\nhf : Pairwise (Disjoint on f)\nhfmeas : \u2200 (i : \u2115), MeasurableSet (f i)\nheq :\n  \u2200 (i : \u2115),\n    \u2191\u2191\u03bc (f i) < \u22a4 \u2192 \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 \u2211' (n : \u2115), \u222b (a : \u03a9) in f n, g a \u2202\u03bc = \u2211' (n : \u2115), \u222b (a : \u03a9) in f n, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc a \u2202\u03bc"}, {"tactic": "exact tsum_congr fun n => heq _ (measure_lt_top _ _)", "annotated_tactic": ["exact <a>tsum_congr</a> fun n => heq _ (<a>measure_lt_top</a> _ _)", [{"full_name": "tsum_congr", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [498, 9], "def_end_pos": [498, 19]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}]], "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\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR\u271d : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ng : \u03a9 \u2192 \u211d\nhg : Integrable g\nhgmeas : StronglyMeasurable g\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc\nR : \u211d\u22650\nhR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R\nhlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc)\nthis :\n  \u2200 (n : \u2115) (s : Set \u03a9),\n    MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\ns : Set \u03a9\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u03a9\nhf : Pairwise (Disjoint on f)\nhfmeas : \u2200 (i : \u2115), MeasurableSet (f i)\nheq :\n  \u2200 (i : \u2115),\n    \u2191\u2191\u03bc (f i) < \u22a4 \u2192 \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc\n\u22a2 \u2211' (n : \u2115), \u222b (a : \u03a9) in f n, g a \u2202\u03bc = \u2211' (n : \u2115), \u222b (a : \u03a9) in f n, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc a \u2202\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.eq_one_of_mul_eq_one_left", "start": [857, 1], "end": [858, 59], "traced_tactics": [{"tactic": "rw [Int.mul_comm, H']", "annotated_tactic": ["rw [<a>Int.mul_comm</a>, H']", [{"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 \u2264 b\nH' : a * b = 1\n\u22a2 b * ?m.91489 H H' = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.measure_inter_add_diff\u2080", "start": [300, 1], "end": [316, 56], "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": "\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 s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) = \u2191\u2191\u03bc s", "state_after": "case refine'_1\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 s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s\n\ncase refine'_2\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 s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t)"}, {"tactic": "rcases exists_measurable_superset \u03bc s with \u27e8s', hsub, hs'm, hs'\u27e9", "annotated_tactic": ["rcases <a>exists_measurable_superset</a> \u03bc s with \u27e8s', hsub, hs'm, hs'\u27e9", [{"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 refine'_1\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 s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s", "state_after": "case refine'_1.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 s : Set \u03b1\nht : NullMeasurableSet t\ns' : Set \u03b1\nhsub : s \u2286 s'\nhs'm : MeasurableSet s'\nhs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s"}, {"tactic": "replace hs'm : NullMeasurableSet s' \u03bc := hs'm.nullMeasurableSet", "annotated_tactic": ["replace hs'm : <a>NullMeasurableSet</a> s' \u03bc := hs'm.nullMeasurableSet", [{"full_name": "MeasureTheory.NullMeasurableSet", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [97, 5], "def_end_pos": [97, 22]}]], "state_before": "case refine'_1.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 s : Set \u03b1\nht : NullMeasurableSet t\ns' : Set \u03b1\nhsub : s \u2286 s'\nhs'm : MeasurableSet s'\nhs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s", "state_after": "case refine'_1.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 s : Set \u03b1\nht : NullMeasurableSet t\ns' : Set \u03b1\nhsub : s \u2286 s'\nhs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s\nhs'm : NullMeasurableSet s'\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s"}, {"tactic": "calc\n  \u03bc (s \u2229 t) + \u03bc (s \\ t) \u2264 \u03bc (s' \u2229 t) + \u03bc (s' \\ t) :=\n    add_le_add (measure_mono <| inter_subset_inter_left _ hsub)\n      (measure_mono <| diff_subset_diff_left hsub)\n  _ = \u03bc (s' \u2229 t \u222a s' \\ t) :=\n    (measure_union\u2080_aux (hs'm.inter ht) (hs'm.diff ht) <|\n        (@disjoint_inf_sdiff _ s' t _).aedisjoint).symm\n  _ = \u03bc s' := (congr_arg \u03bc (inter_union_diff _ _))\n  _ = \u03bc s := hs'", "annotated_tactic": ["calc\n      \u03bc (s \u2229 t) + \u03bc (s \\ t) \u2264 \u03bc (s' \u2229 t) + \u03bc (s' \\ t) :=\n        <a>add_le_add</a> (<a>measure_mono</a> <| <a>inter_subset_inter_left</a> _ hsub)\n          (<a>measure_mono</a> <| <a>diff_subset_diff_left</a> hsub)\n      _ = \u03bc (s' \u2229 t \u222a s' \\ t) :=\n        (<a>measure_union\u2080_aux</a> (hs'm.inter ht) (hs'm.diff ht) <|\n            (@<a>disjoint_inf_sdiff</a> _ s' t _).<a>aedisjoint</a>).<a>symm</a>\n      _ = \u03bc s' := (<a>congr_arg</a> \u03bc (<a>inter_union_diff</a> _ _))\n      _ = \u03bc s := hs'", [{"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.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": "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_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1908, 9], "def_end_pos": [1908, 30]}, {"full_name": "MeasureTheory.measure_union\u2080_aux", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [292, 9], "def_end_pos": [292, 27]}, {"full_name": "disjoint_inf_sdiff", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [122, 9], "def_end_pos": [122, 27]}, {"full_name": "Disjoint.aedisjoint", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [65, 19], "def_end_pos": [65, 45]}, {"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": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Set.inter_union_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 25]}]], "state_before": "case refine'_1.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 s : Set \u03b1\nht : NullMeasurableSet t\ns' : Set \u03b1\nhsub : s \u2286 s'\nhs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s\nhs'm : NullMeasurableSet s'\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s", "state_after": "no goals"}, {"tactic": "calc\n  \u03bc s = \u03bc (s \u2229 t \u222a s \\ t) := by rw [inter_union_diff]\n  _ \u2264 \u03bc (s \u2229 t) + \u03bc (s \\ t) := measure_union_le _ _", "annotated_tactic": ["calc\n      \u03bc s = \u03bc (s \u2229 t \u222a s \\ t) := by rw [<a>inter_union_diff</a>]\n      _ \u2264 \u03bc (s \u2229 t) + \u03bc (s \\ t) := <a>measure_union_le</a> _ _", [{"full_name": "Set.inter_union_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1889, 9], "def_end_pos": [1889, 25]}, {"full_name": "MeasureTheory.measure_union_le", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}]], "state_before": "case refine'_2\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 s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t)", "state_after": "no goals"}, {"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\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 s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 t \u222a s \\ t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "TorusIntegrable.torusIntegrable_zero_radius", "start": [135, 1], "end": [137, 40], "traced_tactics": [{"tactic": "rw [TorusIntegrable, torusMap_zero_radius]", "annotated_tactic": ["rw [<a>TorusIntegrable</a>, <a>torusMap_zero_radius</a>]", [{"full_name": "TorusIntegrable", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [105, 5], "def_end_pos": [105, 20]}, {"full_name": "torusMap_zero_radius", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [95, 9], "def_end_pos": [95, 29]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\n\u22a2 TorusIntegrable f c 0", "state_after": "n : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\n\u22a2 IntegrableOn (fun \u03b8 => f (const (Fin n \u2192 \u211d) c \u03b8)) (Icc 0 fun x => 2 * \u03c0)"}, {"tactic": "apply torusIntegrable_const (f c) c 0", "annotated_tactic": ["apply <a>torusIntegrable_const</a> (f c) c 0", [{"full_name": "TorusIntegrable.torusIntegrable_const", "def_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "def_pos": [115, 9], "def_end_pos": [115, 30]}]], "state_before": "n : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\n\u22a2 IntegrableOn (fun \u03b8 => f (const (Fin n \u2192 \u211d) c \u03b8)) (Icc 0 fun x => 2 * \u03c0)", "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_left_of_right_invariant", "start": [456, 1], "end": [461, 39], "traced_tactics": [{"tactic": "exact\n  (quasiMeasurePreserving_div_left \u03bc.inv g).mono (inv_absolutelyContinuous \u03bc.inv)\n    (absolutelyContinuous_inv \u03bc.inv)", "annotated_tactic": ["exact\n    (<a>quasiMeasurePreserving_div_left</a> \u03bc.inv g).<a>mono</a> (<a>inv_absolutelyContinuous</a> \u03bc.inv)\n      (<a>absolutelyContinuous_inv</a> \u03bc.inv)", [{"full_name": "MeasureTheory.quasiMeasurePreserving_div_left", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [447, 9], "def_end_pos": [447, 40]}, {"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": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.totalDegree_smul_le", "start": [681, 1], "end": [683, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.add_measure", "start": [541, 1], "end": [546, 43], "traced_tactics": [{"tactic": "simp_rw [\u2190 mem\u2112p_one_iff_integrable] at h\u03bc h\u03bd \u22a2", "annotated_tactic": ["simp_rw [\u2190 <a>mem\u2112p_one_iff_integrable</a>] at h\u03bc h\u03bd \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\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 Integrable 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\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1\nh\u03bd : Mem\u2112p f 1\n\u22a2 Mem\u2112p f 1"}, {"tactic": "refine' \u27e8h\u03bc.aestronglyMeasurable.add_measure h\u03bd.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8h\u03bc.aestronglyMeasurable.add_measure h\u03bd.aestronglyMeasurable, _\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\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1\nh\u03bd : Mem\u2112p f 1\n\u22a2 Mem\u2112p f 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 \u03b2\nh\u03bc : Mem\u2112p f 1\nh\u03bd : Mem\u2112p f 1\n\u22a2 snorm f 1 (\u03bc + \u03bd) < \u22a4"}, {"tactic": "rw [snorm_one_add_measure, ENNReal.add_lt_top]", "annotated_tactic": ["rw [<a>snorm_one_add_measure</a>, <a>ENNReal.add_lt_top</a>]", [{"full_name": "MeasureTheory.snorm_one_add_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [688, 9], "def_end_pos": [688, 30]}, {"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [561, 17], "def_end_pos": [561, 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\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1\nh\u03bd : Mem\u2112p f 1\n\u22a2 snorm f 1 (\u03bc + \u03bd) < \u22a4", "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\nh\u03bc : Mem\u2112p f 1\nh\u03bd : Mem\u2112p f 1\n\u22a2 snorm f 1 \u03bc < \u22a4 \u2227 snorm f 1 \u03bd < \u22a4"}, {"tactic": "exact \u27e8h\u03bc.snorm_lt_top, h\u03bd.snorm_lt_top\u27e9", "annotated_tactic": ["exact \u27e8h\u03bc.snorm_lt_top, h\u03bd.snorm_lt_top\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\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1\nh\u03bd : Mem\u2112p f 1\n\u22a2 snorm f 1 \u03bc < \u22a4 \u2227 snorm f 1 \u03bd < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_to_prop", "start": [473, 1], "end": [478, 20], "traced_tactics": [{"tactic": "refine' measurable_to_countable' fun x => _", "annotated_tactic": ["refine' <a>measurable_to_countable'</a> fun x => _", [{"full_name": "measurable_to_countable'", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 33]}]], "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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\n\u22a2 Measurable f", "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 t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\nx : Prop\n\u22a2 MeasurableSet (f \u207b\u00b9' {x})"}, {"tactic": "by_cases hx : x", "annotated_tactic": ["by_cases hx : x", []], "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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\nx : Prop\n\u22a2 MeasurableSet (f \u207b\u00b9' {x})", "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\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\nx : Prop\nhx : x\n\u22a2 MeasurableSet (f \u207b\u00b9' {x})\n\ncase neg\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\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\nx : Prop\nhx : \u00acx\n\u22a2 MeasurableSet (f \u207b\u00b9' {x})"}, {"tactic": "simpa [hx] using h", "annotated_tactic": ["simpa [hx] using h", []], "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\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\nx : Prop\nhx : x\n\u22a2 MeasurableSet (f \u207b\u00b9' {x})", "state_after": "no goals"}, {"tactic": "simpa only [hx, \u2190 preimage_compl, Prop.compl_singleton, not_true, preimage_singleton_false]\n  using h.compl", "annotated_tactic": ["simpa only [hx, \u2190 <a>preimage_compl</a>, <a>Prop.compl_singleton</a>, <a>not_true</a>, <a>preimage_singleton_false</a>]\n      using h.compl", [{"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Prop.compl_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [3022, 17], "def_end_pos": [3022, 37]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "Set.preimage_singleton_false", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [197, 17], "def_end_pos": [197, 41]}]], "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\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 Prop\nh : MeasurableSet (f \u207b\u00b9' {True})\nx : Prop\nhx : \u00acx\n\u22a2 MeasurableSet (f \u207b\u00b9' {x})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.powerset_insert", "start": [639, 1], "end": [654, 36], "traced_tactics": [{"tactic": "ext t", "annotated_tactic": ["ext t", []], "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\ns : Set \u03b1\na : \u03b1\n\u22a2 \ud835\udcab insert a s = \ud835\udcab s \u222a insert a '' \ud835\udcab s", "state_after": "case h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 t \u2208 \ud835\udcab insert a s \u2194 t \u2208 \ud835\udcab s \u222a insert a '' \ud835\udcab s"}, {"tactic": "simp_rw [mem_union, mem_image, mem_powerset_iff]", "annotated_tactic": ["simp_rw [<a>mem_union</a>, <a>mem_image</a>, <a>mem_powerset_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_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Set.mem_powerset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2165, 9], "def_end_pos": [2165, 25]}]], "state_before": "case h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 t \u2208 \ud835\udcab insert a s \u2194 t \u2208 \ud835\udcab s \u222a insert a '' \ud835\udcab s", "state_after": "case h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 t \u2286 insert a s \u2194 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 t \u2286 insert a s \u2194 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t", "state_after": "case h.mp\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 t \u2286 insert a s \u2192 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t\n\ncase h.mpr\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 (t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t) \u2192 t \u2286 insert a s"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h.mp\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 t \u2286 insert a s \u2192 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t", "state_after": "case h.mp\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\n\u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t"}, {"tactic": "by_cases hs : a \u2208 t", "annotated_tactic": ["by_cases hs : a \u2208 t", []], "state_before": "case h.mp\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\n\u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t", "state_after": "case pos\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t\n\ncase neg\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : \u00aca \u2208 t\n\u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case pos\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t", "state_after": "case pos.h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 \u2203 x, x \u2286 s \u2227 insert a x = t"}, {"tactic": "refine' \u27e8t \\ {a}, _, _\u27e9", "annotated_tactic": ["refine' \u27e8t \\ {a}, _, _\u27e9", []], "state_before": "case pos.h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 \u2203 x, x \u2286 s \u2227 insert a x = t", "state_after": "case pos.h.refine'_1\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 t \\ {a} \u2286 s\n\ncase pos.h.refine'_2\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 insert a (t \\ {a}) = t"}, {"tactic": "rw [diff_singleton_subset_iff]", "annotated_tactic": ["rw [<a>diff_singleton_subset_iff</a>]", [{"full_name": "Set.diff_singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1961, 9], "def_end_pos": [1961, 34]}]], "state_before": "case pos.h.refine'_1\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 t \\ {a} \u2286 s", "state_after": "case pos.h.refine'_1\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 t \u2286 insert a s"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case pos.h.refine'_1\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 t \u2286 insert a s", "state_after": "no goals"}, {"tactic": "rw [insert_diff_singleton, insert_eq_of_mem hs]", "annotated_tactic": ["rw [<a>insert_diff_singleton</a>, <a>insert_eq_of_mem</a> hs]", [{"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.insert_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1152, 9], "def_end_pos": [1152, 25]}]], "state_before": "case pos.h.refine'_2\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : a \u2208 t\n\u22a2 insert a (t \\ {a}) = t", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case neg\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : \u00aca \u2208 t\n\u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t", "state_after": "case neg.h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : \u00aca \u2208 t\n\u22a2 t \u2286 s"}, {"tactic": "exact (subset_insert_iff_of_not_mem hs).mp h", "annotated_tactic": ["exact (<a>subset_insert_iff_of_not_mem</a> hs).<a>mp</a> h", [{"full_name": "Set.subset_insert_iff_of_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1185, 9], "def_end_pos": [1185, 37]}, {"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 neg.h\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 insert a s\nhs : \u00aca \u2208 t\n\u22a2 t \u2286 s", "state_after": "no goals"}, {"tactic": "rintro (h | \u27e8s', h\u2081, rfl\u27e9)", "annotated_tactic": ["rintro (h | \u27e8s', h\u2081, rfl\u27e9)", []], "state_before": "case h.mpr\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\n\u22a2 (t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t) \u2192 t \u2286 insert a s", "state_after": "case h.mpr.inl\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 s\n\u22a2 t \u2286 insert a s\n\ncase h.mpr.inr.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\ns : Set \u03b1\na : \u03b1\ns' : Set \u03b1\nh\u2081 : s' \u2286 s\n\u22a2 insert a s' \u2286 insert a s"}, {"tactic": "exact subset_trans h (subset_insert a s)", "annotated_tactic": ["exact <a>subset_trans</a> h (<a>subset_insert</a> a s)", [{"full_name": "subset_trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [643, 7], "def_end_pos": [643, 19]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1123, 9], "def_end_pos": [1123, 22]}]], "state_before": "case h.mpr.inl\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\ns : Set \u03b1\na : \u03b1\nt : Set \u03b1\nh : t \u2286 s\n\u22a2 t \u2286 insert a s", "state_after": "no goals"}, {"tactic": "exact insert_subset_insert h\u2081", "annotated_tactic": ["exact <a>insert_subset_insert</a> h\u2081", [{"full_name": "Set.insert_subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 29]}]], "state_before": "case h.mpr.inr.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\ns : Set \u03b1\na : \u03b1\ns' : Set \u03b1\nh\u2081 : s' \u2286 s\n\u22a2 insert a s' \u2286 insert a s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.le_iff_cmp", "start": [882, 1], "end": [883, 89], "traced_tactics": [{"tactic": "rw [\u2190 cmp_swap]", "annotated_tactic": ["rw [\u2190 <a>cmp_swap</a>]", [{"full_name": "Num.cmp_swap", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [852, 9], "def_end_pos": [852, 17]}]], "state_before": "\u03b1 : Type u_1\nm n : Num\n\u22a2 cmp n m = Ordering.lt \u2194 cmp m n = Ordering.gt", "state_after": "\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap (cmp m n) = Ordering.lt \u2194 cmp m n = Ordering.gt"}, {"tactic": "cases cmp m n <;> exact by decide", "annotated_tactic": ["cases <a>cmp</a> m n <;> exact by decide", [{"full_name": "Num.cmp", "def_path": "Mathlib/Data/Num/Basic.lean", "def_pos": [299, 5], "def_end_pos": [299, 8]}]], "state_before": "\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap (cmp m n) = Ordering.lt \u2194 cmp m n = Ordering.gt", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt \u2194 Ordering.gt = Ordering.gt", "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_top_of_surjective", "start": [640, 1], "end": [641, 43], "traced_tactics": [{"tactic": "rw [map_top, hf.range_eq, restrict_univ]", "annotated_tactic": ["rw [<a>map_top</a>, hf.range_eq, <a>restrict_univ</a>]", [{"full_name": "MeasureTheory.OuterMeasure.map_top", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [633, 9], "def_end_pos": [633, 16]}, {"full_name": "MeasureTheory.OuterMeasure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [573, 9], "def_end_pos": [573, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 \u2191(map f) \u22a4 = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "full_name": "MeasureTheory.zero_trim", "start": [49, 1], "end": [50, 56], "traced_tactics": [{"tactic": "simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m]", "annotated_tactic": ["simp [<a>Measure.trim</a>, @<a>OuterMeasure.toMeasure_zero</a> _ m]", [{"full_name": "MeasureTheory.Measure.trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [32, 5], "def_end_pos": [32, 17]}, {"full_name": "MeasureTheory.OuterMeasure.toMeasure_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4269, 9], "def_end_pos": [4269, 36]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 Measure.trim 0 hm = 0", "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_top", "start": [318, 1], "end": [337, 36], "traced_tactics": [{"tactic": "intro \u03b4 h\u03b4", "annotated_tactic": ["intro \u03b4 h\u03b4", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)"}, {"tactic": "simp only [snorm_exponent_top, snormEssSup] at hfg", "annotated_tactic": ["simp only [<a>snorm_exponent_top</a>, <a>snormEssSup</a>] at hfg", [{"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": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : Tendsto (fun n => essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \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\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : Tendsto (fun n => essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5"}, {"tactic": "specialize hfg (ENNReal.ofReal \u03b4 / 2)\n    (ENNReal.div_pos_iff.2 \u27e8(ENNReal.ofReal_pos.2 h\u03b4).ne.symm, ENNReal.two_ne_top\u27e9)", "annotated_tactic": ["specialize hfg (<a>ENNReal.ofReal</a> \u03b4 / 2)\n      (<a>ENNReal.div_pos_iff</a>.2 \u27e8(<a>ENNReal.ofReal_pos</a>.2 h\u03b4).ne.symm, <a>ENNReal.two_ne_top</a>\u27e9)", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"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.ofReal_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2166, 9], "def_end_pos": [2166, 19]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5"}, {"tactic": "refine' hfg.mono fun n hn => _", "annotated_tactic": ["refine' hfg.mono fun n hn => _", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5"}, {"tactic": "simp only [true_and_iff, gt_iff_lt, ge_iff_le, zero_tsub, zero_le, zero_add, Set.mem_Icc,\n  Pi.sub_apply] at *", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>gt_iff_lt</a>, <a>ge_iff_le</a>, <a>zero_tsub</a>, <a>zero_le</a>, <a>zero_add</a>, <a>Set.mem_Icc</a>,\n    <a>Pi.sub_apply</a>] at *", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"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": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"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.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5"}, {"tactic": "have : essSup (fun x : \u03b1 => (\u2016f n x - g x\u2016\u208a : \u211d\u22650\u221e)) \u03bc < ENNReal.ofReal \u03b4 :=\n  lt_of_le_of_lt hn\n    (ENNReal.half_lt_self (ENNReal.ofReal_pos.2 h\u03b4).ne.symm ENNReal.ofReal_lt_top.ne)", "annotated_tactic": ["have : <a>essSup</a> (fun x : \u03b1 => (\u2016f n x - g x\u2016\u208a : \u211d\u22650\u221e)) \u03bc < <a>ENNReal.ofReal</a> \u03b4 :=\n    <a>lt_of_le_of_lt</a> hn\n      (<a>ENNReal.half_lt_self</a> (<a>ENNReal.ofReal_pos</a>.2 h\u03b4).ne.symm ENNReal.ofReal_lt_top.ne)", [{"full_name": "essSup", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [44, 5], "def_end_pos": [44, 11]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 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": "ENNReal.half_lt_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1803, 19], "def_end_pos": [1803, 31]}, {"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\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5"}, {"tactic": "refine' ((le_of_eq _).trans (ae_lt_of_essSup_lt this).le).trans h\u03b5.le", "annotated_tactic": ["refine' ((<a>le_of_eq</a> _).<a>trans</a> (<a>ae_lt_of_essSup_lt</a> this).<a>le</a>).<a>trans</a> h\u03b5.le", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"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": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}, {"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\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} = \u2191\u2191\u03bc {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} = \u2191\u2191\u03bc {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c", "state_after": "case e_a.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 x \u2208 {x | \u03b4 \u2264 dist (f n x) (g x)} \u2194 x \u2208 {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c"}, {"tactic": "simp only [ENNReal.ofReal_le_iff_le_toReal ENNReal.coe_lt_top.ne, ENNReal.coe_toReal, not_lt,\n  coe_nnnorm, Set.mem_setOf_eq, Set.mem_compl_iff]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_le_iff_le_toReal</a> ENNReal.coe_lt_top.ne, <a>ENNReal.coe_toReal</a>, <a>not_lt</a>,\n    <a>coe_nnnorm</a>, <a>Set.mem_setOf_eq</a>, <a>Set.mem_compl_iff</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": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"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_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 e_a.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 x \u2208 {x | \u03b4 \u2264 dist (f n x) (g x)} \u2194 x \u2208 {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c", "state_after": "case e_a.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 \u03b4 \u2264 dist (f n x) (g x) \u2194 \u03b4 \u2264 \u2016f n x - g x\u2016"}, {"tactic": "rw [\u2190 dist_eq_norm (f n x) (g x)]", "annotated_tactic": ["rw [\u2190 <a>dist_eq_norm</a> (f n x) (g x)]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE\u271d : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_4\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 \u03b4 \u2264 dist (f n x) (g x) \u2194 \u03b4 \u2264 \u2016f n x - g x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/DensityTheorem.lean", "full_name": "IsUnifLocDoublingMeasure.ae_tendsto_average", "start": [166, 1], "end": [172, 58], "traced_tactics": [{"tactic": "filter_upwards [(vitaliFamily \u03bc K).ae_tendsto_average hf] with x hx \u03b9 l w \u03b4 \u03b4lim xmem using\n  hx.comp (tendsto_closedBall_filterAt \u03bc _ _ \u03b4lim xmem)", "annotated_tactic": ["filter_upwards [(<a>vitaliFamily</a> \u03bc K).<a>ae_tendsto_average</a> hf] with x hx \u03b9 l w \u03b4 \u03b4lim xmem using\n    hx.comp (<a>tendsto_closedBall_filterAt</a> \u03bc _ _ \u03b4lim xmem)", [{"full_name": "IsUnifLocDoublingMeasure.vitaliFamily", "def_path": "Mathlib/MeasureTheory/Covering/DensityTheorem.lean", "def_pos": [51, 17], "def_end_pos": [51, 29]}, {"full_name": "VitaliFamily.ae_tendsto_average", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [938, 9], "def_end_pos": [938, 27]}, {"full_name": "IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt", "def_path": "Mathlib/MeasureTheory/Covering/DensityTheorem.lean", "def_pos": [112, 9], "def_end_pos": [112, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2078 : MetricSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : IsUnifLocDoublingMeasure \u03bc\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\ninst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 E\nhf : LocallyIntegrable f\nK : \u211d\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 {\u03b9 : Type u_3} {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, x \u2208 closedBall (w j) (K * \u03b4 j)) \u2192\n          Tendsto (fun j => \u2a0d (y : \u03b1) in closedBall (w j) (\u03b4 j), f y \u2202\u03bc) l (\ud835\udcdd (f x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "full_name": "Function.Periodic.intervalIntegral_add_eq_of_pos", "start": [245, 1], "end": [251, 100], "traced_tactics": [{"tactic": "simp only [integral_of_le, hT.le, le_add_iff_nonneg_right]", "annotated_tactic": ["simp only [<a>integral_of_le</a>, hT.le, <a>le_add_iff_nonneg_right</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": "le_add_iff_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [457, 30], "def_end_pos": [457, 53]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\n\u22a2 \u222b (x : \u211d) in t..t + T, f x = \u222b (x : \u211d) in s..s + T, f x", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\n\u22a2 \u222b (x : \u211d) in Ioc t (t + T), f x = \u222b (x : \u211d) in Ioc s (s + T), f x"}, {"tactic": "haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) \u211d volume :=\n  \u27e8fun c s _ => measure_preimage_add _ _ _\u27e9", "annotated_tactic": ["haveI : <a>VAddInvariantMeasure</a> (<a>AddSubgroup.zmultiples</a> T) \u211d <a>volume</a> :=\n    \u27e8fun c s _ => <a>measure_preimage_add</a> _ _ _\u27e9", [{"full_name": "MeasureTheory.VAddInvariantMeasure", "def_path": "Mathlib/MeasureTheory/Group/Action.lean", "def_pos": [35, 7], "def_end_pos": [35, 27]}, {"full_name": "AddSubgroup.zmultiples", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}, {"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_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\n\u22a2 \u222b (x : \u211d) in Ioc t (t + T), f x = \u222b (x : \u211d) in Ioc s (s + T), f x", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 \u222b (x : \u211d) in Ioc t (t + T), f x = \u222b (x : \u211d) in Ioc s (s + T), f x"}, {"tactic": "apply IsAddFundamentalDomain.set_integral_eq (G := AddSubgroup.zmultiples T)", "annotated_tactic": ["apply <a>IsAddFundamentalDomain.set_integral_eq</a> (G := <a>AddSubgroup.zmultiples</a> T)", [{"full_name": "MeasureTheory.IsAddFundamentalDomain.set_integral_eq", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [461, 3], "def_end_pos": [461, 14]}, {"full_name": "AddSubgroup.zmultiples", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 \u222b (x : \u211d) in Ioc t (t + T), f x = \u222b (x : \u211d) in Ioc s (s + T), f x", "state_after": "case hs\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 IsAddFundamentalDomain { x // x \u2208 zmultiples T } (Ioc t (t + T))\n\ncase ht\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 IsAddFundamentalDomain { x // x \u2208 zmultiples T } (Ioc s (s + T))\n\ncase hf\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 \u2200 (g : { x // x \u2208 zmultiples T }) (x : \u211d), f (g +\u1d65 x) = f x"}, {"tactic": "exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomain_Ioc hT s, hf.map_vadd_zmultiples]", "annotated_tactic": ["exacts [<a>isAddFundamentalDomain_Ioc</a> hT t, <a>isAddFundamentalDomain_Ioc</a> hT s, hf.map_vadd_zmultiples]", [{"full_name": "isAddFundamentalDomain_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [39, 9], "def_end_pos": [39, 35]}, {"full_name": "isAddFundamentalDomain_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/Periodic.lean", "def_pos": [39, 9], "def_end_pos": [39, 35]}]], "state_before": "case hs\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 IsAddFundamentalDomain { x // x \u2208 zmultiples T } (Ioc t (t + T))\n\ncase ht\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 IsAddFundamentalDomain { x // x \u2208 zmultiples T } (Ioc s (s + T))\n\ncase hf\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\nT : \u211d\nhf : Periodic f T\nhT : 0 < T\nt s : \u211d\nthis : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume\n\u22a2 \u2200 (g : { x // x \u2208 zmultiples T }) (x : \u211d), f (g +\u1d65 x) = f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.mkMetric_toOuterMeasure", "start": [464, 1], "end": [466, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_one'", "start": [376, 1], "end": [377, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.ae_ae_eq_of_ae_eq_uncurry", "start": [465, 1], "end": [467, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pairwise/Lattice.lean", "full_name": "Set.biUnion_diff_biUnion_eq", "start": [124, 1], "end": [130, 91], "traced_tactics": [{"tactic": "refine'\n  (biUnion_diff_biUnion_subset f s t).antisymm\n    (iUnion\u2082_subset fun i hi a ha => (mem_diff _).2 \u27e8mem_biUnion hi.1 ha, _\u27e9)", "annotated_tactic": ["refine'\n    (<a>biUnion_diff_biUnion_subset</a> f s t).<a>antisymm</a>\n      (<a>iUnion\u2082_subset</a> fun i hi a ha => (<a>mem_diff</a> _).2 \u27e8<a>mem_biUnion</a> hi.1 ha, _\u27e9)", [{"full_name": "Set.biUnion_diff_biUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2307, 9], "def_end_pos": [2307, 36]}, {"full_name": "HasSubset.Subset.antisymm", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [667, 7], "def_end_pos": [667, 32]}, {"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}, {"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_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}]], "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\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\n\u22a2 (\u22c3 i \u2208 s, f i) \\ \u22c3 i \u2208 t, f i = \u22c3 i \u2208 s \\ t, f i", "state_after": "\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\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\n\u22a2 \u00aca \u2208 \u22c3 x \u2208 t, f x"}, {"tactic": "rw [mem_iUnion\u2082]", "annotated_tactic": ["rw [<a>mem_iUnion\u2082</a>]", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "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\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\n\u22a2 \u00aca \u2208 \u22c3 x \u2208 t, f x", "state_after": "\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\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\n\u22a2 \u00ac\u2203 i j, a \u2208 f i"}, {"tactic": "rintro \u27e8j, hj, haj\u27e9", "annotated_tactic": ["rintro \u27e8j, hj, haj\u27e9", []], "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\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\n\u22a2 \u00ac\u2203 i j, a \u2208 f i", "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\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\nj : \u03b9\nhj : j \u2208 t\nhaj : a \u2208 f j\n\u22a2 False"}, {"tactic": "exact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot \u27e8ha, haj\u27e9", "annotated_tactic": ["exact (h (<a>Or.inl</a> hi.1) (<a>Or.inr</a> hj) (<a>ne_of_mem_of_not_mem</a> hj hi.2).<a>symm</a>).<a>le_bot</a> \u27e8ha, haj\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": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"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]}, {"full_name": "Disjoint.le_bot", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [130, 9], "def_end_pos": [130, 24]}]], "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\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\nj : \u03b9\nhj : j \u2208 t\nhaj : a \u2208 f j\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.ofInt'_toZNum", "start": [1260, 1], "end": [1264, 20], "traced_tactics": [{"tactic": "rw [Nat.cast_succ, Num.add_one, toZNum_succ, ofInt'_toZNum n, Nat.cast_succ, succ_ofInt',\n  ZNum.add_one]", "annotated_tactic": ["rw [<a>Nat.cast_succ</a>, <a>Num.add_one</a>, <a>toZNum_succ</a>, ofInt'_toZNum n, <a>Nat.cast_succ</a>, <a>succ_ofInt'</a>,\n      <a>ZNum.add_one</a>]", [{"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.toZNum_succ", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 20]}, {"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.succ_ofInt'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1244, 9], "def_end_pos": [1244, 20]}, {"full_name": "ZNum.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1157, 9], "def_end_pos": [1157, 16]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNum \u2191(n + 1) = ZNum.ofInt' \u2191(n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_ne_zero_of_mem", "start": [542, 1], "end": [543, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.offDiag_insert", "start": [646, 1], "end": [650, 14], "traced_tactics": [{"tactic": "rw [insert_eq, union_comm, offDiag_union, offDiag_singleton, union_empty, union_right_comm]", "annotated_tactic": ["rw [<a>insert_eq</a>, <a>union_comm</a>, <a>offDiag_union</a>, <a>offDiag_singleton</a>, <a>union_empty</a>, <a>union_right_comm</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.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "Set.offDiag_union", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [631, 9], "def_end_pos": [631, 22]}, {"full_name": "Set.offDiag_singleton", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [605, 9], "def_end_pos": [605, 26]}, {"full_name": "Set.union_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 20]}, {"full_name": "Set.union_right_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 25]}]], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 offDiag (insert a s) = offDiag s \u222a {a} \u00d7\u02e2 s \u222a s \u00d7\u02e2 {a}", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 Disjoint s {a}"}, {"tactic": "rw [disjoint_left]", "annotated_tactic": ["rw [<a>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": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 Disjoint s {a}", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 s \u2192 \u00aca_1 \u2208 {a}"}, {"tactic": "rintro b hb (rfl : b = a)", "annotated_tactic": ["rintro b hb (rfl : b = a)", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 s \u2192 \u00aca_1 \u2208 {a}", "state_after": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\nb : \u03b1\nhb : b \u2208 s\nha : \u00acb \u2208 s\n\u22a2 False"}, {"tactic": "exact ha hb", "annotated_tactic": ["exact ha hb", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\nb : \u03b1\nhb : b \u2208 s\nha : \u00acb \u2208 s\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.Subtype.volume_univ", "start": [1459, 1], "end": [1464, 66], "traced_tactics": [{"tactic": "rw [Subtype.volume_def, comap_apply\u2080 _ _ _ _ MeasurableSet.univ.nullMeasurableSet]", "annotated_tactic": ["rw [<a>Subtype.volume_def</a>, <a>comap_apply\u2080</a> _ _ _ _ MeasurableSet.univ.nullMeasurableSet]", [{"full_name": "MeasureTheory.Measure.Subtype.volume_def", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1455, 9], "def_end_pos": [1455, 27]}, {"full_name": "MeasureTheory.Measure.comap_apply\u2080", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1321, 9], "def_end_pos": [1321, 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\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191volume univ = \u2191\u2191volume 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\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191volume (Subtype.val '' univ) = \u2191\u2191volume s\n\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\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 Injective Subtype.val\n\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\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u2191s), MeasurableSet s_1 \u2192 NullMeasurableSet (Subtype.val '' s_1)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], 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"state_after": "no goals"}, {"tactic": "exact fun t => MeasurableSet.nullMeasurableSet_subtype_coe hs", "annotated_tactic": ["exact fun t => <a>MeasurableSet.nullMeasurableSet_subtype_coe</a> hs", [{"full_name": "MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1398, 9], "def_end_pos": [1398, 52]}]], "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\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u2191s), MeasurableSet s_1 \u2192 NullMeasurableSet (Subtype.val '' s_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_biUnion_finset", "start": [1231, 1], "end": [1234, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "full_name": "List.map_get_sublist", "start": [204, 1], "end": [218, 55], "traced_tactics": [{"tactic": "suffices \u2200 n l', l' = l.drop n \u2192 (\u2200 i \u2208 is, n \u2264 i) \u2192 map (get l) is <+ l'\n  from this 0 l (by simp) (by simp)", "annotated_tactic": ["suffices \u2200 n l', l' = l.drop n \u2192 (\u2200 i \u2208 is, n \u2264 i) \u2192 <a>map</a> (<a>get</a> l) is <+ l'\n    from this 0 l (by simp) (by simp)", [{"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.get", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2261, 5], "def_end_pos": [2261, 13]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\n\u22a2 map (get l) is <+ l", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\n\u22a2 \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l'"}, {"tactic": "intro n l' hl' his", "annotated_tactic": ["intro n l' hl' his", []], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\n\u22a2 \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l'", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\nn : Nat\nl' : List \u03b1\nhl' : l' = drop n l\nhis : \u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i\n\u22a2 map (get l) is <+ l'"}, {"tactic": "induction is generalizing n l' with\n| nil => simp\n| cons hd tl IH =>\n  simp; cases hl'\n  have := IH h.of_cons (hd+1) _ rfl (pairwise_cons.mp h).1\n  specialize his hd (.head _)\n  have := get_cons_drop .. \u25b8 this.cons\u2082 (get l hd)\n  have := Sublist.append (nil_sublist (take hd l |>.drop n)) this\n  rwa [nil_append, \u2190 (drop_append_of_le_length ?_), take_append_drop] at this\n  simp [Nat.min_eq_left (Nat.le_of_lt hd.isLt), his]", "annotated_tactic": ["induction is generalizing n l' with\n  | <a>nil</a> => simp\n  | <a>cons</a> hd tl IH =>\n    simp; cases hl'\n    have := IH h.of_cons (hd+1) _ <a>rfl</a> (pairwise_cons.mp h).1\n    specialize his hd (.head _)\n    have := <a>get_cons_drop</a> .. \u25b8 this.cons\u2082 (<a>get</a> l hd)\n    have := <a>Sublist.append</a> (<a>nil_sublist</a> (<a>take</a> hd l |>.<a>drop</a> n)) this\n    rwa [<a>nil_append</a>, \u2190 (<a>drop_append_of_le_length</a> ?_), <a>take_append_drop</a>] at this\n    simp [<a>Nat.min_eq_left</a> (<a>Nat.le_of_lt</a> hd.isLt), his]", [{"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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "List.get_cons_drop", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [795, 9], "def_end_pos": [795, 22]}, {"full_name": "List.get", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2261, 5], "def_end_pos": [2261, 13]}, {"full_name": "List.Sublist.append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}, {"full_name": "List.nil_sublist", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [365, 17], "def_end_pos": [365, 28]}, {"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.nil_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [92, 17], "def_end_pos": [92, 27]}, {"full_name": "List.drop_append_of_le_length", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [828, 9], "def_end_pos": [828, 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]}, {"full_name": "Nat.min_eq_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [462, 19], "def_end_pos": [462, 30]}, {"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": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\nn : Nat\nl' : List \u03b1\nhl' : l' = drop n l\nhis : \u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i\n\u22a2 map (get l) is <+ l'", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\nthis : \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l'\n\u22a2 l = drop 0 l", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\nis : List (Fin (length l))\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is\nthis : \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l'\n\u22a2 \u2200 (i : Fin (length l)), i \u2208 is \u2192 0 \u2264 \u2191i", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\n\u03b1 : Type u_1\nl : List \u03b1\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) []\nn : Nat\nl' : List \u03b1\nhl' : l' = drop n l\nhis : \u2200 (i : Fin (length l)), i \u2208 [] \u2192 n \u2264 \u2191i\n\u22a2 map (get l) [] <+ l'", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nl' : List \u03b1\nhl' : l' = drop n l\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\n\u22a2 map (get l) (hd :: tl) <+ l'", "state_after": "case cons\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nl' : List \u03b1\nhl' : l' = drop n l\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\n\u22a2 get l hd :: map (get l) tl <+ l'"}, {"tactic": "cases hl'", "annotated_tactic": ["cases hl'", []], "state_before": "case cons\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nl' : List \u03b1\nhl' : l' = drop n l\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\n\u22a2 get l hd :: map (get l) tl <+ l'", "state_after": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\n\u22a2 get l hd :: map (get l) tl <+ drop n l"}, {"tactic": "have := IH h.of_cons (hd+1) _ rfl (pairwise_cons.mp h).1", "annotated_tactic": ["have := IH h.of_cons (hd+1) _ <a>rfl</a> (pairwise_cons.mp h).1", [{"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.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\n\u22a2 get l hd :: map (get l) tl <+ drop n l", "state_after": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\nthis : map (get l) tl <+ drop (\u2191hd + 1) l\n\u22a2 get l hd :: map (get l) tl <+ drop n l"}, {"tactic": "specialize his hd (.head _)", "annotated_tactic": ["specialize his hd (.head _)", []], "state_before": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nhis : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i\nthis : map (get l) tl <+ drop (\u2191hd + 1) l\n\u22a2 get l hd :: map (get l) tl <+ drop n l", "state_after": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\n\u22a2 get l hd :: map (get l) tl <+ drop n l"}, {"tactic": "have := get_cons_drop .. \u25b8 this.cons\u2082 (get l hd)", "annotated_tactic": ["have := <a>get_cons_drop</a> .. \u25b8 this.cons\u2082 (<a>get</a> l hd)", [{"full_name": "List.get_cons_drop", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [795, 9], "def_end_pos": [795, 22]}, {"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 cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\n\u22a2 get l hd :: map (get l) tl <+ drop n l", "state_after": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis\u271d : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\nthis : get l hd :: map (get l) tl <+ drop (\u2191hd) l\n\u22a2 get l hd :: map (get l) tl <+ drop n l"}, {"tactic": "have := Sublist.append (nil_sublist (take hd l |>.drop n)) this", "annotated_tactic": ["have := <a>Sublist.append</a> (<a>nil_sublist</a> (<a>take</a> hd l |>.<a>drop</a> n)) this", [{"full_name": "List.Sublist.append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}, {"full_name": "List.nil_sublist", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [365, 17], "def_end_pos": [365, 28]}, {"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": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis\u271d : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\nthis : get l hd :: map (get l) tl <+ drop (\u2191hd) l\n\u22a2 get l hd :: map (get l) tl <+ drop n l", "state_after": "case cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis\u271d\u00b9 : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\nthis\u271d : get l hd :: map (get l) tl <+ drop (\u2191hd) l\nthis : [] ++ get l hd :: map (get l) tl <+ drop n (take (\u2191hd) l) ++ drop (\u2191hd) l\n\u22a2 get l hd :: map (get l) tl <+ drop n l"}, {"tactic": "rwa [nil_append, \u2190 (drop_append_of_le_length ?_), take_append_drop] at this", "annotated_tactic": ["rwa [<a>nil_append</a>, \u2190 (<a>drop_append_of_le_length</a> ?_), <a>take_append_drop</a>] at this", [{"full_name": "List.nil_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [92, 17], "def_end_pos": [92, 27]}, {"full_name": "List.drop_append_of_le_length", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [828, 9], "def_end_pos": [828, 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 cons.refl\n\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis\u271d\u00b9 : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\nthis\u271d : get l hd :: map (get l) tl <+ drop (\u2191hd) l\nthis : [] ++ get l hd :: map (get l) tl <+ drop n (take (\u2191hd) l) ++ drop (\u2191hd) l\n\u22a2 get l hd :: map (get l) tl <+ drop n l", "state_after": "\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis\u271d\u00b9 : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\nthis\u271d : get l hd :: map (get l) tl <+ drop (\u2191hd) l\nthis : get l hd :: map (get l) tl <+ drop n (take (\u2191hd) l) ++ drop (\u2191hd) l\n\u22a2 n \u2264 length (take (\u2191hd) l)"}, {"tactic": "simp [Nat.min_eq_left (Nat.le_of_lt hd.isLt), his]", "annotated_tactic": ["simp [<a>Nat.min_eq_left</a> (<a>Nat.le_of_lt</a> hd.isLt), his]", [{"full_name": "Nat.min_eq_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [462, 19], "def_end_pos": [462, 30]}, {"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": "\u03b1 : Type u_1\nl : List \u03b1\nhd : Fin (length l)\ntl : List (Fin (length l))\nIH :\n  Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192\n    \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l'\nh : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl)\nn : Nat\nthis\u271d\u00b9 : map (get l) tl <+ drop (\u2191hd + 1) l\nhis : n \u2264 \u2191hd\nthis\u271d : get l hd :: map (get l) tl <+ drop (\u2191hd) l\nthis : get l hd :: map (get l) tl <+ drop n (take (\u2191hd) l) ++ drop (\u2191hd) l\n\u22a2 n \u2264 length (take (\u2191hd) l)", "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_aux\u2081", "start": [109, 1], "end": [135, 50], "traced_tactics": [{"tactic": "simp only [\u2190 set_integral_congr_set_ae (Box.coe_ae_eq_Icc _)]", "annotated_tactic": ["simp only [\u2190 <a>set_integral_congr_set_ae</a> (<a>Box.coe_ae_eq_Icc</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": "BoxIntegral.Box.coe_ae_eq_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [74, 9], "def_end_pos": [74, 22]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \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 \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n        \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)"}, {"tactic": "have A := (Hi.mono_set Box.coe_subset_Icc).hasBoxIntegral \u22a5 rfl", "annotated_tactic": ["have A := (Hi.mono_set <a>Box.coe_subset_Icc</a>).<a>hasBoxIntegral</a> \u22a5 <a>rfl</a>", [{"full_name": "BoxIntegral.Box.coe_subset_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 23]}, {"full_name": "MeasureTheory.IntegrableOn.hasBoxIntegral", "def_path": "Mathlib/Analysis/BoxIntegral/Integrability.lean", "def_pos": [198, 9], "def_end_pos": [198, 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": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)"}, {"tactic": "have B :=\n  hasIntegral_GP_divergence_of_forall_hasDerivWithinAt I f f' (s \u2229 Box.Icc I)\n    (hs.mono (inter_subset_left _ _)) (fun x hx => Hc _ hx.2) fun x hx =>\n    Hd _ \u27e8hx.1, fun h => hx.2 \u27e8h, hx.1\u27e9\u27e9", "annotated_tactic": ["have B :=\n    <a>hasIntegral_GP_divergence_of_forall_hasDerivWithinAt</a> I f f' (s \u2229 <a>Box.Icc</a> I)\n      (hs.mono (<a>inter_subset_left</a> _ _)) (fun x hx => Hc _ hx.2) fun x hx =>\n      Hd _ \u27e8hx.1, fun h => hx.2 \u27e8h, hx.1\u27e9\u27e9", [{"full_name": "BoxIntegral.hasIntegral_GP_divergence_of_forall_hasDerivWithinAt", "def_path": "Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean", "def_pos": [270, 9], "def_end_pos": [270, 61]}, {"full_name": "BoxIntegral.Box.Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [199, 15], "def_end_pos": [199, 18]}, {"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\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)"}, {"tactic": "rw [continuousOn_pi] at Hc", "annotated_tactic": ["rw [<a>continuousOn_pi</a>] at Hc", [{"full_name": "continuousOn_pi", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [618, 9], "def_end_pos": [618, 24]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)"}, {"tactic": "refine' (A.unique B).trans (sum_congr rfl fun i _ => _)", "annotated_tactic": ["refine' (A.unique B).<a>trans</a> (<a>sum_congr</a> <a>rfl</a> fun i _ => _)", [{"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": "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": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n    \u2211 x : Fin (n + 1),\n      ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n        \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)", "state_after": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n        BoxAdditiveMap.volume -\n      BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n        BoxAdditiveMap.volume =\n    (\u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n      \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i"}, {"tactic": "refine' congr_arg\u2082 Sub.sub _ _", "annotated_tactic": ["refine' <a>congr_arg\u2082</a> <a>Sub.sub</a> _ _", [{"full_name": "congr_arg\u2082", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [601, 7], "def_end_pos": [601, 17]}, {"full_name": "Sub.sub", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1270, 3], "def_end_pos": [1270, 6]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n        BoxAdditiveMap.volume -\n      BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n        BoxAdditiveMap.volume =\n    (\u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n      \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i", "state_after": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i\n\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i"}, {"tactic": "have := Box.continuousOn_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))", "annotated_tactic": ["have := <a>Box.continuousOn_face_Icc</a> (Hc i) (<a>Set.right_mem_Icc</a>.2 (I.lower_le_upper i))", [{"full_name": "BoxIntegral.Box.continuousOn_face_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 30]}, {"full_name": "Set.right_mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [214, 9], "def_end_pos": [214, 22]}]], "state_before": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i", "state_after": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i))\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i"}, {"tactic": "have := (this.integrableOn_compact (\u03bc := volume) (Box.isCompact_Icc _)).mono_set\n  Box.coe_subset_Icc", "annotated_tactic": ["have := (this.integrableOn_compact (\u03bc := <a>volume</a>) (<a>Box.isCompact_Icc</a> _)).<a>mono_set</a>\n      <a>Box.coe_subset_Icc</a>", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "BoxIntegral.Box.isCompact_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [216, 19], "def_end_pos": [216, 32]}, {"full_name": "MeasureTheory.IntegrableOn.mono_set", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}, {"full_name": "BoxIntegral.Box.coe_subset_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 23]}]], "state_before": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i))\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i", "state_after": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i))\nthis : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) \u2191(Box.face I i)\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i"}, {"tactic": "exact (this.hasBoxIntegral \u22a5 rfl).integral_eq", "annotated_tactic": ["exact (this.hasBoxIntegral \u22a5 <a>rfl</a>).<a>integral_eq</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": "BoxIntegral.HasIntegral.integral_eq", "def_path": "Mathlib/Analysis/BoxIntegral/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 32]}]], "state_before": "case refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i))\nthis : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) \u2191(Box.face I i)\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i", "state_after": "no goals"}, {"tactic": "have := Box.continuousOn_face_Icc (Hc i) (Set.left_mem_Icc.2 (I.lower_le_upper i))", "annotated_tactic": ["have := <a>Box.continuousOn_face_Icc</a> (Hc i) (<a>Set.left_mem_Icc</a>.2 (I.lower_le_upper i))", [{"full_name": "BoxIntegral.Box.continuousOn_face_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 30]}, {"full_name": "Set.left_mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [191, 9], "def_end_pos": [191, 21]}]], "state_before": "case refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i", "state_after": "case refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i))\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i"}, {"tactic": "have := (this.integrableOn_compact (\u03bc := volume) (Box.isCompact_Icc _)).mono_set\n  Box.coe_subset_Icc", "annotated_tactic": ["have := (this.integrableOn_compact (\u03bc := <a>volume</a>) (<a>Box.isCompact_Icc</a> _)).<a>mono_set</a>\n      <a>Box.coe_subset_Icc</a>", [{"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "BoxIntegral.Box.isCompact_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [216, 19], "def_end_pos": [216, 32]}, {"full_name": "MeasureTheory.IntegrableOn.mono_set", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [127, 9], "def_end_pos": [127, 30]}, {"full_name": "BoxIntegral.Box.coe_subset_Icc", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 23]}]], "state_before": "case refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i))\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i", "state_after": "case refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i))\nthis : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) \u2191(Box.face I i)\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i"}, {"tactic": "exact (this.hasBoxIntegral \u22a5 rfl).integral_eq", "annotated_tactic": ["exact (this.hasBoxIntegral \u22a5 <a>rfl</a>).<a>integral_eq</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": "BoxIntegral.HasIntegral.integral_eq", "def_path": "Mathlib/Analysis/BoxIntegral/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 32]}]], "state_before": "case refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n  HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n    (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n  HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n    (\u2211 i : Fin (n + 1),\n      (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n          BoxAdditiveMap.volume -\n        BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n          BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i))\nthis : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) \u2191(Box.face I i)\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n      BoxAdditiveMap.volume =\n    \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) 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_smul", "start": [387, 1], "end": [390, 75], "traced_tactics": [{"tactic": "simp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, ENNReal.smul_iSup]", "annotated_tactic": ["simp only [<a>mkMetric</a>, <a>mkMetric'</a>, <a>mkMetric'.pre</a>, <a>inducedOuterMeasure</a>, <a>ENNReal.smul_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": "ENNReal.smul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [656, 9], "def_end_pos": [656, 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\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nhc' : c \u2260 0\n\u22a2 mkMetric (c \u2022 m) = c \u2022 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\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nhc' : c \u2260 0\n\u22a2 \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => (c \u2022 m) (diam s)) =\n    c \u2022 \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s))"}, {"tactic": "simp_rw [smul_iSup, smul_boundedBy hc, smul_extend _ hc', Pi.smul_apply]", "annotated_tactic": ["simp_rw [<a>smul_iSup</a>, <a>smul_boundedBy</a> hc, <a>smul_extend</a> _ hc', <a>Pi.smul_apply</a>]", [{"full_name": "MeasureTheory.OuterMeasure.smul_iSup", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [426, 9], "def_end_pos": [426, 18]}, {"full_name": "MeasureTheory.OuterMeasure.smul_boundedBy", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [892, 9], "def_end_pos": [892, 23]}, {"full_name": "MeasureTheory.smul_extend", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1322, 9], "def_end_pos": [1322, 20]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "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\u221e\nhc : c \u2260 \u22a4\nhc' : c \u2260 0\n\u22a2 \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => (c \u2022 m) (diam s)) =\n    c \u2022 \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s))", "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?_eq_some", "start": [600, 1], "end": [604, 35], "traced_tactics": [{"tactic": "cases get?_len_le hn \u25b8 e", "annotated_tactic": ["cases <a>get?_len_le</a> hn \u25b8 e", [{"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]}]], "state_before": "\u03b1\u271d : Type u_1\na : \u03b1\u271d\nl : List \u03b1\u271d\nn : Nat\ne : get? l n = some a\nhn : length l \u2264 n\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rwa [get?_eq_get this, Option.some.injEq] at e", "annotated_tactic": ["rwa [<a>get?_eq_get</a> this, Option.some.injEq] at e", [{"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\u271d : Type u_1\na : \u03b1\u271d\nl : List \u03b1\u271d\nn : Nat\ne : get? l n = some a\nthis : n < length l\n\u22a2 get l { val := n, isLt := this } = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.aemeasurable_limRatio", "start": [420, 1], "end": [422, 54], "traced_tactics": [{"tactic": "apply ENNReal.aemeasurable_of_exist_almost_disjoint_supersets _ _ fun p q hpq => ?_", "annotated_tactic": ["apply <a>ENNReal.aemeasurable_of_exist_almost_disjoint_supersets</a> _ _ fun p q hpq => ?_", [{"full_name": "ENNReal.aemeasurable_of_exist_almost_disjoint_supersets", "def_path": "Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean", "def_pos": [113, 9], "def_end_pos": [113, 64]}]], "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\nh\u03c1 : \u03c1 \u226a \u03bc\n\u22a2 AEMeasurable (limRatio v \u03c1)", "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\nh\u03c1 : \u03c1 \u226a \u03bc\np q : \u211d\u22650\nhpq : p < q\n\u22a2 \u2203 u v_1,\n    MeasurableSet u \u2227\n      MeasurableSet v_1 \u2227 {x | limRatio v \u03c1 x < \u2191p} \u2286 u \u2227 {x | \u2191q < limRatio v \u03c1 x} \u2286 v_1 \u2227 \u2191\u2191\u03bc (u \u2229 v_1) = 0"}, {"tactic": "exact v.exists_measurable_supersets_limRatio h\u03c1 hpq", "annotated_tactic": ["exact v.exists_measurable_supersets_limRatio h\u03c1 hpq", []], "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\nh\u03c1 : \u03c1 \u226a \u03bc\np q : \u211d\u22650\nhpq : p < q\n\u22a2 \u2203 u v_1,\n    MeasurableSet u \u2227\n      MeasurableSet v_1 \u2227 {x | limRatio v \u03c1 x < \u2191p} \u2286 u \u2227 {x | \u2191q < limRatio v \u03c1 x} \u2286 v_1 \u2227 \u2191\u2191\u03bc (u \u2229 v_1) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.diag_singleton", "start": [415, 1], "end": [416, 92], "traced_tactics": [{"tactic": "rw [\u2190 product_sdiff_offDiag, offDiag_singleton, sdiff_empty, singleton_product_singleton]", "annotated_tactic": ["rw [\u2190 <a>product_sdiff_offDiag</a>, <a>offDiag_singleton</a>, <a>sdiff_empty</a>, <a>singleton_product_singleton</a>]", [{"full_name": "Finset.product_sdiff_offDiag", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [381, 9], "def_end_pos": [381, 30]}, {"full_name": "Finset.offDiag_singleton", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [412, 9], "def_end_pos": [412, 26]}, {"full_name": "Finset.sdiff_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2140, 9], "def_end_pos": [2140, 20]}, {"full_name": "Finset.singleton_product_singleton", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [232, 9], "def_end_pos": [232, 36]}]], "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\na : \u03b1\n\u22a2 diag {a} = {(a, a)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nonempty_of_ssubset'", "start": [493, 1], "end": [494, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.piecewise_insert", "start": [2532, 1], "end": [2537, 7], "traced_tactics": [{"tactic": "classical simp only [\u2190 piecewise_coe, coe_insert, \u2190 Set.piecewise_insert]", "annotated_tactic": ["classical simp only [\u2190 <a>piecewise_coe</a>, <a>coe_insert</a>, \u2190 <a>Set.piecewise_insert</a>]", [{"full_name": "Finset.piecewise_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2506, 9], "def_end_pos": [2506, 22]}, {"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Set.piecewise_insert", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 25]}]], "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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 piecewise (insert j s) f g = update (piecewise s f g) j (f j)", "state_after": "\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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 Set.piecewise (\u2191(insert j s)) f g = Set.piecewise (insert j \u2191s) f g"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 Set.piecewise (\u2191(insert j s)) f g = Set.piecewise (insert j \u2191s) f g", "state_after": "case h\n\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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nx\u271d : \u03b1\n\u22a2 Set.piecewise (\u2191(insert j s)) f g x\u271d = Set.piecewise (insert j \u2191s) f g x\u271d"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h\n\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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nx\u271d : \u03b1\n\u22a2 Set.piecewise (\u2191(insert j s)) f g x\u271d = Set.piecewise (insert j \u2191s) f g x\u271d", "state_after": "case h.e_s\n\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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nx\u271d : \u03b1\n\u22a2 \u2191(insert j s) = insert j \u2191s"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e_s\n\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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nx\u271d : \u03b1\n\u22a2 \u2191(insert j s) = insert j \u2191s", "state_after": "no goals"}, {"tactic": "simp only [\u2190 piecewise_coe, coe_insert, \u2190 Set.piecewise_insert]", "annotated_tactic": ["simp only [\u2190 <a>piecewise_coe</a>, <a>coe_insert</a>, \u2190 <a>Set.piecewise_insert</a>]", [{"full_name": "Finset.piecewise_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2506, 9], "def_end_pos": [2506, 22]}, {"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Set.piecewise_insert", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 25]}]], "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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 piecewise (insert j s) f g = update (piecewise s f g) j (f j)", "state_after": "\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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 Set.piecewise (\u2191(insert j s)) f g = Set.piecewise (insert j \u2191s) f g"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable", "start": [75, 1], "end": [107, 23], "traced_tactics": [{"tactic": "rcases ht with \u27e8d, d_count, hd\u27e9", "annotated_tactic": ["rcases ht with \u27e8d, d_count, hd\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nht : IsSeparable t\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "haveI : Encodable d := d_count.toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> d := d_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\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have : \u2200 x : d, \u2203 g : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016g\u2016 \u2264 1 \u2227 g x = \u2016(x : E)\u2016 :=\n  fun x => exists_dual_vector'' \ud835\udd5c (x : E)", "annotated_tactic": ["have : \u2200 x : d, \u2203 g : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016g\u2016 \u2264 1 \u2227 g x = \u2016(x : E)\u2016 :=\n    fun x => <a>exists_dual_vector''</a> \ud835\udd5c (x : E)", [{"full_name": "exists_dual_vector''", "def_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "def_pos": [152, 9], "def_end_pos": [152, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\nthis : \u2200 (x : \u2191d), \u2203 g, \u2016g\u2016 \u2264 1 \u2227 \u2191g \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "choose s hs using this", "annotated_tactic": ["choose s hs using this", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\nthis : \u2200 (x : \u2191d), \u2203 g, \u2016g\u2016 \u2264 1 \u2227 \u2191g \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have A : \u2200 a : E, a \u2208 t \u2192 (\u2200 x, \u27eaa, s x\u27eb = (0 : \ud835\udd5c)) \u2192 a = 0 := by\n  intro a hat ha\n  contrapose! ha\n  have a_pos : 0 < \u2016a\u2016 := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]\n  have a_mem : a \u2208 closure d := hd hat\n  obtain \u27e8x, hx\u27e9 : \u2203 x : d, dist a x < \u2016a\u2016 / 2 := by\n    rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n    exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9\n  use x\n  have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n    have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n    have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx\n    linarith\n  intro h\n  apply lt_irrefl \u2016s x x\u2016\n  calc\n    \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n    _ \u2264 1 * \u2016(x : E) - a\u2016 := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n    _ < \u2016a\u2016 / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx\n    _ < \u2016(x : E)\u2016 := I\n    _ = \u2016s x x\u2016 := by rw [(hs x).2, IsROrC.norm_coe_norm]", "annotated_tactic": ["have A : \u2200 a : E, a \u2208 t \u2192 (\u2200 x, \u27eaa, s x\u27eb = (0 : \ud835\udd5c)) \u2192 a = 0 := by\n    intro a hat ha\n    contrapose! ha\n    have a_pos : 0 < \u2016a\u2016 := by simp only [ha, <a>norm_pos_iff</a>, <a>Ne.def</a>, <a>not_false_iff</a>]\n    have a_mem : a \u2208 <a>closure</a> d := hd hat\n    obtain \u27e8x, hx\u27e9 : \u2203 x : d, <a>dist</a> a x < \u2016a\u2016 / 2 := by\n      rcases <a>Metric.mem_closure_iff</a>.1 a_mem (\u2016a\u2016 / 2) (<a>half_pos</a> a_pos) with \u27e8x, h'x, hx\u27e9\n      exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9\n    use x\n    have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n      have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := <a>norm_le_insert'</a> _ _\n      have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [<a>dist_eq_norm</a>] at hx\n      linarith\n    intro h\n    apply <a>lt_irrefl</a> \u2016s x x\u2016\n    calc\n      \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, <a>sub_zero</a>, <a>ContinuousLinearMap.map_sub</a>]\n      _ \u2264 1 * \u2016(x : E) - a\u2016 := (<a>ContinuousLinearMap.le_of_op_norm_le</a> _ (hs x).1 _)\n      _ < \u2016a\u2016 / 2 := by rw [<a>one_mul</a>]; rwa [<a>dist_eq_norm'</a>] at hx\n      _ < \u2016(x : E)\u2016 := I\n      _ = \u2016s x x\u2016 := by rw [(hs x).2, <a>IsROrC.norm_coe_norm</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]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closure_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 24]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "norm_le_insert'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [605, 7], "def_end_pos": [605, 22]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "ContinuousLinearMap.map_sub", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1376, 19], "def_end_pos": [1376, 26]}, {"full_name": "ContinuousLinearMap.le_of_op_norm_le", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [215, 9], "def_end_pos": [215, 25]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "dist_eq_norm'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [386, 7], "def_end_pos": [386, 20]}, {"full_name": "IsROrC.norm_coe_norm", "def_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "def_pos": [36, 9], "def_end_pos": [36, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have hfs : \u2200 y : d, \u2200\u1d50 x \u2202\u03bc, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := fun y => hf (s y)", "annotated_tactic": ["have hfs : \u2200 y : d, \u2200\u1d50 x \u2202\u03bc, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := fun y => hf (s y)", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have hf' : \u2200\u1d50 x \u2202\u03bc, \u2200 y : d, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := by rwa [ae_all_iff]", "annotated_tactic": ["have hf' : \u2200\u1d50 x \u2202\u03bc, \u2200 y : d, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := by rwa [<a>ae_all_iff</a>]", [{"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\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "filter_upwards [hf', h't] with x hx h'x", "annotated_tactic": ["filter_upwards [hf', h't] with x hx h'x", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\nx : \u03b1\nhx : \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\nh'x : f x \u2208 t\n\u22a2 f x = OfNat.ofNat 0 x"}, {"tactic": "exact A (f x) h'x hx", "annotated_tactic": ["exact A (f x) h'x hx", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\nx : \u03b1\nhx : \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\nh'x : f x \u2208 t\n\u22a2 f x = OfNat.ofNat 0 x", "state_after": "no goals"}, {"tactic": "intro a hat ha", "annotated_tactic": ["intro a hat ha", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : \u2200 (x : \u2191d), \u2191(s x) a = 0\n\u22a2 a = 0"}, {"tactic": "contrapose! ha", "annotated_tactic": ["contrapose! ha", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : \u2200 (x : \u2191d), \u2191(s x) a = 0\n\u22a2 a = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0"}, {"tactic": "have a_pos : 0 < \u2016a\u2016 := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]", "annotated_tactic": ["have a_pos : 0 < \u2016a\u2016 := by simp only [ha, <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\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0"}, {"tactic": "have a_mem : a \u2208 closure d := hd hat", "annotated_tactic": ["have a_mem : a \u2208 <a>closure</a> d := hd hat", [{"full_name": "closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0"}, {"tactic": "obtain \u27e8x, hx\u27e9 : \u2203 x : d, dist a x < \u2016a\u2016 / 2 := by\n  rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n  exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9", "annotated_tactic": ["obtain \u27e8x, hx\u27e9 : \u2203 x : d, <a>dist</a> a x < \u2016a\u2016 / 2 := by\n      rcases <a>Metric.mem_closure_iff</a>.1 a_mem (\u2016a\u2016 / 2) (<a>half_pos</a> a_pos) with \u27e8x, h'x, hx\u27e9\n      exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.mem_closure_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 24]}, {"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\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0"}, {"tactic": "use x", "annotated_tactic": ["use x", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2191(s x) a \u2260 0"}, {"tactic": "have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n  have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n  have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx\n  linarith", "annotated_tactic": ["have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n      have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := <a>norm_le_insert'</a> _ _\n      have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [<a>dist_eq_norm</a>] at hx\n      linarith", [{"full_name": "norm_le_insert'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [605, 7], "def_end_pos": [605, 22]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2191(s x) a \u2260 0", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\n\u22a2 \u2191(s x) a \u2260 0"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\n\u22a2 \u2191(s x) a \u2260 0", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 False"}, {"tactic": "apply lt_irrefl \u2016s x x\u2016", "annotated_tactic": ["apply <a>lt_irrefl</a> \u2016s x x\u2016", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 False", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191(s x) \u2191x\u2016 < \u2016\u2191(s x) \u2191x\u2016"}, {"tactic": "calc\n  \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n  _ \u2264 1 * \u2016(x : E) - a\u2016 := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n  _ < \u2016a\u2016 / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx\n  _ < \u2016(x : E)\u2016 := I\n  _ = \u2016s x x\u2016 := by rw [(hs x).2, IsROrC.norm_coe_norm]", "annotated_tactic": ["calc\n      \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, <a>sub_zero</a>, <a>ContinuousLinearMap.map_sub</a>]\n      _ \u2264 1 * \u2016(x : E) - a\u2016 := (<a>ContinuousLinearMap.le_of_op_norm_le</a> _ (hs x).1 _)\n      _ < \u2016a\u2016 / 2 := by rw [<a>one_mul</a>]; rwa [<a>dist_eq_norm'</a>] at hx\n      _ < \u2016(x : E)\u2016 := I\n      _ = \u2016s x x\u2016 := by rw [(hs x).2, <a>IsROrC.norm_coe_norm</a>]", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "ContinuousLinearMap.map_sub", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1376, 19], "def_end_pos": [1376, 26]}, {"full_name": "ContinuousLinearMap.le_of_op_norm_le", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [215, 9], "def_end_pos": [215, 25]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "dist_eq_norm'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [386, 7], "def_end_pos": [386, 20]}, {"full_name": "IsROrC.norm_coe_norm", "def_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "def_pos": [36, 9], "def_end_pos": [36, 29]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191(s x) \u2191x\u2016 < \u2016\u2191(s x) \u2191x\u2016", "state_after": "no goals"}, {"tactic": "simp only [ha, norm_pos_iff, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [ha, <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\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\n\u22a2 0 < \u2016a\u2016", "state_after": "no goals"}, {"tactic": "rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9", "annotated_tactic": ["rcases <a>Metric.mem_closure_iff</a>.1 a_mem (\u2016a\u2016 / 2) (<a>half_pos</a> a_pos) with \u27e8x, h'x, hx\u27e9", [{"full_name": "Metric.mem_closure_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1925, 9], "def_end_pos": [1925, 24]}, {"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\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\n\u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2", "state_after": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : E\nh'x : x \u2208 d\nhx : dist a x < \u2016a\u2016 / 2\n\u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2"}, {"tactic": "exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9", "annotated_tactic": ["exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : E\nh'x : x \u2208 d\nhx : dist a x < \u2016a\u2016 / 2\n\u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2", "state_after": "no goals"}, {"tactic": "have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _", "annotated_tactic": ["have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := <a>norm_le_insert'</a> _ _", [{"full_name": "norm_le_insert'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [605, 7], "def_end_pos": [605, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016"}, {"tactic": "have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx", "annotated_tactic": ["have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [<a>dist_eq_norm</a>] at hx", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d\u00b9 : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis\u271d : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\nthis : \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d\u00b9 : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis\u271d : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\nthis : \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016", "state_after": "no goals"}, {"tactic": "rwa [dist_eq_norm] at hx", "annotated_tactic": ["rwa [<a>dist_eq_norm</a>] at hx", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\n\u22a2 \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2", "state_after": "no goals"}, {"tactic": "simp only [h, sub_zero, ContinuousLinearMap.map_sub]", "annotated_tactic": ["simp only [h, <a>sub_zero</a>, <a>ContinuousLinearMap.map_sub</a>]", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}, {"full_name": "ContinuousLinearMap.map_sub", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1376, 19], "def_end_pos": [1376, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191(s x) \u2191x\u2016 = \u2016\u2191(s x) (\u2191x - a)\u2016", "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": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 1 * \u2016\u2191x - a\u2016 < \u2016a\u2016 / 2", "state_after": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191x - a\u2016 < \u2016a\u2016 / 2"}, {"tactic": "rwa [dist_eq_norm'] at hx", "annotated_tactic": ["rwa [<a>dist_eq_norm'</a>] at hx", [{"full_name": "dist_eq_norm'", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [386, 7], "def_end_pos": [386, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : 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<a>IsROrC.norm_coe_norm</a>]", [{"full_name": "IsROrC.norm_coe_norm", "def_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "def_pos": [36, 9], "def_end_pos": [36, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191x\u2016 = \u2016\u2191(s x) \u2191x\u2016", "state_after": "no goals"}, {"tactic": "rwa [ae_all_iff]", "annotated_tactic": ["rwa [<a>ae_all_iff</a>]", [{"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\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), 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"state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\nthis : Nonempty \u2191I\n\u22a2 \u2191(restrict s) (\u2a05 i, m \u2191i) = \u2a05 i, \u2191(restrict s) (m \u2191i)"}, {"tactic": "exact restrict_iInf _ _", "annotated_tactic": ["exact <a>restrict_iInf</a> _ _", [{"full_name": "MeasureTheory.OuterMeasure.restrict_iInf", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\nthis : Nonempty \u2191I\n\u22a2 \u2191(restrict s) (\u2a05 i, m \u2191i) = \u2a05 i, \u2191(restrict s) (m \u2191i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.inr_mem_disjSum", "start": [78, 1], "end": [79, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.volume_subtype_coe_eq_zero_of_volume_eq_zero", "start": [1472, 1], "end": [1474, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_map", "start": [318, 1], "end": [321, 50], "traced_tactics": [{"tactic": "simp [noncommProd, Multiset.noncommProd_map]", "annotated_tactic": ["simp [<a>noncommProd</a>, <a>Multiset.noncommProd_map</a>]", [{"full_name": "Finset.noncommProd", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [242, 5], "def_end_pos": [242, 16]}, {"full_name": "Multiset.noncommProd_map", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [188, 9], "def_end_pos": [188, 24]}]], "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 : MonoidHomClass F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ng : F\n\u22a2 \u2191g (noncommProd s f comm) =\n    noncommProd s (fun i => \u2191g (f i))\n      (_ : \u2200 (x : \u03b1), x \u2208 \u2191s \u2192 \u2200 (y : \u03b1), y \u2208 \u2191s \u2192 x \u2260 y \u2192 Commute (\u2191g (f x)) (\u2191g (f y)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_real", "start": [1059, 1], "end": [1062, 65], "traced_tactics": [{"tactic": "rw [\u2190 (volume_preserving_funUnique Unit \u211d).map_eq,\n  \u2190 (hausdorffMeasure_measurePreserving_funUnique Unit \u211d 1).map_eq,\n  \u2190 hausdorffMeasure_pi_real, Fintype.card_unit, Nat.cast_one]", "annotated_tactic": ["rw [\u2190 (<a>volume_preserving_funUnique</a> <a>Unit</a> \u211d).<a>map_eq</a>,\n    \u2190 (<a>hausdorffMeasure_measurePreserving_funUnique</a> <a>Unit</a> \u211d 1).<a>map_eq</a>,\n    \u2190 <a>hausdorffMeasure_pi_real</a>, <a>Fintype.card_unit</a>, <a>Nat.cast_one</a>]", [{"full_name": "MeasureTheory.volume_preserving_funUnique", "def_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "def_pos": [825, 9], "def_end_pos": [825, 36]}, {"full_name": "Unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [129, 8], "def_end_pos": [129, 12]}, {"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_funUnique", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [1044, 9], "def_end_pos": [1044, 53]}, {"full_name": "Unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [129, 8], "def_end_pos": [129, 12]}, {"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_unit", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [369, 9], "def_end_pos": [369, 26]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 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[1] = volume", "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_right", "start": [2575, 1], "end": [2577, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_neg_one", "start": [512, 1], "end": [515, 68], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "R : Type u_1\ninst\u271d : Ring R\nn : \u2115\n\u22a2 \u2191(-1) = \u2191n - 1", "state_after": "case zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191(-1) = \u2191Nat.zero - 1\n\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\n\u22a2 \u2191(-1) = \u2191(Nat.succ n) - 1"}, {"tactic": "dsimp [ZMod, ZMod.cast]", "annotated_tactic": ["dsimp [<a>ZMod</a>, <a>ZMod.cast</a>]", [{"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "ZMod.cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [152, 12], "def_end_pos": [152, 16]}]], "state_before": "case zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191(-1) = \u2191Nat.zero - 1", "state_after": "case zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191(-1) = \u21910 - 1"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191(-1) = \u21910 - 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 nat_cast_val, val_neg_one, Nat.cast_succ, add_sub_cancel]", "annotated_tactic": ["rw [\u2190 <a>nat_cast_val</a>, <a>val_neg_one</a>, <a>Nat.cast_succ</a>, <a>add_sub_cancel</a>]", [{"full_name": "ZMod.nat_cast_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [260, 9], "def_end_pos": [260, 21]}, {"full_name": "ZMod.val_neg_one", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 20]}, {"full_name": "Nat.cast_succ", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}, {"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}]], "state_before": "case succ\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\n\u22a2 \u2191(-1) = \u2191(Nat.succ n) - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondCdf.lean", "full_name": "ProbabilityTheory.inf_gt_condCdfRat", "start": [668, 1], "end": [702, 45], "traced_tactics": [{"tactic": "by_cases ha : a \u2208 condCdfSet \u03c1", "annotated_tactic": ["by_cases ha : a \u2208 <a>condCdfSet</a> \u03c1", [{"full_name": "ProbabilityTheory.condCdfSet", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [541, 5], "def_end_pos": [541, 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\nt : \u211a\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t", "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t"}, {"tactic": "simp_rw [condCdfRat_of_mem \u03c1 a ha]", "annotated_tactic": ["simp_rw [<a>condCdfRat_of_mem</a> \u03c1 a ha]", [{"full_name": "ProbabilityTheory.condCdfRat_of_mem", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [578, 9], "def_end_pos": [578, 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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t", "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 r, ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)"}, {"tactic": "have ha' := hasCondCdf_of_mem_condCdfSet ha", "annotated_tactic": ["have ha' := <a>hasCondCdf_of_mem_condCdfSet</a> ha", [{"full_name": "ProbabilityTheory.hasCondCdf_of_mem_condCdfSet", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [549, 9], "def_end_pos": [549, 37]}]], "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 r, ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)", "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2a05 r, ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)"}, {"tactic": "rw [\u2190 ENNReal.toReal_iInf]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.toReal_iInf</a>]", [{"full_name": "ENNReal.toReal_iInf", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2439, 9], "def_end_pos": [2439, 20]}]], "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2a05 r, ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)", "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 ENNReal.toReal (\u2a05 r, preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)\n\ncase 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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u2191(Ioi t)), preCdf \u03c1 (\u2191i) a \u2260 \u22a4"}, {"tactic": "suffices \u2a05 i : \u21a5(Ioi t), preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a by rw [this]", "annotated_tactic": ["suffices \u2a05 i : \u21a5(<a>Ioi</a> t), <a>preCdf</a> \u03c1 (\u2191i) a = <a>preCdf</a> \u03c1 t a by rw [this]", [{"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": "ProbabilityTheory.preCdf", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [294, 19], "def_end_pos": [294, 25]}]], "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 ENNReal.toReal (\u2a05 r, preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)", "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2a05 i, preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a"}, {"tactic": "rw [\u2190 ha'.iInf_rat_gt_eq]", "annotated_tactic": ["rw [\u2190 ha'.iInf_rat_gt_eq]", []], "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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2a05 i, preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a", "state_after": "no goals"}, {"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\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\nthis : \u2a05 i, preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a\n\u22a2 ENNReal.toReal (\u2a05 r, preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)", "state_after": "no goals"}, {"tactic": "exact fun r => ((ha'.le_one r).trans_lt ENNReal.one_lt_top).ne", "annotated_tactic": ["exact fun r => ((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 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\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u2191(Ioi t)), preCdf \u03c1 (\u2191i) a \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp_rw [condCdfRat_of_not_mem \u03c1 a ha]", "annotated_tactic": ["simp_rw [<a>condCdfRat_of_not_mem</a> \u03c1 a ha]", [{"full_name": "ProbabilityTheory.condCdfRat_of_not_mem", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [574, 9], "def_end_pos": [574, 30]}]], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t", "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 1"}, {"tactic": "have h_bdd : BddBelow (range fun r : \u21a5(Ioi t) => ite ((r : \u211a) < 0) (0 : \u211d) 1) := by\n  refine' \u27e80, fun x hx => _\u27e9\n  obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx\n  dsimp only\n  split_ifs\n  exacts [le_rfl, zero_le_one]", "annotated_tactic": ["have h_bdd : <a>BddBelow</a> (<a>range</a> fun r : \u21a5(<a>Ioi</a> t) => <a>ite</a> ((r : \u211a) < 0) (0 : \u211d) 1) := by\n      refine' \u27e80, fun x hx => _\u27e9\n      obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx\n      dsimp only\n      split_ifs\n      exacts [<a>le_rfl</a>, <a>zero_le_one</a>]", [{"full_name": "BddBelow", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 1"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = 0\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = 1"}, {"tactic": "refine' \u27e80, fun x hx => _\u27e9", "annotated_tactic": ["refine' \u27e80, fun x hx => _\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 BddBelow (range fun r => if \u2191r < 0 then 0 else 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nx : \u211d\nhx : x \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 x"}, {"tactic": "obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx", "annotated_tactic": ["obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx", []], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nx : \u211d\nhx : x \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 x", "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y", "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 if \u2191y < 0 then 0 else 1"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 if \u2191y < 0 then 0 else 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\nh\u271d : \u2191y < 0\n\u22a2 0 \u2264 0\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\nh\u271d : \u00ac\u2191y < 0\n\u22a2 0 \u2264 1"}, {"tactic": "exacts [le_rfl, zero_le_one]", "annotated_tactic": ["exacts [<a>le_rfl</a>, <a>zero_le_one</a>]", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\nh\u271d : \u2191y < 0\n\u22a2 0 \u2264 0\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\nh\u271d : \u00ac\u2191y < 0\n\u22a2 0 \u2264 1", "state_after": "no goals"}, {"tactic": "refine' le_antisymm _ (le_ciInf fun x => _)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>le_ciInf</a> fun x => _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "le_ciInf", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [827, 9], "def_end_pos": [827, 17]}]], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = 0", "state_after": "case pos.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) \u2264 0\n\ncase pos.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\n\u22a2 0 \u2264 if \u2191x < 0 then 0 else 1"}, {"tactic": "refine' (ciInf_le h_bdd \u27e8q, htq\u27e9).trans _", "annotated_tactic": ["refine' (<a>ciInf_le</a> h_bdd \u27e8q, htq\u27e9).<a>trans</a> _", [{"full_name": "ciInf_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [832, 9], "def_end_pos": [832, 17]}, {"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'_1.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) \u2264 0", "state_after": "case pos.refine'_1.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 (if \u2191{ val := q, property := htq } < 0 then 0 else 1) \u2264 0"}, {"tactic": "rw [if_pos]", "annotated_tactic": ["rw [<a>if_pos</a>]", [{"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.refine'_1.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 (if \u2191{ val := q, property := htq } < 0 then 0 else 1) \u2264 0", "state_after": "case pos.refine'_1.intro.intro.hc\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 \u2191{ val := q, property := htq } < 0"}, {"tactic": "rwa [Subtype.coe_mk]", "annotated_tactic": ["rwa [<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": "case pos.refine'_1.intro.intro.hc\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 \u2191{ val := q, property := htq } < 0", "state_after": "no goals"}, {"tactic": "refine' \u27e8t / 2, _, _\u27e9", "annotated_tactic": ["refine' \u27e8t / 2, _, _\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 \u2203 q, t < q \u2227 q < 0", "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 t < t / 2\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 t / 2 < 0"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 t < t / 2", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 t / 2 < 0", "state_after": "no goals"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case pos.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\n\u22a2 0 \u2264 if \u2191x < 0 then 0 else 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\nh\u271d : \u2191x < 0\n\u22a2 0 \u2264 0\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\nh\u271d : \u00ac\u2191x < 0\n\u22a2 0 \u2264 1"}, {"tactic": "exacts [le_rfl, zero_le_one]", "annotated_tactic": ["exacts [<a>le_rfl</a>, <a>zero_le_one</a>]", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\nh\u271d : \u2191x < 0\n\u22a2 0 \u2264 0\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\nh\u271d : \u00ac\u2191x < 0\n\u22a2 0 \u2264 1", "state_after": "no goals"}, {"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": "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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = 1", "state_after": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) \u2264 1\n\ncase neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 1 \u2264 \u2a05 r, if \u2191r < 0 then 0 else 1"}, {"tactic": "refine' (ciInf_le h_bdd \u27e8t + 1, lt_add_one t\u27e9).trans _", "annotated_tactic": ["refine' (<a>ciInf_le</a> h_bdd \u27e8t + 1, <a>lt_add_one</a> t\u27e9).<a>trans</a> _", [{"full_name": "ciInf_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [832, 9], "def_end_pos": [832, 17]}, {"full_name": "lt_add_one", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [20, 7], "def_end_pos": [20, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) \u2264 1", "state_after": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (if \u2191{ val := t + 1, property := (_ : t < t + 1) } < 0 then 0 else 1) \u2264 1"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (if \u2191{ val := t + 1, property := (_ : t < t + 1) } < 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nh\u271d : \u2191{ val := t + 1, property := (_ : t < t + 1) } < 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nh\u271d : \u00ac\u2191{ val := t + 1, property := (_ : t < t + 1) } < 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nh\u271d : \u2191{ val := t + 1, property := (_ : t < t + 1) } < 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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nh\u271d : \u00ac\u2191{ val := t + 1, property := (_ : t < t + 1) } < 0\n\u22a2 1 \u2264 1", "state_after": "no goals"}, {"tactic": "refine' le_ciInf fun x => _", "annotated_tactic": ["refine' <a>le_ciInf</a> fun x => _", [{"full_name": "le_ciInf", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [827, 9], "def_end_pos": [827, 17]}]], "state_before": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 1 \u2264 \u2a05 r, if \u2191r < 0 then 0 else 1", "state_after": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nx : \u2191(Ioi t)\n\u22a2 1 \u2264 if \u2191x < 0 then 0 else 1"}, {"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": "case neg.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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nx : \u2191(Ioi t)\n\u22a2 1 \u2264 if \u2191x < 0 then 0 else 1", "state_after": "case neg.refine'_2.hnc\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nx : \u2191(Ioi t)\n\u22a2 \u00ac\u2191x < 0"}, {"tactic": "rw [not_lt] at h \u22a2", "annotated_tactic": ["rw [<a>not_lt</a>] at h \u22a2", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case neg.refine'_2.hnc\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nx : \u2191(Ioi t)\n\u22a2 \u00ac\u2191x < 0", "state_after": "case neg.refine'_2.hnc\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : 0 \u2264 t\nx : \u2191(Ioi t)\n\u22a2 0 \u2264 \u2191x"}, {"tactic": "exact h.trans (mem_Ioi.mp x.prop).le", "annotated_tactic": ["exact h.trans (mem_Ioi.mp x.prop).<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": "case neg.refine'_2.hnc\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\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : 0 \u2264 t\nx : \u2191(Ioi t)\n\u22a2 0 \u2264 \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_erase", "start": [2360, 1], "end": [2362, 16], "traced_tactics": [{"tactic": "rw [\u2190 sdiff_singleton_eq_erase, sdiff_sdiff_eq_sdiff_union (singleton_subset_iff.2 h), insert_eq,\n  union_comm]", "annotated_tactic": ["rw [\u2190 <a>sdiff_singleton_eq_erase</a>, <a>sdiff_sdiff_eq_sdiff_union</a> (<a>singleton_subset_iff</a>.2 h), <a>insert_eq</a>,\n    <a>union_comm</a>]", [{"full_name": "Finset.sdiff_singleton_eq_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 33]}, {"full_name": "Finset.sdiff_sdiff_eq_sdiff_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2346, 9], "def_end_pos": [2346, 35]}, {"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [772, 9], "def_end_pos": [772, 29]}, {"full_name": "Finset.insert_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 18]}, {"full_name": "Finset.union_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1425, 9], "def_end_pos": [1425, 19]}]], "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\nh : a \u2208 s\n\u22a2 s \\ erase t a = insert a (s \\ t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "full_name": "volume_regionBetween_eq_integral'", "start": [22, 1], "end": [31, 6], "traced_tactics": [{"tactic": "have h : g - f =\u1d50[\u03bc.restrict s] fun x => Real.toNNReal (g x - f x) :=\n  hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm", "annotated_tactic": ["have h : g - f =\u1d50[\u03bc.restrict s] fun x => <a>Real.toNNReal</a> (g x - f x) :=\n    hfg.mono fun x hx => (<a>Real.coe_toNNReal</a> _ <| <a>sub_nonneg</a>.2 hx).<a>symm</a>", [{"full_name": "Real.toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [118, 19], "def_end_pos": [118, 39]}, {"full_name": "Real.coe_toNNReal", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [122, 9], "def_end_pos": [122, 33]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"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\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc)"}, {"tactic": "rw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs,\n  integral_congr_ae h, lintegral_congr_ae,\n  lintegral_coe_eq_integral _ ((integrable_congr h).mp (g_int.sub f_int))]", "annotated_tactic": ["rw [<a>volume_regionBetween_eq_lintegral</a> f_int.aemeasurable g_int.aemeasurable hs,\n    <a>integral_congr_ae</a> h, <a>lintegral_congr_ae</a>,\n    <a>lintegral_coe_eq_integral</a> _ ((<a>integrable_congr</a> h).<a>mp</a> (g_int.sub f_int))]", [{"full_name": "volume_regionBetween_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [553, 9], "def_end_pos": [553, 42]}, {"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.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"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": "MeasureTheory.integrable_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [496, 9], "def_end_pos": [496, 25]}, {"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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 (fun y => ENNReal.ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2191(Real.toNNReal (g a - f a))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 (fun y => ENNReal.ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2191(Real.toNNReal (g a - f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_nonneg_of_ae", "start": [1201, 1], "end": [1205, 44], "traced_tactics": [{"tactic": "have A : CompleteSpace \u211d := by infer_instance", "annotated_tactic": ["have A : <a>CompleteSpace</a> \u211d := by infer_instance", [{"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\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1d50[\u03bc] f\n\u22a2 0 \u2264 \u222b (a : \u03b1), 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 : 0 \u2264\u1d50[\u03bc] f\nA : CompleteSpace \u211d\n\u22a2 0 \u2264 \u222b (a : \u03b1), f a \u2202\u03bc"}, {"tactic": "simp only [integral_def, A, L1.integral_def, dite_true, ge_iff_le]", "annotated_tactic": ["simp only [<a>integral_def</a>, A, <a>L1.integral_def</a>, <a>dite_true</a>, <a>ge_iff_le</a>]", [{"full_name": "MeasureTheory.integral_def", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}, {"full_name": "MeasureTheory.L1.integral_def", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}, {"full_name": "dite_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [79, 17], "def_end_pos": [79, 26]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}]], "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 : 0 \u2264\u1d50[\u03bc] f\nA : CompleteSpace \u211d\n\u22a2 0 \u2264 \u222b (a : \u03b1), 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 : 0 \u2264\u1d50[\u03bc] f\nA : CompleteSpace \u211d\n\u22a2 0 \u2264 if hf : Integrable fun a => f a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0"}, {"tactic": "exact setToFun_nonneg (dominatedFinMeasAdditive_weightedSMul \u03bc)\n  (fun s _ _ => weightedSMul_nonneg s) hf", "annotated_tactic": ["exact <a>setToFun_nonneg</a> (<a>dominatedFinMeasAdditive_weightedSMul</a> \u03bc)\n    (fun s _ _ => <a>weightedSMul_nonneg</a> s) hf", [{"full_name": "MeasureTheory.setToFun_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1478, 9], "def_end_pos": [1478, 24]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_weightedSMul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [243, 9], "def_end_pos": [243, 46]}, {"full_name": "MeasureTheory.weightedSMul_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [248, 9], "def_end_pos": [248, 28]}]], "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 : 0 \u2264\u1d50[\u03bc] f\nA : CompleteSpace \u211d\n\u22a2 0 \u2264 if hf : Integrable fun a => f a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0", "state_after": "no goals"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "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 : 0 \u2264\u1d50[\u03bc] f\n\u22a2 CompleteSpace \u211d", "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", "start": [2350, 1], "end": [2352, 16], "traced_tactics": [{"tactic": "simp_rw [\u2190 sdiff_singleton_eq_erase, insert_eq, sdiff_sdiff_left', sdiff_union_distrib,\n  inter_comm]", "annotated_tactic": ["simp_rw [\u2190 <a>sdiff_singleton_eq_erase</a>, <a>insert_eq</a>, <a>sdiff_sdiff_left'</a>, <a>sdiff_union_distrib</a>,\n    <a>inter_comm</a>]", [{"full_name": "Finset.sdiff_singleton_eq_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 33]}, {"full_name": "Finset.insert_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1484, 9], "def_end_pos": [1484, 18]}, {"full_name": "Finset.sdiff_sdiff_left'", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2334, 9], "def_end_pos": [2334, 26]}, {"full_name": "Finset.sdiff_union_distrib", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2248, 9], "def_end_pos": [2248, 28]}, {"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\u271d t\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nx : \u03b1\n\u22a2 s \\ insert x t = erase (s \\ t) 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.isCaratheodory_iUnion_nat", "start": [999, 1], "end": [1013, 56], "traced_tactics": [{"tactic": "apply (isCaratheodory_iff_le' m).mpr", "annotated_tactic": ["apply (<a>isCaratheodory_iff_le'</a> m).<a>mpr</a>", [{"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le'", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [941, 9], "def_end_pos": [941, 31]}, {"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\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 IsCaratheodory m (\u22c3 i, s i)", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2200 (t : Set \u03b1), \u2191m (t \u2229 \u22c3 i, s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t"}, {"tactic": "intro t", "annotated_tactic": ["intro t", []], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2200 (t : Set \u03b1), \u2191m (t \u2229 \u22c3 i, s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 i, s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t"}, {"tactic": "refine' le_trans (add_le_add_right hp _) _", "annotated_tactic": ["refine' <a>le_trans</a> (<a>add_le_add_right</a> 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]}]], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 \u2191m (t \u2229 \u22c3 i, s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 (\u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t"}, {"tactic": "rw [ENNReal.iSup_add]", "annotated_tactic": ["rw [<a>ENNReal.iSup_add</a>]", [{"full_name": "ENNReal.iSup_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [560, 9], "def_end_pos": [560, 17]}]], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 (\u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 \u2a06 b, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < b), s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t"}, {"tactic": "refine'\n  iSup_le fun n =>\n    le_trans (add_le_add_left _ _) (ge_of_eq (isCaratheodory_iUnion_lt m (fun i _ => h i) _))", "annotated_tactic": ["refine'\n        <a>iSup_le</a> fun n =>\n          <a>le_trans</a> (<a>add_le_add_left</a> _ _) (<a>ge_of_eq</a> (<a>isCaratheodory_iUnion_lt</a> m (fun i _ => h i) _))", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"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": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 17]}, {"full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [972, 9], "def_end_pos": [972, 33]}]], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 \u2a06 b, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < b), s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m (t \\ \u22c3 i, \u22c3 (_ : i < n), s i)"}, {"tactic": "refine' m.mono (diff_subset_diff_right _)", "annotated_tactic": ["refine' m.mono (<a>diff_subset_diff_right</a> _)", [{"full_name": "Set.diff_subset_diff_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1912, 9], "def_end_pos": [1912, 31]}]], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m (t \\ \u22c3 i, \u22c3 (_ : i < n), s i)", "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 \u22c3 i, \u22c3 (_ : i < n), s i \u2286 \u22c3 i, s i"}, {"tactic": "exact iUnion\u2082_subset fun i _ => subset_iUnion _ i", "annotated_tactic": ["exact <a>iUnion\u2082_subset</a> fun i _ => <a>subset_iUnion</a> _ i", [{"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}, {"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\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 \u22c3 i, \u22c3 (_ : i < n), s i \u2286 \u22c3 i, s i", "state_after": "no goals"}, {"tactic": "convert m.iUnion fun i => t \u2229 s i using 1", "annotated_tactic": ["convert m.iUnion fun i => t \u2229 s i using 1", []], "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)", "state_after": "case h.e'_3\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 i, s i) = \u2191m (\u22c3 i, t \u2229 s i)\n\ncase h.e'_4\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) = \u2211' (i : \u2115), \u2191m (t \u2229 s i)"}, {"tactic": "simp [inter_iUnion]", "annotated_tactic": ["simp [<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": "case h.e'_3\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 i, s i) = \u2191m (\u22c3 i, t \u2229 s i)", "state_after": "no goals"}, {"tactic": "simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]", "annotated_tactic": ["simp [<a>ENNReal.tsum_eq_iSup_nat</a>, <a>isCaratheodory_sum</a> m h hd]", [{"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": "MeasureTheory.OuterMeasure.isCaratheodory_sum", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [988, 9], "def_end_pos": [988, 27]}]], "state_before": "case h.e'_4\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) = \u2211' (i : \u2115), \u2191m (t \u2229 s 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.preimage_neg_uIcc", "start": [437, 1], "end": [438, 72], "traced_tactics": [{"tactic": "simp only [\u2190 Icc_min_max, preimage_neg_Icc, min_neg_neg, max_neg_neg]", "annotated_tactic": ["simp only [\u2190 <a>Icc_min_max</a>, <a>preimage_neg_Icc</a>, <a>min_neg_neg</a>, <a>max_neg_neg</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_neg_Icc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [150, 9], "def_end_pos": [150, 25]}, {"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 -[[a, b]] = [[-a, -b]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.isometry_lpMeasSubgroupToLpTrim", "start": [449, 1], "end": [453, 20], "traced_tactics": [{"tactic": "rw [dist_eq_norm, \u2190 lpMeasSubgroupToLpTrim_sub, lpMeasSubgroupToLpTrim_norm_map,\n  dist_eq_norm]", "annotated_tactic": ["rw [<a>dist_eq_norm</a>, \u2190 <a>lpMeasSubgroupToLpTrim_sub</a>, <a>lpMeasSubgroupToLpTrim_norm_map</a>,\n      <a>dist_eq_norm</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": "MeasureTheory.lpMeasSubgroupToLpTrim_sub", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [422, 9], "def_end_pos": [422, 35]}, {"full_name": "MeasureTheory.lpMeasSubgroupToLpTrim_norm_map", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [442, 9], "def_end_pos": [442, 40]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 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\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\nhp : Fact (1 \u2264 p)\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 dist (lpMeasSubgroupToLpTrim F p \u03bc hm f) (lpMeasSubgroupToLpTrim F p \u03bc hm g) = dist f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul", "start": [535, 1], "end": [541, 48], "traced_tactics": [{"tactic": "simp only [IndepFun, kernel.indepFun_iff_measure_inter_preimage_eq_mul, ae_dirac_eq,\n  Filter.eventually_pure, kernel.const_apply]", "annotated_tactic": ["simp only [<a>IndepFun</a>, <a>kernel.indepFun_iff_measure_inter_preimage_eq_mul</a>, <a>ae_dirac_eq</a>,\n    <a>Filter.eventually_pure</a>, <a>kernel.const_apply</a>]", [{"full_name": "ProbabilityTheory.IndepFun", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 13]}, {"full_name": "ProbabilityTheory.kernel.indepFun_iff_measure_inter_preimage_eq_mul", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [662, 9], "def_end_pos": [662, 51]}, {"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 : \u03a9 \u2192 \u03b2\ng : \u03a9 \u2192 \u03b2'\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b2' : MeasurableSpace \u03b2'\n\u22a2 IndepFun f g \u2194\n    \u2200 (s : Set \u03b2) (t : Set \u03b2'),\n      MeasurableSet s \u2192 MeasurableSet t \u2192 \u2191\u2191\u03bc (f \u207b\u00b9' s \u2229 g \u207b\u00b9' t) = \u2191\u2191\u03bc (f \u207b\u00b9' s) * \u2191\u2191\u03bc (g \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.range_toLp", "start": [1792, 1], "end": [1795, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.sum_add_sum_compl", "start": [2070, 1], "end": [2074, 84], "traced_tactics": [{"tactic": "ext1 t ht", "annotated_tactic": ["ext1 t ht", []], "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 \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 ((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) = sum \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\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 \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) t = \u2191\u2191(sum \u03bc) t"}, {"tactic": "simp only [add_apply, sum_apply _ ht]", "annotated_tactic": ["simp only [<a>add_apply</a>, <a>sum_apply</a> _ ht]", [{"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.sum_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1989, 9], "def_end_pos": [1989, 18]}]], "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\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) t = \u2191\u2191(sum \u03bc) 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\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2211' (i : \u2191s), \u2191\u2191(\u03bc \u2191i) t + \u2211' (i : \u2191s\u1d9c), \u2191\u2191(\u03bc \u2191i) t = \u2211' (i : \u03b9), \u2191\u2191(\u03bc i) t"}, {"tactic": "exact tsum_add_tsum_compl (f := fun i => \u03bc i t) ENNReal.summable ENNReal.summable", "annotated_tactic": ["exact <a>tsum_add_tsum_compl</a> (f := fun i => \u03bc i t) <a>ENNReal.summable</a> <a>ENNReal.summable</a>", [{"full_name": "tsum_add_tsum_compl", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 28]}, {"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]}]], "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\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2211' (i : \u2191s), \u2191\u2191(\u03bc \u2191i) t + \u2211' (i : \u2191s\u1d9c), \u2191\u2191(\u03bc \u2191i) t = \u2211' (i : \u03b9), \u2191\u2191(\u03bc i) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq", "start": [236, 1], "end": [238, 61], "traced_tactics": [{"tactic": "rw [toMeasurable_def, dif_pos]", "annotated_tactic": ["rw [<a>toMeasurable_def</a>, <a>dif_pos</a>]", [{"full_name": "MeasureTheory.toMeasurable_def", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}, {"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": "\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\nh : NullMeasurableSet s\n\u22a2 toMeasurable \u03bc s =\u1d50[\u03bc] s", "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 t : Set \u03b1\nh : NullMeasurableSet s\n\u22a2 Exists.choose ?hc =\u1d50[\u03bc] s\n\ncase hc\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\nh : NullMeasurableSet s\n\u22a2 \u2203 t, t \u2287 s \u2227 MeasurableSet t \u2227 t =\u1d50[\u03bc] s"}, {"tactic": "exact (exists_measurable_superset_ae_eq h).choose_spec.2.2", "annotated_tactic": ["exact (<a>exists_measurable_superset_ae_eq</a> h).<a>choose_spec</a>.2.2", [{"full_name": "MeasureTheory.NullMeasurableSet.exists_measurable_superset_ae_eq", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [227, 9], "def_end_pos": [227, 41]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "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\nh : NullMeasurableSet s\n\u22a2 Exists.choose ?hc =\u1d50[\u03bc] s\n\ncase hc\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\nh : NullMeasurableSet s\n\u22a2 \u2203 t, t \u2287 s \u2227 MeasurableSet t \u2227 t =\u1d50[\u03bc] s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "full_name": "MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_const_of_countable_range", "start": [96, 1], "end": [106, 62], "traced_tactics": [{"tactic": "have : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} := by\n  ext1 x\n  simp only [Set.mem_univ, Set.mem_range, true_and_iff, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n    Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply']", "annotated_tactic": ["have : <a>Set.univ</a> = \u22c3 i \u2208 <a>Set.range</a> \u03c4, {x | \u03c4 x = i} := by\n    ext1 x\n    simp only [<a>Set.mem_univ</a>, <a>Set.mem_range</a>, <a>true_and_iff</a>, <a>Set.iUnion_exists</a>, <a>Set.iUnion_iUnion_eq'</a>,\n      <a>Set.mem_iUnion</a>, <a>Set.mem_setOf_eq</a>, <a>exists_apply_eq_apply'</a>]", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"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_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"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]}, {"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_apply_eq_apply'", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [780, 9], "def_end_pos": [780, 31]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\n\u22a2 stoppedValue f \u03c4 =\u1d50[\u03bc] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nthis : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}\n\u22a2 stoppedValue f \u03c4 =\u1d50[\u03bc] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]"}, {"tactic": "nth_rw 1 [\u2190 @Measure.restrict_univ \u03a9 _ \u03bc]", "annotated_tactic": ["nth_rw 1 [\u2190 @<a>Measure.restrict_univ</a> \u03a9 _ \u03bc]", [{"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nthis : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}\n\u22a2 stoppedValue f \u03c4 =\u1d50[\u03bc] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nthis : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc Set.univ] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]"}, {"tactic": "rw [this, ae_eq_restrict_biUnion_iff _ h_countable_range]", "annotated_tactic": ["rw [this, <a>ae_eq_restrict_biUnion_iff</a> _ h_countable_range]", [{"full_name": "MeasureTheory.ae_eq_restrict_biUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2538, 9], "def_end_pos": [2538, 35]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nthis : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}\n\u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc Set.univ] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nthis : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}\n\u22a2 \u2200 (i : \u03b9),\n    i \u2208 Set.range \u03c4 \u2192 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]"}, {"tactic": "exact fun i _ => stoppedValue_ae_eq_restrict_eq h _ h\u03c4_le i", "annotated_tactic": ["exact fun i _ => <a>stoppedValue_ae_eq_restrict_eq</a> h _ h\u03c4_le i", [{"full_name": "MeasureTheory.Martingale.stoppedValue_ae_eq_restrict_eq", "def_path": "Mathlib/Probability/Martingale/OptionalSampling.lean", "def_pos": [83, 9], "def_end_pos": [83, 39]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nthis : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}\n\u22a2 \u2200 (i : \u03b9),\n    i \u2208 Set.range \u03c4 \u2192 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4]", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\n\u22a2 Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nx : \u03a9\n\u22a2 x \u2208 Set.univ \u2194 x \u2208 \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}"}, {"tactic": "simp only [Set.mem_univ, Set.mem_range, true_and_iff, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n  Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply']", "annotated_tactic": ["simp only [<a>Set.mem_univ</a>, <a>Set.mem_range</a>, <a>true_and_iff</a>, <a>Set.iUnion_exists</a>, <a>Set.iUnion_iUnion_eq'</a>,\n      <a>Set.mem_iUnion</a>, <a>Set.mem_setOf_eq</a>, <a>exists_apply_eq_apply'</a>]", [{"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_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"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]}, {"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_apply_eq_apply'", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [780, 9], "def_end_pos": [780, 31]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : CompleteSpace E\n\u03b9 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b9\ninst\u271d\u2074 : TopologicalSpace \u03b9\ninst\u271d\u00b3 : OrderTopology \u03b9\ninst\u271d\u00b2 : FirstCountableTopology \u03b9\n\u2131 : Filtration \u03b9 m\ninst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131\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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n\nh_countable_range : Set.Countable (Set.range \u03c4)\ninst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m))\nx : \u03a9\n\u22a2 x \u2208 Set.univ \u2194 x \u2208 \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Units.lean", "full_name": "Int.isUnit_eq_or_eq_neg", "start": [60, 1], "end": [61, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1_mono", "start": [592, 1], "end": [598, 86], "traced_tactics": [{"tactic": "rw [coeFn_le]", "annotated_tactic": ["rw [<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\nhm : m \u2264 m0\ninst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_8\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 \u2191\u2191(condexpL1 hm \u03bc f) \u2264\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc g)", "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\nhm : m \u2264 m0\ninst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_8\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 condexpL1 hm \u03bc f \u2264 condexpL1 hm \u03bc g"}, {"tactic": "have h_nonneg : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 \u2200 x : E, 0 \u2264 x \u2192 0 \u2264 condexpInd E hm \u03bc s x :=\n  fun s hs h\u03bcs x hx => condexpInd_nonneg hs h\u03bcs.ne x hx", "annotated_tactic": ["have h_nonneg : \u2200 s, <a>MeasurableSet</a> s \u2192 \u03bc s < \u221e \u2192 \u2200 x : E, 0 \u2264 x \u2192 0 \u2264 <a>condexpInd</a> E hm \u03bc s x :=\n    fun s hs h\u03bcs x hx => <a>condexpInd_nonneg</a> hs h\u03bcs.ne x hx", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.condexpInd", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [266, 5], "def_end_pos": [266, 15]}, {"full_name": "MeasureTheory.condexpInd_nonneg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [359, 9], "def_end_pos": [359, 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\nhm : m \u2264 m0\ninst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_8\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 condexpL1 hm \u03bc f \u2264 condexpL1 hm \u03bc g", "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\nhm : m \u2264 m0\ninst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_8\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\nh_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : E), 0 \u2264 x \u2192 0 \u2264 \u2191(condexpInd E hm \u03bc s) x\n\u22a2 condexpL1 hm \u03bc f \u2264 condexpL1 hm \u03bc g"}, {"tactic": "exact setToFun_mono (dominatedFinMeasAdditive_condexpInd E hm \u03bc) h_nonneg hf hg hfg", "annotated_tactic": ["exact <a>setToFun_mono</a> (<a>dominatedFinMeasAdditive_condexpInd</a> E hm \u03bc) h_nonneg hf hg hfg", [{"full_name": "MeasureTheory.setToFun_mono", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 22]}, {"full_name": "MeasureTheory.dominatedFinMeasAdditive_condexpInd", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [329, 9], "def_end_pos": [329, 44]}]], "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\nhm : m \u2264 m0\ninst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_8\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\nh_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : E), 0 \u2264 x \u2192 0 \u2264 \u2191(condexpInd E hm \u03bc s) x\n\u22a2 condexpL1 hm \u03bc f \u2264 condexpL1 hm \u03bc g", "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_Ioo", "start": [232, 1], "end": [234, 94], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ioo, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ioo_subset_Ioi_self <| Ioi_subset_Ioi bot_le)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ioo</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Subset.trans</a> <a>Ioo_subset_Ioi_self</a> <| <a>Ioi_subset_Ioi</a> <a>bot_le</a>)]", [{"full_name": "WithBot.preimage_coe_Ioo", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [179, 9], "def_end_pos": [179, 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.Ioo_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [529, 9], "def_end_pos": [529, 28]}, {"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 '' Ioo a b = Ioo \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_aux6", "start": [604, 1], "end": [621, 35], "traced_tactics": [{"tactic": "have H : \u2200 n : \u2115, (0 : \u211d) < \u230ac ^ n\u230b\u208a := by\n  intro n\n  refine' zero_lt_one.trans_le _\n  simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]", "annotated_tactic": ["have H : \u2200 n : \u2115, (0 : \u211d) < \u230ac ^ n\u230b\u208a := by\n    intro n\n    refine' zero_lt_one.trans_le _\n    simp only [<a>Nat.one_le_cast</a>, <a>Nat.one_le_floor_iff</a>, <a>one_le_pow_of_one_le</a> c_one.le n]", [{"full_name": "Nat.one_le_cast", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [105, 9], "def_end_pos": [105, 20]}, {"full_name": "Nat.one_le_floor_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [202, 9], "def_end_pos": [202, 25]}, {"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": "\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\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) 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 : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "filter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one,\n  strong_law_aux5 X hint hident hnonneg] with \u03c9 h\u03c9 h'\u03c9", "annotated_tactic": ["filter_upwards [<a>strong_law_aux4</a> X hint hindep hident hnonneg c_one,\n    <a>strong_law_aux5</a> X hint hident hnonneg] with \u03c9 h\u03c9 h'\u03c9", [{"full_name": "ProbabilityTheory.strong_law_aux4", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [562, 9], "def_end_pos": [562, 24]}, {"full_name": "ProbabilityTheory.strong_law_aux5", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [576, 9], "def_end_pos": [576, 24]}]], "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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) 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 : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))"}, {"tactic": "rw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d]", "annotated_tactic": ["rw [\u2190 <a>tendsto_sub_nhds_zero_iff</a>, \u2190 <a>Asymptotics.isLittleO_one_iff</a> \u211d]", [{"full_name": "tendsto_sub_nhds_zero_iff", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1192, 3], "def_end_pos": [1192, 14]}, {"full_name": "Asymptotics.isLittleO_one_iff", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1363, 9], "def_end_pos": [1363, 26]}]], "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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) 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 : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1"}, {"tactic": "have L : (fun n : \u2115 => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u230ac ^ n\u230b\u208a * \ud835\udd3c[X 0]) =o[atTop] fun n =>\n    (\u230ac ^ n\u230b\u208a : \u211d) := by\n  have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n    tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)\n  convert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1\n  ext1 n\n  simp only [Function.comp_apply, sub_sub_sub_cancel_left]", "annotated_tactic": ["have L : (fun n : \u2115 => \u2211 i in <a>range</a> \u230ac ^ n\u230b\u208a, X i \u03c9 - \u230ac ^ n\u230b\u208a * \ud835\udd3c[X 0]) =o[<a>atTop</a>] fun n =>\n      (\u230ac ^ n\u230b\u208a : \u211d) := by\n    have A : <a>Tendsto</a> (fun n : \u2115 => \u230ac ^ n\u230b\u208a) <a>atTop</a> <a>atTop</a> :=\n      tendsto_nat_floor_atTop.comp (<a>tendsto_pow_atTop_atTop_of_one_lt</a> c_one)\n    convert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1\n    ext1 n\n    simp only [<a>Function.comp_apply</a>, <a>sub_sub_sub_cancel_left</a>]", [{"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": "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": "tendsto_pow_atTop_atTop_of_one_lt", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 42]}, {"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": "sub_sub_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [997, 3], "def_end_pos": [997, 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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1", "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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1"}, {"tactic": "convert L.mul_isBigO (isBigO_refl (fun n : \u2115 => (\u230ac ^ n\u230b\u208a : \u211d)\u207b\u00b9) atTop) using 1 <;>\n(ext1 n; field_simp [(H n).ne'])", "annotated_tactic": ["convert L.mul_isBigO (<a>isBigO_refl</a> (fun n : \u2115 => (\u230ac ^ n\u230b\u208a : \u211d)\u207b\u00b9) <a>atTop</a>) using 1 <;>\n  (ext1 n; field_simp [(H n).<a>ne'</a>])", [{"full_name": "Asymptotics.isBigO_refl", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [602, 9], "def_end_pos": [602, 20]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1", "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\u22a2 \u2200 (n : \u2115), 0 < \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\nn : \u2115\n\u22a2 0 < \u2191\u230ac ^ n\u230b\u208a"}, {"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\nn : \u2115\n\u22a2 0 < \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\nn : \u2115\n\u22a2 1 \u2264 \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]", "annotated_tactic": ["simp only [<a>Nat.one_le_cast</a>, <a>Nat.one_le_floor_iff</a>, <a>one_le_pow_of_one_le</a> c_one.le n]", [{"full_name": "Nat.one_le_cast", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [105, 9], "def_end_pos": [105, 20]}, {"full_name": "Nat.one_le_floor_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [202, 9], "def_end_pos": [202, 25]}, {"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": "\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\nn : \u2115\n\u22a2 1 \u2264 \u2191\u230ac ^ n\u230b\u208a", "state_after": "no goals"}, {"tactic": "have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n  tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)", "annotated_tactic": ["have A : <a>Tendsto</a> (fun n : \u2115 => \u230ac ^ n\u230b\u208a) <a>atTop</a> <a>atTop</a> :=\n      tendsto_nat_floor_atTop.comp (<a>tendsto_pow_atTop_atTop_of_one_lt</a> c_one)", [{"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": "tendsto_pow_atTop_atTop_of_one_lt", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 42]}]], "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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "convert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1", "annotated_tactic": ["convert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1", []], "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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a", "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\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>\n    (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ x\u230b\u208a * \u222b (a : \u03a9), X 0 a) -\n      ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) x"}, {"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\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>\n    (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ x\u230b\u208a * \u222b (a : \u03a9), X 0 a) -\n      ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) x", "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\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\nn : \u2115\n\u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =\n    (\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) -\n      ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) n"}, {"tactic": "simp only [Function.comp_apply, sub_sub_sub_cancel_left]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>sub_sub_sub_cancel_left</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": "sub_sub_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [997, 3], "def_end_pos": [997, 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\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\nn : \u2115\n\u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =\n    (\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) -\n      ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) n", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "case h.e'_8\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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun _x => 1) = fun x => \u2191\u230ac ^ x\u230b\u208a * (\u2191\u230ac ^ x\u230b\u208a)\u207b\u00b9", "state_after": "case h.e'_8.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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nn : \u2115\n\u22a2 1 = \u2191\u230ac ^ n\u230b\u208a * (\u2191\u230ac ^ n\u230b\u208a)\u207b\u00b9"}, {"tactic": "field_simp [(H n).ne']", "annotated_tactic": ["field_simp [(H n).<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 h.e'_8.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\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nn : \u2115\n\u22a2 1 = \u2191\u230ac ^ n\u230b\u208a * (\u2191\u230ac ^ n\u230b\u208a)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.toFinset_eq_empty", "start": [293, 11], "end": [294, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_add_le'", "start": [842, 1], "end": [853, 33], "traced_tactics": [{"tactic": "rcases eq_or_ne p 0 with (rfl | hp)", "annotated_tactic": ["rcases <a>eq_or_ne</a> p 0 with (rfl | hp)", [{"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\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)", "state_after": "case inl\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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 snorm (f + g) 0 \u03bc \u2264 LpAddConst 0 * (snorm f 0 \u03bc + snorm g 0 \u03bc)\n\ncase inr\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)"}, {"tactic": "rcases lt_or_le p 1 with (h'p | h'p)", "annotated_tactic": ["rcases <a>lt_or_le</a> p 1 with (h'p | h'p)", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case inr\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)", "state_after": "case inr.inl\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)\n\ncase inr.inr\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)"}, {"tactic": "simp only [snorm_exponent_zero, add_zero, mul_zero, le_zero_iff]", "annotated_tactic": ["simp only [<a>snorm_exponent_zero</a>, <a>add_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": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"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 inl\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 g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 snorm (f + g) 0 \u03bc \u2264 LpAddConst 0 * (snorm f 0 \u03bc + snorm g 0 \u03bc)", "state_after": "no goals"}, {"tactic": "simp only [snorm_eq_snorm' hp (h'p.trans ENNReal.one_lt_top).ne]", "annotated_tactic": ["simp only [<a>snorm_eq_snorm'</a> hp (h'p.trans <a>ENNReal.one_lt_top</a>).<a>ne</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": "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 inr.inl\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)", "state_after": "case inr.inl\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 LpAddConst p * (snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc)"}, {"tactic": "convert snorm'_add_le_of_le_one hf ENNReal.toReal_nonneg _", "annotated_tactic": ["convert <a>snorm'_add_le_of_le_one</a> hf <a>ENNReal.toReal_nonneg</a> _", [{"full_name": "MeasureTheory.snorm'_add_le_of_le_one", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [783, 9], "def_end_pos": [783, 32]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case inr.inl\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 LpAddConst p * (snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc)", "state_after": "case h.e'_4.h.e'_5\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1)\n\ncase inr.inl.convert_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 : \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\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 ENNReal.toReal p \u2264 1"}, {"tactic": "have : p \u2208 Set.Ioo (0 : \u211d\u22650\u221e) 1 := \u27e8hp.bot_lt, h'p\u27e9", "annotated_tactic": ["have : p \u2208 <a>Set.Ioo</a> (0 : \u211d\u22650\u221e) 1 := \u27e8hp.bot_lt, h'p\u27e9", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}]], "state_before": "case h.e'_4.h.e'_5\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1)", "state_after": "case h.e'_4.h.e'_5\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\nthis : p \u2208 Set.Ioo 0 1\n\u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1)"}, {"tactic": "simp only [LpAddConst, if_pos this]", "annotated_tactic": ["simp only [<a>LpAddConst</a>, <a>if_pos</a> this]", [{"full_name": "MeasureTheory.LpAddConst", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [816, 5], "def_end_pos": [816, 15]}, {"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.e'_4.h.e'_5\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\nthis : p \u2208 Set.Ioo 0 1\n\u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1)", "state_after": "no goals"}, {"tactic": "simpa using ENNReal.toReal_mono ENNReal.one_ne_top h'p.le", "annotated_tactic": ["simpa using <a>ENNReal.toReal_mono</a> <a>ENNReal.one_ne_top</a> h'p.le", [{"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}, {"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 inr.inl.convert_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 : \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\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 ENNReal.toReal p \u2264 1", "state_after": "no goals"}, {"tactic": "simp [LpAddConst_of_one_le h'p]", "annotated_tactic": ["simp [<a>LpAddConst_of_one_le</a> h'p]", [{"full_name": "MeasureTheory.LpAddConst_of_one_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [820, 9], "def_end_pos": [820, 29]}]], "state_before": "case inr.inr\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)", "state_after": "case inr.inr\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc"}, {"tactic": "exact snorm_add_le hf hg h'p", "annotated_tactic": ["exact <a>snorm_add_le</a> hf hg h'p", [{"full_name": "MeasureTheory.snorm_add_le", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [802, 9], "def_end_pos": [802, 21]}]], "state_before": "case inr.inr\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\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : 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/Data/Finset/Sups.lean", "full_name": "Finset.card_disjSups_le", "start": [488, 1], "end": [489, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_ssubset_insert", "start": [2002, 1], "end": [2004, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "MeasureTheory.Integrable.condexpKernel_ae", "start": [126, 1], "end": [131, 63], "traced_tactics": [{"tactic": "exact Integrable.condDistrib_ae\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.condDistrib_ae</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.condDistrib_ae", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [144, 9], "def_end_pos": [144, 55]}, {"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\u2075 : TopologicalSpace \u03a9\nm m\u03a9 : MeasurableSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NormedAddCommGroup F\nf : \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Integrable f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Floor.lean", "full_name": "Int.measurable_ceil", "start": [36, 1], "end": [38, 69], "traced_tactics": [{"tactic": "simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc", "annotated_tactic": ["simpa only [<a>Int.preimage_ceil_singleton</a>] using <a>measurableSet_Ioc</a>", [{"full_name": "Int.preimage_ceil_singleton", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1311, 9], "def_end_pos": [1311, 32]}, {"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\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 : OpensMeasurableSpace R\nx : R\n\u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308x\u2309})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.join\u2081_rename", "start": [209, 1], "end": [211, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/ZeroOne.lean", "full_name": "ProbabilityTheory.indep_limsup_atBot_self", "start": [162, 1], "end": [174, 31], "traced_tactics": [{"tactic": "let ns : \u03b9 \u2192 Set \u03b9 := Set.Ici", "annotated_tactic": ["let ns : \u03b9 \u2192 <a>Set</a> \u03b9 := <a>Set.Ici</a>", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\n\u22a2 Indep (limsup s atBot) (limsup s atBot)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\n\u22a2 Indep (limsup s atBot) (limsup s atBot)"}, {"tactic": "have hnsp : \u2200 i, BddBelow (ns i) := fun i => bddBelow_Ici", "annotated_tactic": ["have hnsp : \u2200 i, <a>BddBelow</a> (ns i) := fun i => <a>bddBelow_Ici</a>", [{"full_name": "BddBelow", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "bddBelow_Ici", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [538, 9], "def_end_pos": [538, 21]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\n\u22a2 Indep (limsup s atBot) (limsup s atBot)", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 Indep (limsup s atBot) (limsup s atBot)"}, {"tactic": "refine' indep_limsup_self h_le h_indep _ _ hnsp _", "annotated_tactic": ["refine' <a>indep_limsup_self</a> h_le h_indep _ _ hnsp _", [{"full_name": "ProbabilityTheory.indep_limsup_self", "def_path": "Mathlib/Probability/Independence/ZeroOne.lean", "def_pos": [110, 9], "def_end_pos": [110, 26]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 Indep (limsup s atBot) (limsup s atBot)", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 \u2200 (t : Set \u03b9), BddBelow t \u2192 t\u1d9c \u2208 atBot\n\ncase refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => ns a\n\ncase refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a"}, {"tactic": "simp only [mem_atBot_sets, ge_iff_le, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty]", "annotated_tactic": ["simp only [<a>mem_atBot_sets</a>, <a>ge_iff_le</a>, <a>Set.mem_compl_iff</a>, <a>BddBelow</a>, <a>lowerBounds</a>, <a>Set.Nonempty</a>]", [{"full_name": "Filter.mem_atBot_sets", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [172, 9], "def_end_pos": [172, 23]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}, {"full_name": "BddBelow", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 \u2200 (t : Set \u03b9), BddBelow t \u2192 t\u1d9c \u2208 atBot", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 \u2200 (t : Set \u03b9), (\u2203 x, x \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}) \u2192 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t"}, {"tactic": "rintro t \u27e8a, ha\u27e9", "annotated_tactic": ["rintro t \u27e8a, ha\u27e9", []], "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 \u2200 (t : Set \u03b9), (\u2203 x, x \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}) \u2192 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t", "state_after": "case refine'_1.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t"}, {"tactic": "obtain \u27e8b, hb\u27e9 : \u2203 b, b < a := exists_lt a", "annotated_tactic": ["obtain \u27e8b, hb\u27e9 : \u2203 b, b < a := <a>exists_lt</a> a", [{"full_name": "NoMinOrder.exists_lt", "def_path": "Mathlib/Order/Max.lean", "def_pos": [55, 3], "def_end_pos": [55, 12]}]], "state_before": "case refine'_1.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t", "state_after": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t"}, {"tactic": "refine' \u27e8b, fun c hc hct => _\u27e9", "annotated_tactic": ["refine' \u27e8b, fun c hc hct => _\u27e9", []], "state_before": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t", "state_after": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\n\u22a2 False"}, {"tactic": "suffices : \u2200 i \u2208 t, c < i", "annotated_tactic": ["suffices : \u2200 i \u2208 t, c < i", []], "state_before": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\n\u22a2 False", "state_after": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\nthis : \u2200 (i : \u03b9), i \u2208 t \u2192 c < i\n\u22a2 False\n\ncase this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 c < i"}, {"tactic": "exact lt_irrefl c (this c hct)", "annotated_tactic": ["exact <a>lt_irrefl</a> c (this c hct)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\nthis : \u2200 (i : \u03b9), i \u2208 t \u2192 c < i\n\u22a2 False\n\ncase this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 c < i", "state_after": "case this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 c < i"}, {"tactic": "exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))", "annotated_tactic": ["exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))", []], "state_before": "case this\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\nt : Set \u03b9\na : \u03b9\nha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}\nb : \u03b9\nhb : b < a\nc : \u03b9\nhc : c \u2264 b\nhct : c \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 c < i", "state_after": "no goals"}, {"tactic": "exact directed_of_inf fun i j hij k hki => hij.trans hki", "annotated_tactic": ["exact <a>directed_of_inf</a> fun i j hij k hki => hij.trans hki", [{"full_name": "directed_of_inf", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [144, 9], "def_end_pos": [144, 24]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => ns a", "state_after": "no goals"}, {"tactic": "exact fun n => \u27e8n, le_rfl\u27e9", "annotated_tactic": ["exact fun n => \u27e8n, <a>le_rfl</a>\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm m0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\ns : \u03b9 \u2192 MeasurableSpace \u03a9\ninst\u271d\u00b2 : SemilatticeInf \u03b9\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : Nonempty \u03b9\nh_le : \u2200 (n : \u03b9), s n \u2264 m0\nh_indep : iIndep s\nns : \u03b9 \u2192 Set \u03b9 := Set.Ici\nhnsp : \u2200 (i : \u03b9), BddBelow (ns i)\n\u22a2 \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.measurable_of_le", "start": [585, 11], "end": [588, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.measurableSet_exists_tendsto", "start": [926, 1], "end": [951, 74], "traced_tactics": [{"tactic": "rcases l.eq_or_neBot with rfl | hl", "annotated_tactic": ["rcases l.eq_or_neBot with rfl | hl", []], "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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}", "state_after": "case inl\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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\ninst\u271d : IsCountablyGenerated \u22a5\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd c)}\n\ncase inr\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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}"}, {"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": "case inr\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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}", "state_after": "case inr\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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}"}, {"tactic": "rcases l.exists_antitone_basis with \u27e8u, hu\u27e9", "annotated_tactic": ["rcases l.exists_antitone_basis with \u27e8u, hu\u27e9", []], "state_before": "case inr\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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}"}, {"tactic": "simp_rw [\u2190 cauchy_map_iff_exists_tendsto]", "annotated_tactic": ["simp_rw [\u2190 <a>cauchy_map_iff_exists_tendsto</a>]", [{"full_name": "cauchy_map_iff_exists_tendsto", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [462, 9], "def_end_pos": [462, 38]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | Cauchy (map (fun n => f n x) l)}"}, {"tactic": "change MeasurableSet { x | _ \u2227 _ }", "annotated_tactic": ["change <a>MeasurableSet</a> { x | _ \u2227 _ }", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | Cauchy (map (fun n => f n x) l)}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3}"}, {"tactic": "have :\n  \u2200 x,\n    (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l).HasAntitoneBasis fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) :=\n  fun x => hu.map.prod hu.map", "annotated_tactic": ["have :\n    \u2200 x,\n      (<a>map</a> (fun i => f i x) l \u00d7\u02e2 <a>map</a> (fun i => f i x) l).<a>HasAntitoneBasis</a> fun n =>\n        ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) :=\n    fun x => hu.map.prod hu.map", [{"full_name": "Filter.map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1830, 5], "def_end_pos": [1830, 8]}, {"full_name": "Filter.map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1830, 5], "def_end_pos": [1830, 8]}, {"full_name": "Filter.HasAntitoneBasis", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [839, 11], "def_end_pos": [839, 27]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\n\u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3}"}, {"tactic": "simp_rw [and_iff_right (hl.map _),\n  Filter.HasBasis.le_basis_iff (this _).toHasBasis Metric.uniformity_basis_dist_inv_nat_succ,\n  Set.setOf_forall]", "annotated_tactic": ["simp_rw [<a>and_iff_right</a> (hl.map _),\n    <a>Filter.HasBasis.le_basis_iff</a> (this _).toHasBasis <a>Metric.uniformity_basis_dist_inv_nat_succ</a>,\n    <a>Set.setOf_forall</a>]", [{"full_name": "and_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [206, 9], "def_end_pos": [206, 22]}, {"full_name": "Filter.HasBasis.le_basis_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [463, 9], "def_end_pos": [463, 30]}, {"full_name": "Metric.uniformity_basis_dist_inv_nat_succ", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 43]}, {"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\n\u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\n\u22a2 MeasurableSet\n    (\u22c2 i,\n      \u22c2 (_ : True),\n        {x |\n          \u2203 i_2, True \u2227 ((fun n => f n x) '' u i_2) \u00d7\u02e2 ((fun n => f n x) '' u i_2) \u2286 {p | dist p.1 p.2 < 1 / (\u2191i + 1)}})"}, {"tactic": "refine' MeasurableSet.biInter Set.countable_univ fun K _ => _", "annotated_tactic": ["refine' <a>MeasurableSet.biInter</a> <a>Set.countable_univ</a> fun K _ => _", [{"full_name": "MeasurableSet.biInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [164, 9], "def_end_pos": [164, 30]}, {"full_name": "Set.countable_univ", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [112, 9], "def_end_pos": [112, 23]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\n\u22a2 MeasurableSet\n    (\u22c2 i,\n      \u22c2 (_ : True),\n        {x |\n          \u2203 i_2, True \u2227 ((fun n => f n x) '' u i_2) \u00d7\u02e2 ((fun n => f n x) '' u i_2) \u2286 {p | dist p.1 p.2 < 1 / (\u2191i + 1)}})", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\n\u22a2 MeasurableSet\n    {x | \u2203 i, True \u2227 ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}}"}, {"tactic": "simp_rw [Set.setOf_exists, true_and]", "annotated_tactic": ["simp_rw [<a>Set.setOf_exists</a>, <a>true_and</a>]", [{"full_name": "Set.setOf_exists", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [382, 9], "def_end_pos": [382, 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": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\n\u22a2 MeasurableSet\n    {x | \u2203 i, True \u2227 ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\n\u22a2 MeasurableSet (\u22c3 i, {x | ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}})"}, {"tactic": "refine' MeasurableSet.iUnion fun N => _", "annotated_tactic": ["refine' <a>MeasurableSet.iUnion</a> fun N => _", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\n\u22a2 MeasurableSet (\u22c3 i, {x | ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}})", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\nN : \u2115\n\u22a2 MeasurableSet {x | ((fun n => f n x) '' u N) \u00d7\u02e2 ((fun n => f n x) '' u N) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}}"}, {"tactic": "simp_rw [prod_image_image_eq, image_subset_iff, prod_subset_iff, Set.setOf_forall]", "annotated_tactic": ["simp_rw [<a>prod_image_image_eq</a>, <a>image_subset_iff</a>, <a>prod_subset_iff</a>, <a>Set.setOf_forall</a>]", [{"full_name": "Set.prod_image_image_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [302, 9], "def_end_pos": [302, 28]}, {"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.prod_subset_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [100, 9], "def_end_pos": [100, 24]}, {"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\nN : \u2115\n\u22a2 MeasurableSet {x | ((fun n => f n x) '' u N) \u00d7\u02e2 ((fun n => f n x) '' u N) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}}", "state_after": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\nN : \u2115\n\u22a2 MeasurableSet\n    (\u22c2 i \u2208 u N, \u22c2 i_1 \u2208 u N, {x | (i, i_1) \u2208 (fun p => (f p.1 x, f p.2 x)) \u207b\u00b9' {p | dist p.1 p.2 < 1 / (\u2191K + 1)}})"}, {"tactic": "exact\n  MeasurableSet.biInter (to_countable (u N)) fun i _ =>\n    MeasurableSet.biInter (to_countable (u N)) fun j _ =>\n      measurableSet_lt (Measurable.dist (hf i) (hf j)) measurable_const", "annotated_tactic": ["exact\n    <a>MeasurableSet.biInter</a> (<a>to_countable</a> (u N)) fun i _ =>\n      <a>MeasurableSet.biInter</a> (<a>to_countable</a> (u N)) fun j _ =>\n        <a>measurableSet_lt</a> (<a>Measurable.dist</a> (hf i) (hf j)) <a>measurable_const</a>", [{"full_name": "MeasurableSet.biInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [164, 9], "def_end_pos": [164, 30]}, {"full_name": "Set.to_countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [41, 9], "def_end_pos": [41, 21]}, {"full_name": "MeasurableSet.biInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [164, 9], "def_end_pos": [164, 30]}, {"full_name": "Set.to_countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [41, 9], "def_end_pos": [41, 21]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "Measurable.dist", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1717, 9], "def_end_pos": [1717, 24]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case inr.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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\nN : \u2115\n\u22a2 MeasurableSet\n    (\u22c2 i \u2208 u N, \u22c2 i_1 \u2208 u N, {x | (i, i_1) \u2208 (fun p => (f p.1 x, f p.2 x)) \u207b\u00b9' {p | dist p.1 p.2 < 1 / (\u2191K + 1)}})", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\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\u2076 : T2Space \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : PolishSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\ninst\u271d : IsCountablyGenerated \u22a5\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd c)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.castNum_xor", "start": [943, 1], "end": [944, 85], "traced_tactics": [{"tactic": "apply castNum_eq_bitwise PosNum.lxor <;> intros <;> (try cases_type* Bool) <;> rfl", "annotated_tactic": ["apply <a>castNum_eq_bitwise</a> <a>PosNum.lxor</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.lxor", "def_path": "Mathlib/Data/Num/Bitwise.lean", "def_pos": [77, 5], "def_end_pos": [77, 9]}, {"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(m ^^^ n) = \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.lxor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d != b\u271d) (PosNum.lxor m\u271d n\u271d)", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false != false) (PosNum.lxor m\u271d n\u271d)\n\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (false != true) (PosNum.lxor m\u271d n\u271d)\n\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (true != false) (PosNum.lxor m\u271d n\u271d)\n\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (true != true) (PosNum.lxor m\u271d n\u271d)"}, {"tactic": "cases_type* Bool", "annotated_tactic": ["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.lxor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d != b\u271d) (PosNum.lxor m\u271d n\u271d)", "state_after": "case pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false != false) (PosNum.lxor m\u271d n\u271d)\n\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (false != true) (PosNum.lxor m\u271d n\u271d)\n\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (true != false) (PosNum.lxor m\u271d n\u271d)\n\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (true != true) (PosNum.lxor m\u271d n\u271d)"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Multiset.coe_nodupKeys", "start": [43, 1], "end": [44, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM", "start": [1207, 1], "end": [1216, 66], "traced_tactics": [{"tactic": "refine'\n  ContinuousLinearMap.op_norm_extend_le (setToL1SCLM \u03b1 E \u03bc hT) (coeToLp \u03b1 E \u211d)\n    (simpleFunc.denseRange one_ne_top) fun x => le_of_eq _", "annotated_tactic": ["refine'\n        <a>ContinuousLinearMap.op_norm_extend_le</a> (<a>setToL1SCLM</a> \u03b1 E \u03bc hT) (<a>coeToLp</a> \u03b1 E \u211d)\n          (<a>simpleFunc.denseRange</a> <a>one_ne_top</a>) fun x => <a>le_of_eq</a> _", [{"full_name": "ContinuousLinearMap.op_norm_extend_le", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [1764, 9], "def_end_pos": [1764, 26]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [877, 5], "def_end_pos": [877, 16]}, {"full_name": "MeasureTheory.Lp.simpleFunc.coeToLp", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [796, 5], "def_end_pos": [796, 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": "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\nE : Type u_2\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\n\u22a2 \u2016setToL1 hT\u2016 \u2264 \u21911 * \u2016setToL1SCLM \u03b1 E \u03bc hT\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\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\nx : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016x\u2016 = \u21911 * \u2016\u2191(coeToLp \u03b1 E \u211d) x\u2016"}, {"tactic": "rw [NNReal.coe_one, one_mul]", "annotated_tactic": ["rw [<a>NNReal.coe_one</a>, <a>one_mul</a>]", [{"full_name": "NNReal.coe_one", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [176, 19], "def_end_pos": [176, 26]}, {"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\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\nx : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016x\u2016 = \u21911 * \u2016\u2191(coeToLp \u03b1 E \u211d) x\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\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\nx : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016x\u2016 = \u2016\u2191(coeToLp \u03b1 E \u211d) x\u2016"}, {"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\nx : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016x\u2016 = \u2016\u2191(coeToLp \u03b1 E \u211d) x\u2016", "state_after": "no goals"}, {"tactic": "rw [NNReal.coe_one, one_mul]", "annotated_tactic": ["rw [<a>NNReal.coe_one</a>, <a>one_mul</a>]", [{"full_name": "NNReal.coe_one", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [176, 19], "def_end_pos": [176, 26]}, {"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\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\n\u22a2 \u21911 * \u2016setToL1SCLM \u03b1 E \u03bc hT\u2016 = \u2016setToL1SCLM \u03b1 E \u03bc hT\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.integrable_stoppedValue", "start": [975, 1], "end": [978, 83], "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_outerMeasure_self_pos", "start": [577, 1], "end": [582, 28], "traced_tactics": [{"tactic": "refine' zero_lt_one.trans_le _", "annotated_tactic": ["refine' zero_lt_one.trans_le _", []], "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\n\u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080", "state_after": "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\n\u22a2 1 \u2264 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080"}, {"tactic": "rw [Content.outerMeasure_eq_iInf]", "annotated_tactic": ["rw [<a>Content.outerMeasure_eq_iInf</a>]", [{"full_name": "MeasureTheory.Content.outerMeasure_eq_iInf", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [276, 9], "def_end_pos": [276, 29]}]], "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\n\u22a2 1 \u2264 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080", "state_after": "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\n\u22a2 1 \u2264 \u2a05 U, \u2a05 (hU : IsOpen U), \u2a05 (_ : \u2191K\u2080 \u2286 U), Content.innerContent (haarContent K\u2080) { carrier := U, is_open' := hU }"}, {"tactic": "refine' le_iInf\u2082 fun U hU => le_iInf fun hK\u2080 => le_trans _ <| le_iSup\u2082 K\u2080.toCompacts hK\u2080", "annotated_tactic": ["refine' <a>le_iInf\u2082</a> fun U hU => <a>le_iInf</a> fun hK\u2080 => <a>le_trans</a> _ <| <a>le_iSup\u2082</a> K\u2080.toCompacts hK\u2080", [{"full_name": "le_iInf\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [887, 9], "def_end_pos": [887, 17]}, {"full_name": "le_iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [879, 9], "def_end_pos": [879, 16]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_iSup\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [857, 9], "def_end_pos": [857, 17]}]], "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\n\u22a2 1 \u2264 \u2a05 U, \u2a05 (hU : IsOpen U), \u2a05 (_ : \u2191K\u2080 \u2286 U), Content.innerContent (haarContent K\u2080) { carrier := U, is_open' := hU }", "state_after": "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\nU : Set G\nhU : IsOpen U\nhK\u2080 : \u2191K\u2080 \u2286 U\n\u22a2 1 \u2264 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts"}, {"tactic": "exact haarContent_self.ge", "annotated_tactic": ["exact haarContent_self.ge", []], "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\nU : Set G\nhU : IsOpen U\nhK\u2080 : \u2191K\u2080 \u2286 U\n\u22a2 1 \u2264 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts", "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_subset", "start": [176, 9], "end": [177, 73], "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 MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n\u271d) \u2286 i", "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 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) \u2286\n    i"}, {"tactic": "exact someExistsOneDivLT_subset'", "annotated_tactic": ["exact <a>someExistsOneDivLT_subset'</a>", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_subset'", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [141, 17], "def_end_pos": [141, 43]}]], "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 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) \u2286\n    i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.summable_measure_toReal", "start": [2974, 1], "end": [2979, 38], "traced_tactics": [{"tactic": "apply ENNReal.summable_toReal", "annotated_tactic": ["apply <a>ENNReal.summable_toReal</a>", [{"full_name": "ENNReal.summable_toReal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 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\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\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 Summable fun x => ENNReal.toReal (\u2191\u2191\u03bc (f x))", "state_after": "case hsum\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 : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc (f x) \u2260 \u22a4"}, {"tactic": "rw [\u2190 MeasureTheory.measure_iUnion hf\u2082 hf\u2081]", "annotated_tactic": ["rw [\u2190 <a>MeasureTheory.measure_iUnion</a> hf\u2082 hf\u2081]", [{"full_name": "MeasureTheory.measure_iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [272, 9], "def_end_pos": [272, 23]}]], "state_before": "case hsum\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 : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc (f x) \u2260 \u22a4", "state_after": "case hsum\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 : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) \u2260 \u22a4"}, {"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 hsum\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 : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.disjSups_inter_subset_right", "start": [553, 1], "end": [554, 92], "traced_tactics": [{"tactic": "simpa only [disjSups, product_inter, filter_inter_distrib] using image_inter_subset _ _ _", "annotated_tactic": ["simpa only [<a>disjSups</a>, <a>product_inter</a>, <a>filter_inter_distrib</a>] using <a>image_inter_subset</a> _ _ _", [{"full_name": "Finset.disjSups", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [465, 5], "def_end_pos": [465, 13]}, {"full_name": "Finset.product_inter", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [254, 9], "def_end_pos": [254, 22]}, {"full_name": "Finset.filter_inter_distrib", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2881, 9], "def_end_pos": [2881, 29]}, {"full_name": "Finset.image_inter_subset", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 27]}]], "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\u2081 \u2229 t\u2082) \u2286 s \u25cb t\u2081 \u2229 s \u25cb t\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "StieltjesFunction.id_leftLim", "start": [101, 1], "end": [103, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_pi_Ico_toReal", "start": [251, 1], "end": [253, 93], "traced_tactics": [{"tactic": "simp only [volume_pi_Ico, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]", "annotated_tactic": ["simp only [<a>volume_pi_Ico</a>, <a>ENNReal.toReal_prod</a>, <a>ENNReal.toReal_ofReal</a> (<a>sub_nonneg</a>.2 (h _))]", [{"full_name": "Real.volume_pi_Ico", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [245, 9], "def_end_pos": [245, 22]}, {"full_name": "ENNReal.toReal_prod", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2308, 9], "def_end_pos": [2308, 20]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\nh : a \u2264 b\n\u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ico (a i) (b i))) = \u220f i : \u03b9, (b i - a i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to1.exists_enc_dec", "start": [1619, 1], "end": [1630, 94], "traced_tactics": [{"tactic": "letI := Classical.decEq \u0393", "annotated_tactic": ["letI := <a>Classical.decEq</a> \u0393", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [983, 19], "def_end_pos": [983, 24]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a"}, {"tactic": "let n := Fintype.card \u0393", "annotated_tactic": ["let n := <a>Fintype.card</a> \u0393", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a"}, {"tactic": "obtain \u27e8F\u27e9 := Fintype.truncEquivFin \u0393", "annotated_tactic": ["obtain \u27e8F\u27e9 := <a>Fintype.truncEquivFin</a> \u0393", [{"full_name": "Fintype.truncEquivFin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [77, 5], "def_end_pos": [77, 18]}]], "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a"}, {"tactic": "let G : Fin n \u21aa Fin n \u2192 Bool :=\n  \u27e8fun a b \u21a6 a = b, fun a b h \u21a6\n    Bool.of_decide_true <| (congr_fun h b).trans <| Bool.decide_true rfl\u27e9", "annotated_tactic": ["let G : <a>Fin</a> n \u21aa <a>Fin</a> n \u2192 <a>Bool</a> :=\n    \u27e8fun a b \u21a6 a = b, fun a b h \u21a6\n      <a>Bool.of_decide_true</a> <| (<a>congr_fun</a> h b).<a>trans</a> <| <a>Bool.decide_true</a> <a>rfl</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": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Bool", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}, {"full_name": "Bool.of_decide_true", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [134, 9], "def_end_pos": [134, 23]}, {"full_name": "congr_fun", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [42, 7], "def_end_pos": [42, 16]}, {"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": "Bool.decide_true", "def_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "def_pos": [129, 9], "def_end_pos": [129, 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 mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a"}, {"tactic": "let H := (F.toEmbedding.trans G).trans (Equiv.vectorEquivFin _ _).symm.toEmbedding", "annotated_tactic": ["let H := (F.toEmbedding.trans G).<a>trans</a> (<a>Equiv.vectorEquivFin</a> _ _).symm.toEmbedding", [{"full_name": "Function.Embedding.trans", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [141, 15], "def_end_pos": [141, 20]}, {"full_name": "Equiv.vectorEquivFin", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [163, 5], "def_end_pos": [163, 32]}]], "state_before": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n  Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n    (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a"}, {"tactic": "classical\n  let enc := H.setValue default (Vector.replicate n false)\n  exact \u27e8_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2\u27e9", "annotated_tactic": ["classical\n    let enc := H.setValue <a>default</a> (<a>Vector.replicate</a> n <a>false</a>)\n    exact \u27e8_, enc, <a>Function.invFun</a> enc, H.setValue_eq _ _, <a>Function.leftInverse_invFun</a> enc.2\u27e9", [{"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": "Vector.replicate", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"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": "Function.invFun", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [434, 19], "def_end_pos": [434, 25]}, {"full_name": "Function.leftInverse_invFun", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 27]}]], "state_before": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n  Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n    (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "no goals"}, {"tactic": "let enc := H.setValue default (Vector.replicate n false)", "annotated_tactic": ["let enc := H.setValue <a>default</a> (<a>Vector.replicate</a> n <a>false</a>)", [{"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": "Vector.replicate", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [150, 5], "def_end_pos": [150, 14]}, {"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}]], "state_before": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n  Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n    (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n  Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n    (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\nenc : \u0393 \u21aa Vector Bool n := Function.Embedding.setValue H default (Vector.replicate n false)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a"}, {"tactic": "exact \u27e8_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2\u27e9", "annotated_tactic": ["exact \u27e8_, enc, <a>Function.invFun</a> enc, H.setValue_eq _ _, <a>Function.leftInverse_invFun</a> enc.2\u27e9", [{"full_name": "Function.invFun", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [434, 19], "def_end_pos": [434, 25]}, {"full_name": "Function.leftInverse_invFun", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 27]}]], "state_before": "case mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n  { toFun := fun a b => decide (a = b),\n    inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n  Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n    (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\nenc : \u0393 \u21aa Vector Bool n := Function.Embedding.setValue H default (Vector.replicate n false)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.Lp.induction_stronglyMeasurable_aux", "start": [568, 1], "end": [610, 72], "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    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)\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 P f"}, {"tactic": "let f' := (\u27e8f, hf\u27e9 : lpMeas F \u211d m p \u03bc)", "annotated_tactic": ["let f' := (\u27e8f, hf\u27e9 : <a>lpMeas</a> F \u211d m p \u03bc)", [{"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\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    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)\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\n\u22a2 P f"}, {"tactic": "let g := lpMeasToLpTrimLie F \u211d p \u03bc hm f'", "annotated_tactic": ["let g := <a>lpMeasToLpTrimLie</a> F \u211d p \u03bc hm f'", [{"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}]], "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\n\u22a2 P f"}, {"tactic": "have hfg : f' = (lpMeasToLpTrimLie F \u211d p \u03bc hm).symm g := by\n  simp only [LinearIsometryEquiv.symm_apply_apply]", "annotated_tactic": ["have hfg : f' = (<a>lpMeasToLpTrimLie</a> F \u211d p \u03bc hm).<a>symm</a> g := by\n    simp only [<a>LinearIsometryEquiv.symm_apply_apply</a>]", [{"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}, {"full_name": "LinearIsometryEquiv.symm_apply_apply", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [747, 9], "def_end_pos": [747, 25]}]], "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P f"}, {"tactic": "change P \u2191f'", "annotated_tactic": ["change P \u2191f'", []], "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P \u2191f'"}, {"tactic": "rw [hfg]", "annotated_tactic": ["rw [hfg]", []], "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P \u2191f'", "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)"}, {"tactic": "refine'\n  @Lp.induction \u03b1 F m _ p (\u03bc.trim hm) _ hp_ne_top\n    (fun g => P ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm g)) _ _ _ g", "annotated_tactic": ["refine'\n    @<a>Lp.induction</a> \u03b1 F m _ p (\u03bc.trim hm) _ hp_ne_top\n      (fun g => P ((<a>lpMeasToLpTrimLie</a> F \u211d p \u03bc hm).<a>symm</a> g)) _ _ _ g", [{"full_name": "MeasureTheory.Lp.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [924, 9], "def_end_pos": [924, 21]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}]], "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),\n    (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))\n      \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)\n\ncase 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (Function.support f) (Function.support g) \u2192\n      (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp f hf) \u2192\n        (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp g hg) \u2192\n          (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))\n            (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n\ncase refine'_3\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 IsClosed {f | (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) f}"}, {"tactic": "simp only [LinearIsometryEquiv.symm_apply_apply]", "annotated_tactic": ["simp only [<a>LinearIsometryEquiv.symm_apply_apply</a>]", [{"full_name": "LinearIsometryEquiv.symm_apply_apply", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [747, 9], "def_end_pos": [747, 25]}]], "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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\n\u22a2 f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g", "state_after": "no goals"}, {"tactic": "intro b t ht h\u03bct", "annotated_tactic": ["intro b t ht h\u03bct", []], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),\n    (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))\n      \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 P\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm))\n        \u2191(simpleFunc.indicatorConst p ht (_ : \u2191\u2191(Measure.trim \u03bc hm) t \u2260 \u22a4) b))"}, {"tactic": "rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator ht h\u03bct.ne b]", "annotated_tactic": ["rw [@<a>Lp.simpleFunc.coe_indicatorConst</a> _ _ m, <a>lpMeasToLpTrimLie_symm_indicator</a> ht h\u03bct.ne b]", [{"full_name": "MeasureTheory.Lp.simpleFunc.coe_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [700, 9], "def_end_pos": [700, 27]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie_symm_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [535, 9], "def_end_pos": [535, 41]}]], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 P\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm))\n        \u2191(simpleFunc.indicatorConst p ht (_ : \u2191\u2191(Measure.trim \u03bc hm) t \u2260 \u22a4) b))", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)"}, {"tactic": "have h\u03bct' : \u03bc t < \u221e := (le_trim hm).trans_lt h\u03bct", "annotated_tactic": ["have h\u03bct' : \u03bc t < \u221e := (<a>le_trim</a> hm).<a>trans_lt</a> h\u03bct", [{"full_name": "MeasureTheory.le_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [57, 9], "def_end_pos": [57, 16]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh\u03bct' : \u2191\u2191\u03bc t < \u22a4\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)"}, {"tactic": "specialize h_ind b ht h\u03bct'", "annotated_tactic": ["specialize h_ind b ht h\u03bct'", []], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh\u03bct' : \u2191\u2191\u03bc t < \u22a4\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)", "state_after": "case 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\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_add :\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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh\u03bct' : \u2191\u2191\u03bc t < \u22a4\nh_ind : P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)"}, {"tactic": "rwa [Lp.simpleFunc.coe_indicatorConst] at h_ind", "annotated_tactic": ["rwa [<a>Lp.simpleFunc.coe_indicatorConst</a>] at h_ind", [{"full_name": "MeasureTheory.Lp.simpleFunc.coe_indicatorConst", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [700, 9], "def_end_pos": [700, 27]}]], "state_before": "case 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\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_add :\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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh\u03bct' : \u2191\u2191\u03bc t < \u22a4\nh_ind : P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)", "state_after": "no goals"}, {"tactic": "intro f g hf hg h_disj hfP hgP", "annotated_tactic": ["intro f g hf hg h_disj hfP hgP", []], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (Function.support f) (Function.support g) \u2192\n      (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp f hf) \u2192\n        (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp g hg) \u2192\n          (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))\n            (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))"}, {"tactic": "rw [LinearIsometryEquiv.map_add]", "annotated_tactic": ["rw [<a>LinearIsometryEquiv.map_add</a>]", [{"full_name": "LinearIsometryEquiv.map_add", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [955, 9], "def_end_pos": [955, 16]}]], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf) +\n        \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf) +\n        \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P\n    (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))"}, {"tactic": "have h_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p (\u03bc.trim hm)),\n    ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm (Mem\u2112p.toLp f hf) : Lp F p \u03bc) =\n      (mem\u2112p_of_mem\u2112p_trim hm hf).toLp f :=\n  lpMeasToLpTrimLie_symm_toLp hm", "annotated_tactic": ["have h_eq :\n      \u2200 (f : \u03b1 \u2192 F) (hf : <a>Mem\u2112p</a> f p (\u03bc.trim hm)),\n        ((<a>lpMeasToLpTrimLie</a> F \u211d p \u03bc hm).<a>symm</a> (<a>Mem\u2112p.toLp</a> f hf) : <a>Lp</a> F p \u03bc) =\n          (<a>mem\u2112p_of_mem\u2112p_trim</a> hm hf).<a>toLp</a> f :=\n      <a>lpMeasToLpTrimLie_symm_toLp</a> hm", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}, {"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.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.mem\u2112p_of_mem\u2112p_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1035, 9], "def_end_pos": [1035, 28]}, {"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.lpMeasToLpTrimLie_symm_toLp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [548, 9], "def_end_pos": [548, 36]}]], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P\n    (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\nh_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P\n    (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))"}, {"tactic": "rw [h_eq f hf] at hfP \u22a2", "annotated_tactic": ["rw [h_eq f hf] at hfP \u22a2", []], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\nh_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P\n    (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n      \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\nh_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))"}, {"tactic": "rw [h_eq g hg] at hgP \u22a2", "annotated_tactic": ["rw [h_eq g hg] at hgP \u22a2", []], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\nh_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))", "state_after": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p))\nhgP : P (Mem\u2112p.toLp g (_ : Mem\u2112p g p))\nh_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + Mem\u2112p.toLp g (_ : Mem\u2112p g p))"}, {"tactic": "exact\n  h_add (mem\u2112p_of_mem\u2112p_trim hm hf) (mem\u2112p_of_mem\u2112p_trim hm hg)\n    (aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm hf.aestronglyMeasurable)\n    (aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm hg.aestronglyMeasurable)\n    h_disj hfP hgP", "annotated_tactic": ["exact\n      h_add (<a>mem\u2112p_of_mem\u2112p_trim</a> hm hf) (<a>mem\u2112p_of_mem\u2112p_trim</a> hm hg)\n        (<a>aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim</a> hm hf.aestronglyMeasurable)\n        (<a>aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim</a> hm hg.aestronglyMeasurable)\n        h_disj hfP hgP", [{"full_name": "MeasureTheory.mem\u2112p_of_mem\u2112p_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1035, 9], "def_end_pos": [1035, 28]}, {"full_name": "MeasureTheory.mem\u2112p_of_mem\u2112p_trim", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [1035, 9], "def_end_pos": [1035, 28]}, {"full_name": "MeasureTheory.aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [132, 9], "def_end_pos": [132, 60]}, {"full_name": "MeasureTheory.aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [132, 9], "def_end_pos": [132, 60]}]], "state_before": "case 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\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p))\nhgP : P (Mem\u2112p.toLp g (_ : Mem\u2112p g p))\nh_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n    \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + Mem\u2112p.toLp g (_ : Mem\u2112p g p))", "state_after": "no goals"}, {"tactic": "change IsClosed ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm \u207b\u00b9' {g : lpMeas F \u211d m p \u03bc | P \u2191g})", "annotated_tactic": ["change <a>IsClosed</a> ((<a>lpMeasToLpTrimLie</a> F \u211d p \u03bc hm).<a>symm</a> \u207b\u00b9' {g : <a>lpMeas</a> F \u211d m p \u03bc | P \u2191g})", [{"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [477, 19], "def_end_pos": [477, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [736, 5], "def_end_pos": [736, 9]}, {"full_name": "MeasureTheory.lpMeas", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [222, 5], "def_end_pos": [222, 11]}]], "state_before": "case refine'_3\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 IsClosed {f | (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) f}", "state_after": "case refine'_3\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 IsClosed (\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) \u207b\u00b9' {g | P \u2191g})"}, {"tactic": "exact IsClosed.preimage (LinearIsometryEquiv.continuous _) h_closed", "annotated_tactic": ["exact <a>IsClosed.preimage</a> (<a>LinearIsometryEquiv.continuous</a> _) h_closed", [{"full_name": "IsClosed.preimage", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1752, 9], "def_end_pos": [1752, 26]}, {"full_name": "LinearIsometryEquiv.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [676, 19], "def_end_pos": [676, 29]}]], "state_before": "case refine'_3\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    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)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 IsClosed (\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) \u207b\u00b9' {g | P \u2191g})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.ae_eventually_measure_pos", "start": [100, 1], "end": [116, 48], "traced_tactics": [{"tactic": "set s := {x | \u00ac\u2200\u1da0 a in v.filterAt x, 0 < \u03bc a} with hs", "annotated_tactic": ["set s := {x | \u00ac\u2200\u1da0 a in v.filterAt x, 0 < \u03bc a} with hs", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a"}, {"tactic": "simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs", "annotated_tactic": ["simp (config := { zeta := <a>false</a> }) only [<a>not_lt</a>, <a>not_eventually</a>, <a>nonpos_iff_eq_zero</a>] at hs", [{"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": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "Filter.not_eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 23]}, {"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": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a"}, {"tactic": "change \u03bc s = 0", "annotated_tactic": ["change \u03bc s = 0", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "let f : \u03b1 \u2192 Set (Set \u03b1) := fun _ => {a | \u03bc a = 0}", "annotated_tactic": ["let f : \u03b1 \u2192 <a>Set</a> (<a>Set</a> \u03b1) := fun _ => {a | \u03bc a = 0}", [{"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]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "have h : v.FineSubfamilyOn f s := by\n  intro x hx \u03b5 \u03b5pos\n  rw [hs] at hx\n  simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx\n  rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9\n  exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", "annotated_tactic": ["have h : v.FineSubfamilyOn f s := by\n    intro x hx \u03b5 \u03b5pos\n    rw [hs] at hx\n    simp only [<a>frequently_filterAt_iff</a>, <a>exists_prop</a>, <a>gt_iff_lt</a>, <a>mem_setOf_eq</a>] at hx\n    rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9\n    exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", [{"full_name": "VitaliFamily.frequently_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"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\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"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": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s \u2264 0"}, {"tactic": "calc\n  \u03bc s \u2264 \u2211' x : h.index, \u03bc (h.covering x) := h.measure_le_tsum\n  _ = \u2211' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n  _ = 0 := by simp only [tsum_zero, add_zero]", "annotated_tactic": ["calc\n    \u03bc s \u2264 \u2211' x : h.index, \u03bc (h.covering x) := h.measure_le_tsum\n    _ = \u2211' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n    _ = 0 := by simp only [<a>tsum_zero</a>, <a>add_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": "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\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s \u2264 0", "state_after": "no goals"}, {"tactic": "intro x hx \u03b5 \u03b5pos", "annotated_tactic": ["intro x hx \u03b5 \u03b5pos", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\n\u22a2 FineSubfamilyOn v f s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "rw [hs] at hx", "annotated_tactic": ["rw [hs] at hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx", "annotated_tactic": ["simp only [<a>frequently_filterAt_iff</a>, <a>exists_prop</a>, <a>gt_iff_lt</a>, <a>mem_setOf_eq</a>] at hx", [{"full_name": "VitaliFamily.frequently_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"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\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9", "annotated_tactic": ["rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\na : Set \u03b1\na_sets : a \u2208 setsAt v x\nax : a \u2286 closedBall x \u03b5\n\u03bca : \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", "annotated_tactic": ["exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\na : Set \u03b1\na_sets : a \u2208 setsAt v x\nax : a \u2286 closedBall x \u03b5\n\u03bca : \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0", "state_after": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 (fun x => \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x)) = fun x => 0"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 (fun x => \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x)) = fun x => 0", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\nx : \u2191(FineSubfamilyOn.index h)\n\u22a2 \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = 0"}, {"tactic": "exact h.covering_mem x.2", "annotated_tactic": ["exact h.covering_mem x.2", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\nx : \u2191(FineSubfamilyOn.index h)\n\u22a2 \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = 0", "state_after": "no goals"}, {"tactic": "simp only [tsum_zero, add_zero]", "annotated_tactic": ["simp only [<a>tsum_zero</a>, <a>add_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": "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\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), 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_condCdfRat_atBot", "start": [627, 1], "end": [639, 13], "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\n\u22a2 Tendsto (condCdfRat \u03c1 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)\na : \u03b1\n\u22a2 Tendsto (if a \u2208 condCdfSet \u03c1 then fun r => ENNReal.toReal (preCdf \u03c1 r a) else fun r => if r < 0 then 0 else 1) atBot\n    (\ud835\udcdd 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\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (if a \u2208 condCdfSet \u03c1 then fun r => ENNReal.toReal (preCdf \u03c1 r a) else fun r => if r < 0 then 0 else 1) atBot\n    (\ud835\udcdd 0)", "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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => ENNReal.toReal (preCdf \u03c1 r a)) atBot (\ud835\udcdd 0)\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\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => if r < 0 then 0 else 1) atBot (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.zero_toReal</a>, <a>ENNReal.tendsto_toReal_iff</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]}]], "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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => ENNReal.toReal (preCdf \u03c1 r a)) atBot (\ud835\udcdd 0)", "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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\n\ncase pos.hf\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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4\n\ncase pos.hx\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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 0 \u2260 \u22a4"}, {"tactic": "exact (hasCondCdf_of_mem_condCdfSet h).tendsto_atBot_zero", "annotated_tactic": ["exact (<a>hasCondCdf_of_mem_condCdfSet</a> h).<a>tendsto_atBot_zero</a>", [{"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.tendsto_atBot_zero", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [529, 3], "def_end_pos": [529, 21]}]], "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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "have h' := hasCondCdf_of_mem_condCdfSet h", "annotated_tactic": ["have h' := <a>hasCondCdf_of_mem_condCdfSet</a> h", [{"full_name": "ProbabilityTheory.hasCondCdf_of_mem_condCdfSet", "def_path": "Mathlib/Probability/Kernel/CondCdf.lean", "def_pos": [549, 9], "def_end_pos": [549, 37]}]], "state_before": "case pos.hf\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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4", "state_after": "case pos.hf\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\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4"}, {"tactic": "exact fun r => ((h'.le_one r).trans_lt ENNReal.one_lt_top).ne", "annotated_tactic": ["exact fun r => ((h'.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 pos.hf\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\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4", "state_after": "no goals"}, {"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": "case pos.hx\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\nh : a \u2208 condCdfSet \u03c1\n\u22a2 0 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "refine' (tendsto_congr' _).mp tendsto_const_nhds", "annotated_tactic": ["refine' (<a>tendsto_congr'</a> _).<a>mp</a> <a>tendsto_const_nhds</a>", [{"full_name": "Filter.tendsto_congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3005, 9], "def_end_pos": [3005, 23]}, {"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": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 27]}]], "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\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => if r < 0 then 0 else 1) atBot (\ud835\udcdd 0)", "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\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun x => 0) =\u1da0[atBot] fun r => if r < 0 then 0 else 1"}, {"tactic": "rw [EventuallyEq, eventually_atBot]", "annotated_tactic": ["rw [<a>EventuallyEq</a>, <a>eventually_atBot</a>]", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "Filter.eventually_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [184, 9], "def_end_pos": [184, 25]}]], "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\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun x => 0) =\u1da0[atBot] fun r => if r < 0 then 0 else 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\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2203 a, \u2200 (b : \u211a), b \u2264 a \u2192 0 = if b < 0 then 0 else 1"}, {"tactic": "refine' \u27e8-1, fun q hq => (if_pos (hq.trans_lt _)).symm\u27e9", "annotated_tactic": ["refine' \u27e8-1, fun q hq => (<a>if_pos</a> (hq.trans_lt _)).<a>symm</a>\u27e9", [{"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": "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\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\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2203 a, \u2200 (b : \u211a), b \u2264 a \u2192 0 = if b < 0 then 0 else 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\nh : \u00aca \u2208 condCdfSet \u03c1\nq : \u211a\nhq : q \u2264 -1\n\u22a2 -1 < 0"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "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\nh : \u00aca \u2208 condCdfSet \u03c1\nq : \u211a\nhq : q \u2264 -1\n\u22a2 -1 < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Nat.encard_range", "start": [338, 1], "end": [341, 25], "traced_tactics": [{"tactic": "convert encard_coe_eq_coe_finsetCard (Finset.range k) using 1", "annotated_tactic": ["convert <a>encard_coe_eq_coe_finsetCard</a> (<a>Finset.range</a> k) using 1", [{"full_name": "Set.encard_coe_eq_coe_finsetCard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [84, 9], "def_end_pos": [84, 37]}, {"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.102669\ns t : Set \u03b1\nk : \u2115\n\u22a2 encard {i | i < k} = \u2191k", "state_after": "case h.e'_2\n\u03b1 : Type ?u.102669\ns t : Set \u03b1\nk : \u2115\n\u22a2 encard {i | i < k} = encard \u2191(Finset.range k)\n\ncase h.e'_3\n\u03b1 : Type ?u.102669\ns t : Set \u03b1\nk : \u2115\n\u22a2 \u2191k = \u2191(Finset.card (Finset.range k))"}, {"tactic": "rw [Finset.card_range]", "annotated_tactic": ["rw [<a>Finset.card_range</a>]", [{"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}]], "state_before": "case h.e'_3\n\u03b1 : Type ?u.102669\ns t : Set \u03b1\nk : \u2115\n\u22a2 \u2191k = \u2191(Finset.card (Finset.range k))", "state_after": "no goals"}, {"tactic": "rw [Finset.coe_range, Iio_def]", "annotated_tactic": ["rw [<a>Finset.coe_range</a>, <a>Iio_def</a>]", [{"full_name": "Finset.coe_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3042, 9], "def_end_pos": [3042, 18]}, {"full_name": "Set.Iio_def", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "case h.e'_2\n\u03b1 : Type ?u.102669\ns t : Set \u03b1\nk : \u2115\n\u22a2 encard {i | i < k} = encard \u2191(Finset.range k)", "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": [985, 1], "end": [987, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.map\u2082'_mk''", "start": [762, 1], "end": [765, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge", "start": [177, 1], "end": [188, 68], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\n\u22a2 (\u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F) \u2194\n    \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L", "state_after": "case mp\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\n\u22a2 (\u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F) \u2192\n    \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L\n\ncase mpr\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\n\u22a2 (\u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L) \u2192\n    \u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F"}, {"tactic": "intro h G G_open", "annotated_tactic": ["intro h G G_open", []], "state_before": "case mp\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\n\u22a2 (\u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F) \u2192\n    \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L", "state_after": "case mp\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nh : \u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F\nG : Set \u03a9\nG_open : IsOpen G\n\u22a2 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L"}, {"tactic": "exact le_measure_liminf_of_limsup_measure_compl_le\n  G_open.measurableSet (h G\u1d9c (isClosed_compl_iff.mpr G_open))", "annotated_tactic": ["exact <a>le_measure_liminf_of_limsup_measure_compl_le</a>\n      G_open.measurableSet (h G\u1d9c (isClosed_compl_iff.mpr G_open))", [{"full_name": "MeasureTheory.le_measure_liminf_of_limsup_measure_compl_le", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [131, 9], "def_end_pos": [131, 53]}]], "state_before": "case mp\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nh : \u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F\nG : Set \u03a9\nG_open : IsOpen G\n\u22a2 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L", "state_after": "no goals"}, {"tactic": "intro h F F_closed", "annotated_tactic": ["intro h F F_closed", []], "state_before": "case mpr\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\n\u22a2 (\u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L) \u2192\n    \u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F", "state_after": "case mpr\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nh : \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L\nF : Set \u03a9\nF_closed : IsClosed F\n\u22a2 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F"}, {"tactic": "exact limsup_measure_le_of_le_liminf_measure_compl\n  F_closed.measurableSet (h F\u1d9c (isOpen_compl_iff.mpr F_closed))", "annotated_tactic": ["exact <a>limsup_measure_le_of_le_liminf_measure_compl</a>\n      F_closed.measurableSet (h F\u1d9c (isOpen_compl_iff.mpr F_closed))", [{"full_name": "MeasureTheory.limsup_measure_le_of_le_liminf_measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "def_pos": [159, 9], "def_end_pos": [159, 53]}]], "state_before": "case mpr\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nh : \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L\nF : Set \u03a9\nF_closed : IsClosed F\n\u22a2 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.ae_inf_principal_eq_mk", "start": [70, 1], "end": [71, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.StronglyMeasurable.mono", "start": [369, 11], "end": [376, 38], "traced_tactics": [{"tactic": "let f_approx : \u2115 \u2192 @SimpleFunc \u03b1 m \u03b2 := fun n =>\n  @SimpleFunc.mk \u03b1 m \u03b2\n    (hf.approx n)\n    (fun x => h_mono _ (SimpleFunc.measurableSet_fiber' _ x))\n    (SimpleFunc.finite_range (hf.approx n))", "annotated_tactic": ["let f_approx : \u2115 \u2192 @<a>SimpleFunc</a> \u03b1 m \u03b2 := fun n =>\n    @<a>SimpleFunc.mk</a> \u03b1 m \u03b2\n      (hf.approx n)\n      (fun x => h_mono _ (<a>SimpleFunc.measurableSet_fiber'</a> _ x))\n      (<a>SimpleFunc.finite_range</a> (hf.approx n))", [{"full_name": "MeasureTheory.SimpleFunc", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [44, 11], "def_end_pos": [44, 21]}, {"full_name": "MeasureTheory.SimpleFunc.mk", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [44, 29], "def_end_pos": [44, 74]}, {"full_name": "MeasureTheory.SimpleFunc.measurableSet_fiber'", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [46, 3], "def_end_pos": [46, 23]}, {"full_name": "MeasureTheory.SimpleFunc.finite_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [76, 9], "def_end_pos": [76, 21]}]], "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 g : \u03b1 \u2192 \u03b2\nm m' : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : StronglyMeasurable f\nh_mono : m' \u2264 m\n\u22a2 StronglyMeasurable f", "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 g : \u03b1 \u2192 \u03b2\nm m' : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : StronglyMeasurable f\nh_mono : m' \u2264 m\nf_approx : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 :=\n  fun n =>\n    { toFun := \u2191(StronglyMeasurable.approx hf n),\n      measurableSet_fiber' := (_ : \u2200 (x : \u03b2), MeasurableSet (\u2191(StronglyMeasurable.approx hf n) \u207b\u00b9' {x})),\n      finite_range' := (_ : Set.Finite (range \u2191(StronglyMeasurable.approx hf n))) }\n\u22a2 StronglyMeasurable f"}, {"tactic": "exact \u27e8f_approx, hf.tendsto_approx\u27e9", "annotated_tactic": ["exact \u27e8f_approx, hf.tendsto_approx\u27e9", []], "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 g : \u03b1 \u2192 \u03b2\nm m' : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : StronglyMeasurable f\nh_mono : m' \u2264 m\nf_approx : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 :=\n  fun n =>\n    { toFun := \u2191(StronglyMeasurable.approx hf n),\n      measurableSet_fiber' := (_ : \u2200 (x : \u03b2), MeasurableSet (\u2191(StronglyMeasurable.approx hf n) \u207b\u00b9' {x})),\n      finite_range' := (_ : Set.Finite (range \u2191(StronglyMeasurable.approx hf n))) }\n\u22a2 StronglyMeasurable f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IsROrC.interval_integral_ofReal", "start": [651, 8], "end": [653, 67], "traced_tactics": [{"tactic": "simp only [intervalIntegral, integral_ofReal, IsROrC.ofReal_sub]", "annotated_tactic": ["simp only [<a>intervalIntegral</a>, <a>integral_ofReal</a>, <a>IsROrC.ofReal_sub</a>]", [{"full_name": "intervalIntegral", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [447, 5], "def_end_pos": [447, 21]}, {"full_name": "integral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1191, 9], "def_end_pos": [1191, 24]}, {"full_name": "IsROrC.ofReal_sub", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 19]}]], "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\n\ud835\udd5c : Type u_6\ninst\u271d : IsROrC \ud835\udd5c\na b : \u211d\n\u03bc : Measure \u211d\nf : \u211d \u2192 \u211d\n\u22a2 \u222b (x : \u211d) in a..b, \u2191(f x) \u2202\u03bc = \u2191(\u222b (x : \u211d) in a..b, f x \u2202\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_range_ite", "start": [1030, 1], "end": [1032, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.measure_eq_top_of_setLintegral_ne_top", "start": [858, 1], "end": [860, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.Mem\u2112p.add", "start": [1234, 1], "end": [1235, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/StrongLaw.lean", "full_name": "ProbabilityTheory.strong_law_aux2", "start": [528, 1], "end": [540, 75], "traced_tactics": [{"tactic": "obtain \u27e8v, -, v_pos, v_lim\u27e9 :\n    \u2203 v : \u2115 \u2192 \u211d, StrictAnti v \u2227 (\u2200 n : \u2115, 0 < v n) \u2227 Tendsto v atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d)", "annotated_tactic": ["obtain \u27e8v, -, v_pos, v_lim\u27e9 :\n      \u2203 v : \u2115 \u2192 \u211d, <a>StrictAnti</a> v \u2227 (\u2200 n : \u2115, 0 < v n) \u2227 <a>Tendsto</a> v <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": "\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\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    (fun n =>\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) =o[atTop]\n      fun n => \u2191\u230ac ^ n\u230b\u208a", "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 : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    (fun n =>\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) =o[atTop]\n      fun n => \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "have := fun i => strong_law_aux1 X hint hindep hident hnonneg c_one (v_pos i)", "annotated_tactic": ["have := fun i => <a>strong_law_aux1</a> X hint hindep hident hnonneg c_one (v_pos i)", [{"full_name": "ProbabilityTheory.strong_law_aux1", "def_path": "Mathlib/Probability/StrongLaw.lean", "def_pos": [416, 9], "def_end_pos": [416, 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 : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    (fun n =>\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) =o[atTop]\n      fun n => \u2191\u230ac ^ n\u230b\u208a", "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 : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    (fun n =>\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) =o[atTop]\n      fun n => \u2191\u230ac ^ n\u230b\u208a"}, {"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 : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n    (fun n =>\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) =o[atTop]\n      fun n => \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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n =>\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) =o[atTop]\n    fun n => \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "apply Asymptotics.isLittleO_iff.2 fun \u03b5 \u03b5pos => ?_", "annotated_tactic": ["apply <a>Asymptotics.isLittleO_iff</a>.2 fun \u03b5 \u03b5pos => ?_", [{"full_name": "Asymptotics.isLittleO_iff", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [190, 9], "def_end_pos": [190, 22]}]], "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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n =>\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) =o[atTop]\n    fun n => \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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n          \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n      \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016"}, {"tactic": "obtain \u27e8i, hi\u27e9 : \u2203 i, v i < \u03b5 := ((tendsto_order.1 v_lim).2 \u03b5 \u03b5pos).exists", "annotated_tactic": ["obtain \u27e8i, hi\u27e9 : \u2203 i, v i < \u03b5 := ((<a>tendsto_order</a>.1 v_lim).2 \u03b5 \u03b5pos).<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": "\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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n          \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n      \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016", "state_after": "case 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 : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n          \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n      \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016"}, {"tactic": "filter_upwards [h\u03c9 i] with n hn", "annotated_tactic": ["filter_upwards [h\u03c9 i] with n hn", []], "state_before": "case 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 : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n    \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n          \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n      \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016", "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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\nn : \u2115\nhn :\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    v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2016\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\u2016 \u2264\n    \u03b5 * \u2016\u2191\u230ac ^ n\u230b\u208a\u2016"}, {"tactic": "simp only [Real.norm_eq_abs, LatticeOrderedGroup.abs_abs, Nat.abs_cast]", "annotated_tactic": ["simp only [<a>Real.norm_eq_abs</a>, <a>LatticeOrderedGroup.abs_abs</a>, <a>Nat.abs_cast</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": "LatticeOrderedGroup.abs_abs", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [440, 30], "def_end_pos": [440, 37]}, {"full_name": "Nat.abs_cast", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [132, 9], "def_end_pos": [132, 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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\nn : \u2115\nhn :\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    v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2016\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\u2016 \u2264\n    \u03b5 * \u2016\u2191\u230ac ^ n\u230b\u208a\u2016", "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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\nn : \u2115\nhn :\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    v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 |\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| \u2264\n    \u03b5 * \u2191\u230ac ^ n\u230b\u208a"}, {"tactic": "exact hn.le.trans (mul_le_mul_of_nonneg_right hi.le (Nat.cast_nonneg _))", "annotated_tactic": ["exact hn.le.trans (<a>mul_le_mul_of_nonneg_right</a> hi.le (<a>Nat.cast_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": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "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\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n  \u2200 (i : \u2115),\n    \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          v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n  \u2200 (i : \u2115),\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        v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\nn : \u2115\nhn :\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    v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 |\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| \u2264\n    \u03b5 * \u2191\u230ac ^ n\u230b\u208a", "state_after": "no goals"}]}, {"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": [1590, 1], "end": [1619, 53], "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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": "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : p = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have hp1 : 1 \u2264 p.toReal := by\n  rw [\u2190 ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_one]\n  exact hp", "annotated_tactic": ["have hp1 : 1 \u2264 p.toReal := by\n    rw [\u2190 <a>ENNReal.ofReal_le_iff_le_toReal</a> hp_top, <a>ENNReal.ofReal_one</a>]\n    exact hp", [{"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.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 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 : \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\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_cau' : \u2200 N n m : \u2115, N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p.toReal \u03bc < B N := by\n  intro N n m hn hm\n  specialize h_cau N n m hn hm\n  rwa [snorm_eq_snorm' (zero_lt_one.trans_le hp).ne.symm hp_top] at h_cau", "annotated_tactic": ["have h_cau' : \u2200 N n m : \u2115, N \u2264 n \u2192 N \u2264 m \u2192 <a>snorm'</a> (f n - f m) p.toReal \u03bc < B N := by\n    intro N n m hn hm\n    specialize h_cau N n m hn hm\n    rwa [<a>snorm_eq_snorm'</a> (zero_lt_one.trans_le hp).ne.symm hp_top] at h_cau", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\nh_cau' : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "exact ae_tendsto_of_cauchy_snorm' hf hp1 hB h_cau'", "annotated_tactic": ["exact <a>ae_tendsto_of_cauchy_snorm'</a> hf hp1 hB h_cau'", [{"full_name": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1547, 9], "def_end_pos": [1547, 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 : \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\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\nh_cau' : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) (ENNReal.toReal 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": "no goals"}, {"tactic": "simp_rw [hp_top] at *", "annotated_tactic": ["simp_rw [hp_top] at *", []], "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : p = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_cau_ae : \u2200\u1d50 x \u2202\u03bc, \u2200 N n m, N \u2264 n \u2192 N \u2264 m \u2192 (\u2016(f n - f m) x\u2016\u208a : \u211d\u22650\u221e) < B N := by\n  simp_rw [ae_all_iff]\n  exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)", "annotated_tactic": ["have h_cau_ae : \u2200\u1d50 x \u2202\u03bc, \u2200 N n m, N \u2264 n \u2192 N \u2264 m \u2192 (\u2016(f n - f m) x\u2016\u208a : \u211d\u22650\u221e) < B N := by\n      simp_rw [<a>ae_all_iff</a>]\n      exact fun N n m hnN hmN => <a>ae_lt_of_essSup_lt</a> (h_cau N n m hnN hmN)", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "ae_lt_of_essSup_lt", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [98, 9], "def_end_pos": [98, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "simp_rw [snorm_exponent_top, snormEssSup] at h_cau", "annotated_tactic": ["simp_rw [<a>snorm_exponent_top</a>, <a>snormEssSup</a>] at h_cau", [{"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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _", "annotated_tactic": ["refine' h_cau_ae.mono fun x hx => <a>cauchySeq_tendsto_of_complete</a> _", [{"full_name": "cauchySeq_tendsto_of_complete", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [468, 9], "def_end_pos": [468, 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 : \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\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \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": "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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 CauchySeq fun n => f n x"}, {"tactic": "refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _", "annotated_tactic": ["refine' <a>cauchySeq_of_le_tendsto_0</a> (fun n => (B n).<a>toReal</a>) _ _", [{"full_name": "cauchySeq_of_le_tendsto_0", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1563, 9], "def_end_pos": [1563, 34]}, {"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\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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 CauchySeq fun n => f n x", "state_after": "case pos.refine'_1\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N\n\ncase pos.refine'_2\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 Tendsto (fun n => ENNReal.toReal (B n)) atTop (\ud835\udcdd 0)"}, {"tactic": "simp_rw [ae_all_iff]", "annotated_tactic": ["simp_rw [<a>ae_all_iff</a>]", [{"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\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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N", "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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\n\u22a2 \u2200 (i i_1 i_2 : \u2115), i \u2264 i_1 \u2192 i \u2264 i_2 \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2016(f i_1 - f i_2) a\u2016\u208a < B i"}, {"tactic": "exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)", "annotated_tactic": ["exact fun N n m hnN hmN => <a>ae_lt_of_essSup_lt</a> (h_cau N n m hnN hmN)", [{"full_name": "ae_lt_of_essSup_lt", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [98, 9], "def_end_pos": [98, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp : True\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N\nhp_top : True\n\u22a2 \u2200 (i i_1 i_2 : \u2115), i \u2264 i_1 \u2192 i \u2264 i_2 \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2016(f i_1 - f i_2) a\u2016\u208a < B i", "state_after": "no goals"}, {"tactic": "intro n m N hnN hmN", "annotated_tactic": ["intro n m N hnN hmN", []], "state_before": "case pos.refine'_1\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\n\u22a2 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N"}, {"tactic": "specialize hx N n m hnN hmN", "annotated_tactic": ["specialize hx N n m hnN hmN", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\n\u22a2 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\nhx : \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N"}, {"tactic": "rw [dist_eq_norm, \u2190 ENNReal.toReal_ofReal (norm_nonneg _),\n  ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]", "annotated_tactic": ["rw [<a>dist_eq_norm</a>, \u2190 <a>ENNReal.toReal_ofReal</a> (<a>norm_nonneg</a> _),\n        <a>ENNReal.toReal_le_toReal</a> <a>ENNReal.ofReal_ne_top</a> (<a>ENNReal.ne_top_of_tsum_ne_top</a> hB N)]", [{"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.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_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.ne_top_of_tsum_ne_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [893, 19], "def_end_pos": [893, 40]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\nhx : \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\nhx : \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 ENNReal.ofReal \u2016f n x - f m x\u2016 \u2264 B N"}, {"tactic": "rw [\u2190 ofReal_norm_eq_coe_nnnorm] at hx", "annotated_tactic": ["rw [\u2190 <a>ofReal_norm_eq_coe_nnnorm</a>] at hx", [{"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 pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\nhx : \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 ENNReal.ofReal \u2016f n x - f m x\u2016 \u2264 B N", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\nhx : ENNReal.ofReal \u2016(f n - f m) x\u2016 < B N\n\u22a2 ENNReal.ofReal \u2016f n x - f m x\u2016 \u2264 B N"}, {"tactic": "exact hx.le", "annotated_tactic": ["exact hx.le", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nn m N : \u2115\nhnN : N \u2264 n\nhmN : N \u2264 m\nhx : ENNReal.ofReal \u2016(f n - f m) x\u2016 < B N\n\u22a2 ENNReal.ofReal \u2016f n x - f m x\u2016 \u2264 B N", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.zero_toReal</a>]", [{"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}]], "state_before": "case pos.refine'_2\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 Tendsto (fun n => ENNReal.toReal (B n)) atTop (\ud835\udcdd 0)", "state_after": "case pos.refine'_2\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 Tendsto (fun n => ENNReal.toReal (B n)) atTop (\ud835\udcdd (ENNReal.toReal 0))"}, {"tactic": "exact\n  Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)\n    (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)", "annotated_tactic": ["exact\n        <a>Tendsto.comp</a> (g := <a>ENNReal.toReal</a>) (<a>ENNReal.tendsto_toReal</a> <a>ENNReal.zero_ne_top</a>)\n          (<a>ENNReal.tendsto_atTop_zero_of_tsum_ne_top</a> hB)", [{"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"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": "ENNReal.tendsto_atTop_zero_of_tsum_ne_top", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [954, 9], "def_end_pos": [954, 42]}]], "state_before": "case pos.refine'_2\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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp hp_top : True\nh_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N\nx : \u03b1\nhx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N\n\u22a2 Tendsto (fun n => ENNReal.toReal (B n)) atTop (\ud835\udcdd (ENNReal.toReal 0))", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_one]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.ofReal_le_iff_le_toReal</a> hp_top, <a>ENNReal.ofReal_one</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.ofReal_one", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [248, 17], "def_end_pos": [248, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\n\u22a2 1 \u2264 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\n\u22a2 1 \u2264 p"}, {"tactic": "exact hp", "annotated_tactic": ["exact hp", []], "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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\n\u22a2 1 \u2264 p", "state_after": "no goals"}, {"tactic": "intro N n m hn hm", "annotated_tactic": ["intro N n m hn hm", []], "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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N"}, {"tactic": "specialize h_cau N n m hn hm", "annotated_tactic": ["specialize h_cau N n m hn hm", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 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\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nh_cau : snorm (f n - f m) p \u03bc < B N\n\u22a2 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N"}, {"tactic": "rwa [snorm_eq_snorm' (zero_lt_one.trans_le hp).ne.symm hp_top] at h_cau", "annotated_tactic": ["rwa [<a>snorm_eq_snorm'</a> (zero_lt_one.trans_le hp).ne.symm hp_top] at h_cau", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [88, 9], "def_end_pos": [88, 24]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d 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 : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nhp_top : \u00acp = \u22a4\nhp1 : 1 \u2264 ENNReal.toReal p\nN n m : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\nh_cau : snorm (f n - f m) p \u03bc < B N\n\u22a2 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.mkMetric_le_liminf_tsum", "start": [539, 1], "end": [559, 21], "traced_tactics": [{"tactic": "haveI : \u2200 n, Encodable (\u03b9 n) := fun n => Encodable.ofCountable _", "annotated_tactic": ["haveI : \u2200 n, <a>Encodable</a> (\u03b9 n) := fun n => <a>Encodable.ofCountable</a> _", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [45, 7], "def_end_pos": [45, 16]}, {"full_name": "Encodable.ofCountable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [458, 19], "def_end_pos": [458, 30]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l"}, {"tactic": "simp only [mkMetric_apply]", "annotated_tactic": ["simp only [<a>mkMetric_apply</a>]", [{"full_name": "MeasureTheory.Measure.mkMetric_apply", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [503, 9], "def_end_pos": [503, 23]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u22a2 \u2a06 r,\n      \u2a06 (_ : 0 < r),\n        \u2a05 t,\n          \u2a05 (_ : s \u2286 iUnion t),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l"}, {"tactic": "refine' iSup\u2082_le fun \u03b5 h\u03b5 => _", "annotated_tactic": ["refine' <a>iSup\u2082_le</a> fun \u03b5 h\u03b5 => _", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u22a2 \u2a06 r,\n      \u2a06 (_ : 0 < r),\n        \u2a05 t,\n          \u2a05 (_ : s \u2286 iUnion t),\n            \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l"}, {"tactic": "refine' le_of_forall_le_of_dense fun c hc => _", "annotated_tactic": ["refine' <a>le_of_forall_le_of_dense</a> fun c hc => _", [{"full_name": "le_of_forall_le_of_dense", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1329, 9], "def_end_pos": [1329, 33]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l", "state_after": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c"}, {"tactic": "rcases ((frequently_lt_of_liminf_lt (by isBoundedDefault) hc).and_eventually\n      ((hr.eventually (gt_mem_nhds h\u03b5)).and (ht.and hst))).exists with\n  \u27e8n, hn, hrn, htn, hstn\u27e9", "annotated_tactic": ["rcases ((<a>frequently_lt_of_liminf_lt</a> (by isBoundedDefault) hc).<a>and_eventually</a>\n        ((hr.eventually (<a>gt_mem_nhds</a> h\u03b5)).<a>and</a> (ht.and hst))).<a>exists</a> with\n    \u27e8n, hn, hrn, htn, hstn\u27e9", [{"full_name": "Filter.frequently_lt_of_liminf_lt", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 35]}, {"full_name": "Filter.Frequently.and_eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 34]}, {"full_name": "gt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 20]}, {"full_name": "Filter.Eventually.and", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1103, 19], "def_end_pos": [1103, 33]}, {"full_name": "Filter.Frequently.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1302, 9], "def_end_pos": [1302, 26]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c", "state_after": "case intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c"}, {"tactic": "set u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b", "annotated_tactic": ["set u : \u2115 \u2192 <a>Set</a> X := fun j => \u22c3 b \u2208 <a>decode\u2082</a> (\u03b9 n) j, t n b", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Encodable.decode\u2082", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [188, 5], "def_end_pos": [188, 12]}]], "state_before": "case intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c", "state_after": "case intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c"}, {"tactic": "refine' iInf\u2082_le_of_le u (by rwa [iUnion_decode\u2082]) _", "annotated_tactic": ["refine' <a>iInf\u2082_le_of_le</a> u (by rwa [<a>iUnion_decode\u2082</a>]) _", [{"full_name": "iInf\u2082_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [870, 9], "def_end_pos": [870, 23]}, {"full_name": "Encodable.iUnion_decode\u2082", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [35, 9], "def_end_pos": [35, 23]}]], "state_before": "case intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c", "state_after": "case intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 \u2a05 (_ : \u2200 (n : \u2115), diam (u n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c"}, {"tactic": "refine' iInf_le_of_le (fun j => _) _", "annotated_tactic": ["refine' <a>iInf_le_of_le</a> (fun j => _) _", [{"full_name": "iInf_le_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [853, 9], "def_end_pos": [853, 22]}]], "state_before": "case intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 \u2a05 (_ : \u2200 (n : \u2115), diam (u n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c", "state_after": "case intro.intro.intro.intro.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\nj : \u2115\n\u22a2 diam (u j) \u2264 \u03b5\n\ncase intro.intro.intro.intro.refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) l fun n => \u2211' (i : \u03b9 n), m (diam (t n i))", "state_after": "no goals"}, {"tactic": "rwa [iUnion_decode\u2082]", "annotated_tactic": ["rwa [<a>iUnion_decode\u2082</a>]", [{"full_name": "Encodable.iUnion_decode\u2082", "def_path": "Mathlib/Logic/Encodable/Lattice.lean", "def_pos": [35, 9], "def_end_pos": [35, 23]}]], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 s \u2286 iUnion u", "state_after": "no goals"}, {"tactic": "rw [EMetric.diam_iUnion_mem_option]", "annotated_tactic": ["rw [<a>EMetric.diam_iUnion_mem_option</a>]", [{"full_name": "EMetric.diam_iUnion_mem_option", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [928, 9], "def_end_pos": [928, 31]}]], "state_before": "case intro.intro.intro.intro.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\nj : \u2115\n\u22a2 diam (u j) \u2264 \u03b5", "state_after": "case intro.intro.intro.intro.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\nj : \u2115\n\u22a2 \u2a06 i \u2208 decode\u2082 (\u03b9 n) j, diam (t n i) \u2264 \u03b5"}, {"tactic": "exact iSup\u2082_le fun _ _ => (htn _).trans hrn.le", "annotated_tactic": ["exact <a>iSup\u2082_le</a> fun _ _ => (htn _).<a>trans</a> hrn.le", [{"full_name": "iSup\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}, {"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.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\nj : \u2115\n\u22a2 \u2a06 i \u2208 decode\u2082 (\u03b9 n) j, diam (t n i) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "calc\n  (\u2211' j : \u2115, \u2a06 _ : (u j).Nonempty, m (diam (u j))) = _ :=\n    tsum_iUnion_decode\u2082 (fun t : Set X => \u2a06 _ : t.Nonempty, m (diam t)) (by simp) _\n  _ \u2264 \u2211' i : \u03b9 n, m (diam (t n i)) := (ENNReal.tsum_le_tsum fun b => iSup_le fun _ => le_rfl)\n  _ \u2264 c := hn.le", "annotated_tactic": ["calc\n      (\u2211' j : \u2115, \u2a06 _ : (u j).<a>Nonempty</a>, m (<a>diam</a> (u j))) = _ :=\n        <a>tsum_iUnion_decode\u2082</a> (fun t : <a>Set</a> X => \u2a06 _ : t.Nonempty, m (<a>diam</a> t)) (by simp) _\n      _ \u2264 \u2211' i : \u03b9 n, m (<a>diam</a> (t n i)) := (<a>ENNReal.tsum_le_tsum</a> fun b => <a>iSup_le</a> fun _ => <a>le_rfl</a>)\n      _ \u2264 c := hn.le", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "EMetric.diam", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [881, 19], "def_end_pos": [881, 23]}, {"full_name": "tsum_iUnion_decode\u2082", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 28]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "EMetric.diam", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [881, 19], "def_end_pos": [881, 23]}, {"full_name": "EMetric.diam", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [881, 19], "def_end_pos": [881, 23]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"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.intro.refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 i, t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n\u22a2 (fun t => \u2a06 (_ : Set.Nonempty t), m (diam t)) \u2205 = 0", "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": [273, 1], "end": [277, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "full_name": "Nat.div_gcd_pos_of_pos_left", "start": [98, 1], "end": [99, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "Continuous.ae_eq_iff_eq", "start": [147, 1], "end": [149, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.Realizer.ofEquiv_\u03c3", "start": [200, 1], "end": [200, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "generateFrom_pi_eq", "start": [103, 1], "end": [128, 57], "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": "\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 MeasurableSpace.pi = generateFrom (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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 MeasurableSpace.pi = generateFrom (Set.pi univ '' Set.pi univ C)"}, {"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 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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 MeasurableSpace.pi = generateFrom (Set.pi univ '' Set.pi univ C)", "state_after": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 MeasurableSpace.pi \u2264 generateFrom (Set.pi univ '' Set.pi univ C)\n\ncase intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 generateFrom (Set.pi univ '' Set.pi univ C) \u2264 MeasurableSpace.pi"}, {"tactic": "refine' iSup_le _", "annotated_tactic": ["refine' <a>iSup_le</a> _", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 MeasurableSpace.pi \u2264 generateFrom (Set.pi univ '' Set.pi univ C)", "state_after": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 \u2200 (i : \u03b9),\n    MeasurableSpace.comap (fun b => b i) ((fun i => generateFrom (C i)) i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C)"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 \u2200 (i : \u03b9),\n    MeasurableSpace.comap (fun b => b i) ((fun i => generateFrom (C i)) i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C)", "state_after": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\n\u22a2 MeasurableSpace.comap (fun b => b i) ((fun i => generateFrom (C i)) i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C)"}, {"tactic": "rw [comap_generateFrom]", "annotated_tactic": ["rw [<a>comap_generateFrom</a>]", [{"full_name": "MeasurableSpace.comap_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [198, 9], "def_end_pos": [198, 27]}]], "state_before": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\n\u22a2 MeasurableSpace.comap (fun b => b i) ((fun i => generateFrom (C i)) i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C)", "state_after": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\n\u22a2 generateFrom ((preimage fun b => b i) '' C i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C)"}, {"tactic": "apply generateFrom_le", "annotated_tactic": ["apply <a>generateFrom_le</a>", [{"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}]], "state_before": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\n\u22a2 generateFrom ((preimage fun b => b i) '' C i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C)", "state_after": "case intro.a.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\n\u22a2 \u2200 (t : Set ((a : \u03b9) \u2192 (fun i => \u03b1 i) a)), t \u2208 (preimage fun b => b i) '' C i \u2192 MeasurableSet t"}, {"tactic": "rintro _ \u27e8s, hs, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8s, hs, rfl\u27e9", []], "state_before": "case intro.a.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\n\u22a2 \u2200 (t : Set ((a : \u03b9) \u2192 (fun i => \u03b1 i) a)), t \u2208 (preimage fun b => b i) '' C i \u2192 MeasurableSet t", "state_after": "case intro.a.h.intro.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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s)"}, {"tactic": "choose t h1t h2t using hC", "annotated_tactic": ["choose t h1t h2t using hC", []], "state_before": "case intro.a.h.intro.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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s)", "state_after": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\n\u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s)"}, {"tactic": "simp_rw [eval_preimage, \u2190 h2t]", "annotated_tactic": ["simp_rw [<a>eval_preimage</a>, \u2190 h2t]", [{"full_name": "Set.eval_preimage", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [890, 9], "def_end_pos": [890, 22]}]], "state_before": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\n\u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s)", "state_after": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\n\u22a2 MeasurableSet (Set.pi univ (update (fun i => \u22c3 n, t i n) i s))"}, {"tactic": "rw [\u2190 @iUnion_const _ \u2115 _ s]", "annotated_tactic": ["rw [\u2190 @<a>iUnion_const</a> _ \u2115 _ s]", [{"full_name": "Set.iUnion_const", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [588, 7], "def_end_pos": [588, 19]}]], "state_before": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\n\u22a2 MeasurableSet (Set.pi univ (update (fun i => \u22c3 n, t i n) i s))", "state_after": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\n\u22a2 MeasurableSet (Set.pi univ (update (fun i => \u22c3 n, t i n) i (\u22c3 x, s)))"}, {"tactic": "rw [this, \u2190 iUnion_univ_pi]", "annotated_tactic": ["rw [this, \u2190 <a>iUnion_univ_pi</a>]", [{"full_name": "Set.iUnion_univ_pi", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2180, 9], "def_end_pos": [2180, 23]}]], "state_before": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\n\u22a2 MeasurableSet (Set.pi univ (update (fun i => \u22c3 n, t i n) i (\u22c3 x, s)))", "state_after": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\n\u22a2 MeasurableSet (\u22c3 x, Set.pi univ fun i_1 => update (fun i' => t i' (x i_1)) i s i_1)"}, {"tactic": "apply MeasurableSet.iUnion", "annotated_tactic": ["apply <a>MeasurableSet.iUnion</a>", [{"full_name": "MeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [115, 19], "def_end_pos": [115, 39]}]], "state_before": "case intro.a.h.intro.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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\n\u22a2 MeasurableSet (\u22c3 x, Set.pi univ fun i_1 => update (fun i' => t i' (x i_1)) i s i_1)", "state_after": "case intro.a.h.intro.intro.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\n\u22a2 \u2200 (b : \u03b9 \u2192 \u2115), MeasurableSet (Set.pi univ fun i_1 => update (fun i' => t i' (b i_1)) i s i_1)"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case intro.a.h.intro.intro.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\n\u22a2 \u2200 (b : \u03b9 \u2192 \u2115), MeasurableSet (Set.pi univ fun i_1 => update (fun i' => t i' (b i_1)) i s i_1)", "state_after": "case intro.a.h.intro.intro.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\n\u22a2 MeasurableSet (Set.pi univ fun i_1 => update (fun i' => t i' (n i_1)) i s i_1)"}, {"tactic": "apply measurableSet_generateFrom", "annotated_tactic": ["apply <a>measurableSet_generateFrom</a>", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}]], "state_before": "case intro.a.h.intro.intro.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\n\u22a2 MeasurableSet (Set.pi univ fun i_1 => update (fun i' => t i' (n i_1)) i s i_1)", "state_after": "case intro.a.h.intro.intro.h.ht\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\n\u22a2 (Set.pi univ fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) \u2208 Set.pi univ '' Set.pi univ C"}, {"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": "case intro.a.h.intro.intro.h.ht\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\n\u22a2 (Set.pi univ fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) \u2208 Set.pi univ '' Set.pi univ C", "state_after": "case intro.a.h.intro.intro.h.ht.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\n\u22a2 (fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) \u2208 Set.pi univ C"}, {"tactic": "intro j _", "annotated_tactic": ["intro j _", []], "state_before": "case intro.a.h.intro.intro.h.ht.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\n\u22a2 (fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) \u2208 Set.pi univ C", "state_after": "case intro.a.h.intro.intro.h.ht.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 (fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) j \u2208 C j"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case intro.a.h.intro.intro.h.ht.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 (fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) j \u2208 C j", "state_after": "case intro.a.h.intro.intro.h.ht.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j"}, {"tactic": "by_cases h : j = i", "annotated_tactic": ["by_cases h : j = i", []], "state_before": "case intro.a.h.intro.intro.h.ht.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j", "state_after": "case pos\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j\n\ncase neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "case pos\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j\n\ncase neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j", "state_after": "case pos\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))\nval\u271d : Encodable \u03b9\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\ns : Set ((fun i => \u03b1 i) j)\nhs : s \u2208 C j\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) j (\u22c3 x, s)) =\n    Set.pi univ fun k => \u22c3 j_1, update (fun i' => t i' j_1) j s k\n\u22a2 update (fun i' => t i' (n j)) j s j \u2208 C j\n\ncase neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j"}, {"tactic": "rwa [update_same]", "annotated_tactic": ["rwa [<a>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\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))\nval\u271d : Encodable \u03b9\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\ns : Set ((fun i => \u03b1 i) j)\nhs : s \u2208 C j\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) j (\u22c3 x, s)) =\n    Set.pi univ fun k => \u22c3 j_1, update (fun i' => t i' j_1) j s k\n\u22a2 update (fun i' => t i' (n j)) j s j \u2208 C j\n\ncase neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j", "state_after": "case neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j"}, {"tactic": "rw [update_noteq h]", "annotated_tactic": ["rw [<a>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\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j", "state_after": "case neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 t j (n j) \u2208 C j"}, {"tactic": "apply h1t", "annotated_tactic": ["apply h1t", []], "state_before": "case neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nthis :\n  Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k\nn : \u03b9 \u2192 \u2115\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 t j (n j) \u2208 C j", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\n\u22a2 Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k", "state_after": "case h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) \u2194\n    x\u271d \u2208 Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k"}, {"tactic": "simp_rw [mem_univ_pi]", "annotated_tactic": ["simp_rw [<a>mem_univ_pi</a>]", [{"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 h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) \u2194\n    x\u271d \u2208 Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k", "state_after": "case h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (\u2200 (i_1 : \u03b9), x\u271d i_1 \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i_1) \u2194\n    \u2200 (i_1 : \u03b9), x\u271d i_1 \u2208 \u22c3 j, update (fun i' => t i' j) i s i_1"}, {"tactic": "apply forall_congr'", "annotated_tactic": ["apply <a>forall_congr'</a>", [{"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}]], "state_before": "case h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (\u2200 (i_1 : \u03b9), x\u271d i_1 \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i_1) \u2194\n    \u2200 (i_1 : \u03b9), x\u271d i_1 \u2208 \u22c3 j, update (fun i' => t i' j) i s i_1", "state_after": "case h.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 \u2200 (a : \u03b9), x\u271d a \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) a \u2194 x\u271d a \u2208 \u22c3 j, update (fun i' => t i' j) i s a"}, {"tactic": "intro i'", "annotated_tactic": ["intro i'", []], "state_before": "case h.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 \u2200 (a : \u03b9), x\u271d a \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) a \u2194 x\u271d a \u2208 \u22c3 j, update (fun i' => t i' j) i s a", "state_after": "case h.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'"}, {"tactic": "by_cases h : i' = i", "annotated_tactic": ["by_cases h : i' = i", []], "state_before": "case h.h\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'", "state_after": "case pos\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\nh : i' = i\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'\n\ncase neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\nh : \u00aci' = i\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "case pos\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\nh : i' = i\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'", "state_after": "case pos\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))\nval\u271d : Encodable \u03b9\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\ns : Set ((fun i => \u03b1 i) i')\nhs : s \u2208 C i'\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i' (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i' s i'"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\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))\nval\u271d : Encodable \u03b9\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\ns : Set ((fun i => \u03b1 i) i')\nhs : s \u2208 C i'\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i' (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i' s i'", "state_after": "no goals"}, {"tactic": "rw [\u2190 Ne.def] at h", "annotated_tactic": ["rw [\u2190 <a>Ne.def</a>] at h", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "case neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\nh : \u00aci' = i\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'", "state_after": "case neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\nh : i' \u2260 i\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case neg\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))\nval\u271d : Encodable \u03b9\ni : \u03b9\ns : Set ((fun i => \u03b1 i) i)\nhs : s \u2208 C i\nt : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i)\nh1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i\nh2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni' : \u03b9\nh : i' \u2260 i\n\u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i'", "state_after": "no goals"}, {"tactic": "apply generateFrom_le", "annotated_tactic": ["apply <a>generateFrom_le</a>", [{"full_name": "MeasurableSpace.generateFrom_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [384, 9], "def_end_pos": [384, 24]}]], "state_before": "case intro.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 generateFrom (Set.pi univ '' Set.pi univ C) \u2264 MeasurableSpace.pi", "state_after": "case intro.a.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 \u2200 (t : Set ((i : \u03b9) \u2192 \u03b1 i)), t \u2208 Set.pi univ '' Set.pi univ C \u2192 MeasurableSet t"}, {"tactic": "rintro _ \u27e8s, hs, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8s, hs, rfl\u27e9", []], "state_before": "case intro.a.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\n\u22a2 \u2200 (t : Set ((i : \u03b9) \u2192 \u03b1 i)), t \u2208 Set.pi univ '' Set.pi univ C \u2192 MeasurableSet t", "state_after": "case intro.a.h.intro.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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\n\u22a2 MeasurableSet (Set.pi univ s)"}, {"tactic": "rw [univ_pi_eq_iInter]", "annotated_tactic": ["rw [<a>univ_pi_eq_iInter</a>]", [{"full_name": "Set.univ_pi_eq_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2168, 9], "def_end_pos": [2168, 26]}]], "state_before": "case intro.a.h.intro.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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\n\u22a2 MeasurableSet (Set.pi univ s)", "state_after": "case intro.a.h.intro.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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\n\u22a2 MeasurableSet (\u22c2 i, eval i \u207b\u00b9' s i)"}, {"tactic": "apply MeasurableSet.iInter", "annotated_tactic": ["apply <a>MeasurableSet.iInter</a>", [{"full_name": "MeasurableSet.iInter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 29]}]], "state_before": "case intro.a.h.intro.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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\n\u22a2 MeasurableSet (\u22c2 i, eval i \u207b\u00b9' s i)", "state_after": "case intro.a.h.intro.intro.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\n\u22a2 \u2200 (b : \u03b9), MeasurableSet (eval b \u207b\u00b9' s b)"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case intro.a.h.intro.intro.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\n\u22a2 \u2200 (b : \u03b9), MeasurableSet (eval b \u207b\u00b9' s b)", "state_after": "case intro.a.h.intro.intro.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\ni : \u03b9\n\u22a2 MeasurableSet (eval i \u207b\u00b9' s i)"}, {"tactic": "apply @measurable_pi_apply _ _ (fun i => generateFrom (C i))", "annotated_tactic": ["apply @<a>measurable_pi_apply</a> _ _ (fun i => <a>generateFrom</a> (C i))", [{"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [896, 9], "def_end_pos": [896, 28]}, {"full_name": "MeasurableSpace.generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [363, 5], "def_end_pos": [363, 17]}]], "state_before": "case intro.a.h.intro.intro.h\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\ni : \u03b9\n\u22a2 MeasurableSet (eval i \u207b\u00b9' s i)", "state_after": "case intro.a.h.intro.intro.h.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\ni : \u03b9\n\u22a2 MeasurableSet (s i)"}, {"tactic": "exact measurableSet_generateFrom (hs i (mem_univ i))", "annotated_tactic": ["exact <a>measurableSet_generateFrom</a> (hs i (<a>mem_univ</a> i))", [{"full_name": "MeasurableSpace.measurableSet_generateFrom", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 35]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "case intro.a.h.intro.intro.h.a\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))\nhC : \u2200 (i : \u03b9), IsCountablySpanning (C i)\nval\u271d : Encodable \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ C\ni : \u03b9\n\u22a2 MeasurableSet (s i)", "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_independent_measurableSpace", "start": [82, 1], "end": [104, 78], "traced_tactics": [{"tactic": "revert g", "annotated_tactic": ["revert g", []], "state_before": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_meas_g : Measurable g\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * 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\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\n\u22a2 \u2200 {g : \u03a9 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc"}, {"tactic": "have h_measM_f : Measurable f := h_meas_f.mono hMf le_rfl", "annotated_tactic": ["have h_measM_f : <a>Measurable</a> f := h_meas_f.mono hMf <a>le_rfl</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"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\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\n\u22a2 \u2200 {g : \u03a9 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * 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\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 {g : \u03a9 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc"}, {"tactic": "apply @Measurable.ennreal_induction _ Mg", "annotated_tactic": ["apply @<a>Measurable.ennreal_induction</a> _ Mg", [{"full_name": "Measurable.ennreal_induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 37]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 {g : \u03a9 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set \u03a9\u2984,\n    MeasurableSet s \u2192\n      \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator s (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c) \u03c9 \u2202\u03bc\n\ncase h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 \u2983f_1 g : \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    Disjoint (Function.support f_1) (Function.support g) \u2192\n      Measurable f_1 \u2192\n        Measurable g \u2192\n          \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f_1 \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f_1 \u03c9 \u2202\u03bc \u2192\n            \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc \u2192\n              \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f_1 + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f_1 + g) \u03c9 \u2202\u03bc\n\ncase h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 \u2983f_1 : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    (\u2200 (n : \u2115), Measurable (f_1 n)) \u2192\n      Monotone f_1 \u2192\n        (\u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f_1 n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f_1 n \u03c9 \u2202\u03bc) \u2192\n          \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f_1 n x) \u03c9 \u2202\u03bc =\n            (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f_1 n x) \u03c9 \u2202\u03bc"}, {"tactic": "intro c s h_s", "annotated_tactic": ["intro c s h_s", []], "state_before": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set \u03a9\u2984,\n    MeasurableSet s \u2192\n      \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator s (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c) \u03c9 \u2202\u03bc", "state_after": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03a9\nh_s : MeasurableSet s\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator s (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c) \u03c9 \u2202\u03bc"}, {"tactic": "apply lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator hMf _ (hMg _ h_s) _ h_meas_f", "annotated_tactic": ["apply <a>lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator</a> hMf _ (hMg _ h_s) _ h_meas_f", [{"full_name": "ProbabilityTheory.lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator", "def_path": "Mathlib/Probability/Integration.lean", "def_pos": [45, 9], "def_end_pos": [45, 69]}]], "state_before": "case h_ind\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03a9\nh_s : MeasurableSet s\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator s (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c) \u03c9 \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03a9\nh_s : MeasurableSet s\n\u22a2 IndepSets {s | MeasurableSet s} {s}"}, {"tactic": "apply indepSets_of_indepSets_of_le_right h_ind", "annotated_tactic": ["apply <a>indepSets_of_indepSets_of_le_right</a> h_ind", [{"full_name": "ProbabilityTheory.indepSets_of_indepSets_of_le_right", "def_path": "Mathlib/Probability/Independence/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 43]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03a9\nh_s : MeasurableSet s\n\u22a2 IndepSets {s | MeasurableSet s} {s}", "state_after": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03a9\nh_s : MeasurableSet s\n\u22a2 {s} \u2286 {s | MeasurableSet s}"}, {"tactic": "rwa [singleton_subset_iff]", "annotated_tactic": ["rwa [<a>singleton_subset_iff</a>]", [{"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "state_before": "\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03a9\nh_s : MeasurableSet s\n\u22a2 {s} \u2286 {s | MeasurableSet s}", "state_after": "no goals"}, {"tactic": "intro f' g _ h_measMg_f' _ h_ind_f' h_ind_g'", "annotated_tactic": ["intro f' g _ h_measMg_f' _ h_ind_f' h_ind_g'", []], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 \u2983f_1 g : \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    Disjoint (Function.support f_1) (Function.support g) \u2192\n      Measurable f_1 \u2192\n        Measurable g \u2192\n          \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f_1 \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f_1 \u03c9 \u2202\u03bc \u2192\n            \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc \u2192\n              \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f_1 + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f_1 + g) \u03c9 \u2202\u03bc", "state_after": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_measMg_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc\nh_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f' + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc"}, {"tactic": "have h_measM_f' : Measurable f' := h_measMg_f'.mono hMg le_rfl", "annotated_tactic": ["have h_measM_f' : <a>Measurable</a> f' := h_measMg_f'.mono hMg <a>le_rfl</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_measMg_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc\nh_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f' + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc", "state_after": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_measMg_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc\nh_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f' + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc"}, {"tactic": "simp_rw [Pi.add_apply, left_distrib]", "annotated_tactic": ["simp_rw [<a>Pi.add_apply</a>, <a>left_distrib</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "left_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [78, 9], "def_end_pos": [78, 21]}]], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_measMg_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc\nh_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f' + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc", "state_after": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_measMg_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc\nh_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 + f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 + g \u03c9 \u2202\u03bc"}, {"tactic": "rw [lintegral_add_left h_measM_f', lintegral_add_left (h_measM_f.mul h_measM_f'), left_distrib,\n  h_ind_f', h_ind_g']", "annotated_tactic": ["rw [<a>lintegral_add_left</a> h_measM_f', <a>lintegral_add_left</a> (h_measM_f.mul h_measM_f'), <a>left_distrib</a>,\n      h_ind_f', h_ind_g']", [{"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]}, {"full_name": "left_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [78, 9], "def_end_pos": [78, 21]}]], "state_before": "case h_add\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' g : \u03a9 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (Function.support f') (Function.support g)\nh_measMg_f' : Measurable f'\na\u271d : Measurable g\nh_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc\nh_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc\nh_measM_f' : Measurable f'\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 + f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 + g \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro f' h_meas_f' h_mono_f' h_ind_f'", "annotated_tactic": ["intro f' h_meas_f' h_mono_f' h_ind_f'", []], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\n\u22a2 \u2200 \u2983f_1 : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\u2984,\n    (\u2200 (n : \u2115), Measurable (f_1 n)) \u2192\n      Monotone f_1 \u2192\n        (\u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f_1 n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f_1 n \u03c9 \u2202\u03bc) \u2192\n          \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f_1 n x) \u03c9 \u2202\u03bc =\n            (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f_1 n x) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc"}, {"tactic": "have h_measM_f' : \u2200 n, Measurable (f' n) := fun n => (h_meas_f' n).mono hMg le_rfl", "annotated_tactic": ["have h_measM_f' : \u2200 n, <a>Measurable</a> (f' n) := fun n => (h_meas_f' n).<a>mono</a> hMg <a>le_rfl</a>", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Measurable.mono", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 24]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc"}, {"tactic": "simp_rw [ENNReal.mul_iSup]", "annotated_tactic": ["simp_rw [<a>ENNReal.mul_iSup</a>]", [{"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}]], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2a06 i, f \u03c9 * f' i \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), \u2a06 n, f' n \u03c9 \u2202\u03bc"}, {"tactic": "rw [lintegral_iSup, lintegral_iSup h_measM_f' h_mono_f', ENNReal.mul_iSup]", "annotated_tactic": ["rw [<a>lintegral_iSup</a>, <a>lintegral_iSup</a> h_measM_f' h_mono_f', <a>ENNReal.mul_iSup</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.lintegral_iSup", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [345, 9], "def_end_pos": [345, 23]}, {"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}]], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2a06 i, f \u03c9 * f' i \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), \u2a06 n, f' n \u03c9 \u2202\u03bc", "state_after": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03a9), f a * f' n a \u2202\u03bc = \u2a06 i, (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (a : \u03a9), f' i a \u2202\u03bc\n\ncase h_iSup.hf\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03c9 => f \u03c9 * f' n \u03c9\n\ncase h_iSup.h_mono\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 Monotone fun i \u03c9 => f \u03c9 * f' i \u03c9"}, {"tactic": "simp_rw [\u2190 h_ind_f']", "annotated_tactic": ["simp_rw [\u2190 h_ind_f']", []], "state_before": "case h_iSup\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u2a06 n, \u222b\u207b (a : \u03a9), f a * f' n a \u2202\u03bc = \u2a06 i, (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (a : \u03a9), f' i a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact fun n => h_measM_f.mul (h_measM_f' n)", "annotated_tactic": ["exact fun n => h_measM_f.mul (h_measM_f' n)", []], "state_before": "case h_iSup.hf\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03c9 => f \u03c9 * f' n \u03c9", "state_after": "no goals"}, {"tactic": "exact fun n m (h_le : n \u2264 m) a => mul_le_mul_left' (h_mono_f' h_le a) _", "annotated_tactic": ["exact fun n m (h_le : n \u2264 m) a => <a>mul_le_mul_left'</a> (h_mono_f' h_le a) _", [{"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_iSup.h_mono\n\u03a9 : Type u_1\nm\u03a9\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\nf : \u03a9 \u2192 \u211d\u22650\u221e\nX Y : \u03a9 \u2192 \u211d\nMf Mg m\u03a9 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\nhMf : Mf \u2264 m\u03a9\nhMg : Mg \u2264 m\u03a9\nh_ind : Indep Mf Mg\nh_meas_f : Measurable f\nh_measM_f : Measurable f\nf' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\nh_meas_f' : \u2200 (n : \u2115), Measurable (f' n)\nh_mono_f' : Monotone f'\nh_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc\nh_measM_f' : \u2200 (n : \u2115), Measurable (f' n)\n\u22a2 Monotone fun i \u03c9 => f \u03c9 * f' i \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.zero_shiftRight", "start": [458, 1], "end": [459, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.erase_union_singleton", "start": [633, 1], "end": [640, 64], "traced_tactics": [{"tactic": "by_cases h' : x = a", "annotated_tactic": ["by_cases h' : x = a", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : x = a\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s\n\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : \u00acx = a\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s"}, {"tactic": "subst a", "annotated_tactic": ["subst a", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : x = a\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns : Finmap \u03b2\nx : \u03b1\nb : \u03b2 x\nh : lookup x s = some b\n\u22a2 lookup x (erase x s \u222a singleton x b) = lookup x s"}, {"tactic": "rw [lookup_union_right not_mem_erase_self, lookup_singleton_eq, h]", "annotated_tactic": ["rw [<a>lookup_union_right</a> <a>not_mem_erase_self</a>, <a>lookup_singleton_eq</a>, h]", [{"full_name": "Finmap.lookup_union_right", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [588, 9], "def_end_pos": [588, 27]}, {"full_name": "Finmap.not_mem_erase_self", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [439, 9], "def_end_pos": [439, 27]}, {"full_name": "Finmap.lookup_singleton_eq", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [304, 9], "def_end_pos": [304, 28]}]], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns : Finmap \u03b2\nx : \u03b1\nb : \u03b2 x\nh : lookup x s = some b\n\u22a2 lookup x (erase x s \u222a singleton x b) = lookup x s", "state_after": "no goals"}, {"tactic": "have : x \u2209 singleton a b := by rwa [mem_singleton]", "annotated_tactic": ["have : x \u2209 <a>singleton</a> a b := by rwa [<a>mem_singleton</a>]", [{"full_name": "Finmap.singleton", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [237, 5], "def_end_pos": [237, 14]}, {"full_name": "Finmap.mem_singleton", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [247, 9], "def_end_pos": [247, 22]}]], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : \u00acx = a\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : \u00acx = a\nthis : \u00acx \u2208 singleton a b\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s"}, {"tactic": "rw [lookup_union_left_of_not_in this, lookup_erase_ne h']", "annotated_tactic": ["rw [<a>lookup_union_left_of_not_in</a> this, <a>lookup_erase_ne</a> h']", [{"full_name": "Finmap.lookup_union_left_of_not_in", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [592, 9], "def_end_pos": [592, 36]}, {"full_name": "Finmap.lookup_erase_ne", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [451, 9], "def_end_pos": [451, 24]}]], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : \u00acx = a\nthis : \u00acx \u2208 singleton a b\n\u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s", "state_after": "no goals"}, {"tactic": "rwa [mem_singleton]", "annotated_tactic": ["rwa [<a>mem_singleton</a>]", [{"full_name": "Finmap.mem_singleton", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [247, 9], "def_end_pos": [247, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Finmap \u03b2\nh : lookup a s = some b\nx : \u03b1\nh' : \u00acx = a\n\u22a2 \u00acx \u2208 singleton a b", "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.bisim_aux", "start": [431, 9], "end": [465, 26], "traced_tactics": [{"tactic": "intro x", "annotated_tactic": ["intro x", []], "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 : Cofix F), r x x\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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n\u22a2 \u2200 (y : Cofix F), r x y \u2192 x = y"}, {"tactic": "apply Quot.inductionOn (motive := _) x", "annotated_tactic": ["apply <a>Quot.inductionOn</a> (motive := _) x", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n\u22a2 \u2200 (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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = y"}, {"tactic": "clear x", "annotated_tactic": ["clear x", []], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\n\u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = y"}, {"tactic": "intro x y", "annotated_tactic": ["intro 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\n\u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n\u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y"}, {"tactic": "apply Quot.inductionOn (motive := _) y", "annotated_tactic": ["apply <a>Quot.inductionOn</a> (motive := _) y", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n\u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a"}, {"tactic": "clear y", "annotated_tactic": ["clear 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\n\u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a"}, {"tactic": "intro y rxy", "annotated_tactic": ["intro 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\n\u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y", "state_after": "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y"}, {"tactic": "let r' x y := r (Quot.mk _ x) (Quot.mk _ y)", "annotated_tactic": ["let r' 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 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y", "state_after": "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := 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    Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n    h _ _ r'ab\n  have h\u2081 : \u2200 u v : q.P.M, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n    intro u v cuv\n    apply Quot.sound\n    simp only\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; apply Quot.inductionOn (motive := _) c; clear c\n        intro c d; apply Quot.inductionOn (motive := _) d; 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, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map,\n    abs_map, h\u2080]\n  rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]", "annotated_tactic": ["have : <a>IsPrecongr</a> r' := by\n    intro a b r'ab\n    have h\u2080 :\n      <a>Quot.mk</a> r <$> <a>Quot.mk</a> <a>Mcongr</a> <$> <a>abs</a> (<a>PFunctor.M.dest</a> a) =\n        <a>Quot.mk</a> r <$> <a>Quot.mk</a> <a>Mcongr</a> <$> <a>abs</a> (<a>PFunctor.M.dest</a> b) :=\n      h _ _ r'ab\n    have h\u2081 : \u2200 u v : q.P.M, <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      simp only\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; apply <a>Quot.inductionOn</a> (motive := _) c; clear c\n          intro c d; apply <a>Quot.inductionOn</a> (motive := _) d; 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>PFunctor.comp_map</a> _ _ f, <a>PFunctor.comp_map</a> _ _ (<a>Quot.mk</a> r), <a>abs_map</a>, <a>abs_map</a>,\n      <a>abs_map</a>, h\u2080]\n    rw [<a>PFunctor.comp_map</a> _ _ f, <a>PFunctor.comp_map</a> _ _ (<a>Quot.mk</a> r), <a>abs_map</a>, <a>abs_map</a>, <a>abs_map</a>]", [{"full_name": "QPF.IsPrecongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [384, 5], "def_end_pos": [384, 15]}, {"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": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 11]}, {"full_name": "QPF.abs", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "PFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [290, 5], "def_end_pos": [290, 9]}, {"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": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 11]}, {"full_name": "QPF.abs", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "PFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [290, 5], "def_end_pos": [290, 9]}, {"full_name": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 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.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"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": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 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": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"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.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"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_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"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.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"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_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y", "state_after": "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := 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 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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := 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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 IsPrecongr r'", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)"}, {"tactic": "have h\u2080 :\n  Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n    Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n  h _ _ r'ab", "annotated_tactic": ["have h\u2080 :\n      <a>Quot.mk</a> r <$> <a>Quot.mk</a> <a>Mcongr</a> <$> <a>abs</a> (<a>PFunctor.M.dest</a> a) =\n        <a>Quot.mk</a> r <$> <a>Quot.mk</a> <a>Mcongr</a> <$> <a>abs</a> (<a>PFunctor.M.dest</a> b) :=\n      h _ _ r'ab", [{"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": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 11]}, {"full_name": "QPF.abs", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "PFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [290, 5], "def_end_pos": [290, 9]}, {"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": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 11]}, {"full_name": "QPF.abs", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "PFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [290, 5], "def_end_pos": [290, 9]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)"}, {"tactic": "have h\u2081 : \u2200 u v : q.P.M, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n  intro u v cuv\n  apply Quot.sound\n  simp only\n  rw [Quot.sound cuv]\n  apply h'", "annotated_tactic": ["have h\u2081 : \u2200 u v : q.P.M, <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      simp only\n      rw [<a>Quot.sound</a> cuv]\n      apply h'", [{"full_name": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)"}, {"tactic": "let f : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (by\n      intro c; apply Quot.inductionOn (motive := _) c; clear c\n      intro c d; apply Quot.inductionOn (motive := _) d; 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; apply <a>Quot.inductionOn</a> (motive := _) c; clear c\n          intro c d; apply <a>Quot.inductionOn</a> (motive := _) d; 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.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}, {"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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest 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": "QPF.Mcongr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [389, 5], "def_end_pos": [389, 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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), 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 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)"}, {"tactic": "rw [\u2190 this, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map,\n  abs_map, h\u2080]", "annotated_tactic": ["rw [\u2190 this, <a>PFunctor.comp_map</a> _ _ f, <a>PFunctor.comp_map</a> _ _ (<a>Quot.mk</a> r), <a>abs_map</a>, <a>abs_map</a>,\n      <a>abs_map</a>, h\u2080]", [{"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"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.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"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_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), 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 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), 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 f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =\n    abs ((f \u2218 Quot.mk r \u2218 Quot.mk Mcongr) <$> PFunctor.M.dest b)"}, {"tactic": "rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]", "annotated_tactic": ["rw [<a>PFunctor.comp_map</a> _ _ f, <a>PFunctor.comp_map</a> _ _ (<a>Quot.mk</a> r), <a>abs_map</a>, <a>abs_map</a>, <a>abs_map</a>]", [{"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"full_name": "PFunctor.comp_map", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [73, 19], "def_end_pos": [73, 27]}, {"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.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"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_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), 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 f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =\n    abs ((f \u2218 Quot.mk r \u2218 Quot.mk Mcongr) <$> PFunctor.M.dest b)", "state_after": "no goals"}, {"tactic": "intro u v cuv", "annotated_tactic": ["intro u v cuv", []], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n\u22a2 \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 Quot.mk r' u = Quot.mk r' v", "state_after": "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 r' u v"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 r' u v", "state_after": "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v)", "state_after": "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a b : Cofix F), r a b \u2192 Quot.lift (Quot.mk r') h\u2081 a = Quot.lift (Quot.mk r') h\u2081 b", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n\u22a2 \u2200 (b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b"}, {"tactic": "apply Quot.inductionOn (motive := _) c", "annotated_tactic": ["apply <a>Quot.inductionOn</a> (motive := _) c", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n\u22a2 \u2200 (b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b", "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\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 (motive := _) d", "annotated_tactic": ["apply <a>Quot.inductionOn</a> (motive := _) d", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}]], "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 : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\n\u22a2 \u2200 (a : PFunctor.M (P F)),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\n\u22a2 \u2200 (a : PFunctor.M (P F)),\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\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": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\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\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : QPF F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 r' c d", "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.dvd_add_iff_right", "start": [881, 11], "end": [885, 80], "traced_tactics": [{"tactic": "rw [Nat.mul_sub_left_distrib, \u2190 he, Nat.add_sub_cancel_left]", "annotated_tactic": ["rw [<a>Nat.mul_sub_left_distrib</a>, \u2190 he, <a>Nat.add_sub_cancel_left</a>]", [{"full_name": "Nat.mul_sub_left_distrib", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [716, 19], "def_end_pos": [716, 39]}, {"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": "k m n : Nat\nh : k \u2223 m\nd : Nat\nx\u271d : k \u2223 k * d + n\ne : Nat\nhe : k * d + n = k * e\n\u22a2 n = k * (e - d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_of_integrable_trim", "start": [1220, 1], "end": [1226, 48], "traced_tactics": [{"tactic": "obtain \u27e8hf_meas_ae, hf\u27e9 := hf_int", "annotated_tactic": ["obtain \u27e8hf_meas_ae, hf\u27e9 := hf_int", []], "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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_int : Integrable f\n\u22a2 Integrable f", "state_after": "case 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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : HasFiniteIntegral f\n\u22a2 Integrable f"}, {"tactic": "refine' \u27e8aestronglyMeasurable_of_aestronglyMeasurable_trim hm hf_meas_ae, _\u27e9", "annotated_tactic": ["refine' \u27e8<a>aestronglyMeasurable_of_aestronglyMeasurable_trim</a> hm hf_meas_ae, _\u27e9", [{"full_name": "aestronglyMeasurable_of_aestronglyMeasurable_trim", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1560, 9], "def_end_pos": [1560, 65]}]], "state_before": "case 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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : HasFiniteIntegral f\n\u22a2 Integrable f", "state_after": "case 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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : HasFiniteIntegral f\n\u22a2 HasFiniteIntegral f"}, {"tactic": "rw [HasFiniteIntegral] at hf \u22a2", "annotated_tactic": ["rw [<a>HasFiniteIntegral</a>] at hf \u22a2", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 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\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : HasFiniteIntegral f\n\u22a2 HasFiniteIntegral f", "state_after": "case 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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202Measure.trim \u03bc' hm < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc' < \u22a4"}, {"tactic": "rwa [lintegral_trim_ae hm _] at hf", "annotated_tactic": ["rwa [<a>lintegral_trim_ae</a> hm _] at hf", [{"full_name": "MeasureTheory.lintegral_trim_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1615, 9], "def_end_pos": [1615, 26]}]], "state_before": "case 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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202Measure.trim \u03bc' hm < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc' < \u22a4", "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\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202Measure.trim \u03bc' hm < \u22a4\n\u22a2 AEMeasurable fun a => \u2191\u2016f a\u2016\u208a"}, {"tactic": "exact AEStronglyMeasurable.ennnorm hf_meas_ae", "annotated_tactic": ["exact <a>AEStronglyMeasurable.ennnorm</a> hf_meas_ae", [{"full_name": "MeasureTheory.AEStronglyMeasurable.ennnorm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1491, 19], "def_end_pos": [1491, 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\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\nH : Type u_5\ninst\u271d : NormedAddCommGroup H\nm0 : MeasurableSpace \u03b1\n\u03bc' : Measure \u03b1\nf : \u03b1 \u2192 H\nhm : m \u2264 m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm)\nhf : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202Measure.trim \u03bc' hm < \u22a4\n\u22a2 AEMeasurable fun a => \u2191\u2016f a\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "full_name": "MeasureTheory.Measure.lintegral_join", "start": [126, 1], "end": [147, 42], "traced_tactics": [{"tactic": "simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral,\n  join_apply (SimpleFunc.measurableSet_preimage _ _)]", "annotated_tactic": ["simp_rw [<a>lintegral_eq_iSup_eapprox_lintegral</a> hf, <a>SimpleFunc.lintegral</a>,\n    <a>join_apply</a> (<a>SimpleFunc.measurableSet_preimage</a> _ _)]", [{"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.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}, {"full_name": "MeasureTheory.Measure.join_apply", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [110, 9], "def_end_pos": [110, 19]}, {"full_name": "MeasureTheory.SimpleFunc.measurableSet_preimage", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [195, 9], "def_end_pos": [195, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202join m = \u222b\u207b (\u03bc : Measure \u03b1), \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2202m", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2a06 n,\n      \u2211 x in SimpleFunc.range (SimpleFunc.eapprox f n),\n        x * \u222b\u207b (\u03bc : Measure \u03b1), \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {x}) \u2202m =\n    \u222b\u207b (\u03bc : Measure \u03b1),\n      \u2a06 n, \u2211 x in SimpleFunc.range (SimpleFunc.eapprox f n), x * \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {x}) \u2202m"}, {"tactic": "intro s f hf hm", "annotated_tactic": ["intro s f hf hm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),\n    (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192\n      (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192\n        \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m"}, {"tactic": "rw [lintegral_iSup _ hm]", "annotated_tactic": ["rw [<a>lintegral_iSup</a> _ hm]", [{"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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2a06 n, \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03bc => \u2211 r in s n, r * f n r \u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2a06 n, \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03bc => \u2211 r in s n, r * f n r \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03bc => \u2211 r in s n, r * f n r \u03bc\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2a06 n, \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2a06 n, \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m", "state_after": "case e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 (fun n => \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m) = fun n => \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 (fun n => \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m) = fun n => \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m", "state_after": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\nn : \u2115\n\u22a2 \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m"}, {"tactic": "rw [lintegral_finset_sum (s n)]", "annotated_tactic": ["rw [<a>lintegral_finset_sum</a> (s n)]", [{"full_name": "MeasureTheory.lintegral_finset_sum", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [665, 9], "def_end_pos": [665, 29]}]], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\nn : \u2115\n\u22a2 \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m", "state_after": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\nn : \u2115\n\u22a2 \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2211 b in s n, \u222b\u207b (a : Measure \u03b1), b * f n b a \u2202m\n\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\nn : \u2115\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b \u2208 s n \u2192 Measurable fun a => b * f n b a"}, {"tactic": "refine'\n  this (fun n => SimpleFunc.range (SimpleFunc.eapprox f n))\n    (fun n r \u03bc => \u03bc (SimpleFunc.eapprox f n \u207b\u00b9' {r})) _ _", "annotated_tactic": ["refine'\n      this (fun n => <a>SimpleFunc.range</a> (<a>SimpleFunc.eapprox</a> f n))\n        (fun n r \u03bc => \u03bc (<a>SimpleFunc.eapprox</a> f n \u207b\u00b9' {r})) _ _", [{"full_name": "MeasureTheory.SimpleFunc.range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [103, 15], "def_end_pos": [103, 20]}, {"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.eapprox", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [873, 5], "def_end_pos": [873, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nthis :\n  \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),\n    (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192\n      (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192\n        \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m\n\u22a2 \u2a06 n,\n      \u2211 x in SimpleFunc.range (SimpleFunc.eapprox f n),\n        x * \u222b\u207b (\u03bc : Measure \u03b1), \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {x}) \u2202m =\n    \u222b\u207b (\u03bc : Measure \u03b1),\n      \u2a06 n, \u2211 x in SimpleFunc.range (SimpleFunc.eapprox f n), x * \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {x}) \u2202m", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nthis :\n  \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),\n    (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192\n      (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192\n        \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m\n\u22a2 \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable ((fun n r \u03bc => \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {r})) n r)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nthis :\n  \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),\n    (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192\n      (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192\n        \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m\n\u22a2 Monotone fun n \u03bc =>\n    \u2211 r in (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) n,\n      r * (fun n r \u03bc => \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {r})) n r \u03bc"}, {"tactic": "exact fun n r => measurable_coe (SimpleFunc.measurableSet_preimage _ _)", "annotated_tactic": ["exact fun n r => <a>measurable_coe</a> (<a>SimpleFunc.measurableSet_preimage</a> _ _)", [{"full_name": "MeasureTheory.Measure.measurable_coe", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [52, 9], "def_end_pos": [52, 23]}, {"full_name": "MeasureTheory.SimpleFunc.measurableSet_preimage", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [195, 9], "def_end_pos": [195, 31]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nthis :\n  \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),\n    (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192\n      (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192\n        \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m\n\u22a2 \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable ((fun n r \u03bc => \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {r})) n r)", "state_after": "no goals"}, {"tactic": "exact fun n m h \u03bc => SimpleFunc.lintegral_mono (SimpleFunc.monotone_eapprox _ h) le_rfl", "annotated_tactic": ["exact fun n m h \u03bc => <a>SimpleFunc.lintegral_mono</a> (<a>SimpleFunc.monotone_eapprox</a> _ h) <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": "MeasureTheory.SimpleFunc.monotone_eapprox", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [892, 9], "def_end_pos": [892, 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\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nthis :\n  \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),\n    (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192\n      (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192\n        \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m\n\u22a2 Monotone fun n \u03bc =>\n    \u2211 r in (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) n,\n      r * (fun n r \u03bc => \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {r})) n r \u03bc", "state_after": "no goals"}, {"tactic": "exact fun n => Finset.measurable_sum _ fun r _ => (hf _ _).const_mul _", "annotated_tactic": ["exact fun n => <a>Finset.measurable_sum</a> _ fun r _ => (hf _ _).<a>const_mul</a> _", [{"full_name": "Finset.measurable_sum", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [935, 3], "def_end_pos": [935, 14]}, {"full_name": "Measurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [106, 9], "def_end_pos": [106, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun \u03bc => \u2211 r in s n, r * f n r \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [lintegral_const_mul _ (hf _ _)]", "annotated_tactic": ["simp_rw [<a>lintegral_const_mul</a> _ (hf _ _)]", [{"full_name": "MeasureTheory.lintegral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [671, 9], "def_end_pos": [671, 28]}]], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\nn : \u2115\n\u22a2 \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2211 b in s n, \u222b\u207b (a : Measure \u03b1), b * f n b a \u2202m", "state_after": "no goals"}, {"tactic": "exact fun r _ => (hf _ _).const_mul _", "annotated_tactic": ["exact fun r _ => (hf _ _).<a>const_mul</a> _", [{"full_name": "Measurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [106, 9], "def_end_pos": [106, 29]}]], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nm : Measure (Measure \u03b1)\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\ns : \u2115 \u2192 Finset \u211d\u22650\u221e\nf : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)\nhm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc\nn : \u2115\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b \u2208 s n \u2192 Measurable fun a => b * f n b 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.bind_bind", "start": [188, 1], "end": [193, 51], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nm : Measure \u03b1\nf : \u03b1 \u2192 Measure \u03b2\ng : \u03b2 \u2192 Measure \u03b3\nhf : Measurable f\nhg : Measurable g\n\u22a2 bind (bind m f) g = bind m fun a => bind (f a) g", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nm : Measure \u03b1\nf : \u03b1 \u2192 Measure \u03b2\ng : \u03b2 \u2192 Measure \u03b3\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(bind (bind m f) g) s = \u2191\u2191(bind m fun a => bind (f a) g) s"}, {"tactic": "erw [bind_apply hs hg, bind_apply hs ((measurable_bind' hg).comp hf),\n  lintegral_bind hf ((measurable_coe hs).comp hg)]", "annotated_tactic": ["erw [<a>bind_apply</a> hs hg, <a>bind_apply</a> hs ((<a>measurable_bind'</a> hg).<a>comp</a> hf),\n    <a>lintegral_bind</a> hf ((<a>measurable_coe</a> hs).<a>comp</a> hg)]", [{"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.bind_apply", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [174, 9], "def_end_pos": [174, 19]}, {"full_name": "MeasureTheory.Measure.measurable_bind'", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [179, 9], "def_end_pos": [179, 25]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}, {"full_name": "MeasureTheory.Measure.lintegral_bind", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [183, 9], "def_end_pos": [183, 23]}, {"full_name": "MeasureTheory.Measure.measurable_coe", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [52, 9], "def_end_pos": [52, 23]}, {"full_name": "Measurable.comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [557, 19], "def_end_pos": [557, 34]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nm : Measure \u03b1\nf : \u03b1 \u2192 Measure \u03b2\ng : \u03b2 \u2192 Measure \u03b3\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(bind (bind m f) g) s = \u2191\u2191(bind m fun a => bind (f a) g) s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nm : Measure \u03b1\nf : \u03b1 \u2192 Measure \u03b2\ng : \u03b2 \u2192 Measure \u03b3\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (x : \u03b2), ((fun \u03bc => \u2191\u2191\u03bc s) \u2218 g) x \u2202f a \u2202m = \u222b\u207b (a : \u03b1), \u2191\u2191(((fun m => bind m g) \u2218 f) a) s \u2202m"}, {"tactic": "conv_rhs => enter [2, a]; erw [bind_apply hs hg]", "annotated_tactic": ["conv_rhs => enter [2, a]; erw [<a>bind_apply</a> hs hg]", [{"full_name": "MeasureTheory.Measure.bind_apply", "def_path": "Mathlib/MeasureTheory/Measure/GiryMonad.lean", "def_pos": [174, 9], "def_end_pos": [174, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nm : Measure \u03b1\nf : \u03b1 \u2192 Measure \u03b2\ng : \u03b2 \u2192 Measure \u03b3\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (x : \u03b2), ((fun \u03bc => \u2191\u2191\u03bc s) \u2218 g) x \u2202f a \u2202m = \u222b\u207b (a : \u03b1), \u2191\u2191(((fun m => bind m g) \u2218 f) a) s \u2202m", "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_injective_image", "start": [177, 1], "end": [183, 31], "traced_tactics": [{"tactic": "by_cases hs : s.Finite", "annotated_tactic": ["by_cases hs : s.Finite", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\n\u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : Set.Finite s\n\u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : \u00acSet.Finite s\n\u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s"}, {"tactic": "rw [count_apply_infinite hs]", "annotated_tactic": ["rw [<a>count_apply_infinite</a> hs]", [{"full_name": "MeasureTheory.Measure.count_apply_infinite", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [74, 9], "def_end_pos": [74, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : \u00acSet.Finite s\n\u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : \u00acSet.Finite s\n\u22a2 \u2191\u2191count (f '' s) = \u22a4"}, {"tactic": "rw [\u2190 finite_image_iff <| hf.injOn _] at hs", "annotated_tactic": ["rw [\u2190 <a>finite_image_iff</a> <| hf.injOn _] at hs", [{"full_name": "Set.finite_image_iff", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1054, 9], "def_end_pos": [1054, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : \u00acSet.Finite s\n\u22a2 \u2191\u2191count (f '' s) = \u22a4", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : \u00acSet.Finite (f '' s)\n\u22a2 \u2191\u2191count (f '' s) = \u22a4"}, {"tactic": "rw [count_apply_infinite hs]", "annotated_tactic": ["rw [<a>count_apply_infinite</a> hs]", [{"full_name": "MeasureTheory.Measure.count_apply_infinite", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [74, 9], "def_end_pos": [74, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : \u00acSet.Finite (f '' s)\n\u22a2 \u2191\u2191count (f '' s) = \u22a4", "state_after": "no goals"}, {"tactic": "exact count_injective_image' hf hs.measurableSet (Finite.image f hs).measurableSet", "annotated_tactic": ["exact <a>count_injective_image'</a> hf hs.measurableSet (<a>Finite.image</a> f hs).<a>measurableSet</a>", [{"full_name": "MeasureTheory.Measure.count_injective_image'", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [165, 9], "def_end_pos": [165, 31]}, {"full_name": "Set.Finite.image", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [862, 9], "def_end_pos": [862, 21]}, {"full_name": "Set.Finite.measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [305, 9], "def_end_pos": [305, 33]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : MeasurableSingletonClass \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Function.Injective f\ns : Set \u03b2\nhs : Set.Finite s\n\u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.AnalyticSet.iUnion", "start": [274, 1], "end": [287, 48], "traced_tactics": [{"tactic": "choose \u03b2 h\u03b2 h'\u03b2 f f_cont f_range using fun n =>\n  analyticSet_iff_exists_polishSpace_range.1 (hs n)", "annotated_tactic": ["choose \u03b2 h\u03b2 h'\u03b2 f f_cont f_range using fun n =>\n    <a>analyticSet_iff_exists_polishSpace_range</a>.1 (hs n)", [{"full_name": "MeasureTheory.analyticSet_iff_exists_polishSpace_range", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [194, 9], "def_end_pos": [194, 49]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u22a2 AnalyticSet (\u22c3 n, s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u22a2 AnalyticSet (\u22c3 n, s n)"}, {"tactic": "let \u03b3 := \u03a3n, \u03b2 n", "annotated_tactic": ["let \u03b3 := \u03a3n, \u03b2 n", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u22a2 AnalyticSet (\u22c3 n, s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\n\u22a2 AnalyticSet (\u22c3 n, s n)"}, {"tactic": "let F : \u03b3 \u2192 \u03b1 := fun \u27e8n, x\u27e9 \u21a6 f n x", "annotated_tactic": ["let F : \u03b3 \u2192 \u03b1 := fun \u27e8n, x\u27e9 \u21a6 f n x", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\n\u22a2 AnalyticSet (\u22c3 n, s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\n\u22a2 AnalyticSet (\u22c3 n, s n)"}, {"tactic": "have F_cont : Continuous F := continuous_sigma f_cont", "annotated_tactic": ["have F_cont : <a>Continuous</a> F := <a>continuous_sigma</a> f_cont", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "continuous_sigma", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1577, 9], "def_end_pos": [1577, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\n\u22a2 AnalyticSet (\u22c3 n, s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\n\u22a2 AnalyticSet (\u22c3 n, s n)"}, {"tactic": "have F_range : range F = \u22c3 n, s n := by\n  simp only [range_sigma_eq_iUnion_range, f_range]", "annotated_tactic": ["have F_range : <a>range</a> F = \u22c3 n, s n := by\n    simp only [<a>range_sigma_eq_iUnion_range</a>, f_range]", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Set.range_sigma_eq_iUnion_range", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1403, 9], "def_end_pos": [1403, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\n\u22a2 AnalyticSet (\u22c3 n, s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\nF_range : range F = \u22c3 n, s n\n\u22a2 AnalyticSet (\u22c3 n, s n)"}, {"tactic": "rw [\u2190 F_range]", "annotated_tactic": ["rw [\u2190 F_range]", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\nF_range : range F = \u22c3 n, s n\n\u22a2 AnalyticSet (\u22c3 n, s n)", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\nF_range : range F = \u22c3 n, s n\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": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\nF_range : range F = \u22c3 n, s n\n\u22a2 AnalyticSet (range F)", "state_after": "no goals"}, {"tactic": "simp only [range_sigma_eq_iUnion_range, f_range]", "annotated_tactic": ["simp only [<a>range_sigma_eq_iUnion_range</a>, f_range]", [{"full_name": "Set.range_sigma_eq_iUnion_range", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1403, 9], "def_end_pos": [1403, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 :=\n  fun x =>\n    match x with\n    | { fst := n, snd := x } => f n x\nF_cont : Continuous F\n\u22a2 range F = \u22c3 n, s 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.posOfAux_eq", "start": [326, 1], "end": [326, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/WF.lean", "full_name": "WellFounded.fix_eq_fixC", "start": [109, 18], "end": [111, 32], "traced_tactics": [{"tactic": "funext \u03b1 C r hwf F x", "annotated_tactic": ["funext \u03b1 C r hwf F x", []], "state_before": "\u22a2 @fix = @WellFounded.fixC", "state_after": "case h.h.h.h.h.h\n\u03b1 : Sort u_1\nC : \u03b1 \u2192 Sort u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhwf : WellFounded r\nF : (x : \u03b1) \u2192 ((y : \u03b1) \u2192 r y x \u2192 C y) \u2192 C x\nx : \u03b1\n\u22a2 fix hwf F x = WellFounded.fixC hwf F x"}, {"tactic": "rw [fix, fixF_eq_fixFC, fixC]", "annotated_tactic": ["rw [<a>fix</a>, <a>fixF_eq_fixFC</a>, <a>fixC</a>]", [{"full_name": "WellFounded.fix", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [71, 19], "def_end_pos": [71, 22]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.WF.0.WellFounded.fixF_eq_fixFC", "def_path": "lake-packages/std/Std/WF.lean", "def_pos": [100, 26], "def_end_pos": [100, 39]}, {"full_name": "_private.\u00ablake-packages\u00bb.std.Std.WF.0.WellFounded.fixC", "def_path": "lake-packages/std/Std/WF.lean", "def_pos": [105, 13], "def_end_pos": [105, 17]}]], "state_before": "case h.h.h.h.h.h\n\u03b1 : Sort u_1\nC : \u03b1 \u2192 Sort u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhwf : WellFounded r\nF : (x : \u03b1) \u2192 ((y : \u03b1) \u2192 r y x \u2192 C y) \u2192 C x\nx : \u03b1\n\u22a2 fix hwf F x = WellFounded.fixC hwf F x", "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\u2082", "start": [328, 1], "end": [330, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_countable", "start": [1799, 1], "end": [1814, 7], "traced_tactics": [{"tactic": "have hi : Countable { x // x \u2208 s } := Iff.mpr countable_coe_iff hs", "annotated_tactic": ["have hi : <a>Countable</a> { x // x \u2208 s } := <a>Iff.mpr</a> <a>countable_coe_iff</a> hs", [{"full_name": "Countable", "def_path": "Mathlib/Data/Countable/Defs.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}, {"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": "Set.countable_coe_iff", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [36, 9], "def_end_pos": [36, 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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a", "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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a"}, {"tactic": "rw [set_integral_eq_subtype' hs.measurableSet, integral_countable' hf']", "annotated_tactic": ["rw [<a>set_integral_eq_subtype'</a> hs.measurableSet, <a>integral_countable'</a> hf']", [{"full_name": "MeasureTheory.set_integral_eq_subtype'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1648, 9], "def_end_pos": [1648, 33]}, {"full_name": "MeasureTheory.integral_countable'", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1781, 9], "def_end_pos": [1781, 28]}]], "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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a", "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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\n\u22a2 \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191(Measure.comap Subtype.val \u03bc) {a}) \u2022 f \u2191a =\n    \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a"}, {"tactic": "congr 1 with a : 1", "annotated_tactic": ["congr 1 with a : 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 : \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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\n\u22a2 \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191(Measure.comap Subtype.val \u03bc) {a}) \u2022 f \u2191a =\n    \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a", "state_after": "case e_f.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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\na : \u2191s\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.comap Subtype.val \u03bc) {a}) \u2022 f \u2191a = ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a"}, {"tactic": "rw [Measure.comap_apply Subtype.val Subtype.coe_injective\n  (fun s' hs' => MeasurableSet.subtype_image (Countable.measurableSet hs) hs') _\n  (MeasurableSet.singleton a)]", "annotated_tactic": ["rw [<a>Measure.comap_apply</a> <a>Subtype.val</a> <a>Subtype.coe_injective</a>\n    (fun s' hs' => <a>MeasurableSet.subtype_image</a> (<a>Countable.measurableSet</a> hs) hs') _\n    (<a>MeasurableSet.singleton</a> a)]", [{"full_name": "MeasureTheory.Measure.comap_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1335, 9], "def_end_pos": [1335, 20]}, {"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": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}, {"full_name": "MeasurableSet.subtype_image", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [596, 9], "def_end_pos": [596, 36]}, {"full_name": "Set.Countable.measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [314, 9], "def_end_pos": [314, 36]}, {"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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\na : \u2191s\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.comap Subtype.val \u03bc) {a}) \u2022 f \u2191a = ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a", "state_after": "case e_f.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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\na : \u2191s\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (Subtype.val '' {a})) \u2022 f \u2191a = ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_f.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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\nhf' : Integrable fun x => f \u2191x\na : \u2191s\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (Subtype.val '' {a})) \u2022 f \u2191a = ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a", "state_after": "no goals"}, {"tactic": "rw [\u2190 map_comap_subtype_coe, integrable_map_measure] at hf", "annotated_tactic": ["rw [\u2190 <a>map_comap_subtype_coe</a>, <a>integrable_map_measure</a>] at hf", [{"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.integrable_map_measure", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [610, 9], "def_end_pos": [610, 31]}]], "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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 Integrable fun x => f \u2191x", "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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable (f \u2218 Subtype.val)\nhi : Countable { x // x \u2208 s }\n\u22a2 Integrable fun x => f \u2191x\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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val \u03bc))\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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEMeasurable Subtype.val\n\ncase hs\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 MeasurableSet s"}, {"tactic": "apply hf", "annotated_tactic": ["apply hf", []], "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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable (f \u2218 Subtype.val)\nhi : Countable { x // x \u2208 s }\n\u22a2 Integrable fun x => f \u2191x\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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val \u03bc))\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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEMeasurable Subtype.val\n\ncase hs\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 MeasurableSet s", "state_after": "case hg\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val \u03bc))\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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEMeasurable Subtype.val\n\ncase hs\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 MeasurableSet s"}, {"tactic": "exact Integrable.aestronglyMeasurable hf", "annotated_tactic": ["exact <a>Integrable.aestronglyMeasurable</a> hf", [{"full_name": "MeasureTheory.Integrable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [457, 9], "def_end_pos": [457, 40]}]], "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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val \u03bc))", "state_after": "no goals"}, {"tactic": "exact Measurable.aemeasurable measurable_subtype_coe", "annotated_tactic": ["exact <a>Measurable.aemeasurable</a> <a>measurable_subtype_coe</a>", [{"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "measurable_subtype_coe", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 31]}]], "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\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 AEMeasurable Subtype.val", "state_after": "no goals"}, {"tactic": "exact Countable.measurableSet hs", "annotated_tactic": ["exact <a>Countable.measurableSet</a> hs", [{"full_name": "Set.Countable.measurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [314, 9], "def_end_pos": [314, 36]}]], "state_before": "case hs\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\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : Set.Countable s\nhf : Integrable f\nhi : Countable { x // x \u2208 s }\n\u22a2 MeasurableSet s", "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.neg_toSimpleFunc", "start": [653, 1], "end": [657, 25], "traced_tactics": [{"tactic": "filter_upwards [toSimpleFunc_eq_toFun (-f), toSimpleFunc_eq_toFun f,\n  Lp.coeFn_neg (f : Lp E p \u03bc)] with _", "annotated_tactic": ["filter_upwards [<a>toSimpleFunc_eq_toFun</a> (-f), <a>toSimpleFunc_eq_toFun</a> f,\n    <a>Lp.coeFn_neg</a> (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_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [228, 9], "def_end_pos": [228, 18]}, {"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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc (-f)) =\u1d50[\u03bc] -\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191\u2191\u2191(-f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(-\u2191f) a\u271d = (-\u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = (-\u2191(toSimpleFunc f)) a\u271d"}, {"tactic": "simp only [Pi.neg_apply, AddSubgroup.coe_neg]", "annotated_tactic": ["simp only [<a>Pi.neg_apply</a>, <a>AddSubgroup.coe_neg</a>]", [{"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "AddSubgroup.coe_neg", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [712, 3], "def_end_pos": [712, 14]}]], "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191\u2191\u2191(-f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(-\u2191f) a\u271d = (-\u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = (-\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = -\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = -\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(-\u2191\u2191f) a\u271d = -\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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_cons", "start": [911, 1], "end": [912, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.restrict_restrict", "start": [199, 1], "end": [201, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Option.lean", "full_name": "Finset.map_some_eraseNone", "start": [148, 1], "end": [150, 64], "traced_tactics": [{"tactic": "rw [map_eq_image, Embedding.some_apply, image_some_eraseNone]", "annotated_tactic": ["rw [<a>map_eq_image</a>, <a>Embedding.some_apply</a>, <a>image_some_eraseNone</a>]", [{"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [342, 9], "def_end_pos": [342, 21]}, {"full_name": "Function.Embedding.some_apply", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [211, 3], "def_end_pos": [211, 8]}, {"full_name": "Finset.image_some_eraseNone", "def_path": "Mathlib/Data/Finset/Option.lean", "def_pos": [143, 9], "def_end_pos": [143, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq (Option \u03b1)\ns : Finset (Option \u03b1)\n\u22a2 map Embedding.some (\u2191eraseNone s) = erase s none", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "List.perm_of_nodup_nodup_toFinset_eq", "start": [3364, 1], "end": [3367, 45], "traced_tactics": [{"tactic": "rw [\u2190 Multiset.coe_eq_coe]", "annotated_tactic": ["rw [\u2190 <a>Multiset.coe_eq_coe</a>]", [{"full_name": "Multiset.coe_eq_coe", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 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\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 l ~ l'", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 \u2191l = \u2191l'"}, {"tactic": "exact Multiset.Nodup.toFinset_inj hl hl' h", "annotated_tactic": ["exact <a>Multiset.Nodup.toFinset_inj</a> hl hl' h", [{"full_name": "Multiset.Nodup.toFinset_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3194, 9], "def_end_pos": [3194, 27]}]], "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\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 \u2191l = \u2191l'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.vars_bind\u2081", "start": [357, 1], "end": [386, 48], "traced_tactics": [{"tactic": "calc\n  (bind\u2081 f \u03c6).vars = (\u03c6.support.sum fun x : \u03c3 \u2192\u2080 \u2115 => (bind\u2081 f) (monomial x (coeff x \u03c6))).vars :=\n    by rw [\u2190 AlgHom.map_sum, \u2190 \u03c6.as_sum]\n  _ \u2264 \u03c6.support.biUnion fun i : \u03c3 \u2192\u2080 \u2115 => ((bind\u2081 f) (monomial i (coeff i \u03c6))).vars :=\n    (vars_sum_subset _ _)\n  _ = \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => vars (C (coeff d \u03c6) * \u220f i in d.support, f i ^ d i) := by\n    simp only [bind\u2081_monomial]\n  _ \u2264 \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => d.support.biUnion fun i => vars (f i) := ?_\n  _ \u2264 \u03c6.vars.biUnion fun i : \u03c3 => vars (f i) := ?_", "annotated_tactic": ["calc\n    (<a>bind\u2081</a> f \u03c6).<a>vars</a> = (\u03c6.support.sum fun x : \u03c3 \u2192\u2080 \u2115 => (<a>bind\u2081</a> f) (<a>monomial</a> x (<a>coeff</a> x \u03c6))).<a>vars</a> :=\n      by rw [\u2190 <a>AlgHom.map_sum</a>, \u2190 \u03c6.as_sum]\n    _ \u2264 \u03c6.support.biUnion fun i : \u03c3 \u2192\u2080 \u2115 => ((<a>bind\u2081</a> f) (<a>monomial</a> i (<a>coeff</a> i \u03c6))).<a>vars</a> :=\n      (<a>vars_sum_subset</a> _ _)\n    _ = \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => <a>vars</a> (<a>C</a> (<a>coeff</a> d \u03c6) * \u220f i in d.support, f i ^ d i) := by\n      simp only [<a>bind\u2081_monomial</a>]\n    _ \u2264 \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => d.support.biUnion fun i => <a>vars</a> (f i) := ?_\n    -- proof below\n    _ \u2264 \u03c6.vars.biUnion fun i : \u03c3 => <a>vars</a> (f i) := ?_", [{"full_name": "MvPolynomial.bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [69, 5], "def_end_pos": [69, 10]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [69, 5], "def_end_pos": [69, 10]}, {"full_name": "MvPolynomial.monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [167, 5], "def_end_pos": [167, 13]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "AlgHom.map_sum", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [269, 19], "def_end_pos": [269, 26]}, {"full_name": "MvPolynomial.bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [69, 5], "def_end_pos": [69, 10]}, {"full_name": "MvPolynomial.monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [167, 5], "def_end_pos": [167, 13]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.vars_sum_subset", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [416, 9], "def_end_pos": [416, 24]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [180, 5], "def_end_pos": [180, 6]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "MvPolynomial.bind\u2081_monomial", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [338, 9], "def_end_pos": [338, 23]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}]], "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\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 vars (\u2191(bind\u2081 f) \u03c6) \u2286 Finset.biUnion (vars \u03c6) fun i => vars (f i)", "state_after": "case calc_1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i)) \u2264\n    Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)\n\ncase calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2264\n    Finset.biUnion (vars \u03c6) fun i => vars (f i)"}, {"tactic": "rw [\u2190 AlgHom.map_sum, \u2190 \u03c6.as_sum]", "annotated_tactic": ["rw [\u2190 <a>AlgHom.map_sum</a>, \u2190 \u03c6.as_sum]", [{"full_name": "AlgHom.map_sum", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [269, 19], "def_end_pos": [269, 26]}]], "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\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 vars (\u2191(bind\u2081 f) \u03c6) = vars (\u2211 x in support \u03c6, \u2191(bind\u2081 f) (\u2191(monomial x) (coeff x \u03c6)))", "state_after": "no goals"}, {"tactic": "simp only [bind\u2081_monomial]", "annotated_tactic": ["simp only [<a>bind\u2081_monomial</a>]", [{"full_name": "MvPolynomial.bind\u2081_monomial", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [338, 9], "def_end_pos": [338, 23]}]], "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\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun i => vars (\u2191(bind\u2081 f) (\u2191(monomial i) (coeff i \u03c6)))) =\n    Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i)", "state_after": "no goals"}, {"tactic": "apply Finset.biUnion_mono", "annotated_tactic": ["apply <a>Finset.biUnion_mono</a>", [{"full_name": "Finset.biUnion_mono", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3679, 9], "def_end_pos": [3679, 21]}]], "state_before": "case calc_1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i)) \u2264\n    Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)", "state_after": "case calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2200 (a : \u03c3 \u2192\u2080 \u2115),\n    a \u2208 support \u03c6 \u2192 vars (\u2191C (coeff a \u03c6) * \u220f i in a.support, f i ^ \u2191a i) \u2286 Finset.biUnion a.support fun i => vars (f i)"}, {"tactic": "intro d _hd", "annotated_tactic": ["intro d _hd", []], "state_before": "case calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2200 (a : \u03c3 \u2192\u2080 \u2115),\n    a \u2208 support \u03c6 \u2192 vars (\u2191C (coeff a \u03c6) * \u220f i in a.support, f i ^ \u2191a i) \u2286 Finset.biUnion a.support fun i => vars (f i)", "state_after": "case calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i) \u2286 Finset.biUnion d.support fun i => vars (f i)"}, {"tactic": "calc\n  vars (C (coeff d \u03c6) * \u220f i : \u03c3 in d.support, f i ^ d i) \u2264\n      (C (coeff d \u03c6)).vars \u222a (\u220f i : \u03c3 in d.support, f i ^ d i).vars :=\n    vars_mul _ _\n  _ \u2264 (\u220f i : \u03c3 in d.support, f i ^ d i).vars := by\n    simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, Finset.Subset.refl]\n  _ \u2264 d.support.biUnion fun i : \u03c3 => vars (f i ^ d i) := (vars_prod _)\n  _ \u2264 d.support.biUnion fun i : \u03c3 => (f i).vars := ?_", "annotated_tactic": ["calc\n      <a>vars</a> (<a>C</a> (<a>coeff</a> d \u03c6) * \u220f i : \u03c3 in d.support, f i ^ d i) \u2264\n          (<a>C</a> (<a>coeff</a> d \u03c6)).<a>vars</a> \u222a (\u220f i : \u03c3 in d.support, f i ^ d i).<a>vars</a> :=\n        <a>vars_mul</a> _ _\n      _ \u2264 (\u220f i : \u03c3 in d.support, f i ^ d i).<a>vars</a> := by\n        simp only [<a>Finset.empty_union</a>, <a>vars_C</a>, <a>Finset.le_iff_subset</a>, <a>Finset.Subset.refl</a>]\n      _ \u2264 d.support.biUnion fun i : \u03c3 => <a>vars</a> (f i ^ d i) := (<a>vars_prod</a> _)\n      _ \u2264 d.support.biUnion fun i : \u03c3 => (f i).<a>vars</a> := ?_", [{"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [180, 5], "def_end_pos": [180, 6]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "MvPolynomial.C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [180, 5], "def_end_pos": [180, 6]}, {"full_name": "MvPolynomial.coeff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [580, 5], "def_end_pos": [580, 10]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.vars_mul", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [342, 9], "def_end_pos": [342, 17]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "Finset.empty_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1474, 9], "def_end_pos": [1474, 20]}, {"full_name": "MvPolynomial.vars_C", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [295, 9], "def_end_pos": [295, 15]}, {"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.refl", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}, {"full_name": "MvPolynomial.vars_prod", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [382, 9], "def_end_pos": [382, 18]}, {"full_name": "MvPolynomial.vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [274, 5], "def_end_pos": [274, 9]}]], "state_before": "case calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i) \u2286 Finset.biUnion d.support fun i => vars (f i)", "state_after": "case calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 (Finset.biUnion d.support fun i => vars (f i ^ \u2191d i)) \u2264 Finset.biUnion d.support fun i => vars (f i)"}, {"tactic": "apply Finset.biUnion_mono", "annotated_tactic": ["apply <a>Finset.biUnion_mono</a>", [{"full_name": "Finset.biUnion_mono", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3679, 9], "def_end_pos": [3679, 21]}]], "state_before": "case calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 (Finset.biUnion d.support fun i => vars (f i ^ \u2191d i)) \u2264 Finset.biUnion d.support fun i => vars (f i)", "state_after": "case calc_1.h.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 \u2200 (a : \u03c3), a \u2208 d.support \u2192 vars (f a ^ \u2191d a) \u2286 vars (f a)"}, {"tactic": "intro i _hi", "annotated_tactic": ["intro i _hi", []], "state_before": "case calc_1.h.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 \u2200 (a : \u03c3), a \u2208 d.support \u2192 vars (f a ^ \u2191d a) \u2286 vars (f a)", "state_after": "case calc_1.h.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\ni : \u03c3\n_hi : i \u2208 d.support\n\u22a2 vars (f i ^ \u2191d i) \u2286 vars (f i)"}, {"tactic": "apply vars_pow", "annotated_tactic": ["apply <a>vars_pow</a>", [{"full_name": "MvPolynomial.vars_pow", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [370, 9], "def_end_pos": [370, 17]}]], "state_before": "case calc_1.h.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\ni : \u03c3\n_hi : i \u2208 d.support\n\u22a2 vars (f i ^ \u2191d i) \u2286 vars (f i)", "state_after": "no goals"}, {"tactic": "simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, Finset.Subset.refl]", "annotated_tactic": ["simp only [<a>Finset.empty_union</a>, <a>vars_C</a>, <a>Finset.le_iff_subset</a>, <a>Finset.Subset.refl</a>]", [{"full_name": "Finset.empty_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1474, 9], "def_end_pos": [1474, 20]}, {"full_name": "MvPolynomial.vars_C", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [295, 9], "def_end_pos": [295, 15]}, {"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.refl", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}]], "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\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 vars (\u2191C (coeff d \u03c6)) \u222a vars (\u220f i in d.support, f i ^ \u2191d i) \u2264 vars (\u220f i in d.support, f i ^ \u2191d i)", "state_after": "no goals"}, {"tactic": "intro j", "annotated_tactic": ["intro j", []], "state_before": "case calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2264\n    Finset.biUnion (vars \u03c6) fun i => vars (f i)", "state_after": "case calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\n\u22a2 (j \u2208 Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2192\n    j \u2208 Finset.biUnion (vars \u03c6) fun i => vars (f i)"}, {"tactic": "simp_rw [Finset.mem_biUnion]", "annotated_tactic": ["simp_rw [<a>Finset.mem_biUnion</a>]", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}]], "state_before": "case calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\n\u22a2 (j \u2208 Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2192\n    j \u2208 Finset.biUnion (vars \u03c6) fun i => vars (f i)", "state_after": "case calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\n\u22a2 (\u2203 a, a \u2208 support \u03c6 \u2227 \u2203 a_1, a_1 \u2208 a.support \u2227 j \u2208 vars (f a_1)) \u2192 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a)"}, {"tactic": "rintro \u27e8d, hd, \u27e8i, hi, hj\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8d, hd, \u27e8i, hi, hj\u27e9\u27e9", []], "state_before": "case calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\n\u22a2 (\u2203 a, a \u2208 support \u03c6 \u2227 \u2203 a_1, a_1 \u2208 a.support \u2227 j \u2208 vars (f a_1)) \u2192 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a)", "state_after": "case calc_2.intro.intro.intro.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6\ni : \u03c3\nhi : i \u2208 d.support\nhj : j \u2208 vars (f i)\n\u22a2 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a)"}, {"tactic": "exact \u27e8i, (mem_vars _).mpr \u27e8d, hd, hi\u27e9, hj\u27e9", "annotated_tactic": ["exact \u27e8i, (<a>mem_vars</a> _).<a>mpr</a> \u27e8d, hd, hi\u27e9, hj\u27e9", [{"full_name": "MvPolynomial.mem_vars", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [306, 9], "def_end_pos": [306, 17]}, {"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 calc_2.intro.intro.intro.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6\ni : \u03c3\nhi : i \u2208 d.support\nhj : j \u2208 vars (f i)\n\u22a2 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a)", "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.foldlM_eq_foldlM_data", "start": [42, 1], "end": [45, 43], "traced_tactics": [{"tactic": "simp [foldlM, foldlM_eq_foldlM_data.aux]", "annotated_tactic": ["simp [<a>foldlM</a>, <a>foldlM_eq_foldlM_data.aux</a>]", [{"full_name": "Array.foldlM", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [196, 5], "def_end_pos": [196, 11]}, {"full_name": "Array.foldlM_eq_foldlM_data.aux", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [31, 9], "def_end_pos": [31, 34]}]], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\n\u03b1 : Type u_3\ninst\u271d : Monad m\nf : \u03b2 \u2192 \u03b1 \u2192 m \u03b2\ninit : \u03b2\narr : Array \u03b1\n\u22a2 foldlM f init arr 0 (size arr) = List.foldlM f init arr.data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.decode_encodeBool", "start": [222, 1], "end": [222, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.coe_orderIsoOfFin_apply", "start": [160, 1], "end": [162, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Condexp.lean", "full_name": "ProbabilityTheory.aestronglyMeasurable'_integral_condexpKernel", "start": [109, 1], "end": [117, 49], "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 : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable' m (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' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \u03bc"}, {"tactic": "have h := 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 : m \u2293 m\u03a9 \u2264 m\u03a9))", "annotated_tactic": ["have h := <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> : m \u2293 m\u03a9 \u2264 m\u03a9))", [{"full_name": "ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib", "def_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "def_pos": [101, 9], "def_end_pos": [101, 51]}, {"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' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \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\nh :\n  AEStronglyMeasurable' (MeasurableSpace.comap id (m \u2293 m\u03a9))\n    (fun a => \u222b (y : \u03a9), f (id a, y).2 \u2202\u2191(condDistrib id id \u03bc) (id a)) \u03bc\n\u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \u03bc"}, {"tactic": "rw [MeasurableSpace.comap_id] at h", "annotated_tactic": ["rw [<a>MeasurableSpace.comap_id</a>] at h", [{"full_name": "MeasurableSpace.comap_id", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [117, 9], "def_end_pos": [117, 17]}]], "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\nh :\n  AEStronglyMeasurable' (MeasurableSpace.comap id (m \u2293 m\u03a9))\n    (fun a => \u222b (y : \u03a9), f (id a, y).2 \u2202\u2191(condDistrib id id \u03bc) (id a)) \u03bc\n\u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \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\nh : AEStronglyMeasurable' (m \u2293 m\u03a9) (fun a => \u222b (y : \u03a9), f (id a, y).2 \u2202\u2191(condDistrib id id \u03bc) (id a)) \u03bc\n\u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \u03bc"}, {"tactic": "exact AEStronglyMeasurable'.mono h inf_le_left", "annotated_tactic": ["exact <a>AEStronglyMeasurable'.mono</a> h <a>inf_le_left</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.mono", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [64, 9], "def_end_pos": [64, 13]}, {"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [388, 9], "def_end_pos": [388, 20]}]], "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\nh : AEStronglyMeasurable' (m \u2293 m\u03a9) (fun a => \u222b (y : \u03a9), f (id a, y).2 \u2202\u2191(condDistrib id id \u03bc) (id a)) \u03bc\n\u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \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_of_tendsto_normalize_testAgainstNN_of_tendsto_mass", "start": [466, 1], "end": [471, 95], "traced_tactics": [{"tactic": "rw [tendsto_iff_forall_testAgainstNN_tendsto]", "annotated_tactic": ["rw [<a>tendsto_iff_forall_testAgainstNN_tendsto</a>]", [{"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": "\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))\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc)", "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 (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))"}, {"tactic": "exact fun f =>\n  tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass \u03bcs_lim mass_lim f", "annotated_tactic": ["exact fun f =>\n    <a>tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass</a> \u03bcs_lim mass_lim f", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass", "def_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "def_pos": [419, 9], "def_end_pos": [419, 81]}]], "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))\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bit_decomp", "start": [136, 1], "end": [137, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurableSet_setOf", "start": [1136, 9], "end": [1138, 50], "traced_tactics": [{"tactic": "simpa only [preimage_singleton_true]", "annotated_tactic": ["simpa only [<a>preimage_singleton_true</a>]", [{"full_name": "Set.preimage_singleton_true", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [194, 17], "def_end_pos": [194, 40]}]], "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\ninst\u271d : MeasurableSpace \u03b1\np : \u03b1 \u2192 Prop\nh : MeasurableSet {a | p a}\n\u22a2 MeasurableSet (p \u207b\u00b9' {True})", "state_after": "no goals"}, {"tactic": "simpa using h (measurableSet_singleton True)", "annotated_tactic": ["simpa using h (<a>measurableSet_singleton</a> <a>True</a>)", [{"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}, {"full_name": "True", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [176, 11], "def_end_pos": [176, 15]}]], "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\ninst\u271d : MeasurableSpace \u03b1\np : \u03b1 \u2192 Prop\nh : Measurable p\n\u22a2 MeasurableSet {a | p a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Disintegration.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.integral_condKernel", "start": [441, 1], "end": [445, 57], "traced_tactics": [{"tactic": "rw [measure_eq_compProd \u03c1] at hf", "annotated_tactic": ["rw [<a>measure_eq_compProd</a> \u03c1] at hf", [{"full_name": "ProbabilityTheory.measure_eq_compProd", "def_path": "Mathlib/Probability/Kernel/Disintegration.lean", "def_pos": [369, 9], "def_end_pos": [369, 28]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : AEStronglyMeasurable f \u03c1\n\u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1)", "state_after": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n\u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1)"}, {"tactic": "exact AEStronglyMeasurable.integral_kernel_compProd hf", "annotated_tactic": ["exact <a>AEStronglyMeasurable.integral_kernel_compProd</a> hf", [{"full_name": "MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [71, 9], "def_end_pos": [71, 75]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n\u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_getI", "start": [1094, 1], "end": [1095, 14], "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_sup_eq", "start": [341, 1], "end": [347, 43], "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\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = prehaar (\u2191K\u2080) U K\u2081 + 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\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) + \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U)"}, {"tactic": "rw [div_add_div_same]", "annotated_tactic": ["rw [<a>div_add_div_same</a>]", [{"full_name": "div_add_div_same", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 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\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) + \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\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = (\u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)) / \u2191(index (\u2191K\u2080) U)"}, {"tactic": "refine congr_arg (fun x : \u211d => x / index K\u2080 U) ?_", "annotated_tactic": ["refine <a>congr_arg</a> (fun x : \u211d => x / <a>index</a> K\u2080 U) ?_", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"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\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = (\u2191(index (\u2191K\u2081) U) + \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\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) = \u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)"}, {"tactic": "exact_mod_cast index_union_eq K\u2081 K\u2082 hU h", "annotated_tactic": ["exact_mod_cast <a>index_union_eq</a> K\u2081 K\u2082 hU h", [{"full_name": "MeasureTheory.Measure.haar.index_union_eq", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 23]}]], "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\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) = \u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.some_mul_some", "start": [742, 1], "end": [742, 94], "traced_tactics": [{"tactic": "simp [mul_def]", "annotated_tactic": ["simp [<a>mul_def</a>]", [{"full_name": "Part.mul_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [701, 9], "def_end_pos": [701, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Mul \u03b1\na b : \u03b1\n\u22a2 some a * some b = some (a * b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.withDensity_smul_measure", "start": [105, 1], "end": [109, 45], "traced_tactics": [{"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity (r \u2022 \u03bc) f = r \u2022 withDensity \u03bc f", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity (r \u2022 \u03bc) f) s = \u2191\u2191(r \u2022 withDensity \u03bc f) s"}, {"tactic": "rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,\n  smul_eq_mul, set_lintegral_smul_measure]", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ hs, <a>Measure.coe_smul</a>, <a>Pi.smul_apply</a>, <a>withDensity_apply</a> _ hs,\n    <a>smul_eq_mul</a>, <a>set_lintegral_smul_measure</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.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": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "MeasureTheory.set_lintegral_smul_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [591, 7], "def_end_pos": [591, 33]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity (r \u2022 \u03bc) f) s = \u2191\u2191(r \u2022 withDensity \u03bc 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.toList_sorted", "start": [652, 1], "end": [653, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.Nonempty.cons_induction", "start": [1275, 1], "end": [1283, 28], "traced_tactics": [{"tactic": "induction' s using Finset.cons_induction with a t ha h", "annotated_tactic": ["induction' s using <a>Finset.cons_induction</a> with a t ha h", [{"full_name": "Finset.cons_induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1221, 9], "def_end_pos": [1221, 23]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 p s hs", "state_after": "case empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\nhs : Finset.Nonempty \u2205\n\u22a2 p \u2205 hs\n\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t\u271d u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nt : Finset \u03b1\nha : \u00aca \u2208 t\nh : \u2200 (hs : Finset.Nonempty t), p t hs\nhs : Finset.Nonempty (cons a t ha)\n\u22a2 p (cons a t ha) hs"}, {"tactic": "obtain rfl | ht := t.eq_empty_or_nonempty", "annotated_tactic": ["obtain rfl | ht := t.eq_empty_or_nonempty", []], "state_before": "case cons\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t\u271d u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nt : Finset \u03b1\nha : \u00aca \u2208 t\nh : \u2200 (hs : Finset.Nonempty t), p t hs\nhs : Finset.Nonempty (cons a t ha)\n\u22a2 p (cons a t ha) hs", "state_after": "case cons.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nha : \u00aca \u2208 \u2205\nh : \u2200 (hs : Finset.Nonempty \u2205), p \u2205 hs\nhs : Finset.Nonempty (cons a \u2205 ha)\n\u22a2 p (cons a \u2205 ha) hs\n\ncase cons.inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t\u271d u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nt : Finset \u03b1\nha : \u00aca \u2208 t\nh : \u2200 (hs : Finset.Nonempty t), p t hs\nhs : Finset.Nonempty (cons a t ha)\nht : Finset.Nonempty t\n\u22a2 p (cons a t ha) hs"}, {"tactic": "exact (not_nonempty_empty hs).elim", "annotated_tactic": ["exact (<a>not_nonempty_empty</a> hs).<a>elim</a>", [{"full_name": "Finset.not_nonempty_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [554, 9], "def_end_pos": [554, 27]}, {"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 empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\nhs : Finset.Nonempty \u2205\n\u22a2 p \u2205 hs", "state_after": "no goals"}, {"tactic": "exact h\u2080 a", "annotated_tactic": ["exact h\u2080 a", []], "state_before": "case cons.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nha : \u00aca \u2208 \u2205\nh : \u2200 (hs : Finset.Nonempty \u2205), p \u2205 hs\nhs : Finset.Nonempty (cons a \u2205 ha)\n\u22a2 p (cons a \u2205 ha) hs", "state_after": "no goals"}, {"tactic": "exact h\u2081 t ha ht (h ht)", "annotated_tactic": ["exact h\u2081 t ha ht (h ht)", []], "state_before": "case cons.inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t\u271d u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n  \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n    p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nt : Finset \u03b1\nha : \u00aca \u2208 t\nh : \u2200 (hs : Finset.Nonempty t), p t hs\nhs : Finset.Nonempty (cons a t ha)\nht : Finset.Nonempty t\n\u22a2 p (cons a t ha) hs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.disjSups_assoc", "start": [578, 1], "end": [583, 86], "traced_tactics": [{"tactic": "refine' associative_of_commutative_of_le disjSups_comm _", "annotated_tactic": ["refine' <a>associative_of_commutative_of_le</a> <a>disjSups_comm</a> _", [{"full_name": "associative_of_commutative_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 41]}, {"full_name": "Finset.disjSups_comm", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [559, 9], "def_end_pos": [559, 22]}]], "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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns t u v : Finset \u03b1\n\u22a2 \u2200 (s t u : Finset \u03b1), s \u25cb t \u25cb u = s \u25cb (t \u25cb u)", "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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns t u v : Finset \u03b1\n\u22a2 \u2200 (a b c : Finset \u03b1), a \u25cb b \u25cb c \u2264 a \u25cb (b \u25cb c)"}, {"tactic": "simp only [le_eq_subset, disjSups_subset_iff, mem_disjSups]", "annotated_tactic": ["simp only [<a>le_eq_subset</a>, <a>disjSups_subset_iff</a>, <a>mem_disjSups</a>]", [{"full_name": "Finset.le_eq_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [393, 9], "def_end_pos": [393, 21]}, {"full_name": "Finset.disjSups_subset_iff", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [515, 9], "def_end_pos": [515, 28]}, {"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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns t u v : Finset \u03b1\n\u22a2 \u2200 (a b c : Finset \u03b1), a \u25cb b \u25cb c \u2264 a \u25cb (b \u25cb c)", "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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns t u v : Finset \u03b1\n\u22a2 \u2200 (a b c : Finset \u03b1) (a_1 : \u03b1),\n    (\u2203 a_2, a_2 \u2208 a \u2227 \u2203 b_1, b_1 \u2208 b \u2227 Disjoint a_2 b_1 \u2227 a_2 \u2294 b_1 = a_1) \u2192\n      \u2200 (b_1 : \u03b1),\n        b_1 \u2208 c \u2192\n          Disjoint a_1 b_1 \u2192\n            \u2203 a_5,\n              a_5 \u2208 a \u2227\n                \u2203 b_2, (\u2203 a, a \u2208 b \u2227 \u2203 b, b \u2208 c \u2227 Disjoint a b \u2227 a \u2294 b = b_2) \u2227 Disjoint a_5 b_2 \u2227 a_5 \u2294 b_2 = a_1 \u2294 b_1"}, {"tactic": "rintro s t u _ \u27e8a, ha, b, hb, hab, rfl\u27e9 c hc habc", "annotated_tactic": ["rintro s t u _ \u27e8a, ha, b, hb, hab, rfl\u27e9 c hc habc", []], "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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns t u v : Finset \u03b1\n\u22a2 \u2200 (a b c : Finset \u03b1) (a_1 : \u03b1),\n    (\u2203 a_2, a_2 \u2208 a \u2227 \u2203 b_1, b_1 \u2208 b \u2227 Disjoint a_2 b_1 \u2227 a_2 \u2294 b_1 = a_1) \u2192\n      \u2200 (b_1 : \u03b1),\n        b_1 \u2208 c \u2192\n          Disjoint a_1 b_1 \u2192\n            \u2203 a_5,\n              a_5 \u2208 a \u2227\n                \u2203 b_2, (\u2203 a, a \u2208 b \u2227 \u2203 b, b \u2208 c \u2227 Disjoint a b \u2227 a \u2294 b = b_2) \u2227 Disjoint a_5 b_2 \u2227 a_5 \u2294 b_2 = a_1 \u2294 b_1", "state_after": "case intro.intro.intro.intro.intro\nF : 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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns\u271d t\u271d u\u271d v s t u : Finset \u03b1\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nhab : Disjoint a b\nc : \u03b1\nhc : c \u2208 u\nhabc : Disjoint (a \u2294 b) c\n\u22a2 \u2203 a_1,\n    a_1 \u2208 s \u2227 \u2203 b_1, (\u2203 a, a \u2208 t \u2227 \u2203 b, b \u2208 u \u2227 Disjoint a b \u2227 a \u2294 b = b_1) \u2227 Disjoint a_1 b_1 \u2227 a_1 \u2294 b_1 = a \u2294 b \u2294 c"}, {"tactic": "rw [disjoint_sup_left] at habc", "annotated_tactic": ["rw [<a>disjoint_sup_left</a>] at habc", [{"full_name": "disjoint_sup_left", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [185, 9], "def_end_pos": [185, 26]}]], "state_before": "case intro.intro.intro.intro.intro\nF : 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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns\u271d t\u271d u\u271d v s t u : Finset \u03b1\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nhab : Disjoint a b\nc : \u03b1\nhc : c \u2208 u\nhabc : Disjoint (a \u2294 b) c\n\u22a2 \u2203 a_1,\n    a_1 \u2208 s \u2227 \u2203 b_1, (\u2203 a, a \u2208 t \u2227 \u2203 b, b \u2208 u \u2227 Disjoint a b \u2227 a \u2294 b = b_1) \u2227 Disjoint a_1 b_1 \u2227 a_1 \u2294 b_1 = a \u2294 b \u2294 c", "state_after": "case intro.intro.intro.intro.intro\nF : 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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns\u271d t\u271d u\u271d v s t u : Finset \u03b1\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nhab : Disjoint a b\nc : \u03b1\nhc : c \u2208 u\nhabc : Disjoint a c \u2227 Disjoint b c\n\u22a2 \u2203 a_1,\n    a_1 \u2208 s \u2227 \u2203 b_1, (\u2203 a, a \u2208 t \u2227 \u2203 b, b \u2208 u \u2227 Disjoint a b \u2227 a \u2294 b = b_1) \u2227 Disjoint a_1 b_1 \u2227 a_1 \u2294 b_1 = a \u2294 b \u2294 c"}, {"tactic": "exact \u27e8a, ha, _, \u27e8b, hb, c, hc, habc.2, rfl\u27e9, hab.sup_right habc.1, sup_assoc.symm\u27e9", "annotated_tactic": ["exact \u27e8a, ha, _, \u27e8b, hb, c, hc, habc.2, <a>rfl</a>\u27e9, hab.sup_right habc.1, sup_assoc.symm\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.intro\nF : 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 : DistribLattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns\u271d t\u271d u\u271d v s t u : Finset \u03b1\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 t\nhab : Disjoint a b\nc : \u03b1\nhc : c \u2208 u\nhabc : Disjoint a c \u2227 Disjoint b c\n\u22a2 \u2203 a_1,\n    a_1 \u2208 s \u2227 \u2203 b_1, (\u2203 a, a \u2208 t \u2227 \u2203 b, b \u2208 u \u2227 Disjoint a b \u2227 a \u2294 b = b_1) \u2227 Disjoint a_1 b_1 \u2227 a_1 \u2294 b_1 = a \u2294 b \u2294 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_smul_measure_of_ne_zero_of_ne_top", "start": [642, 9], "end": [649, 26], "traced_tactics": [{"tactic": "simp_rw [snorm_eq_snorm' hp_ne_zero hp_ne_top]", "annotated_tactic": ["simp_rw [<a>snorm_eq_snorm'</a> hp_ne_zero hp_ne_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": "\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f 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\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm' f (ENNReal.toReal p) (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc"}, {"tactic": "rw [snorm'_smul_measure ENNReal.toReal_nonneg]", "annotated_tactic": ["rw [<a>snorm'_smul_measure</a> <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "MeasureTheory.snorm'_smul_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [629, 9], "def_end_pos": [629, 28]}, {"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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm' f (ENNReal.toReal p) (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal 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\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 c ^ (1 / ENNReal.toReal p) * snorm' f (ENNReal.toReal p) \u03bc =\n    c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc"}, {"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\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 c ^ (1 / ENNReal.toReal p) * snorm' f (ENNReal.toReal p) \u03bc =\n    c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc", "state_after": "case e_a.e_a\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 1 / ENNReal.toReal p = ENNReal.toReal (1 / p)"}, {"tactic": "simp_rw [one_div]", "annotated_tactic": ["simp_rw [<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 e_a.e_a\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 1 / ENNReal.toReal p = ENNReal.toReal (1 / p)", "state_after": "case e_a.e_a\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 (ENNReal.toReal p)\u207b\u00b9 = ENNReal.toReal p\u207b\u00b9"}, {"tactic": "rw [ENNReal.toReal_inv]", "annotated_tactic": ["rw [<a>ENNReal.toReal_inv</a>]", [{"full_name": "ENNReal.toReal_inv", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2381, 9], "def_end_pos": [2381, 19]}]], "state_before": "case e_a.e_a\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\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 (ENNReal.toReal p)\u207b\u00b9 = ENNReal.toReal p\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_singleton_inter", "start": [562, 1], "end": [564, 31], "traced_tactics": [{"tactic": "rw [\u2190Nat.cast_le (\u03b1 := \u2115\u221e), (toFinite _).cast_ncard_eq, Nat.cast_one]", "annotated_tactic": ["rw [\u2190<a>Nat.cast_le</a> (\u03b1 := \u2115\u221e), (<a>toFinite</a> _).<a>cast_ncard_eq</a>, <a>Nat.cast_one</a>]", [{"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}, {"full_name": "Set.toFinite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [82, 9], "def_end_pos": [82, 17]}, {"full_name": "Set.Finite.cast_ncard_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [477, 9], "def_end_pos": [477, 29]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 ncard ({a} \u2229 s) \u2264 1", "state_after": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 encard ({a} \u2229 s) \u2264 1"}, {"tactic": "apply encard_singleton_inter", "annotated_tactic": ["apply <a>encard_singleton_inter</a>", [{"full_name": "Set.encard_singleton_inter", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [240, 9], "def_end_pos": [240, 31]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Set \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 encard ({a} \u2229 s) \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.Measure.mkMetric'_toOuterMeasure", "start": [458, 1], "end": [460, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.fderiv_integral_of_tendsto_ae", "start": [728, 1], "end": [733, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "full_name": "iUnion_Ici_eq_Ici_iInf", "start": [245, 1], "end": [248, 38], "traced_tactics": [{"tactic": "simp only [\u2190 IsGLB.biUnion_Ici_eq_Ici (@isGLB_iInf _ _ _ f) has_least_elem, mem_range,\n  iUnion_exists, iUnion_iUnion_eq']", "annotated_tactic": ["simp only [\u2190 <a>IsGLB.biUnion_Ici_eq_Ici</a> (@<a>isGLB_iInf</a> _ _ _ f) has_least_elem, <a>mem_range</a>,\n    <a>iUnion_exists</a>, <a>iUnion_iUnion_eq'</a>]", [{"full_name": "IsGLB.biUnion_Ici_eq_Ici", "def_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "def_pos": [217, 9], "def_end_pos": [217, 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\nhas_least_elem : \u2a05 i, f i \u2208 range f\n\u22a2 \u22c3 i, Ici (f i) = Ici (\u2a05 i, f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.Submartingale.sup", "start": [267, 11], "end": [277, 58], "traced_tactics": [{"tactic": "refine' \u27e8fun i => @StronglyMeasurable.sup _ _ _ _ (\u2131 i) _ _ _ (hf.adapted i) (hg.adapted i),\n  fun i j hij => _, fun i => Integrable.sup (hf.integrable _) (hg.integrable _)\u27e9", "annotated_tactic": ["refine' \u27e8fun i => @<a>StronglyMeasurable.sup</a> _ _ _ _ (\u2131 i) _ _ _ (hf.adapted i) (hg.adapted i),\n    fun i j hij => _, fun i => <a>Integrable.sup</a> (hf.integrable _) (hg.integrable _)\u27e9", [{"full_name": "MeasureTheory.StronglyMeasurable.sup", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [547, 19], "def_end_pos": [547, 22]}, {"full_name": "MeasureTheory.Integrable.sup", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\n\u22a2 Submartingale (f \u2294 g) \u2131 \u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (f \u2294 g) i \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]"}, {"tactic": "refine' EventuallyLE.sup_le _ _", "annotated_tactic": ["refine' <a>EventuallyLE.sup_le</a> _ _", [{"full_name": "Filter.EventuallyLE.sup_le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1806, 9], "def_end_pos": [1806, 28]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (f \u2294 g) i \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]", "state_after": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (fun i_1 => f i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]\n\ncase refine'_2\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (fun i_1 => g i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]"}, {"tactic": "exact EventuallyLE.trans (hf.2.1 i j hij)\n  (condexp_mono (hf.integrable _) (Integrable.sup (hf.integrable j) (hg.integrable j))\n    (eventually_of_forall fun x => le_max_left _ _))", "annotated_tactic": ["exact <a>EventuallyLE.trans</a> (hf.2.1 i j hij)\n      (<a>condexp_mono</a> (hf.integrable _) (<a>Integrable.sup</a> (hf.integrable j) (hg.integrable j))\n        (<a>eventually_of_forall</a> fun x => <a>le_max_left</a> _ _))", [{"full_name": "Filter.EventuallyLE.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 27]}, {"full_name": "MeasureTheory.condexp_mono", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 21]}, {"full_name": "MeasureTheory.Integrable.sup", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}, {"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_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case refine'_1\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (fun i_1 => f i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]", "state_after": "no goals"}, {"tactic": "exact EventuallyLE.trans (hg.2.1 i j hij)\n  (condexp_mono (hg.integrable _) (Integrable.sup (hf.integrable j) (hg.integrable j))\n    (eventually_of_forall fun x => le_max_right _ _))", "annotated_tactic": ["exact <a>EventuallyLE.trans</a> (hg.2.1 i j hij)\n      (<a>condexp_mono</a> (hg.integrable _) (<a>Integrable.sup</a> (hf.integrable j) (hg.integrable j))\n        (<a>eventually_of_forall</a> fun x => <a>le_max_right</a> _ _))", [{"full_name": "Filter.EventuallyLE.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 27]}, {"full_name": "MeasureTheory.condexp_mono", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 21]}, {"full_name": "MeasureTheory.Integrable.sup", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}, {"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_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case refine'_2\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (fun i_1 => g i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ioc_eq_cons_Ioo", "start": [639, 1], "end": [640, 52], "traced_tactics": [{"tactic": "classical rw [cons_eq_insert, Ioo_insert_right h]", "annotated_tactic": ["classical rw [<a>cons_eq_insert</a>, <a>Ioo_insert_right</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.Ioo_insert_right", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [595, 9], "def_end_pos": [595, 25]}]], "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 < b\n\u22a2 Ioc a b = cons b (Ioo a b) (_ : \u00acb \u2208 Ioo a b)", "state_after": "no goals"}, {"tactic": "rw [cons_eq_insert, Ioo_insert_right h]", "annotated_tactic": ["rw [<a>cons_eq_insert</a>, <a>Ioo_insert_right</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.Ioo_insert_right", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [595, 9], "def_end_pos": [595, 25]}]], "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 < b\n\u22a2 Ioc a b = cons b (Ioo a b) (_ : \u00acb \u2208 Ioo a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/SuccPred.lean", "full_name": "Nat.cast_int_covby_iff", "start": [92, 1], "end": [94, 25], "traced_tactics": [{"tactic": "rw [Nat.covby_iff_succ_eq, Int.covby_iff_succ_eq]", "annotated_tactic": ["rw [<a>Nat.covby_iff_succ_eq</a>, <a>Int.covby_iff_succ_eq</a>]", [{"full_name": "Nat.covby_iff_succ_eq", "def_path": "Mathlib/Data/Nat/SuccPred.lean", "def_pos": [82, 19], "def_end_pos": [82, 36]}, {"full_name": "Int.covby_iff_succ_eq", "def_path": "Mathlib/Data/Int/SuccPred.lean", "def_pos": [76, 19], "def_end_pos": [76, 36]}]], "state_before": "a b : \u2115\n\u22a2 \u2191a \u22d6 \u2191b \u2194 a \u22d6 b", "state_after": "a b : \u2115\n\u22a2 \u2191a + 1 = \u2191b \u2194 a + 1 = b"}, {"tactic": "exact Int.coe_nat_inj'", "annotated_tactic": ["exact <a>Int.coe_nat_inj'</a>", [{"full_name": "Int.coe_nat_inj'", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 21]}]], "state_before": "a b : \u2115\n\u22a2 \u2191a + 1 = \u2191b \u2194 a + 1 = b", "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_right", "start": [114, 1], "end": [115, 90], "traced_tactics": [{"tactic": "rw [\u2190 mkRat_mul_left (d := d) a0]", "annotated_tactic": ["rw [\u2190 <a>mkRat_mul_left</a> (d := d) a0]", [{"full_name": "Rat.mkRat_mul_left", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [110, 9], "def_end_pos": [110, 23]}]], "state_before": "n : Int\nd a : Nat\na0 : a \u2260 0\n\u22a2 mkRat (n * \u2191a) (d * a) = mkRat n d", "state_after": "n : Int\nd a : Nat\na0 : a \u2260 0\n\u22a2 mkRat (n * \u2191a) (d * a) = mkRat (\u2191a * n) (a * d)"}, {"tactic": "congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm]", "annotated_tactic": ["congr 1 <;> [apply <a>Int.mul_comm</a>; apply <a>Nat.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]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "n : Int\nd a : Nat\na0 : a \u2260 0\n\u22a2 mkRat (n * \u2191a) (d * a) = mkRat (\u2191a * n) (a * 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_funUnique", "start": [825, 1], "end": [827, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DList/Defs.lean", "full_name": "Std.DList.toList_push", "start": [84, 1], "end": [85, 54], "traced_tactics": [{"tactic": "cases' l with _ l_invariant", "annotated_tactic": ["cases' l with _ l_invariant", []], "state_before": "\u03b1 : Type u\nx : \u03b1\nl : DList \u03b1\n\u22a2 toList (push l x) = toList l ++ [x]", "state_after": "case mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (push { apply := apply\u271d, invariant := l_invariant } x) =\n    toList { apply := apply\u271d, invariant := l_invariant } ++ [x]"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (push { apply := apply\u271d, invariant := l_invariant } x) =\n    toList { apply := apply\u271d, invariant := l_invariant } ++ [x]", "state_after": "case mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 apply\u271d [x] = apply\u271d [] ++ [x]"}, {"tactic": "rw [l_invariant]", "annotated_tactic": ["rw [l_invariant]", []], "state_before": "case mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 apply\u271d [x] = apply\u271d [] ++ [x]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.add_succ", "start": [219, 1], "end": [222, 56], "traced_tactics": [{"tactic": "simp [zero_add]", "annotated_tactic": ["simp [<a>zero_add</a>]", [{"full_name": "Num.zero_add", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [211, 9], "def_end_pos": [211, 17]}]], "state_before": "\u03b1 : Type u_1\nn : Num\n\u22a2 0 + succ n = succ (0 + n)", "state_after": "no goals"}, {"tactic": "rw [PosNum.add_one, add_zero]", "annotated_tactic": ["rw [<a>PosNum.add_one</a>, <a>add_zero</a>]", [{"full_name": "PosNum.add_one", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [84, 9], "def_end_pos": [84, 16]}, {"full_name": "Num.add_zero", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [208, 9], "def_end_pos": [208, 17]}]], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 pos (p + 1) = succ (pos p + 0)", "state_after": "\u03b1 : Type u_1\np : PosNum\n\u22a2 pos (PosNum.succ p) = succ (pos p)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\np : PosNum\n\u22a2 pos (PosNum.succ p) = succ (pos p)", "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_const", "start": [344, 1], "end": [350, 54], "traced_tactics": [{"tactic": "classical\ncalc\n  (const \u03b1 y).integral \u03bc = \u2211 z in {y}, (\u03bc (const \u03b1 y \u207b\u00b9' {z})).toReal \u2022 z :=\n    integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)\n  _ = (\u03bc univ).toReal \u2022 y := by simp [Set.preimage]", "annotated_tactic": ["classical\n  calc\n    (<a>const</a> \u03b1 y).<a>integral</a> \u03bc = \u2211 z in {y}, (\u03bc (<a>const</a> \u03b1 y \u207b\u00b9' {z})).<a>toReal</a> \u2022 z :=\n      <a>integral_eq_sum_of_subset</a> <| (<a>filter_subset</a> _ _).<a>trans</a> (<a>range_const_subset</a> _ _)\n    _ = (\u03bc <a>univ</a>).<a>toReal</a> \u2022 y := by simp [<a>Set.preimage</a>]", [{"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.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [309, 5], "def_end_pos": [309, 13]}, {"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"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.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2701, 9], "def_end_pos": [2701, 22]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "MeasureTheory.SimpleFunc.range_const_subset", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [167, 9], "def_end_pos": [167, 27]}, {"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": "Set.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ny : F\n\u22a2 integral \u03bc (const \u03b1 y) = ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 y", "state_after": "no goals"}, {"tactic": "calc\n  (const \u03b1 y).integral \u03bc = \u2211 z in {y}, (\u03bc (const \u03b1 y \u207b\u00b9' {z})).toReal \u2022 z :=\n    integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)\n  _ = (\u03bc univ).toReal \u2022 y := by simp [Set.preimage]", "annotated_tactic": ["calc\n    (<a>const</a> \u03b1 y).<a>integral</a> \u03bc = \u2211 z in {y}, (\u03bc (<a>const</a> \u03b1 y \u207b\u00b9' {z})).<a>toReal</a> \u2022 z :=\n      <a>integral_eq_sum_of_subset</a> <| (<a>filter_subset</a> _ _).<a>trans</a> (<a>range_const_subset</a> _ _)\n    _ = (\u03bc <a>univ</a>).<a>toReal</a> \u2022 y := by simp [<a>Set.preimage</a>]", [{"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.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [309, 5], "def_end_pos": [309, 13]}, {"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"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.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2701, 9], "def_end_pos": [2701, 22]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "MeasureTheory.SimpleFunc.range_const_subset", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [167, 9], "def_end_pos": [167, 27]}, {"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": "Set.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ny : F\n\u22a2 integral \u03bc (const \u03b1 y) = ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 y", "state_after": "no goals"}, {"tactic": "simp [Set.preimage]", "annotated_tactic": ["simp [<a>Set.preimage</a>]", [{"full_name": "Set.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [47, 5], "def_end_pos": [47, 13]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type u_5\nF' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ny : F\n\u22a2 \u2211 z in {y}, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(const \u03b1 y) \u207b\u00b9' {z})) \u2022 z = ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 y", "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_nonneg", "start": [491, 1], "end": [513, 32], "traced_tactics": [{"tactic": "have h : AEStronglyMeasurable' m (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) \u03bc :=\n  aeStronglyMeasurable'_condexpL2 _ _", "annotated_tactic": ["have h : <a>AEStronglyMeasurable'</a> m (<a>condexpL2</a> \u211d \u211d hm (<a>indicatorConstLp</a> 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) \u03bc :=\n    <a>aeStronglyMeasurable'_condexpL2</a> _ _", [{"full_name": "MeasureTheory.AEStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [49, 5], "def_end_pos": [49, 26]}, {"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}, {"full_name": "MeasureTheory.aeStronglyMeasurable'_condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [79, 9], "def_end_pos": [79, 40]}]], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))"}, {"tactic": "refine' EventuallyLE.trans_eq _ h.ae_eq_mk.symm", "annotated_tactic": ["refine' <a>EventuallyLE.trans_eq</a> _ h.ae_eq_mk.symm", [{"full_name": "Filter.EventuallyLE.trans_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 30]}]], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h"}, {"tactic": "refine' @ae_le_of_ae_le_trim _ _ _ _ _ _ hm (0 : \u03b1 \u2192 \u211d) _ _", "annotated_tactic": ["refine' @<a>ae_le_of_ae_le_trim</a> _ _ _ _ _ _ hm (0 : \u03b1 \u2192 \u211d) _ _", [{"full_name": "MeasureTheory.ae_le_of_ae_le_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [81, 9], "def_end_pos": [81, 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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h"}, {"tactic": "refine' ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite _ _", "annotated_tactic": ["refine' <a>ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite</a> _ _", [{"full_name": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "def_pos": [314, 9], "def_end_pos": [314, 63]}]], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192\n        IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) s_1\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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192\n        0 \u2264\n          \u222b (x : \u03b1) in s_1,\n            AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm"}, {"tactic": "rintro t - -", "annotated_tactic": ["rintro t - -", []], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192\n        IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) s_1", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) t"}, {"tactic": "refine @Integrable.integrableOn _ _ m _ _ _ _ ?_", "annotated_tactic": ["refine @<a>Integrable.integrableOn</a> _ _ m _ _ _ _ ?_", [{"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\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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) t", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)"}, {"tactic": "refine' Integrable.trim hm _ _", "annotated_tactic": ["refine' <a>Integrable.trim</a> hm _ _", [{"full_name": "MeasureTheory.Integrable.trim", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1212, 9], "def_end_pos": [1212, 24]}]], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)", "state_after": "case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)\n\ncase refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 StronglyMeasurable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)"}, {"tactic": "rw [integrable_congr h.ae_eq_mk.symm]", "annotated_tactic": ["rw [<a>integrable_congr</a> h.ae_eq_mk.symm]", [{"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.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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)", "state_after": "case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))"}, {"tactic": "exact integrable_condexpL2_indicator hm hs h\u03bcs _", "annotated_tactic": ["exact <a>integrable_condexpL2_indicator</a> hm hs h\u03bcs _", [{"full_name": "MeasureTheory.integrable_condexpL2_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [366, 9], "def_end_pos": [366, 39]}]], "state_before": "case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))", "state_after": "no goals"}, {"tactic": "exact h.stronglyMeasurable_mk", "annotated_tactic": ["exact h.stronglyMeasurable_mk", []], "state_before": "case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 StronglyMeasurable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)", "state_after": "no goals"}, {"tactic": "intro t ht h\u03bct", "annotated_tactic": ["intro t ht h\u03bct", []], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : 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\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192\n        0 \u2264\n          \u222b (x : \u03b1) in s_1,\n            AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 0 \u2264\n    \u222b (x : \u03b1) in t,\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm"}, {"tactic": "rw [\u2190 set_integral_trim hm h.stronglyMeasurable_mk ht]", "annotated_tactic": ["rw [\u2190 <a>set_integral_trim</a> hm h.stronglyMeasurable_mk ht]", [{"full_name": "MeasureTheory.set_integral_trim", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [621, 9], "def_end_pos": [621, 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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 0 \u2264\n    \u222b (x : \u03b1) in t,\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm", "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202\u03bc"}, {"tactic": "have h_ae :\n  \u2200\u1d50 x \u2202\u03bc, x \u2208 t \u2192 h.mk _ x = (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) x := by\n  filter_upwards [h.ae_eq_mk] with x hx\n  exact fun _ => hx.symm", "annotated_tactic": ["have h_ae :\n      \u2200\u1d50 x \u2202\u03bc, x \u2208 t \u2192 h.mk _ x = (<a>condexpL2</a> \u211d \u211d hm (<a>indicatorConstLp</a> 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) x := by\n      filter_upwards [h.ae_eq_mk] with x hx\n      exact fun _ => hx.symm", [{"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"full_name": "MeasureTheory.indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [744, 5], "def_end_pos": [744, 21]}]], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h 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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh_ae :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n\u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202\u03bc"}, {"tactic": "rw [set_integral_congr_ae (hm t ht) h_ae,\n  set_integral_condexpL2_indicator ht hs ((le_trim hm).trans_lt h\u03bct).ne h\u03bcs]", "annotated_tactic": ["rw [<a>set_integral_congr_ae</a> (hm t ht) h_ae,\n      <a>set_integral_condexpL2_indicator</a> ht hs ((<a>le_trim</a> hm).<a>trans_lt</a> h\u03bct).<a>ne</a> h\u03bcs]", [{"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.set_integral_condexpL2_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [467, 9], "def_end_pos": [467, 41]}, {"full_name": "MeasureTheory.le_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [57, 9], "def_end_pos": [57, 16]}, {"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": "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh_ae :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n\u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h 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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh_ae :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2229 t))"}, {"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\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\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh_ae :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2229 t))", "state_after": "no goals"}, {"tactic": "filter_upwards [h.ae_eq_mk] with x hx", "annotated_tactic": ["filter_upwards [h.ae_eq_mk] with x hx", []], "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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x", "state_after": "case h\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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nx : \u03b1\nhx :\n  \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x =\n    AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x\n\u22a2 x \u2208 t \u2192\n    AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n      \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x"}, {"tactic": "exact fun _ => hx.symm", "annotated_tactic": ["exact fun _ => hx.symm", []], "state_before": "case h\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\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nx : \u03b1\nhx :\n  \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x =\n    AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x\n\u22a2 x \u2208 t \u2192\n    AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n      \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.shiftRight_eq_div_pow", "start": [440, 1], "end": [444, 55], "traced_tactics": [{"tactic": "rw [shiftRight_coe_nat, Nat.shiftRight_eq_div_pow _ _]", "annotated_tactic": ["rw [<a>shiftRight_coe_nat</a>, <a>Nat.shiftRight_eq_div_pow</a> _ _]", [{"full_name": "Int.shiftRight_coe_nat", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [383, 9], "def_end_pos": [383, 27]}, {"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": "m n : \u2115\n\u22a2 \u2191m >>> \u2191n = \u2191m / \u2191(2 ^ n)", "state_after": "m n : \u2115\n\u22a2 \u2191(m / 2 ^ n) = \u2191m / \u2191(2 ^ n)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "m n : \u2115\n\u22a2 \u2191(m / 2 ^ n) = \u2191m / \u2191(2 ^ n)", "state_after": "no goals"}, {"tactic": "rw [shiftRight_negSucc, negSucc_ediv, Nat.shiftRight_eq_div_pow]", "annotated_tactic": ["rw [<a>shiftRight_negSucc</a>, <a>negSucc_ediv</a>, <a>Nat.shiftRight_eq_div_pow</a>]", [{"full_name": "Int.shiftRight_negSucc", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [392, 9], "def_end_pos": [392, 27]}, {"full_name": "Int.negSucc_ediv", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [27, 9], "def_end_pos": [27, 21]}, {"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": "m n : \u2115\n\u22a2 -[m+1] >>> \u2191n = -[m+1] / \u2191(2 ^ n)", "state_after": "m n : \u2115\n\u22a2 -[m / 2 ^ n+1] = -(div \u2191m \u2191(2 ^ n) + 1)\n\ncase H\nm n : \u2115\n\u22a2 0 < \u2191(2 ^ n)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "m n : \u2115\n\u22a2 -[m / 2 ^ n+1] = -(div \u2191m \u2191(2 ^ n) + 1)\n\ncase H\nm n : \u2115\n\u22a2 0 < \u2191(2 ^ n)", "state_after": "case H\nm n : \u2115\n\u22a2 0 < \u2191(2 ^ n)"}, {"tactic": "exact ofNat_lt_ofNat_of_lt (pow_pos (by decide) _)", "annotated_tactic": ["exact <a>ofNat_lt_ofNat_of_lt</a> (<a>pow_pos</a> (by decide) _)", [{"full_name": "Int.ofNat_lt_ofNat_of_lt", "def_path": "Mathlib/Init/Data/Int/Order.lean", "def_pos": [41, 30], "def_end_pos": [41, 50]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 16]}]], "state_before": "case H\nm n : \u2115\n\u22a2 0 < \u2191(2 ^ n)", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "m n : \u2115\n\u22a2 0 < 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.preimage_const_mul_Iic_of_neg", "start": [650, 1], "end": [652, 64], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Iic_of_neg a h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Iic_of_neg</a> a 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_Iic_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [564, 9], "def_end_pos": [564, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a c : \u03b1\nh : c < 0\n\u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Iic a = Ici (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.uIcc_of_not_le", "start": [1033, 1], "end": [1034, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.casesNil_append1", "start": [315, 11], "end": [317, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "full_name": "Finset.Nat.antidiagonal_succ_succ'", "start": [73, 1], "end": [83, 6], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\n\u22a2 \u00ac(n + 2, 0) \u2208\n      map\n        (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n          { toFun := Nat.succ, inj' := Nat.succ_injective })\n        (antidiagonal n)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\n\u22a2 \u00ac(0, n + 2) \u2208\n      cons (n + 2, 0)\n        (map\n          (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n            { toFun := Nat.succ, inj' := Nat.succ_injective })\n          (antidiagonal n))\n        (_ :\n          \u00ac(n + 2, 0) \u2208\n              map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n))", "state_after": "no goals"}, {"tactic": "simp_rw [antidiagonal_succ (n + 1), antidiagonal_succ', Finset.map_cons, map_map]", "annotated_tactic": ["simp_rw [<a>antidiagonal_succ</a> (n + 1), <a>antidiagonal_succ'</a>, <a>Finset.map_cons</a>, <a>map_map</a>]", [{"full_name": "Finset.Nat.antidiagonal_succ", "def_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "def_pos": [51, 9], "def_end_pos": [51, 26]}, {"full_name": "Finset.Nat.antidiagonal_succ'", "def_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "def_pos": [62, 9], "def_end_pos": [62, 27]}, {"full_name": "Finset.map_cons", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [237, 9], "def_end_pos": [237, 17]}, {"full_name": "Finset.map_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [140, 9], "def_end_pos": [140, 16]}]], "state_before": "n : \u2115\n\u22a2 antidiagonal (n + 2) =\n    cons (0, n + 2)\n      (cons (n + 2, 0)\n        (map\n          (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n            { toFun := Nat.succ, inj' := Nat.succ_injective })\n          (antidiagonal n))\n        (_ :\n          \u00ac(n + 2, 0) \u2208\n              map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n)))\n      (_ :\n        \u00ac(0, n + 2) \u2208\n            cons (n + 2, 0)\n              (map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n))\n              (_ :\n                \u00ac(n + 2, 0) \u2208\n                    map\n                      (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                        { toFun := Nat.succ, inj' := Nat.succ_injective })\n                      (antidiagonal n)))", "state_after": "n : \u2115\n\u22a2 cons (0, n + 1 + 1)\n      (cons (\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0))\n        (map\n          (Embedding.trans (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n            (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n          (antidiagonal n))\n        (_ :\n          \u00ac\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0) \u2208\n              map\n                (Embedding.trans\n                  (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n                  (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n                (antidiagonal n)))\n      (_ :\n        \u00ac(0, n + 1 + 1) \u2208\n            cons (\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0))\n              (map\n                (Embedding.trans\n                  (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n                  (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n                (antidiagonal n))\n              (_ :\n                \u00ac\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0) \u2208\n                    map\n                      (Embedding.trans\n                        (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n                        (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n                      (antidiagonal n))) =\n    cons (0, n + 2)\n      (cons (n + 2, 0)\n        (map\n          (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n            { toFun := Nat.succ, inj' := Nat.succ_injective })\n          (antidiagonal n))\n        (_ :\n          \u00ac(n + 2, 0) \u2208\n              map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n)))\n      (_ :\n        \u00ac(0, n + 2) \u2208\n            cons (n + 2, 0)\n              (map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n))\n              (_ :\n                \u00ac(n + 2, 0) \u2208\n                    map\n                      (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                        { toFun := Nat.succ, inj' := Nat.succ_injective })\n                      (antidiagonal n)))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\n\u22a2 cons (0, n + 1 + 1)\n      (cons (\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0))\n        (map\n          (Embedding.trans (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n            (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n          (antidiagonal n))\n        (_ :\n          \u00ac\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0) \u2208\n              map\n                (Embedding.trans\n                  (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n                  (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n                (antidiagonal n)))\n      (_ :\n        \u00ac(0, n + 1 + 1) \u2208\n            cons (\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0))\n              (map\n                (Embedding.trans\n                  (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n                  (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n                (antidiagonal n))\n              (_ :\n                \u00ac\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0) \u2208\n                    map\n                      (Embedding.trans\n                        (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })\n                        (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))\n                      (antidiagonal n))) =\n    cons (0, n + 2)\n      (cons (n + 2, 0)\n        (map\n          (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n            { toFun := Nat.succ, inj' := Nat.succ_injective })\n          (antidiagonal n))\n        (_ :\n          \u00ac(n + 2, 0) \u2208\n              map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n)))\n      (_ :\n        \u00ac(0, n + 2) \u2208\n            cons (n + 2, 0)\n              (map\n                (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                  { toFun := Nat.succ, inj' := Nat.succ_injective })\n                (antidiagonal n))\n              (_ :\n                \u00ac(n + 2, 0) \u2208\n                    map\n                      (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }\n                        { toFun := Nat.succ, inj' := Nat.succ_injective })\n                      (antidiagonal n)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_smul_const", "start": [1257, 1], "end": [1266, 17], "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\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c"}, {"tactic": "exact ((1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c).smulRight c).integral_comp_comm hf", "annotated_tactic": ["exact ((1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c).<a>smulRight</a> c).<a>integral_comp_comm</a> hf", [{"full_name": "ContinuousLinearMap.smulRight", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1201, 5], "def_end_pos": [1201, 14]}, {"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\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c", "state_after": "no goals"}, {"tactic": "by_cases hc : c = 0", "annotated_tactic": ["by_cases hc : c = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : c = 0\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : \u00acc = 0\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c"}, {"tactic": "rw [integral_undef hf, integral_undef, zero_smul]", "annotated_tactic": ["rw [<a>integral_undef</a> hf, <a>integral_undef</a>, <a>zero_smul</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]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : \u00acc = 0\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : \u00acc = 0\n\u22a2 \u00acIntegrable fun x => f x \u2022 c"}, {"tactic": "rw [integrable_smul_const hc]", "annotated_tactic": ["rw [<a>integrable_smul_const</a> hc]", [{"full_name": "MeasureTheory.integrable_smul_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 30]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : \u00acc = 0\n\u22a2 \u00acIntegrable fun x => f x \u2022 c", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : \u00acc = 0\n\u22a2 \u00acIntegrable fun x => f x"}, {"tactic": "simp_rw [hf]", "annotated_tactic": ["simp_rw [hf]", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : \u00acc = 0\n\u22a2 \u00acIntegrable fun x => f x", "state_after": "no goals"}, {"tactic": "simp only [hc, integral_zero, smul_zero]", "annotated_tactic": ["simp only [hc, <a>integral_zero</a>, <a>smul_zero</a>]", [{"full_name": "MeasureTheory.integral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [850, 9], "def_end_pos": [850, 22]}, {"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\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c\u271d : Type u_6\ninst\u271d\u2078 : IsROrC \ud835\udd5c\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F\np : \u211d\u22650\u221e\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d F\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhf : \u00acIntegrable f\nhc : c = 0\n\u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c", "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.rev_le_rev", "start": [130, 9], "end": [132, 29], "traced_tactics": [{"tactic": "simp only [le_def, val_rev, Nat.sub_le_sub_iff_left (Nat.succ_le.2 j.is_lt)]", "annotated_tactic": ["simp only [<a>le_def</a>, <a>val_rev</a>, <a>Nat.sub_le_sub_iff_left</a> (<a>Nat.succ_le</a>.2 j.is_lt)]", [{"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": "Fin.val_rev", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [125, 17], "def_end_pos": [125, 24]}, {"full_name": "Nat.sub_le_sub_iff_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [439, 19], "def_end_pos": [439, 38]}, {"full_name": "Nat.succ_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [221, 9], "def_end_pos": [221, 16]}]], "state_before": "n : Nat\ni j : Fin n\n\u22a2 rev i \u2264 rev j \u2194 j \u2264 i", "state_after": "n : Nat\ni j : Fin n\n\u22a2 Nat.succ \u2191j \u2264 \u2191i + 1 \u2194 \u2191j \u2264 \u2191i"}, {"tactic": "exact Nat.succ_le_succ_iff", "annotated_tactic": ["exact <a>Nat.succ_le_succ_iff</a>", [{"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": "n : Nat\ni j : Fin n\n\u22a2 Nat.succ \u2191j \u2264 \u2191i + 1 \u2194 \u2191j \u2264 \u2191i", "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.lex_wf", "start": [233, 1], "end": [235, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_le_encard_diff_add_encard", "start": [224, 1], "end": [225, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.snorm_indicator_const", "start": [610, 1], "end": [612, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Equiv.piCongrLeft_symm_preimage_univ_pi", "start": [944, 1], "end": [946, 76], "traced_tactics": [{"tactic": "simpa [f.surjective.range_eq] using piCongrLeft_symm_preimage_pi f univ t", "annotated_tactic": ["simpa [f.surjective.range_eq] using <a>piCongrLeft_symm_preimage_pi</a> f <a>univ</a> t", [{"full_name": "Equiv.piCongrLeft_symm_preimage_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [940, 9], "def_end_pos": [940, 37]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\nf : \u03b9' \u2243 \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (\u2191(piCongrLeft \u03b1 f).symm \u207b\u00b9' pi univ fun i' => t (\u2191f i')) = pi univ t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_smul", "start": [929, 1], "end": [931, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae", "start": [192, 1], "end": [237, 20], "traced_tactics": [{"tactic": "have h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d) (_ : 0 < \u03b5), \u2203 k : \u2115, 2 * (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := by\n  intro \u03b5 h\u03b5\n  obtain \u27e8k, h_k\u27e9 : \u2203 k : \u2115, (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := exists_pow_lt_of_lt_one h\u03b5 (by norm_num)\n  refine' \u27e8k + 1, (le_of_eq _).trans_lt h_k\u27e9\n  rw [pow_add]; ring", "annotated_tactic": ["have h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d) (_ : 0 < \u03b5), \u2203 k : \u2115, 2 * (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := by\n    intro \u03b5 h\u03b5\n    obtain \u27e8k, h_k\u27e9 : \u2203 k : \u2115, (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := <a>exists_pow_lt_of_lt_one</a> h\u03b5 (by norm_num)\n    refine' \u27e8k + 1, (<a>le_of_eq</a> _).<a>trans_lt</a> h_k\u27e9\n    rw [<a>pow_add</a>]; ring", [{"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": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "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\n\u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "set ns := ExistsSeqTendstoAe.seqTendstoAeSeq hfg", "annotated_tactic": ["set ns := <a>ExistsSeqTendstoAe.seqTendstoAeSeq</a> hfg", [{"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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n\u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "use ns", "annotated_tactic": ["use ns", []], "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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n\u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "let S := fun k => { x | (2 : \u211d)\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x) }", "annotated_tactic": ["let S := fun k => { x | (2 : \u211d)\u207b\u00b9 ^ k \u2264 <a>dist</a> (f (ns k) x) (g x) }", [{"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}]], "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "have h\u03bcS_le : \u2200 k, \u03bc (S k) \u2264 (2 : \u211d\u22650\u221e)\u207b\u00b9 ^ k :=\n  fun k => ExistsSeqTendstoAe.seqTendstoAeSeq_spec hfg k (ns k) le_rfl", "annotated_tactic": ["have h\u03bcS_le : \u2200 k, \u03bc (S k) \u2264 (2 : \u211d\u22650\u221e)\u207b\u00b9 ^ k :=\n    fun k => <a>ExistsSeqTendstoAe.seqTendstoAeSeq_spec</a> hfg k (ns k) <a>le_rfl</a>", [{"full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [172, 9], "def_end_pos": [172, 29]}, {"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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "set s := Filter.atTop.limsup S with hs", "annotated_tactic": ["set s := Filter.atTop.limsup S with hs", []], "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "have h\u03bcs : \u03bc s = 0 := by\n  refine' measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum h\u03bcS_le) _)\n  simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv]", "annotated_tactic": ["have h\u03bcs : \u03bc s = 0 := by\n    refine' <a>measure_limsup_eq_zero</a> (<a>ne_of_lt</a> <| <a>lt_of_le_of_lt</a> (<a>ENNReal.tsum_le_tsum</a> h\u03bcS_le) _)\n    simp only [<a>ENNReal.tsum_geometric</a>, <a>ENNReal.one_sub_inv_two</a>, <a>inv_inv</a>]", [{"full_name": "MeasureTheory.measure_limsup_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [585, 9], "def_end_pos": [585, 31]}, {"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": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}, {"full_name": "ENNReal.tsum_geometric", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [304, 9], "def_end_pos": [304, 31]}, {"full_name": "ENNReal.one_sub_inv_two", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1820, 9], "def_end_pos": [1820, 24]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "have h_tendsto : \u2200 x \u2208 s\u1d9c, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) := by\n  refine' fun x hx => Metric.tendsto_atTop.mpr fun \u03b5 h\u03b5 => _\n  rw [hs, limsup_eq_iInf_iSup_of_nat] at hx\n  simp only [Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,\n    Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx\n  obtain \u27e8N, hNx\u27e9 := hx\n  obtain \u27e8k, hk_lt_\u03b5\u27e9 := h_lt_\u03b5_real \u03b5 h\u03b5\n  refine' \u27e8max N (k - 1), fun n hn_ge => lt_of_le_of_lt _ hk_lt_\u03b5\u27e9\n  specialize hNx n ((le_max_left _ _).trans hn_ge)\n  have h_inv_n_le_k : (2 : \u211d)\u207b\u00b9 ^ n \u2264 2 * (2 : \u211d)\u207b\u00b9 ^ k := by\n    rw [mul_comm, \u2190 inv_mul_le_iff' (zero_lt_two' \u211d)]\n    conv_lhs =>\n      congr\n      rw [\u2190 pow_one (2 : \u211d)\u207b\u00b9]\n    rw [\u2190 pow_add, add_comm]\n    exact pow_le_pow_of_le_one (one_div (2 : \u211d) \u25b8 one_half_pos.le) (inv_le_one one_le_two)\n      ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n        (add_le_add_right hn_ge 1))\n  exact le_trans hNx.le h_inv_n_le_k", "annotated_tactic": ["have h_tendsto : \u2200 x \u2208 s\u1d9c, <a>Tendsto</a> (fun i => f (ns i) x) <a>atTop</a> (\ud835\udcdd (g x)) := by\n    refine' fun x hx => Metric.tendsto_atTop.mpr fun \u03b5 h\u03b5 => _\n    rw [hs, <a>limsup_eq_iInf_iSup_of_nat</a>] at hx\n    simp only [<a>Set.iSup_eq_iUnion</a>, <a>Set.iInf_eq_iInter</a>, <a>Set.compl_iInter</a>, <a>Set.compl_iUnion</a>,\n      <a>Set.mem_iUnion</a>, <a>Set.mem_iInter</a>, <a>Set.mem_compl_iff</a>, <a>Set.mem_setOf_eq</a>, <a>not_le</a>] at hx\n    obtain \u27e8N, hNx\u27e9 := hx\n    obtain \u27e8k, hk_lt_\u03b5\u27e9 := h_lt_\u03b5_real \u03b5 h\u03b5\n    refine' \u27e8<a>max</a> N (k - 1), fun n hn_ge => <a>lt_of_le_of_lt</a> _ hk_lt_\u03b5\u27e9\n    specialize hNx n ((<a>le_max_left</a> _ _).<a>trans</a> hn_ge)\n    have h_inv_n_le_k : (2 : \u211d)\u207b\u00b9 ^ n \u2264 2 * (2 : \u211d)\u207b\u00b9 ^ k := by\n      rw [<a>mul_comm</a>, \u2190 <a>inv_mul_le_iff'</a> (<a>zero_lt_two'</a> \u211d)]\n      conv_lhs =>\n        congr\n        rw [\u2190 <a>pow_one</a> (2 : \u211d)\u207b\u00b9]\n      rw [\u2190 <a>pow_add</a>, <a>add_comm</a>]\n      exact <a>pow_le_pow_of_le_one</a> (<a>one_div</a> (2 : \u211d) \u25b8 one_half_pos.le) (<a>inv_le_one</a> <a>one_le_two</a>)\n        ((le_tsub_add.trans (<a>add_le_add_right</a> (<a>le_max_right</a> _ _) 1)).<a>trans</a>\n          (<a>add_le_add_right</a> hn_ge 1))\n    exact <a>le_trans</a> hNx.le h_inv_n_le_k", [{"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.limsup_eq_iInf_iSup_of_nat", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [796, 9], "def_end_pos": [796, 35]}, {"full_name": "Set.iSup_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [191, 9], "def_end_pos": [191, 23]}, {"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.compl_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [615, 9], "def_end_pos": [615, 21]}, {"full_name": "Set.compl_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [604, 9], "def_end_pos": [604, 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_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"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]}, {"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_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "inv_mul_le_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 24]}, {"full_name": "zero_lt_two'", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [89, 7], "def_end_pos": [89, 19]}, {"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": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "pow_le_pow_of_le_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "inv_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [305, 9], "def_end_pos": [305, 19]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"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_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"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": "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\n\u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\n\u22a2 StrictMono ns \u2227 \u2191\u2191\u03bc {a | \u00acTendsto (fun i => f (ns i) a) atTop (\ud835\udcdd (g a))} = 0"}, {"tactic": "refine' \u27e8ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono hfg, measure_mono_null (fun x => _) h\u03bcs\u27e9", "annotated_tactic": ["refine' \u27e8<a>ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono</a> hfg, <a>measure_mono_null</a> (fun x => _) h\u03bcs\u27e9", [{"full_name": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [181, 9], "def_end_pos": [181, 35]}, {"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}]], "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\n\u22a2 StrictMono ns \u2227 \u2191\u2191\u03bc {a | \u00acTendsto (fun i => f (ns i) a) atTop (\ud835\udcdd (g a))} = 0", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\nx : \u03b1\n\u22a2 x \u2208 {a | \u00acTendsto (fun i => f (ns i) a) atTop (\ud835\udcdd (g a))} \u2192 x \u2208 s"}, {"tactic": "rw [Set.mem_setOf_eq, \u2190 @Classical.not_not (x \u2208 s), not_imp_not]", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>, \u2190 @<a>Classical.not_not</a> (x \u2208 s), <a>not_imp_not</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": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "not_imp_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\nx : \u03b1\n\u22a2 x \u2208 {a | \u00acTendsto (fun i => f (ns i) a) atTop (\ud835\udcdd (g a))} \u2192 x \u2208 s", "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\nx : \u03b1\n\u22a2 \u00acx \u2208 s \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))"}, {"tactic": "exact h_tendsto x", "annotated_tactic": ["exact h_tendsto x", []], "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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nh_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))\nx : \u03b1\n\u22a2 \u00acx \u2208 s \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "state_after": "no goals"}, {"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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5", "state_after": "\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5"}, {"tactic": "obtain \u27e8k, h_k\u27e9 : \u2203 k : \u2115, (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := exists_pow_lt_of_lt_one h\u03b5 (by norm_num)", "annotated_tactic": ["obtain \u27e8k, h_k\u27e9 : \u2203 k : \u2115, (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := <a>exists_pow_lt_of_lt_one</a> h\u03b5 (by norm_num)", [{"full_name": "exists_pow_lt_of_lt_one", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [232, 9], "def_end_pos": [232, 32]}]], "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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5", "state_after": "case intro\n\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nh_k : 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5"}, {"tactic": "refine' \u27e8k + 1, (le_of_eq _).trans_lt h_k\u27e9", "annotated_tactic": ["refine' \u27e8k + 1, (<a>le_of_eq</a> _).<a>trans_lt</a> h_k\u27e9", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"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\n\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nh_k : 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5", "state_after": "case intro\n\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nh_k : 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 2 * 2\u207b\u00b9 ^ (k + 1) = 2\u207b\u00b9 ^ k"}, {"tactic": "rw [pow_add]", "annotated_tactic": ["rw [<a>pow_add</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "case intro\n\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nh_k : 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 2 * 2\u207b\u00b9 ^ (k + 1) = 2\u207b\u00b9 ^ k", "state_after": "case intro\n\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nh_k : 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 2 * (2\u207b\u00b9 ^ k * 2\u207b\u00b9 ^ 1) = 2\u207b\u00b9 ^ k"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case intro\n\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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nh_k : 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 2 * (2\u207b\u00b9 ^ k * 2\u207b\u00b9 ^ 1) = 2\u207b\u00b9 ^ k", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "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\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 2\u207b\u00b9 < 1", "state_after": "no goals"}, {"tactic": "refine' measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum h\u03bcS_le) _)", "annotated_tactic": ["refine' <a>measure_limsup_eq_zero</a> (<a>ne_of_lt</a> <| <a>lt_of_le_of_lt</a> (<a>ENNReal.tsum_le_tsum</a> h\u03bcS_le) _)", [{"full_name": "MeasureTheory.measure_limsup_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [585, 9], "def_end_pos": [585, 31]}, {"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": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [827, 19], "def_end_pos": [827, 31]}]], "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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\n\u22a2 \u2191\u2191\u03bc s = 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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\n\u22a2 \u2211' (a : \u2115), 2\u207b\u00b9 ^ a < \u22a4"}, {"tactic": "simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv]", "annotated_tactic": ["simp only [<a>ENNReal.tsum_geometric</a>, <a>ENNReal.one_sub_inv_two</a>, <a>inv_inv</a>]", [{"full_name": "ENNReal.tsum_geometric", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [304, 9], "def_end_pos": [304, 31]}, {"full_name": "ENNReal.one_sub_inv_two", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1820, 9], "def_end_pos": [1820, 24]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\n\u22a2 \u2211' (a : \u2115), 2\u207b\u00b9 ^ a < \u22a4", "state_after": "no goals"}, {"tactic": "refine' fun x hx => Metric.tendsto_atTop.mpr fun \u03b5 h\u03b5 => _", "annotated_tactic": ["refine' fun x hx => Metric.tendsto_atTop.mpr fun \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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x))", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\nhx : x \u2208 s\u1d9c\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5"}, {"tactic": "rw [hs, limsup_eq_iInf_iSup_of_nat] at hx", "annotated_tactic": ["rw [hs, <a>limsup_eq_iInf_iSup_of_nat</a>] at hx", [{"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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\nhx : x \u2208 s\u1d9c\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\nhx : x \u2208 (\u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), S i)\u1d9c\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5"}, {"tactic": "simp only [Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,\n  Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx", "annotated_tactic": ["simp only [<a>Set.iSup_eq_iUnion</a>, <a>Set.iInf_eq_iInter</a>, <a>Set.compl_iInter</a>, <a>Set.compl_iUnion</a>,\n      <a>Set.mem_iUnion</a>, <a>Set.mem_iInter</a>, <a>Set.mem_compl_iff</a>, <a>Set.mem_setOf_eq</a>, <a>not_le</a>] at hx", [{"full_name": "Set.iSup_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [191, 9], "def_end_pos": [191, 23]}, {"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.compl_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [615, 9], "def_end_pos": [615, 21]}, {"full_name": "Set.compl_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [604, 9], "def_end_pos": [604, 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_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"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": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\nhx : x \u2208 (\u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), S i)\u1d9c\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhx : \u2203 i, \u2200 (i_1 : \u2115), i_1 \u2265 i \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i_1) x) (g x) < 2\u207b\u00b9 ^ i_1\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5"}, {"tactic": "obtain \u27e8N, hNx\u27e9 := hx", "annotated_tactic": ["obtain \u27e8N, hNx\u27e9 := hx", []], "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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhx : \u2203 i, \u2200 (i_1 : \u2115), i_1 \u2265 i \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i_1) x) (g x) < 2\u207b\u00b9 ^ i_1\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5", "state_after": "case intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5"}, {"tactic": "obtain \u27e8k, hk_lt_\u03b5\u27e9 := h_lt_\u03b5_real \u03b5 h\u03b5", "annotated_tactic": ["obtain \u27e8k, hk_lt_\u03b5\u27e9 := h_lt_\u03b5_real \u03b5 h\u03b5", []], "state_before": "case intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5", "state_after": "case intro.intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i\nk : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5"}, {"tactic": "refine' \u27e8max N (k - 1), fun n hn_ge => lt_of_le_of_lt _ hk_lt_\u03b5\u27e9", "annotated_tactic": ["refine' \u27e8<a>max</a> N (k - 1), fun n hn_ge => <a>lt_of_le_of_lt</a> _ hk_lt_\u03b5\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_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\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i\nk : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5", "state_after": "case intro.intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i\nk : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\n\u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k"}, {"tactic": "specialize hNx n ((le_max_left _ _).trans hn_ge)", "annotated_tactic": ["specialize hNx n ((<a>le_max_left</a> _ _).<a>trans</a> hn_ge)", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i\nk : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\n\u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k", "state_after": "case intro.intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k"}, {"tactic": "have h_inv_n_le_k : (2 : \u211d)\u207b\u00b9 ^ n \u2264 2 * (2 : \u211d)\u207b\u00b9 ^ k := by\n  rw [mul_comm, \u2190 inv_mul_le_iff' (zero_lt_two' \u211d)]\n  conv_lhs =>\n    congr\n    rw [\u2190 pow_one (2 : \u211d)\u207b\u00b9]\n  rw [\u2190 pow_add, add_comm]\n  exact pow_le_pow_of_le_one (one_div (2 : \u211d) \u25b8 one_half_pos.le) (inv_le_one one_le_two)\n    ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n      (add_le_add_right hn_ge 1))", "annotated_tactic": ["have h_inv_n_le_k : (2 : \u211d)\u207b\u00b9 ^ n \u2264 2 * (2 : \u211d)\u207b\u00b9 ^ k := by\n      rw [<a>mul_comm</a>, \u2190 <a>inv_mul_le_iff'</a> (<a>zero_lt_two'</a> \u211d)]\n      conv_lhs =>\n        congr\n        rw [\u2190 <a>pow_one</a> (2 : \u211d)\u207b\u00b9]\n      rw [\u2190 <a>pow_add</a>, <a>add_comm</a>]\n      exact <a>pow_le_pow_of_le_one</a> (<a>one_div</a> (2 : \u211d) \u25b8 one_half_pos.le) (<a>inv_le_one</a> <a>one_le_two</a>)\n        ((le_tsub_add.trans (<a>add_le_add_right</a> (<a>le_max_right</a> _ _) 1)).<a>trans</a>\n          (<a>add_le_add_right</a> hn_ge 1))", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "inv_mul_le_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 24]}, {"full_name": "zero_lt_two'", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [89, 7], "def_end_pos": [89, 19]}, {"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": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "pow_le_pow_of_le_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "inv_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [305, 9], "def_end_pos": [305, 19]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"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_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}]], "state_before": "case intro.intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k", "state_after": "case intro.intro\n\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\nh_inv_n_le_k : 2\u207b\u00b9 ^ n \u2264 2 * 2\u207b\u00b9 ^ k\n\u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k"}, {"tactic": "exact le_trans hNx.le h_inv_n_le_k", "annotated_tactic": ["exact <a>le_trans</a> hNx.le h_inv_n_le_k", [{"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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\nh_inv_n_le_k : 2\u207b\u00b9 ^ n \u2264 2 * 2\u207b\u00b9 ^ k\n\u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k", "state_after": "no goals"}, {"tactic": "rw [mul_comm, \u2190 inv_mul_le_iff' (zero_lt_two' \u211d)]", "annotated_tactic": ["rw [<a>mul_comm</a>, \u2190 <a>inv_mul_le_iff'</a> (<a>zero_lt_two'</a> \u211d)]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "inv_mul_le_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 24]}, {"full_name": "zero_lt_two'", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [89, 7], "def_end_pos": [89, 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 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nhfg : TendstoInMeasure \u03bc f atTop g\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 ^ n \u2264 2 * 2\u207b\u00b9 ^ k", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 * 2\u207b\u00b9 ^ n \u2264 2\u207b\u00b9 ^ k"}, {"tactic": "conv_lhs =>\n  congr\n  rw [\u2190 pow_one (2 : \u211d)\u207b\u00b9]", "annotated_tactic": ["conv_lhs =>\n        congr\n        rw [\u2190 <a>pow_one</a> (2 : \u211d)\u207b\u00b9]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 * 2\u207b\u00b9 ^ n \u2264 2\u207b\u00b9 ^ k", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 ^ 1 * 2\u207b\u00b9 ^ n \u2264 2\u207b\u00b9 ^ k"}, {"tactic": "rw [\u2190 pow_add, add_comm]", "annotated_tactic": ["rw [\u2190 <a>pow_add</a>, <a>add_comm</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 ^ 1 * 2\u207b\u00b9 ^ n \u2264 2\u207b\u00b9 ^ k", "state_after": "\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 ^ (n + 1) \u2264 2\u207b\u00b9 ^ k"}, {"tactic": "exact pow_le_pow_of_le_one (one_div (2 : \u211d) \u25b8 one_half_pos.le) (inv_le_one one_le_two)\n  ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n    (add_le_add_right hn_ge 1))", "annotated_tactic": ["exact <a>pow_le_pow_of_le_one</a> (<a>one_div</a> (2 : \u211d) \u25b8 one_half_pos.le) (<a>inv_le_one</a> <a>one_le_two</a>)\n        ((le_tsub_add.trans (<a>add_le_add_right</a> (<a>le_max_right</a> _ _) 1)).<a>trans</a>\n          (<a>add_le_add_right</a> hn_ge 1))", [{"full_name": "pow_le_pow_of_le_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "inv_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [305, 9], "def_end_pos": [305, 19]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 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": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"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_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\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\nh_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5\nns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)}\nh\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k\ns : Set \u03b1 := limsup S atTop\nhs : s = limsup S atTop\nh\u03bcs : \u2191\u2191\u03bc s = 0\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN k : \u2115\nhk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5\nn : \u2115\nhn_ge : n \u2265 max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 ^ (n + 1) \u2264 2\u207b\u00b9 ^ k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coeFn_compMeasurePreserving", "start": [956, 1], "end": [958, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.lintegral_compProd", "start": [432, 1], "end": [439, 47], "traced_tactics": [{"tactic": "let g := Function.curry f", "annotated_tactic": ["let g := <a>Function.curry</a> f", [{"full_name": "Function.curry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [212, 5], "def_end_pos": [212, 10]}]], "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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), f (b, c) \u2202\u2191\u03b7 (a, b) \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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), f (b, c) \u2202\u2191\u03b7 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "change \u222b\u207b bc, f bc \u2202(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b b, \u222b\u207b c, g b c \u2202\u03b7 (a, b) \u2202\u03ba a", "annotated_tactic": ["change \u222b\u207b bc, f bc \u2202(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b b, \u222b\u207b c, g b c \u2202\u03b7 (a, b) \u2202\u03ba a", []], "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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), f (b, c) \u2202\u2191\u03b7 (a, b) \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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), g b c \u2202\u2191\u03b7 (a, b) \u2202\u2191\u03ba a"}, {"tactic": "rw [\u2190 lintegral_compProd']", "annotated_tactic": ["rw [\u2190 <a>lintegral_compProd'</a>]", [{"full_name": "ProbabilityTheory.kernel.lintegral_compProd'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [368, 9], "def_end_pos": [368, 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\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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), g b c \u2202\u2191\u03b7 (a, b) \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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), g bc.1 bc.2 \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a\n\ncase hf\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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 Measurable (Function.uncurry fun b c => g b c)"}, {"tactic": "simp_rw [Function.curry_apply]", "annotated_tactic": ["simp_rw [<a>Function.curry_apply</a>]", [{"full_name": "Function.curry_apply", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [822, 9], "def_end_pos": [822, 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\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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), g bc.1 bc.2 \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a", "state_after": "no goals"}, {"tactic": "simp_rw [Function.uncurry_curry]", "annotated_tactic": ["simp_rw [<a>Function.uncurry_curry</a>]", [{"full_name": "Function.uncurry_curry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [226, 9], "def_end_pos": [226, 22]}]], "state_before": "case hf\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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 Measurable (Function.uncurry fun b c => g b c)", "state_after": "case hf\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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 Measurable f"}, {"tactic": "exact hf", "annotated_tactic": ["exact hf", []], "state_before": "case hf\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\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f\n\u22a2 Measurable f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.IsStoppingTime.add_const_nat", "start": [285, 1], "end": [296, 13], "traced_tactics": [{"tactic": "refine' isStoppingTime_of_measurableSet_eq fun j => _", "annotated_tactic": ["refine' <a>isStoppingTime_of_measurableSet_eq</a> fun j => _", [{"full_name": "MeasureTheory.isStoppingTime_of_measurableSet_eq", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [239, 9], "def_end_pos": [239, 43]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni : \u2115\n\u22a2 IsStoppingTime f fun \u03c9 => \u03c4 \u03c9 + i", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}"}, {"tactic": "by_cases hij : i \u2264 j", "annotated_tactic": ["by_cases hij : i \u2264 j", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : i \u2264 j\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}\n\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : \u00aci \u2264 j\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}"}, {"tactic": "simp_rw [eq_comm, \u2190 Nat.sub_eq_iff_eq_add hij, eq_comm]", "annotated_tactic": ["simp_rw [<a>eq_comm</a>, \u2190 <a>Nat.sub_eq_iff_eq_add</a> hij, <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]}, {"full_name": "Nat.sub_eq_iff_eq_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [387, 19], "def_end_pos": [387, 36]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : i \u2264 j\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : i \u2264 j\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 = j - i}"}, {"tactic": "exact f.mono (j.sub_le i) _ (h\u03c4.measurableSet_eq (j - i))", "annotated_tactic": ["exact f.mono (j.sub_le i) _ (h\u03c4.measurableSet_eq (j - i))", []], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : i \u2264 j\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 = j - i}", "state_after": "no goals"}, {"tactic": "rw [not_le] at hij", "annotated_tactic": ["rw [<a>not_le</a>] at hij", [{"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\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : \u00aci \u2264 j\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}"}, {"tactic": "convert @MeasurableSet.empty _ (f.1 j)", "annotated_tactic": ["convert @<a>MeasurableSet.empty</a> _ (f.1 j)", [{"full_name": "MeasurableSet.empty", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j}", "state_after": "case h.e'_3\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u22a2 {\u03c9 | \u03c4 \u03c9 + i = j} = \u2205"}, {"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "case h.e'_3\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u22a2 {\u03c9 | \u03c4 \u03c9 + i = j} = \u2205", "state_after": "case h.e'_3.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 + i = j} \u2194 \u03c9 \u2208 \u2205"}, {"tactic": "simp only [Set.mem_empty_iff_false, iff_false_iff]", "annotated_tactic": ["simp only [<a>Set.mem_empty_iff_false</a>, <a>iff_false_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": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case h.e'_3.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 + i = j} \u2194 \u03c9 \u2208 \u2205", "state_after": "case h.e'_3.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u03c9 : \u03a9\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 + i = j}"}, {"tactic": "rintro (hx : \u03c4 \u03c9 + i = j)", "annotated_tactic": ["rintro (hx : \u03c4 \u03c9 + i = j)", []], "state_before": "case h.e'_3.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u03c9 : \u03a9\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 + i = j}", "state_after": "case h.e'_3.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u03c9 : \u03a9\nhx : \u03c4 \u03c9 + i = j\n\u22a2 False"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case h.e'_3.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\nf : Filtration \u2115 m\n\u03c4 : \u03a9 \u2192 \u2115\nh\u03c4 : IsStoppingTime f \u03c4\ni j : \u2115\nhij : j < i\n\u03c9 : \u03a9\nhx : \u03c4 \u03c9 + i = j\n\u22a2 False", "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_lt", "start": [753, 1], "end": [753, 96], "traced_tactics": [{"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "a b : USize\nh : 0 < b\n\u22a2 \u2191b.1 > 0", "state_after": "a b : USize\nh : 0 < toNat b\n\u22a2 \u2191b.1 > 0"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "a b : USize\nh : 0 < toNat b\n\u22a2 \u2191b.1 > 0", "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_lintegral_lt_of_sigmaFinite", "start": [1576, 1], "end": [1591, 87], "traced_tactics": [{"tactic": "set s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)", "annotated_tactic": ["set s : \u2115 \u2192 <a>Set</a> \u03b1 := <a>disjointed</a> (<a>spanningSets</a> \u03bc)", [{"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]}, {"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\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "have : \u2200 n, \u03bc (s n) < \u221e := fun n =>\n  (measure_mono <| disjointed_subset _ _).trans_lt (measure_spanningSets_lt_top \u03bc n)", "annotated_tactic": ["have : \u2200 n, \u03bc (s n) < \u221e := fun n =>\n    (<a>measure_mono</a> <| <a>disjointed_subset</a> _ _).<a>trans_lt</a> (<a>measure_spanningSets_lt_top</a> \u03bc n)", [{"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": "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\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "obtain \u27e8\u03b4, \u03b4pos, \u03b4sum\u27e9 : \u2203 \u03b4 : \u2115 \u2192 \u211d\u22650, (\u2200 i, 0 < \u03b4 i) \u2227 (\u2211' i, \u03bc (s i) * \u03b4 i) < \u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, \u03b4pos, \u03b4sum\u27e9 : \u2203 \u03b4 : \u2115 \u2192 \u211d\u22650, (\u2200 i, 0 < \u03b4 i) \u2227 (\u2211' i, \u03bc (s i) * \u03b4 i) < \u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u2115), 0 < \u03b4 i) \u2227 \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "exact ENNReal.exists_pos_tsum_mul_lt_of_countable \u03b50 _ fun n => (this n).ne", "annotated_tactic": ["exact <a>ENNReal.exists_pos_tsum_mul_lt_of_countable</a> \u03b50 _ fun n => (this n).<a>ne</a>", [{"full_name": "ENNReal.exists_pos_tsum_mul_lt_of_countable", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [542, 9], "def_end_pos": [542, 44]}, {"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 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u2115), 0 < \u03b4 i) \u2227 \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g 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\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "set N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc", "annotated_tactic": ["set N : \u03b1 \u2192 \u2115 := <a>spanningSetsIndex</a> \u03bc", [{"full_name": "MeasureTheory.spanningSetsIndex", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3344, 5], "def_end_pos": [3344, 22]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g 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\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "have hN_meas : Measurable N := measurable_spanningSetsIndex \u03bc", "annotated_tactic": ["have hN_meas : <a>Measurable</a> N := <a>measurable_spanningSetsIndex</a> \u03bc", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "MeasureTheory.measurable_spanningSetsIndex", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3348, 9], "def_end_pos": [3348, 37]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g 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\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "have hNs : \u2200 n, N \u207b\u00b9' {n} = s n := preimage_spanningSetsIndex_singleton \u03bc", "annotated_tactic": ["have hNs : \u2200 n, N \u207b\u00b9' {n} = s n := <a>preimage_spanningSetsIndex_singleton</a> \u03bc", [{"full_name": "MeasureTheory.preimage_spanningSetsIndex_singleton", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3353, 9], "def_end_pos": [3353, 45]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g 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\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5"}, {"tactic": "refine' \u27e8\u03b4 \u2218 N, fun x => \u03b4pos _, measurable_from_nat.comp hN_meas, _\u27e9", "annotated_tactic": ["refine' \u27e8\u03b4 \u2218 N, fun x => \u03b4pos _, measurable_from_nat.comp hN_meas, _\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g 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\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u222b\u207b (x : \u03b1), \u2191((\u03b4 \u2218 N) x) \u2202\u03bc < \u03b5"}, {"tactic": "erw [lintegral_comp measurable_from_nat.coe_nnreal_ennreal hN_meas]", "annotated_tactic": ["erw [<a>lintegral_comp</a> measurable_from_nat.coe_nnreal_ennreal hN_meas]", [{"full_name": "MeasureTheory.lintegral_comp", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1311, 9], "def_end_pos": [1311, 23]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u222b\u207b (x : \u03b1), \u2191((\u03b4 \u2218 N) 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\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u222b\u207b (a : \u2115), \u2191(\u03b4 a) \u2202Measure.map N \u03bc < \u03b5"}, {"tactic": "simpa [hNs, lintegral_countable', measurable_spanningSetsIndex, mul_comm] using \u03b4sum", "annotated_tactic": ["simpa [hNs, <a>lintegral_countable'</a>, <a>measurable_spanningSetsIndex</a>, <a>mul_comm</a>] using \u03b4sum", [{"full_name": "MeasureTheory.lintegral_countable'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1471, 9], "def_end_pos": [1471, 29]}, {"full_name": "MeasureTheory.measurable_spanningSetsIndex", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3348, 9], "def_end_pos": [3348, 37]}, {"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\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u222b\u207b (a : \u2115), \u2191(\u03b4 a) \u2202Measure.map N \u03bc < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_pair_lt", "start": [307, 1], "end": [309, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Nat.Primrec.casesOn1", "start": [108, 1], "end": [109, 44], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "f : \u2115 \u2192 \u2115\nm : \u2115\nhf : Nat.Primrec f\n\u22a2 \u2200 (n : \u2115), Nat.rec m (fun y IH => f (unpair (Nat.pair y IH)).1) n = Nat.casesOn n m f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.inl_mem_disjSum", "start": [73, 1], "end": [74, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.prod_subset_prod_iff", "start": [410, 1], "end": [421, 28], "traced_tactics": [{"tactic": "cases' (s \u00d7\u02e2 t).eq_empty_or_nonempty with h h", "annotated_tactic": ["cases' (s \u00d7\u02e2 t).<a>eq_empty_or_nonempty</a> 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\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\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "case inl\n\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\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n\ncase inr\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205"}, {"tactic": "have st : s.Nonempty \u2227 t.Nonempty := by rwa [prod_nonempty_iff] at h", "annotated_tactic": ["have st : s.Nonempty \u2227 t.Nonempty := by rwa [<a>prod_nonempty_iff</a>] at h", [{"full_name": "Set.prod_nonempty_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [346, 9], "def_end_pos": [346, 26]}]], "state_before": "case inr\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "case inr\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205"}, {"tactic": "refine' \u27e8fun H => Or.inl \u27e8_, _\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8fun H => <a>Or.inl</a> \u27e8_, _\u27e9, _\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 inr\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "case inr.refine'_1\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s \u2286 s\u2081\n\ncase inr.refine'_2\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 t \u2286 t\u2081\n\ncase inr.refine'_3\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\n\u22a2 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 \u2192 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081"}, {"tactic": "simp [h, prod_eq_empty_iff.1 h]", "annotated_tactic": ["simp [h, <a>prod_eq_empty_iff</a>.1 h]", [{"full_name": "Set.prod_eq_empty_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [351, 9], "def_end_pos": [351, 26]}]], "state_before": "case inl\n\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\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205", "state_after": "no goals"}, {"tactic": "rwa [prod_nonempty_iff] at h", "annotated_tactic": ["rwa [<a>prod_nonempty_iff</a>] at h", [{"full_name": "Set.prod_nonempty_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [346, 9], "def_end_pos": [346, 26]}]], "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\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 Set.Nonempty s \u2227 Set.Nonempty t", "state_after": "no goals"}, {"tactic": "have := image_subset (Prod.fst : \u03b1 \u00d7 \u03b2 \u2192 \u03b1) H", "annotated_tactic": ["have := <a>image_subset</a> (<a>Prod.fst</a> : \u03b1 \u00d7 \u03b2 \u2192 \u03b1) H", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"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 inr.refine'_1\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s \u2286 s\u2081", "state_after": "case inr.refine'_1\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\nthis : Prod.fst '' s \u00d7\u02e2 t \u2286 Prod.fst '' s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s \u2286 s\u2081"}, {"tactic": "rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this", "annotated_tactic": ["rwa [<a>fst_image_prod</a> _ st.2, <a>fst_image_prod</a> _ (h.mono H).<a>snd</a>] at this", [{"full_name": "Set.fst_image_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [383, 9], "def_end_pos": [383, 23]}, {"full_name": "Set.fst_image_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [383, 9], "def_end_pos": [383, 23]}, {"full_name": "Set.Nonempty.snd", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [342, 9], "def_end_pos": [342, 21]}]], "state_before": "case inr.refine'_1\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\nthis : Prod.fst '' s \u00d7\u02e2 t \u2286 Prod.fst '' s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s \u2286 s\u2081", "state_after": "no goals"}, {"tactic": "have := image_subset (Prod.snd : \u03b1 \u00d7 \u03b2 \u2192 \u03b2) H", "annotated_tactic": ["have := <a>image_subset</a> (<a>Prod.snd</a> : \u03b1 \u00d7 \u03b2 \u2192 \u03b2) H", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"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 inr.refine'_2\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 t \u2286 t\u2081", "state_after": "case inr.refine'_2\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\nthis : Prod.snd '' s \u00d7\u02e2 t \u2286 Prod.snd '' s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 t \u2286 t\u2081"}, {"tactic": "rwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this", "annotated_tactic": ["rwa [<a>snd_image_prod</a> st.1, <a>snd_image_prod</a> (h.mono H).<a>fst</a>] at this", [{"full_name": "Set.snd_image_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [397, 9], "def_end_pos": [397, 23]}, {"full_name": "Set.snd_image_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [397, 9], "def_end_pos": [397, 23]}, {"full_name": "Set.Nonempty.fst", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [339, 9], "def_end_pos": [339, 21]}]], "state_before": "case inr.refine'_2\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\nthis : Prod.snd '' s \u00d7\u02e2 t \u2286 Prod.snd '' s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 t \u2286 t\u2081", "state_after": "no goals"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case inr.refine'_3\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\n\u22a2 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 \u2192 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081", "state_after": "case inr.refine'_3\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081"}, {"tactic": "simp only [st.1.ne_empty, st.2.ne_empty, or_false_iff] at H", "annotated_tactic": ["simp only [st.1.<a>ne_empty</a>, st.2.<a>ne_empty</a>, <a>or_false_iff</a>] at 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": "Set.Nonempty.ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [622, 8], "def_end_pos": [622, 25]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [188, 9], "def_end_pos": [188, 21]}]], "state_before": "case inr.refine'_3\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081", "state_after": "case inr.refine'_3\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u2286 s\u2081 \u2227 t \u2286 t\u2081\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081"}, {"tactic": "exact prod_mono H.1 H.2", "annotated_tactic": ["exact <a>prod_mono</a> H.1 H.2", [{"full_name": "Set.prod_mono", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [78, 9], "def_end_pos": [78, 18]}]], "state_before": "case inr.refine'_3\n\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\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u2286 s\u2081 \u2227 t \u2286 t\u2081\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.snormEssSup_indicator_const_le", "start": [568, 1], "end": [573, 85], "traced_tactics": [{"tactic": "by_cases h\u03bc0 : \u03bc = 0", "annotated_tactic": ["by_cases h\u03bc0 : \u03bc = 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\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\n\u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a", "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\nh\u03bc0 : \u03bc = 0\n\u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a\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\nh\u03bc0 : \u00ac\u03bc = 0\n\u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a"}, {"tactic": "rw [h\u03bc0, snormEssSup_measure_zero]", "annotated_tactic": ["rw [h\u03bc0, <a>snormEssSup_measure_zero</a>]", [{"full_name": "MeasureTheory.snormEssSup_measure_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [238, 9], "def_end_pos": [238, 33]}]], "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\nh\u03bc0 : \u03bc = 0\n\u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a", "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\nh\u03bc0 : \u03bc = 0\n\u22a2 0 \u2264 \u2191\u2016c\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": "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\nh\u03bc0 : \u03bc = 0\n\u22a2 0 \u2264 \u2191\u2016c\u2016\u208a", "state_after": "no goals"}, {"tactic": "exact (snormEssSup_indicator_le s fun _ => c).trans (snormEssSup_const c h\u03bc0).le", "annotated_tactic": ["exact (<a>snormEssSup_indicator_le</a> s fun _ => c).<a>trans</a> (<a>snormEssSup_const</a> c h\u03bc0).<a>le</a>", [{"full_name": "MeasureTheory.snormEssSup_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [561, 9], "def_end_pos": [561, 33]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.snormEssSup_const", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [305, 9], "def_end_pos": [305, 26]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "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\nh\u03bc0 : \u00ac\u03bc = 0\n\u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a", "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_ae", "start": [129, 1], "end": [139, 13], "traced_tactics": [{"tactic": "have hg : AEStronglyMeasurable g \u03bc := aestronglyMeasurable_of_tendsto_ae _ hf hfg", "annotated_tactic": ["have hg : <a>AEStronglyMeasurable</a> g \u03bc := <a>aestronglyMeasurable_of_tendsto_ae</a> _ hf hfg", [{"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\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 TendstoInMeasure \u03bc f atTop 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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 TendstoInMeasure \u03bc f atTop g"}, {"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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 TendstoInMeasure \u03bc f atTop 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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 TendstoInMeasure \u03bc (fun i => AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc)) atTop\n    (AEStronglyMeasurable.mk g hg)"}, {"tactic": "refine' tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable\n  (fun i => (hf i).stronglyMeasurable_mk) hg.stronglyMeasurable_mk _", "annotated_tactic": ["refine' <a>tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable</a>\n    (fun i => (hf i).<a>stronglyMeasurable_mk</a>) hg.stronglyMeasurable_mk _", [{"full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "def_pos": [108, 9], "def_end_pos": [108, 61]}, {"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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 TendstoInMeasure \u03bc (fun i => AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc)) atTop\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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \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,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g hg x))"}, {"tactic": "have hf_eq_ae : \u2200\u1d50 x \u2202\u03bc, \u2200 n, (hf n).mk (f n) x = f n x :=\n  ae_all_iff.mpr fun n => (hf n).ae_eq_mk.symm", "annotated_tactic": ["have hf_eq_ae : \u2200\u1d50 x \u2202\u03bc, \u2200 n, (hf n).<a>mk</a> (f n) x = f n x :=\n    ae_all_iff.mpr fun n => (hf n).ae_eq_mk.symm", [{"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\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \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,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g hg x))", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\nhf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g hg x))"}, {"tactic": "filter_upwards [hf_eq_ae, hg.ae_eq_mk, hfg] with x hxf hxg hxfg", "annotated_tactic": ["filter_upwards [hf_eq_ae, hg.ae_eq_mk, hfg] with x hxf hxg hxfg", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\nhf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n      (\ud835\udcdd (AEStronglyMeasurable.mk g hg x))", "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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\nhf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nx : \u03b1\nhxf : \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nhxg : g x = AEStronglyMeasurable.mk g hg x\nhxfg : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n    (\ud835\udcdd (AEStronglyMeasurable.mk g hg x))"}, {"tactic": "rw [\u2190 hxg, funext fun n => hxf n]", "annotated_tactic": ["rw [\u2190 hxg, <a>funext</a> fun n => hxf n]", [{"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\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nE : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\nhf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nx : \u03b1\nhxf : \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nhxg : g x = AEStronglyMeasurable.mk g hg x\nhxfg : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop\n    (\ud835\udcdd (AEStronglyMeasurable.mk g hg x))", "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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\nhf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nx : \u03b1\nhxf : \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nhxg : g x = AEStronglyMeasurable.mk g hg x\nhxfg : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))"}, {"tactic": "exact hxfg", "annotated_tactic": ["exact hxfg", []], "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\u00b9 : MetricSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nhg : AEStronglyMeasurable g \u03bc\nhf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nx : \u03b1\nhxf : \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x\nhxg : g x = AEStronglyMeasurable.mk g hg x\nhxfg : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 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/Pointwise.lean", "full_name": "Finset.inter_mul_union_subset", "start": [764, 1], "end": [765, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_neg", "start": [2177, 11], "end": [2179, 41], "traced_tactics": [{"tactic": "simp_rw [\u2190 image_neg]", "annotated_tactic": ["simp_rw [\u2190 <a>image_neg</a>]", [{"full_name": "Finset.image_neg", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : AddGroup \u03b2\ninst\u271d\u00b9 : DistribMulAction \u03b1 \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\ns : Finset \u03b1\nt : Finset \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\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : AddGroup \u03b2\ninst\u271d\u00b9 : DistribMulAction \u03b1 \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 s \u2022 image (fun x => -x) t = image (fun x => -x) (s \u2022 t)"}, {"tactic": "exact image_image\u2082_right_comm smul_neg", "annotated_tactic": ["exact <a>image_image\u2082_right_comm</a> <a>smul_neg</a>", [{"full_name": "Finset.image_image\u2082_right_comm", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [440, 9], "def_end_pos": [440, 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\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : AddGroup \u03b2\ninst\u271d\u00b9 : DistribMulAction \u03b1 \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 s \u2022 image (fun x => -x) t = image (fun x => -x) (s \u2022 t)", "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_eq_empty", "start": [715, 1], "end": [716, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.le_measure_compl_liminf_of_limsup_measure_le", "start": [110, 1], "end": [128, 30], "traced_tactics": [{"tactic": "rcases L.eq_or_neBot with rfl | hne", "annotated_tactic": ["rcases L.eq_or_neBot with rfl | hne", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L", "state_after": "case inl\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) \u22a5 \u2264 \u2191\u2191\u03bc E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) \u22a5\n\ncase inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L"}, {"tactic": "have meas_Ec : \u03bc E\u1d9c = 1 - \u03bc E := by\n  simpa only [measure_univ] using measure_compl E_mble (measure_lt_top \u03bc E).ne", "annotated_tactic": ["have meas_Ec : \u03bc E\u1d9c = 1 - \u03bc E := by\n    simpa only [<a>measure_univ</a>] using <a>measure_compl</a> E_mble (<a>measure_lt_top</a> \u03bc E).<a>ne</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": "MeasureTheory.measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [303, 9], "def_end_pos": [303, 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]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L"}, {"tactic": "have meas_i_Ec : \u2200 i, \u03bcs i E\u1d9c = 1 - \u03bcs i E := by\n  intro i\n  simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (\u03bcs i) E).ne", "annotated_tactic": ["have meas_i_Ec : \u2200 i, \u03bcs i E\u1d9c = 1 - \u03bcs i E := by\n    intro i\n    simpa only [<a>measure_univ</a>] using <a>measure_compl</a> E_mble (<a>measure_lt_top</a> (\u03bcs i) E).<a>ne</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": "MeasureTheory.measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [303, 9], "def_end_pos": [303, 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]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L"}, {"tactic": "simp_rw [meas_Ec, meas_i_Ec]", "annotated_tactic": ["simp_rw [meas_Ec, meas_i_Ec]", []], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L"}, {"tactic": "have obs :\n  (L.liminf fun i : \u03b9 => 1 - \u03bcs i E) = L.liminf ((fun x => 1 - x) \u2218 fun i : \u03b9 => \u03bcs i E) := rfl", "annotated_tactic": ["have obs :\n    (L.liminf fun i : \u03b9 => 1 - \u03bcs i E) = L.liminf ((fun x => 1 - x) \u2218 fun i : \u03b9 => \u03bcs i E) := <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\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L"}, {"tactic": "rw [obs]", "annotated_tactic": ["rw [obs]", []], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L"}, {"tactic": "have := antitone_const_tsub.map_limsup_of_continuousAt (F := L)\n  (fun i => \u03bcs i E) (ENNReal.continuous_sub_left ENNReal.one_ne_top).continuousAt", "annotated_tactic": ["have := antitone_const_tsub.map_limsup_of_continuousAt (F := L)\n    (fun i => \u03bcs i E) (<a>ENNReal.continuous_sub_left</a> <a>ENNReal.one_ne_top</a>).<a>continuousAt</a>", [{"full_name": "ENNReal.continuous_sub_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [445, 9], "def_end_pos": [445, 28]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "Continuous.continuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1709, 9], "def_end_pos": [1709, 32]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\nthis : 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L"}, {"tactic": "simp_rw [\u2190 this]", "annotated_tactic": ["simp_rw [\u2190 this]", []], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\nthis : 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L", "state_after": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\nthis : 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L"}, {"tactic": "exact antitone_const_tsub h", "annotated_tactic": ["exact <a>antitone_const_tsub</a> h", [{"full_name": "antitone_const_tsub", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [130, 9], "def_end_pos": [130, 28]}]], "state_before": "case inr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\nmeas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E\nobs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\nthis : 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L\n\u22a2 1 - \u2191\u2191\u03bc E \u2264 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L", "state_after": "no goals"}, {"tactic": "simp only [liminf_bot, le_top]", "annotated_tactic": ["simp only [<a>liminf_bot</a>, <a>le_top</a>]", [{"full_name": "Filter.liminf_bot", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [722, 17], "def_end_pos": [722, 27]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case inl\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) \u22a5 \u2264 \u2191\u2191\u03bc E\n\u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) \u22a5", "state_after": "no goals"}, {"tactic": "simpa only [measure_univ] using measure_compl E_mble (measure_lt_top \u03bc E).ne", "annotated_tactic": ["simpa only [<a>measure_univ</a>] using <a>measure_compl</a> E_mble (<a>measure_lt_top</a> \u03bc E).<a>ne</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": "MeasureTheory.measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [303, 9], "def_end_pos": [303, 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]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\n\u22a2 \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\n\u22a2 \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E"}, {"tactic": "simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (\u03bcs i) E).ne", "annotated_tactic": ["simpa only [<a>measure_univ</a>] using <a>measure_compl</a> E_mble (<a>measure_lt_top</a> (\u03bcs i) E).<a>ne</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": "MeasureTheory.measure_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [303, 9], "def_end_pos": [303, 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]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\n\u03bc : Measure \u03a9\n\u03bcs : \u03b9 \u2192 Measure \u03a9\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i)\nE : Set \u03a9\nE_mble : MeasurableSet E\nh : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E\nhne : NeBot L\nmeas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E", "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_Ici_of_neg", "start": [656, 1], "end": [658, 64], "traced_tactics": [{"tactic": "simpa only [mul_comm] using preimage_mul_const_Ici_of_neg a h", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using <a>preimage_mul_const_Ici_of_neg</a> a 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_Ici_of_neg", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [570, 9], "def_end_pos": [570, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d a c : \u03b1\nh : c < 0\n\u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Ici a = Iic (a / c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/FinEnum.lean", "full_name": "FinEnum.pi.mem_enum", "start": [253, 1], "end": [254, 89], "traced_tactics": [{"tactic": "simp [pi.enum]", "annotated_tactic": ["simp [<a>pi.enum</a>]", [{"full_name": "FinEnum.pi.enum", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [249, 5], "def_end_pos": [249, 12]}]], "state_before": "\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u v)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nf : (a : \u03b1) \u2192 \u03b2 a\n\u22a2 f \u2208 enum \u03b2", "state_after": "\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u v)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nf : (a : \u03b1) \u2192 \u03b2 a\n\u22a2 \u2203 a, (a \u2208 pi (toList \u03b1) fun x => toList (\u03b2 x)) \u2227 (fun x => a x (_ : x \u2208 toList \u03b1)) = f"}, {"tactic": "refine' \u27e8fun a _ => f a, mem_pi _ _, rfl\u27e9", "annotated_tactic": ["refine' \u27e8fun a _ => f a, <a>mem_pi</a> _ _, <a>rfl</a>\u27e9", [{"full_name": "FinEnum.mem_pi", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [230, 9], "def_end_pos": [230, 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": "\u03b1 : Type u\n\u03b2\u271d : \u03b1 \u2192 Type v\n\u03b2 : \u03b1 \u2192 Type (max u v)\ninst\u271d\u00b9 : FinEnum \u03b1\ninst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a)\nf : (a : \u03b1) \u2192 \u03b2 a\n\u22a2 \u2203 a, (a \u2208 pi (toList \u03b1) fun x => toList (\u03b2 x)) \u2227 (fun x => a x (_ : x \u2208 toList \u03b1)) = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "Finset.aestronglyMeasurable_prod'", "start": [1427, 1], "end": [1431, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean", "full_name": "MvQPF.liftR_map", "start": [378, 1], "end": [385, 38], "traced_tactics": [{"tactic": "rw [LiftR_def]", "annotated_tactic": ["rw [<a>LiftR_def</a>]", [{"full_name": "MvFunctor.LiftR_def", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [142, 9], "def_end_pos": [142, 18]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 LiftR' R (f <$$> x) (g <$$> x)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 \u2203 u, (prod.fst \u229a subtypeVal R) <$$> u = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> u = g <$$> x"}, {"tactic": "exists h <$$> x", "annotated_tactic": ["exists h <$$> x", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 \u2203 u, (prod.fst \u229a subtypeVal R) <$$> u = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> u = g <$$> x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (prod.fst \u229a subtypeVal R) <$$> h <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x"}, {"tactic": "rw [MvFunctor.map_map, comp_assoc, hh, \u2190 comp_assoc, fst_prod_mk, comp_assoc, fst_diag]", "annotated_tactic": ["rw [<a>MvFunctor.map_map</a>, <a>comp_assoc</a>, hh, \u2190 <a>comp_assoc</a>, <a>fst_prod_mk</a>, <a>comp_assoc</a>, <a>fst_diag</a>]", [{"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "TypeVec.comp_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "TypeVec.comp_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "TypeVec.fst_prod_mk", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [557, 9], "def_end_pos": [557, 20]}, {"full_name": "TypeVec.comp_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "TypeVec.fst_diag", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [571, 9], "def_end_pos": [571, 17]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (prod.fst \u229a subtypeVal R) <$$> h <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x"}, {"tactic": "rw [MvFunctor.map_map, comp_assoc, hh, \u2190 comp_assoc, snd_prod_mk, comp_assoc, snd_diag]", "annotated_tactic": ["rw [<a>MvFunctor.map_map</a>, <a>comp_assoc</a>, hh, \u2190 <a>comp_assoc</a>, <a>snd_prod_mk</a>, <a>comp_assoc</a>, <a>snd_diag</a>]", [{"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "TypeVec.comp_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "TypeVec.comp_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "TypeVec.snd_prod_mk", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [564, 9], "def_end_pos": [564, 20]}, {"full_name": "TypeVec.comp_assoc", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "TypeVec.snd_diag", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [577, 9], "def_end_pos": [577, 17]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (g \u229a TypeVec.id) <$$> x = g <$$> x"}, {"tactic": "dsimp [LiftR']", "annotated_tactic": ["dsimp [<a>LiftR'</a>]", [{"full_name": "MvFunctor.LiftR'", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [99, 5], "def_end_pos": [99, 11]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (g \u229a TypeVec.id) <$$> x = g <$$> x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 f <$$> x = f <$$> x \u2227 g <$$> x = g <$$> x"}, {"tactic": "constructor <;> rfl", "annotated_tactic": ["constructor <;> rfl", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec.{u_1} n\nF' : TypeVec.{u_1} n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 f <$$> x = f <$$> x \u2227 g <$$> x = g <$$> x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "full_name": "iUnion_Iic_eq_Iic_iSup", "start": [251, 1], "end": [253, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "full_name": "Std.RBNode.Ordered.balRight", "start": [274, 11], "end": [284, 27], "traced_tactics": [{"tactic": "unfold balRight", "annotated_tactic": ["unfold <a>balRight</a>", [{"full_name": "Std.RBNode.balRight", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [337, 5], "def_end_pos": [337, 13]}]], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n\u22a2 Ordered cmp (balRight l v r)", "state_after": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n\u22a2 Ordered cmp\n    (match r with\n    | node red b y c => node red l v (node black b y c)\n    | r =>\n      match l with\n      | node black a x b => balance1 (node red a x b) v r\n      | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n      | l => node red l v r)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n\u22a2 Ordered cmp\n    (match r with\n    | node red b y c => node red l v (node black b y c)\n    | r =>\n      match l with\n      | node black a x b => balance1 (node red a x b) v r\n      | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n      | l => node red l v r)", "state_after": "case h_1\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nhl : Ordered cmp l\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) (node red a\u271d x\u271d b\u271d)\nhr : Ordered cmp (node red a\u271d x\u271d b\u271d)\n\u22a2 Ordered cmp (node red l v (node black a\u271d x\u271d b\u271d))\n\ncase h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl\u271d : RBNode \u03b1\nx\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\n\u22a2 Ordered cmp\n    (match l with\n    | node black a x b => balance1 (node red a x b) v r\n    | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n    | l => node red l v r)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl\u271d : RBNode \u03b1\nx\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\n\u22a2 Ordered cmp\n    (match l with\n    | node black a x b => balance1 (node red a x b) v r\n    | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n    | l => node red l v r)", "state_after": "case h_2.h_1\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node black a\u271d x\u271d b\u271d)\nhl : Ordered cmp (node black a\u271d x\u271d b\u271d)\n\u22a2 Ordered cmp (balance1 (node red a\u271d x\u271d b\u271d) v r)\n\ncase h_2.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\ny\u271d : \u03b1\nc\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nhl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\n\u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r))\n\ncase h_2.h_3\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), l = node black a x b \u2192 False\nx\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red a x (node black b y c) \u2192 False\n\u22a2 Ordered cmp (node red l v r)"}, {"tactic": "exact \u27e8lv, vr, hl, hr\u27e9", "annotated_tactic": ["exact \u27e8lv, vr, hl, hr\u27e9", []], "state_before": "case h_1\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nhl : Ordered cmp l\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) (node red a\u271d x\u271d b\u271d)\nhr : Ordered cmp (node red a\u271d x\u271d b\u271d)\n\u22a2 Ordered cmp (node red l v (node black a\u271d x\u271d b\u271d))", "state_after": "no goals"}, {"tactic": "exact hl.balance1 lv vr hr", "annotated_tactic": ["exact hl.balance1 lv vr hr", []], "state_before": "case h_2.h_1\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node black a\u271d x\u271d b\u271d)\nhl : Ordered cmp (node black a\u271d x\u271d b\u271d)\n\u22a2 Ordered cmp (balance1 (node red a\u271d x\u271d b\u271d) v r)", "state_after": "no goals"}, {"tactic": "have \u27e8yv, _, cv\u27e9 := lv.2.2", "annotated_tactic": ["have \u27e8yv, _, cv\u27e9 := lv.2.2", []], "state_before": "case h_2.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\ny\u271d : \u03b1\nc\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nhl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\n\u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r))", "state_after": "case h_2.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\ny\u271d : \u03b1\nc\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nhl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nyv : (fun x => cmpLT cmp x v) y\u271d\nleft\u271d : All (fun x => cmpLT cmp x v) b\u271d\ncv : All (fun x => cmpLT cmp x v) c\u271d\n\u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r))"}, {"tactic": "have \u27e8ax, \u27e8xy, xb, _\u27e9, ha, by_, yc, hb, hc\u27e9 := hl", "annotated_tactic": ["have \u27e8ax, \u27e8xy, xb, _\u27e9, ha, by_, yc, hb, hc\u27e9 := hl", []], "state_before": "case h_2.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\ny\u271d : \u03b1\nc\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nhl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nyv : (fun x => cmpLT cmp x v) y\u271d\nleft\u271d : All (fun x => cmpLT cmp x v) b\u271d\ncv : All (fun x => cmpLT cmp x v) c\u271d\n\u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r))", "state_after": "case h_2.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\ny\u271d : \u03b1\nc\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nhl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nyv : (fun x => cmpLT cmp x v) y\u271d\nleft\u271d : All (fun x => cmpLT cmp x v) b\u271d\ncv : All (fun x => cmpLT cmp x v) c\u271d\nax : All (fun x => cmpLT cmp x x\u271d) a\u271d\nxy : (fun x => cmpLT cmp x\u271d x) y\u271d\nxb : All (fun x => cmpLT cmp x\u271d x) b\u271d\nright\u271d : All (fun x => cmpLT cmp x\u271d x) c\u271d\nha : Ordered cmp a\u271d\nby_ : All (fun x => cmpLT cmp x y\u271d) b\u271d\nyc : All (fun x => cmpLT cmp y\u271d x) c\u271d\nhb : Ordered cmp b\u271d\nhc : Ordered cmp c\u271d\n\u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r))"}, {"tactic": "exact \u27e8balance1_All.2 \u27e8xy, (xy.trans_r ax).setRed, by_\u27e9, \u27e8yv, yc, yv.trans_l vr\u27e9,\n  (Ordered.setRed.2 ha).balance1 ax.setRed xb hb, cv, vr, hc, hr\u27e9", "annotated_tactic": ["exact \u27e8<a>balance1_All</a>.2 \u27e8xy, (xy.trans_r ax).<a>setRed</a>, by_\u27e9, \u27e8yv, yc, yv.trans_l vr\u27e9,\n      (<a>Ordered.setRed</a>.2 ha).<a>balance1</a> ax.setRed xb hb, cv, vr, hc, hr\u27e9", [{"full_name": "Std.RBNode.balance1_All", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [58, 17], "def_end_pos": [58, 29]}, {"full_name": "Std.RBNode.All.setRed", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [228, 19], "def_end_pos": [228, 29]}, {"full_name": "Std.RBNode.Ordered.setRed", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [232, 19], "def_end_pos": [232, 33]}, {"full_name": "Std.RBNode.Ordered.balance1", "def_path": "lake-packages/std/Std/Data/RBMap/WF.lean", "def_pos": [46, 19], "def_end_pos": [46, 35]}]], "state_before": "case h_2.h_2\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nr : RBNode \u03b1\nvr : All (fun x => cmpLT cmp v x) r\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d a\u271d : RBNode \u03b1\nx\u271d : \u03b1\nb\u271d : RBNode \u03b1\ny\u271d : \u03b1\nc\u271d : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nhl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d))\nyv : (fun x => cmpLT cmp x v) y\u271d\nleft\u271d : All (fun x => cmpLT cmp x v) b\u271d\ncv : All (fun x => cmpLT cmp x v) c\u271d\nax : All (fun x => cmpLT cmp x x\u271d) a\u271d\nxy : (fun x => cmpLT cmp x\u271d x) y\u271d\nxb : All (fun x => cmpLT cmp x\u271d x) b\u271d\nright\u271d : All (fun x => cmpLT cmp x\u271d x) c\u271d\nha : Ordered cmp a\u271d\nby_ : All (fun x => cmpLT cmp x y\u271d) b\u271d\nyc : All (fun x => cmpLT cmp y\u271d x) c\u271d\nhb : Ordered cmp b\u271d\nhc : Ordered cmp c\u271d\n\u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r))", "state_after": "no goals"}, {"tactic": "exact \u27e8lv, vr, hl, hr\u27e9", "annotated_tactic": ["exact \u27e8lv, vr, hl, hr\u27e9", []], "state_before": "case h_2.h_3\n\u03b1 : Type u_1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\nlv : All (fun x => cmpLT cmp x v) l\nvr : All (fun x => cmpLT cmp v x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nl\u271d\u00b9 : RBNode \u03b1\nx\u271d\u00b2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False\nl\u271d : RBNode \u03b1\nx\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), l = node black a x b \u2192 False\nx\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red a x (node black b y c) \u2192 False\n\u22a2 Ordered cmp (node red l v r)", "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_const", "start": [1113, 1], "end": [1119, 41], "traced_tactics": [{"tactic": "simp_rw [Integrable, aestronglyMeasurable_smul_const_iff (f := f) hc, and_congr_right_iff,\n  HasFiniteIntegral, nnnorm_smul, ENNReal.coe_mul]", "annotated_tactic": ["simp_rw [<a>Integrable</a>, <a>aestronglyMeasurable_smul_const_iff</a> (f := f) hc, <a>and_congr_right_iff</a>,\n    <a>HasFiniteIntegral</a>, <a>nnnorm_smul</a>, <a>ENNReal.coe_mul</a>]", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "aestronglyMeasurable_smul_const_iff", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1780, 9], "def_end_pos": [1780, 51]}, {"full_name": "and_congr_right_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [219, 17], "def_end_pos": [219, 36]}, {"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}, {"full_name": "nnnorm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [99, 9], "def_end_pos": [99, 20]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\n\u22a2 (Integrable fun x => f x \u2022 c) \u2194 Integrable 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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\n\u22a2 AEStronglyMeasurable f \u03bc \u2192 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a * \u2191\u2016c\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4)"}, {"tactic": "intro _", "annotated_tactic": ["intro _", []], "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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\n\u22a2 AEStronglyMeasurable f \u03bc \u2192 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a * \u2191\u2016c\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4)", "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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a * \u2191\u2016c\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "rw [lintegral_mul_const' _ _ ENNReal.coe_ne_top, ENNReal.mul_lt_top_iff]", "annotated_tactic": ["rw [<a>lintegral_mul_const'</a> _ _ <a>ENNReal.coe_ne_top</a>, <a>ENNReal.mul_lt_top_iff</a>]", [{"full_name": "MeasureTheory.lintegral_mul_const'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [737, 9], "def_end_pos": [737, 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_lt_top_iff", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [626, 9], "def_end_pos": [626, 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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a * \u2191\u2016c\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4", "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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2227 \u2191\u2016c\u2016\u208a < \u22a4 \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = 0 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "have : \u2200 x : \u211d\u22650\u221e, x = 0 \u2192 x < \u221e := by simp", "annotated_tactic": ["have : \u2200 x : \u211d\u22650\u221e, x = 0 \u2192 x < \u221e := by simp", []], "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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2227 \u2191\u2016c\u2016\u208a < \u22a4 \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = 0 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4", "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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\nthis : \u2200 (x : \u211d\u22650\u221e), x = 0 \u2192 x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2227 \u2191\u2016c\u2016\u208a < \u22a4 \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = 0 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "simp [hc, or_iff_left_of_imp (this _)]", "annotated_tactic": ["simp [hc, <a>or_iff_left_of_imp</a> (this _)]", [{"full_name": "or_iff_left_of_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [291, 9], "def_end_pos": [291, 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\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\nthis : \u2200 (x : \u211d\u22650\u221e), x = 0 \u2192 x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2227 \u2191\u2016c\u2016\u208a < \u22a4 \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = 0 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4", "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\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b4\ninst\u271d\u2075 : NormedAddCommGroup \u03b2\ninst\u271d\u2074 : NormedAddCommGroup \u03b3\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nE : Type u_6\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \u03b1 \u2192 \ud835\udd5c\nc : E\nhc : c \u2260 0\na\u271d : AEStronglyMeasurable f \u03bc\n\u22a2 \u2200 (x : \u211d\u22650\u221e), x = 0 \u2192 x < \u22a4", "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_Ioc", "start": [221, 1], "end": [223, 94], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ioc_subset_Ioi_self <| Ioi_subset_Ioi bot_le)]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ioc</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>,\n    <a>inter_eq_self_of_subset_left</a> (<a>Subset.trans</a> <a>Ioc_subset_Ioi_self</a> <| <a>Ioi_subset_Ioi</a> <a>bot_le</a>)]", [{"full_name": "WithBot.preimage_coe_Ioc", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [175, 9], "def_end_pos": [175, 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.Ioc_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [526, 9], "def_end_pos": [526, 28]}, {"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 '' Ioc a b = Ioc \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Function.update_comp_eq_of_not_mem_range", "start": [1721, 1], "end": [1723, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "MeasurableSpace.induction_on_inter", "start": [745, 1], "end": [765, 17], "traced_tactics": [{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\n\u22a2 MeasurableSet = DynkinSystem.GenerateHas s", "state_after": "no goals"}, {"tactic": "rwa [eq] at ht", "annotated_tactic": ["rwa [eq] at ht", []], "state_before": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 DynkinSystem.GenerateHas s t", "state_after": "no goals"}, {"tactic": "rw [eq]", "annotated_tactic": ["rw [eq]", []], "state_before": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt\u271d : Set \u03b1\nht\u271d : MeasurableSet t\u271d\nthis : DynkinSystem.GenerateHas s t\u271d\nt : Set \u03b1\nht : DynkinSystem.GenerateHas s t\n\u22a2 MeasurableSet t", "state_after": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt\u271d : Set \u03b1\nht\u271d : MeasurableSet t\u271d\nthis : DynkinSystem.GenerateHas s t\u271d\nt : Set \u03b1\nht : DynkinSystem.GenerateHas s t\n\u22a2 DynkinSystem.GenerateHas s t"}, {"tactic": "exact ht", "annotated_tactic": ["exact ht", []], "state_before": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt\u271d : Set \u03b1\nht\u271d : MeasurableSet t\u271d\nthis : DynkinSystem.GenerateHas s t\u271d\nt : Set \u03b1\nht : DynkinSystem.GenerateHas s t\n\u22a2 DynkinSystem.GenerateHas s t", "state_after": "no goals"}, {"tactic": "rw [eq]", "annotated_tactic": ["rw [eq]", []], "state_before": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt : Set \u03b1\nht\u271d : MeasurableSet t\nthis : DynkinSystem.GenerateHas s t\nf : \u2115 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\nht : \u2200 (i : \u2115), DynkinSystem.GenerateHas s (f i)\ni : \u2115\n\u22a2 MeasurableSet (f i)", "state_after": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt : Set \u03b1\nht\u271d : MeasurableSet t\nthis : DynkinSystem.GenerateHas s t\nf : \u2115 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\nht : \u2200 (i : \u2115), DynkinSystem.GenerateHas s (f i)\ni : \u2115\n\u22a2 DynkinSystem.GenerateHas s (f i)"}, {"tactic": "exact ht _", "annotated_tactic": ["exact ht _", []], "state_before": "\u03b1 : Type u_1\nC : Set \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nm : MeasurableSpace \u03b1\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : C \u2205\nh_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t\nh_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c\nh_union :\n  \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i)\neq : MeasurableSet = DynkinSystem.GenerateHas s\nt : Set \u03b1\nht\u271d : MeasurableSet t\nthis : DynkinSystem.GenerateHas s t\nf : \u2115 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\nht : \u2200 (i : \u2115), DynkinSystem.GenerateHas s (f i)\ni : \u2115\n\u22a2 DynkinSystem.GenerateHas s (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.tr_eval", "start": [923, 1], "end": [928, 33], "traced_tactics": [{"tactic": "cases' mem_eval.1 ab with ab b0", "annotated_tactic": ["cases' <a>mem_eval</a>.1 ab with ab b0", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab : b\u2081 \u2208 eval f\u2081 a\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082", "state_after": "case intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2081 \u2208 eval f\u2081 a\u2081\nab : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082"}, {"tactic": "rcases tr_reaches H aa ab with \u27e8b\u2082, bb, ab\u27e9", "annotated_tactic": ["rcases <a>tr_reaches</a> H aa ab with \u27e8b\u2082, bb, ab\u27e9", [{"full_name": "Turing.tr_reaches", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [895, 9], "def_end_pos": [895, 19]}]], "state_before": "case intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2081 \u2208 eval f\u2081 a\u2081\nab : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082", "state_after": "case intro.intro.intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082"}, {"tactic": "refine' \u27e8_, bb, mem_eval.2 \u27e8ab, _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8_, bb, <a>mem_eval</a>.2 \u27e8ab, _\u27e9\u27e9", [{"full_name": "Turing.mem_eval", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [830, 9], "def_end_pos": [830, 17]}]], "state_before": "case intro.intro.intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082", "state_after": "case intro.intro.intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 f\u2082 b\u2082 = none"}, {"tactic": "have := H bb", "annotated_tactic": ["have := H bb", []], "state_before": "case intro.intro.intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 f\u2082 b\u2082 = none", "state_after": "case intro.intro.intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nthis :\n  match f\u2081 b\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\n\u22a2 f\u2082 b\u2082 = none"}, {"tactic": "rwa [b0] at this", "annotated_tactic": ["rwa [b0] at this", []], "state_before": "case intro.intro.intro\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 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nthis :\n  match f\u2081 b\u2081 with\n  | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n  | none => f\u2082 b\u2082 = none\n\u22a2 f\u2082 b\u2082 = none", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.trim_add", "start": [1744, 1], "end": [1745, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "image_circleMap_Ioc", "start": [154, 1], "end": [155, 91], "traced_tactics": [{"tactic": "rw [\u2190 range_circleMap, \u2190 (periodic_circleMap c R).image_Ioc Real.two_pi_pos 0, zero_add]", "annotated_tactic": ["rw [\u2190 <a>range_circleMap</a>, \u2190 (<a>periodic_circleMap</a> c R).<a>image_Ioc</a> <a>Real.two_pi_pos</a> 0, <a>zero_add</a>]", [{"full_name": "range_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [142, 9], "def_end_pos": [142, 24]}, {"full_name": "periodic_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [93, 9], "def_end_pos": [93, 27]}, {"full_name": "Function.Periodic.image_Ioc", "def_path": "Mathlib/Algebra/Periodic.lean", "def_pos": [306, 9], "def_end_pos": [306, 27]}, {"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": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR : \u211d\n\u22a2 circleMap c R '' Ioc 0 (2 * \u03c0) = sphere c |R|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_true_of_mem", "start": [2761, 1], "end": [2761, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haarMeasure_self", "start": [632, 1], "end": [636, 71], "traced_tactics": [{"tactic": "haveI : LocallyCompactSpace G := K\u2080.locallyCompactSpace_of_group", "annotated_tactic": ["haveI : <a>LocallyCompactSpace</a> G := K\u2080.locallyCompactSpace_of_group", [{"full_name": "LocallyCompactSpace", "def_path": "Mathlib/Topology/Compactness/LocallyCompact.lean", "def_pos": [73, 7], "def_end_pos": [73, 26]}]], "state_before": "G : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1", "state_after": "G : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1"}, {"tactic": "rw [haarMeasure_apply K\u2080.isCompact.measurableSet, ENNReal.div_self]", "annotated_tactic": ["rw [<a>haarMeasure_apply</a> K\u2080.isCompact.measurableSet, <a>ENNReal.div_self</a>]", [{"full_name": "MeasureTheory.Measure.haarMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [611, 9], "def_end_pos": [611, 26]}, {"full_name": "ENNReal.div_self", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1746, 19], "def_end_pos": [1746, 27]}]], "state_before": "G : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1", "state_after": "case h0\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 0\n\ncase hI\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 \u22a4"}, {"tactic": "rw [\u2190 pos_iff_ne_zero]", "annotated_tactic": ["rw [\u2190 <a>pos_iff_ne_zero</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]}]], "state_before": "case h0\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 0", "state_after": "case h0\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080"}, {"tactic": "exact haarContent_outerMeasure_self_pos", "annotated_tactic": ["exact <a>haarContent_outerMeasure_self_pos</a>", [{"full_name": "MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 42]}]], "state_before": "case h0\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080", "state_after": "no goals"}, {"tactic": "exact (Content.outerMeasure_lt_top_of_isCompact _ K\u2080.isCompact).ne", "annotated_tactic": ["exact (<a>Content.outerMeasure_lt_top_of_isCompact</a> _ K\u2080.isCompact).<a>ne</a>", [{"full_name": "MeasureTheory.Content.outerMeasure_lt_top_of_isCompact", "def_path": "Mathlib/MeasureTheory/Measure/Content.lean", "def_pos": [308, 9], "def_end_pos": [308, 41]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case hI\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "full_name": "MeasureTheory.AEDisjoint.union_right", "start": [118, 1], "end": [119, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_spec", "start": [1019, 1], "end": [1031, 100], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\n\u22a2 valMinAbs x = y \u2192 x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n", "state_after": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 x = \u2191(valMinAbs x) \u2227 valMinAbs x * 2 \u2208 Set.Ioc (-\u2191n) \u2191n"}, {"tactic": "exact \u27e8x.coe_valMinAbs.symm, x.valMinAbs_mem_Ioc\u27e9", "annotated_tactic": ["exact \u27e8x.coe_valMinAbs.symm, x.valMinAbs_mem_Ioc\u27e9", []], "state_before": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 x = \u2191(valMinAbs x) \u2227 valMinAbs x * 2 \u2208 Set.Ioc (-\u2191n) \u2191n", "state_after": "no goals"}, {"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": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 valMinAbs x = y", "state_after": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 valMinAbs x - y = 0"}, {"tactic": "apply @Int.eq_zero_of_abs_lt_dvd n", "annotated_tactic": ["apply @<a>Int.eq_zero_of_abs_lt_dvd</a> n", [{"full_name": "Int.eq_zero_of_abs_lt_dvd", "def_path": "Mathlib/Data/Int/Order/Lemmas.lean", "def_pos": [57, 9], "def_end_pos": [57, 30]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 valMinAbs x - y = 0", "state_after": "case h1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 \u2191n \u2223 valMinAbs x - y\n\ncase h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| < \u2191n"}, {"tactic": "rw [\u2190 mul_lt_mul_right (@zero_lt_two \u2124 _ _ _ _ _)]", "annotated_tactic": ["rw [\u2190 <a>mul_lt_mul_right</a> (@<a>zero_lt_two</a> \u2124 _ _ _ _ _)]", [{"full_name": "mul_lt_mul_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}, {"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 h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| < \u2191n", "state_after": "case h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| * 2 < \u2191n * 2"}, {"tactic": "nth_rw 1 [\u2190 abs_eq_self.2 (@zero_le_two \u2124 _ _ _ _)]", "annotated_tactic": ["nth_rw 1 [\u2190 <a>abs_eq_self</a>.2 (@<a>zero_le_two</a> \u2124 _ _ _ _)]", [{"full_name": "abs_eq_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"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 h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| * 2 < \u2191n * 2", "state_after": "case h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| * |2| < \u2191n * 2"}, {"tactic": "rw [\u2190 abs_mul, sub_mul, abs_lt]", "annotated_tactic": ["rw [\u2190 <a>abs_mul</a>, <a>sub_mul</a>, <a>abs_lt</a>]", [{"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [372, 7], "def_end_pos": [372, 14]}, {"full_name": "abs_lt", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [196, 9], "def_end_pos": [196, 15]}]], "state_before": "case h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| * |2| < \u2191n * 2", "state_after": "case h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 -(\u2191n * 2) < valMinAbs x * 2 - y * 2 \u2227 valMinAbs x * 2 - y * 2 < \u2191n * 2"}, {"tactic": "constructor <;> linarith only [x.valMinAbs_mem_Ioc.1, x.valMinAbs_mem_Ioc.2, h.2.1, h.2.2]", "annotated_tactic": ["constructor <;> linarith only [x.valMinAbs_mem_Ioc.1, x.valMinAbs_mem_Ioc.2, h.2.1, h.2.2]", []], "state_before": "case h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 -(\u2191n * 2) < valMinAbs x * 2 - y * 2 \u2227 valMinAbs x * 2 - y * 2 < \u2191n * 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 int_cast_zmod_eq_zero_iff_dvd, Int.cast_sub, coe_valMinAbs, h.1, sub_self]", "annotated_tactic": ["rw [\u2190 <a>int_cast_zmod_eq_zero_iff_dvd</a>, <a>Int.cast_sub</a>, <a>coe_valMinAbs</a>, h.1, <a>sub_self</a>]", [{"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]}, {"full_name": "Int.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "ZMod.coe_valMinAbs", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [971, 9], "def_end_pos": [971, 22]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case h1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 \u2191n \u2223 valMinAbs x - y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_const'", "start": [632, 1], "end": [635, 52], "traced_tactics": [{"tactic": "simp only [setToSimpleFunc, range_const, Set.mem_singleton, preimage_const_of_mem,\n  sum_singleton, \u2190 Function.const_def, coe_const]", "annotated_tactic": ["simp only [<a>setToSimpleFunc</a>, <a>range_const</a>, <a>Set.mem_singleton</a>, <a>preimage_const_of_mem</a>,\n    <a>sum_singleton</a>, \u2190 <a>Function.const_def</a>, <a>coe_const</a>]", [{"full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [281, 5], "def_end_pos": [281, 20]}, {"full_name": "MeasureTheory.SimpleFunc.range_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [163, 9], "def_end_pos": [163, 20]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}, {"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": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "Function.const_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 18]}, {"full_name": "MeasureTheory.SimpleFunc.coe_const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [158, 9], "def_end_pos": [158, 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\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 : Measure \u03b1\ninst\u271d : Nonempty \u03b1\nT : Set \u03b1 \u2192 F \u2192L[\u211d] F'\nx : F\nm : MeasurableSpace \u03b1\n\u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x", "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.rev_lt_rev", "start": [143, 9], "end": [144, 46], "traced_tactics": [{"tactic": "rw [\u2190 Fin.not_le, \u2190 Fin.not_le, rev_le_rev]", "annotated_tactic": ["rw [\u2190 <a>Fin.not_le</a>, \u2190 <a>Fin.not_le</a>, <a>rev_le_rev</a>]", [{"full_name": "Fin.not_le", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [83, 27], "def_end_pos": [83, 33]}, {"full_name": "Fin.not_le", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [83, 27], "def_end_pos": [83, 33]}, {"full_name": "Fin.rev_le_rev", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [130, 17], "def_end_pos": [130, 27]}]], "state_before": "n : Nat\ni j : Fin n\n\u22a2 rev i < rev j \u2194 j < i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.gcd_greatest", "start": [403, 1], "end": [406, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.mem_toSubtype_iff", "start": [554, 1], "end": [556, 75], "traced_tactics": [{"tactic": "rw [toSubtype_apply, Part.mem_mk_iff, exists_subtype_mk_eq_iff, eq_comm]", "annotated_tactic": ["rw [<a>toSubtype_apply</a>, <a>Part.mem_mk_iff</a>, <a>exists_subtype_mk_eq_iff</a>, <a>eq_comm</a>]", [{"full_name": "PFun.toSubtype_apply", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [544, 9], "def_end_pos": [544, 24]}, {"full_name": "Part.mem_mk_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [110, 9], "def_end_pos": [110, 19]}, {"full_name": "exists_subtype_mk_eq_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [145, 9], "def_end_pos": [145, 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 : 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 : \u03b1 \u2192. \u03b2\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : Subtype p\n\u22a2 b \u2208 toSubtype p f a \u2194 \u2191b = f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Function.Semiconj.bijOn_range", "start": [1694, 1], "end": [1697, 70], "traced_tactics": [{"tactic": "rw [\u2190 image_univ]", "annotated_tactic": ["rw [\u2190 <a>image_univ</a>]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [718, 9], "def_end_pos": [718, 19]}]], "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\nfa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns t : Set \u03b1\nh : Semiconj f fa fb\nha : Bijective fa\nhf : Injective f\n\u22a2 BijOn fb (range f) (range f)", "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\nfa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns t : Set \u03b1\nh : Semiconj f fa fb\nha : Bijective fa\nhf : Injective f\n\u22a2 BijOn fb (f '' univ) (f '' univ)"}, {"tactic": "exact h.bijOn_image (bijective_iff_bijOn_univ.1 ha) (hf.injOn univ)", "annotated_tactic": ["exact h.bijOn_image (<a>bijective_iff_bijOn_univ</a>.1 ha) (hf.injOn <a>univ</a>)", [{"full_name": "Set.bijective_iff_bijOn_univ", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 33]}, {"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\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\nfa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns t : Set \u03b1\nh : Semiconj f fa fb\nha : Bijective fa\nhf : Injective f\n\u22a2 BijOn fb (f '' univ) (f '' univ)", "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.sign_mul_natAbs", "start": [534, 1], "end": [537, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "full_name": "BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "start": [34, 1], "end": [39, 27], "traced_tactics": [{"tactic": "apply IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal", "annotated_tactic": ["apply <a>IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal</a>", [{"full_name": "IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 70]}]], "state_before": "X : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (x : X), \u2191(\u2191f x) \u2202\u03bc < \u22a4", "state_after": "case f_bdd\nX : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\n\u22a2 \u2203 c, \u2200 (x : X), \u2191(\u2191f x) \u2264 \u2191c"}, {"tactic": "refine \u27e8nndist f 0, fun x \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>nndist</a> f 0, fun x \u21a6 ?_\u27e9", [{"full_name": "NNDist.nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [294, 3], "def_end_pos": [294, 9]}]], "state_before": "case f_bdd\nX : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\n\u22a2 \u2203 c, \u2200 (x : X), \u2191(\u2191f x) \u2264 \u2191c", "state_after": "case f_bdd\nX : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\nx : X\n\u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0)"}, {"tactic": "have key := BoundedContinuousFunction.NNReal.upper_bound f x", "annotated_tactic": ["have key := <a>BoundedContinuousFunction.NNReal.upper_bound</a> f x", [{"full_name": "BoundedContinuousFunction.NNReal.upper_bound", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [1432, 9], "def_end_pos": [1432, 27]}]], "state_before": "case f_bdd\nX : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\nx : X\n\u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0)", "state_after": "case f_bdd\nX : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\nx : X\nkey : \u2191f x \u2264 nndist f 0\n\u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0)"}, {"tactic": "rwa [ENNReal.coe_le_coe]", "annotated_tactic": ["rwa [<a>ENNReal.coe_le_coe</a>]", [{"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 f_bdd\nX : Type u_1\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : OpensMeasurableSpace X\n\u03bc : Measure X\ninst\u271d : IsFiniteMeasure \u03bc\nf : X \u2192\u1d47 \u211d\u22650\nx : X\nkey : \u2191f x \u2264 nndist f 0\n\u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0)", "state_after": "no goals"}]}]